Advances in Optics Reviews. Vol. 5

Advances in Optics: Reviews Book Series, Volume 5 Sergey Y. Yurish Editor Advances in Optics: Reviews Book Series, Volume 5 International Frequency Sensor Association Publishing Sergey Y. Yurish Editor Advances in Optics: Reviews Book Series, Vol. 5 Published by International Frequency Sensor Association (IFSA) Publishing, S. L., 2021 E-mail (for print book orders and customer service enquires): ifsa.books@sensorsportal.com Visit our Home Page on http://www.sensorsportal.com Advances in Optics: Reviews, Vol. 5 is an open access book which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the authors. This is in accordance with the BOAI definition of open access. Neither the authors nor International Frequency Sensor Association Publishing accept any responsibility or liability for loss or damage occasioned to any person or property through using the material, instructions, methods or ideas contained herein, or acting or refraining from acting as a result of such use. ISBN: Print: 978-84-09-34834-3 e-Book: 978-84-09-34835-0 BN-20211215-XX BIC: TTB Acknowledgments As Editor I would like to express my undying gratitude to all authors, editorial staff, reviewers and others who actively participated in this book. We want also to express our gratitude to all their families, friends and colleagues for their help and understanding. Contents Contents Contributors................................................................................................................... 17 Preface ............................................................................................................................ 21 1. Physical Optics........................................................................................................... 23 1.1. Wave Equations of the Electromagnetic Field ................................................................. 23 1.1.1. The Maxwell’s Equations ...................................................................................................... 23 1.1.2. Waves in a Linear, Isotropic, and Homogeneous Medium .................................................... 24 1.1.3. Wave Speed of Light and the Index of Refraction ................................................................. 25 1.1.4. Scalar Time Harmonic Waves ............................................................................................... 25 1.1.5. The Energy Law of the Electromagnetic Field: Pointing Vector .......................................... 26 1.1.6. Vectorial Nature of the Electromagnetic Field ..................................................................... 27 1.1.7. Electromagnetic Theory of Radiation ................................................................................... 29 1.1.7.1. Radiation from an Accelerating Charge ..................................................................... 29 1.1.7.2. Radiation from an Oscillating Dipole ......................................................................... 30 1.1.8. Evanescent Waves ................................................................................................................. 30 1.1.9. Laws of Reflection and Refraction ........................................................................................ 33 1.1.9.1. Laws of Reflection and Refraction for Homogeneous Dielectric Media .................... 33 1.1.9.2. Fresnel’s Formulas ..................................................................................................... 35 1.2. Scalar Theory of Diffraction ............................................................................................ 37 1.2.1. The Rayleigh-Sommerfeld Formulation of Diffraction ......................................................... 38 1.2.2. Paraxial Approximations for Diffraction .............................................................................. 39 1.2.2.1. Fresnel Diffraction ..................................................................................................... 40 1.2.2.2. Fraunhofer Diffraction ................................................................................................ 41 1.2.3. Diffraction with Nonmonochromatic Light ........................................................................... 43 1.3. Coherence ......................................................................................................................... 44 1.3.1. Correlation Function: Definition .......................................................................................... 45 1.3.2. The Concept of Coherence of Light....................................................................................... 45 1.4. Fourier Theory and Fourier Optics ................................................................................... 50 1.4.1. Fourier Theory...................................................................................................................... 50 1.4.1.1. Analysis of Periodic Functions: Fourier Series .......................................................... 50 1.4.1.2. Analysis of Non-periodic Functions: Fourier Transform ............................................ 51 1.4.2. Fourier Optics....................................................................................................................... 52 1.4.2.1. The Concept of Angular Spectrum ............................................................................. 52 1.4.2.2. Fourier Transform of a Positive Lens ......................................................................... 54 1.4.2.3. Fourier Optics for Analyzing Diffraction Gratings ..................................................... 59 1.4.2.4. Fourier Optics for Imaging under Coherent Illumination ........................................... 64 1.4.2.5. Fourier Optics for Imaging under Incoherent Illumination ......................................... 67 1.4.2.6. Fourier Transform Spectroscopy ................................................................................ 68 1.5. Optical Interferometry ...................................................................................................... 69 1.5.1. Superposition of Waves ......................................................................................................... 70 1.5.1.1. General Principle ........................................................................................................ 70 1.5.1.2. Superposition of Waves with Same Frequencies ........................................................ 70 1.5.1.3. Standing Waves .......................................................................................................... 72 1.5.1.4. Superposition of Waves with Diffident Frequencies: Group Velocity, Phase Velocity ............................................................................................................ 72 1.5.2. Superposition of Light Waves: Optical Interferometry ......................................................... 75 1.5.2.1. Interference Phenomena of Monochromatic Light ..................................................... 76 7 Advances in Optics: Reviews. Book Series, Vol. 5 1.5.2.2. Interference Phenomena of Partial Coherent Light ..................................................... 81 1.5.2.3. Michelson Stellar Interferometer ................................................................................85 1.6. Polarization of Light ......................................................................................................... 86 1.6.1. Generation of Polarization Light .......................................................................................... 86 1.6.1.1. Polarization by Reflection from Dielectric Surfaces ...................................................87 1.6.1.2. Polarization through Diattenuation.............................................................................88 1.6.1.3. Polarization by Scattering ........................................................................................... 89 1.6.1.4. Polarization by Birefringence ..................................................................................... 89 1.6.1.5. Phase Shift by Birefringence ...................................................................................... 93 1.6.1.6. Optical Activity ..........................................................................................................94 1.6.2. Matrix Treatment for Completely Polarized Light ................................................................ 95 1.6.2.1. Mathematical Representation of Completely Polarized Light: Jones Vector ..............95 1.6.2.2. Mathematical Representation of Polarizers: Jones Matrices ....................................... 98 1.6.3. Basic Theory of Polarization of Stochastic Light Fields ..................................................... 100 1.6.3.1. Correlation Matrix of a Light Field .......................................................................... 101 1.6.3.2. Completely Polarized, Unpolarized, and Partially Polarized Light ........................... 101 References.............................................................................................................................. 104 Appendix. Derivation of the Rayleigh-Sommerfeld Formula ................................................ 104 2. CPU Technology Trend and Optical Interconnects ............................................. 109 2.1. Introduction..................................................................................................................... 109 2.2. CPU Design and Fabrication .......................................................................................... 110 2.3. Smartphone SoC Trend ................................................................................................... 118 2.4. x86 CPU Technology and Trend .................................................................................... 125 2.5. Supercomputer Architecture ........................................................................................... 136 2.6. Optical Network-on-chip Architecture ........................................................................... 143 2.7. Key Elements toward Optically Interconnected CPU ..................................................... 148 2.8. Conclusion ...................................................................................................................... 160 References.............................................................................................................................. 161 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera ............................. 163 3.1. Introduction..................................................................................................................... 163 3.2. Polarimetry with a Full Poincaré Beam and a CCD Camera .......................................... 164 3.3. Noise in CCD Images ..................................................................................................... 167 3.4. Strategies for Obtaining the Mueller Matrix ................................................................... 172 3.4.1. Center Point Strategy .......................................................................................................... 172 3.4.2. Nearest Neighbor Strategy .................................................................................................. 173 3.5. Results from the Simulation............................................................................................ 174 3.6. Conclusions..................................................................................................................... 176 Acknowledgements ................................................................................................................ 176 References.............................................................................................................................. 176 4. The Optical Bidirectional Lambertian Conductance Law with Applications ... 179 4.1. Prologue .......................................................................................................................... 179 4.2. Introduction to Solar Concentrators and Their Characterization .................................... 185 4.3. Theory of the Bidirectional Lambertian Irradiation Method (BLIM) ............................. 188 4.4. Experimental ................................................................................................................... 192 8 Contents 4.4.1. Optical Elements under Test ............................................................................................... 192 4.4.2. Simulation Methods ............................................................................................................ 194 4.4.3. Measurement of the Inverse Radiance ................................................................................ 196 4.4.4. Test of the Optical Bidirectional Lambertian Conductance (OBLC) Law .......................... 198 4.5. Optical Conductance Simulation Results ....................................................................... 202 4.5.1. Nonimaging Solar Concentrators (CPC) ............................................................................ 202 4.5.2. Light Cones (LC) ................................................................................................................ 206 4.5.3. Solar Tunnels (ST) .............................................................................................................. 210 4.6. Exit Radiance Simulation Results .................................................................................. 213 4.6.1. Nonimaging Solar Concentrators (CPC) ............................................................................ 213 4.6.2. Light Cones (LC) ................................................................................................................ 215 4.6.3. Solar Tunnels (ST) .............................................................................................................. 221 4.7. Short Discussion of Practical Applications. ................................................................... 222 4.7.1. Net Flux Crossing a Solar Tunnel ....................................................................................... 223 4.7.2. Light Sources from CPCs and LCs ..................................................................................... 225 4.7.3. Bent Solar Tunnels .............................................................................................................. 226 4.8. Some Insights into the Law of Optical Lambertian Conductance .................................. 229 4.9. Conclusions .................................................................................................................... 230 Acknowledgements ............................................................................................................... 232 References ............................................................................................................................. 232 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging .................... 237 5.1. Introduction .................................................................................................................... 237 5.2. Theory of Aperiodic Zone Plates ................................................................................... 239 5.2.1. Aperiodic Sequence............................................................................................................. 239 5.2.2. Method of Constructing Aperiodic Zone Plates .................................................................. 239 5.2.3. Theory of Diffraction .......................................................................................................... 241 5.3. Aperiodic Zone Plates for Optical Tweezers .................................................................. 243 5.3.1. Fractal Zone Plate .............................................................................................................. 243 5.3.2. Fibonacci Zone Plate .......................................................................................................... 249 5.3.3. Thue-Morse Zone Plate....................................................................................................... 250 5.4. Aperiodic Zone Plates for Optical Imaging .................................................................... 252 5.4.1. Clear Imaging ..................................................................................................................... 252 5.4.2. Colourful Imaging............................................................................................................... 257 5.5. Conclusion ..................................................................................................................... 262 Acknowledgments ................................................................................................................. 262 References ............................................................................................................................. 263 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures .................................................................. 267 6.1. Introduction .................................................................................................................... 267 6.2. Theoretical Basis ............................................................................................................ 268 6.2.1. Paraxial Matrix Transformations and Axial Chromatic Aberration ................................... 268 6.2.1.1. Paraxial Matrix Transformations from Object to Image Space in a Single Optical Surface .......................................................................................................... 268 9 Advances in Optics: Reviews. Book Series, Vol. 5 6.2.1.2. Exact Paraxial Matrix Transformations in a Centered Optical System of Conical Surfaces. Invention of Exact Analytical Dependences between Constructive Parameters and Paraxial Characteristics .................................................................... 270 6.2.1.3. Axial Image Chromatic Aberration .......................................................................... 274 6.2.1.4. Simultaneous Correction of Axial Image Chromatic Aberration and Guarantee of Preset Focal Length in a Centered Optical System ................................................ 276 6.2.2. Rigorous Analytical Function of the On-axes Spherical Aberration in a Centered Optical System Containing Conical Surfaces...................................................................... 277 6.2.2.1. Sagittal Radius and Sagittal Radius-coefficient in a Single Conical Surface ............ 277 6.2.2.2. Connections between the Parameters of the Tangential Rays and the Paraxial Matrix Transformations on a Single Conic Surface ................................................... 282 6.2.2.3. Paraxial Matrix Coefficients Linked to the Real Tangential Rays in a System of Centered Conic Surfaces ....................................................................................... 285 6.2.2.4. Geometrical Interpretation of the Paraxial Image Function and Its Connection with On-axis Spherical Aberration ............................................................................287 6.3. Simple Examples of Analytical Aberrations Correction for an Infinitely Distant Object............................................................................................................................. 288 6.3.1. Analytical Correction of Axial Chromatic Aberration Together with Preset Focal Length in a System of Three Surfaces (Case of the so Called Doublets) ............................. 289 6.3.2. Analytical Correction of On-axis Spherical Aberration in a Systems of Two and Three Surfaces.............................................................................................................. 291 6.3.2.1. Systems of Two Surfaces .......................................................................................... 291 6.3.2.2. Multi-conical Surfaces in the Centered Optical Systems .......................................... 292 6.3.2.3. Examples of On-axis Spherical Aberration Correction in the Systems of Two Surfaces ........................................................................................................ 293 6.3.2.4. Examples of On-axis Spherical Aberration Correction in the Systems of Three Surfaces ...................................................................................................... 297 6.4. Conclusions..................................................................................................................... 298 References.............................................................................................................................. 299 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry ........................................................................................................ 301 7.1. Introduction..................................................................................................................... 301 7.2. Relevance ........................................................................................................................ 303 7.3. State of the Art ................................................................................................................ 305 7.4. Time-resolved Photothermal Common-path Interferometry Scheme ............................. 311 7.5. The Prospect of Further Increasing of the TPCI Scheme Sensitivity .............................. 322 7.6. Conclusions..................................................................................................................... 328 Acknowledgements ................................................................................................................ 328 References.............................................................................................................................. 329 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) ......................................................................................... 333 8.1. Introduction..................................................................................................................... 333 8.2. Description of Light-sheet .............................................................................................. 335 8.2.1. Angular Spectrum ............................................................................................................... 335 8.2.1.1. Airy Light-sheet ........................................................................................................ 338 8.2.1.2. Bessel Pincer-sheet ...................................................................................................339 8.2.1.3. Gaussian Light-sheet ................................................................................................ 340 10 Contents 8.2.2. The BSCs (Beam Shape Coefficients) of Arbitrary Laser Light-sheet ................................. 341 8.2.2.1. Expansion of the Incident and Scattering Fields ....................................................... 341 8.2.2.2. The Derivation of General BSCs of Laser Light-sheet ............................................. 343 8.2.2.3. Particular Structural Beam Laser Light-sheet ........................................................... 349 8.3 Light Scattering of Laser Light-sheet by Dielectric Sphere of Arbitrary Size ............... 351 8.3.1. Far-field Scattering Intensity .............................................................................................. 351 8.3.2. Scattering, Extinction and Absorption Efficiencies ............................................................. 352 8.3.3. Radiation Force and Torque ............................................................................................... 353 8.4. Numerical Results and Discussions ............................................................................... 354 8.4.1. Airy light-sheet.................................................................................................................... 354 8.4.1.1. Far-field Scattering Intensity .................................................................................... 355 8.4.1.2. Scattering, Extinction and Absorption Efficiencies .................................................. 357 8.4.1.3. Radiation Force and Torque ..................................................................................... 360 8.4.2. Bessel Pincer Light Sheet.................................................................................................... 365 8.4.2.1. Far-field Scattering Intensity .................................................................................... 365 8.4.2.2. Scattering, Extinction and Absorption Cross Section ............................................... 369 8.4.2.3. Radiation Force and Torque ..................................................................................... 373 8.4.3. Gaussian Light-sheet .......................................................................................................... 383 8.4.3.1. Far-field Scattering Intensity .................................................................................... 383 8.4.3.2. Scattering, Extinction and Absorption Efficiencies .................................................. 385 8.4.3.3. Radiation Force and Torque ..................................................................................... 387 8.5. Conclusions .................................................................................................................... 389 Acknowledgements ............................................................................................................... 390 References ............................................................................................................................. 391 9. Laser Characterizations for Optical Wireless Communications ....................... 395 9.1. Introduction to Optical Wireless Communications ........................................................ 395 9.2. Underwater Wireless Optical Communication ............................................................... 397 9.3. Chaos Based Secure Optical Communications .............................................................. 398 9.4. Vertical-cavity Surface-emitting Lasers (VCSELs) ....................................................... 399 9.4.1. Optical and Electrical Properties of VCSELs ..................................................................... 400 9.4.1.1. Light-current (L-I) Curve Characteristics ................................................................. 400 9.4.1.2. Polarization Switching.............................................................................................. 402 9.5. OWC Based on VCSELs ................................................................................................ 403 9.6. Nonlinearity of VCSELs ................................................................................................ 404 9.7. VCSELs-based Chaos Dynamics ................................................................................... 405 9.7.1. Chaos Synchronization in VCSEL ....................................................................................... 406 9.8. Theoretical Analysis of Lasers Dynamics ...................................................................... 407 9.8.1. Rate Equations Describing Carriers and Photons Density Dynamics ................................ 407 9.9. Conclusions .................................................................................................................... 409 Acknowledgements ............................................................................................................... 410 References ............................................................................................................................. 410 10. Compact Solar-pumped Lasers ............................................................................ 417 10.1. Introduction .................................................................................................................. 417 10.1.1. Solar-pumped Lasers and Their Possible Applications .................................................... 417 10.1.2. Brief History of SPLs ........................................................................................................ 417 10.1.3. Stance and Orientation of This Work ................................................................................ 418 11 Advances in Optics: Reviews. Book Series, Vol. 5 10.2. Background of the Concept Design .............................................................................. 418 10.2.1. Preferable Oscillation Wavelength of SPLs ...................................................................... 418 10.2.2. Incident Power Density Required to Oscillate SPLs ......................................................... 420 10.2.3. Preferable Shape of LMs for SPLs .................................................................................... 422 10.2.4. Theoretical Limit of Solar Concentration ......................................................................... 424 10.2.5. Size of Images of the Sun and Concentration Ratios by Solar Concentrators ................... 425 10.2.5.1. Convex Lens ........................................................................................................... 425 10.2.5.2. Parabolic Concave Mirror ....................................................................................... 426 10.2.6. Merits of Miniaturization of SPLs ..................................................................................... 427 10.2.7. Direct Solar Radiation and Diffuse Solar Radiation ......................................................... 429 10.3. Compact Solar-Pumped Fiber Laser (SPFL) System .................................................... 432 10.3.1. Choice of Laser Mediums (LMs) ....................................................................................... 432 10.3.1.1. Shape of LMs.......................................................................................................... 432 10.3.1.2. Glass LMs ............................................................................................................... 433 10.3.2. Solar Concentrator............................................................................................................ 434 10.3.3. SPL Resonator Based on a Double Cladding Optical Fiber ............................................. 436 10.3.4. Outdoor Oscillation Performance of the Compact SPFL .................................................. 437 10.4. Improvement of Spectral Matching Efficiency by Cr Codoping .................................. 439 10.5. Compact Solar-Pumped Micro-Rod Laser (μSPL) System .......................................... 441 10.6. Continuous Oscillation of μSPL for Over 6.5 Hours Tracking the Sun ........................ 446 10.7. Effect of Cr Content in a Cr-Codoped Nd:YAG Transparent Ceramic Laser Rod for μSPLs................................................................................................................ 450 10.8. For Improvement of Mode-matching Efficiency .......................................................... 453 10.9. Evaluation of Energy Transfer Efficiency from Cr 3+ to Nd3+ in μSPL in Outdoor Operation ....................................................................................................................... 457 10.10. Coordinated Solar Tracking of an Array of μSPLs to Harvest Larger Amount of Solar Energy .............................................................................................................. 461 10.11. SPL-PV Combined System and Its Application to Optical Wireless Power Transmission .................................................................................................................. 463 10.12. Conclusion .................................................................................................................. 464 Acknowledgments ................................................................................................................. 465 References.............................................................................................................................. 465 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser ................................................................................. 473 11.1. Introduction................................................................................................................... 473 11.2. Laser-material Interactions ........................................................................................... 474 11.2.1. Gaussian Beam Propagation ............................................................................................ 474 11.2.2. Ablation Mechanisms ........................................................................................................ 477 11.2.3. Simulations ........................................................................................................................ 478 11.3. Experimental Details ..................................................................................................... 479 Acknowledgements ................................................................................................................ 484 References.............................................................................................................................. 484 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence .................................................... 487 12.1. Introduction................................................................................................................... 487 12.2. Quantum Cascade Lasers .............................................................................................. 488 12.3. Grazing Angle Reflection-Absorption Infrared Spectroscopy (RAIRS) ....................... 489 12.3.1. Design of the Optical Setup ............................................................................................... 489 12 Contents 12.3.2. Advantages over Conventional Methods ........................................................................... 490 12.4. Sample Preparation for Trace Detection Analyses ....................................................... 491 12.4.1. Deposition Techniques ...................................................................................................... 491 12.4.2. Effect of the Substrate: Specular and Diffuse Reflectance Measurements ........................ 492 12.4.3. Sample Preparation of HEs and APIs ............................................................................... 492 12.5. Trace Detection Analyses Using QCL-GAP ................................................................ 494 12.5.1. Detection of HEs ............................................................................................................... 494 12.5.2. Detection of API................................................................................................................ 498 12.6. Improving Detection Performance ............................................................................... 499 12.6.1. Development of Spectral Library ...................................................................................... 499 12.6.2. MVA Routines ................................................................................................................... 503 12.6.2.1. Data Preprocessing ................................................................................................. 503 12.6.2.2. Data Exploration..................................................................................................... 504 12.6.2.3. Data Discrimination................................................................................................ 506 12.6.3. Fourier Transform Data Preprocessing Algorithm........................................................... 507 12.7. Conclusions .................................................................................................................. 508 Acknowledgments ................................................................................................................. 508 References ............................................................................................................................. 508 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers .................................................................................................................. 511 13.1. Introduction .................................................................................................................. 511 13.2. Laser Dynamics ............................................................................................................ 512 13.3. Classification of Lasers ................................................................................................ 512 13.4. Lasers and Systems ...................................................................................................... 512 13.4.1. Semiconductor Lasers ....................................................................................................... 513 13.4.2. Fiber Lasers ...................................................................................................................... 513 13.4.3. Lasers in Optical Fiber Communication Systems ............................................................. 513 13.4.3.1. Lasers and Optical Communication System Technological Evolution ................... 514 13.4.4. Lasers in Optical Fiber Sensing Systems .......................................................................... 515 13.4.4.1. Lasers and Optical Sensing System Technological Evolution ................................ 516 13.5. Linewidth Compression ............................................................................................... 518 13.5.1. Whispering Gallery Mode (WGM) .................................................................................... 519 13.5.2. Parallel Feedback Mechanism .......................................................................................... 522 13.5.3. Optical Self-Injection Feedback........................................................................................ 524 13.5.4. Microfibers ....................................................................................................................... 526 13.5.5. Electrical Feedback Control ............................................................................................. 527 13.5.6. Rayleigh Backscattering ................................................................................................... 529 13.6. Linewidth Measurement ............................................................................................... 538 13.6.1. Heterodyne Detection ....................................................................................................... 538 13.6.2. Delayed Self-Heterodyne Detection .................................................................................. 539 13.6.3. Delayed Self-Homodyne Detection ................................................................................... 540 13.6.4. Amplitude Difference Comparison of Coherent Envelope ................................................ 541 13.7. Noise Characterization ................................................................................................. 544 13.7.1. Digital Cross Correlation ................................................................................................. 544 13.7.2. Michelson Interferometer .................................................................................................. 545 13.7.3. Cross-Spectrum Method.................................................................................................... 546 13.8. Wavelength Switching and Wavelength Tuning .......................................................... 547 13 Advances in Optics: Reviews. Book Series, Vol. 5 13.8.1. Mechanical Switching/Tuning ........................................................................................... 547 13.8.2. Acousto-optic Switching .................................................................................................... 560 13.8.3. All Optical Tuning ............................................................................................................. 563 13.8.4. Thermal Tuning ................................................................................................................. 565 13.9. Conclusion .................................................................................................................... 569 Acknowledgements ................................................................................................................ 569 References.............................................................................................................................. 570 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers .................................................................................... 575 14.1. Introduction................................................................................................................... 575 14.2. Experimental Setup ....................................................................................................... 577 14.3. DSR-like Square Pulses ................................................................................................ 578 14.4. Coherence Characterization of Square Pulses with MZI Technique............................. 581 14.5. Coherence Characterization of Square Pulses with DFT Technique ............................ 584 14.6. Conclusion .................................................................................................................... 588 References.............................................................................................................................. 588 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers ........................................................................................................... 593 15.1. Introduction................................................................................................................... 593 15.2. Fourier Series ................................................................................................................ 594 15.2.1. Fourier Series Coefficients................................................................................................ 594 15.2.2. Formula for Obtaining Fourier Series of Functions ......................................................... 596 15.2.3. Fourier Series of Even and Odd Functions ....................................................................... 597 15.2.4. From Fourier Series to Fourier Transform ....................................................................... 597 15.2.5. Obtaining Fourier Series of Current Functions ................................................................ 598 15.2.5.1. Periodic Function Built from Delta-like Function  a ( x)   ( x),  a  x  a, P  2a ................................................................................ 598 15.2.5.2. The Saw-tooth Function Built from f ( x)  x, 0  x  a ..................................... 598 15.2.5.3. Periodic Function Defined from the Heaviside Function ........................................ 600 15.2.5.4. The Square Wave Function SW ( x) .......................................................................600 15.2.5.5. The Periodic Function Constructed from x .......................................................... 601 15.2.5.6. Fourier Series of a Function Corresponding to a Displaced Graph ......................... 602 15.2.5.7. Fourier Series of a Function Corresponding to a Symmetrised Graph .................... 603 15.2.5.8. The Periodic Function Built from f ( x)  x2 , 0  x  a, P  a ......................... 603 15.2.5.9. The Ramp Function Built from f ( x)  0,  a  x  0, f ( x)  x,0  x  a, 2a ............................................................................................................................605 15.2.5.10. The Parabolic Function f ( x)  x2 , a  x  a, P  2a .................................606 15.2.5.11. The Periodic Function Built from f ( x)  x3 , 0  x  a, P  a ....................... 608 15.2.6. Polynomial Equivalent to an Individual Series in a Fourier Series .................................. 609 15.2.7. Obtaining Bernoulli Polynomials as Fourier Series ......................................................... 610 15.2.8. Obtaining Powered of Pi and Even Positive Zeta Functions ............................................ 615 15.2.9. Fourier Series of Polynomials........................................................................................... 615 15.3. Remarks and Conclusion .............................................................................................. 616 Acknowledgements ................................................................................................................ 617 References.............................................................................................................................. 617 14 Contents 16. Obtaining Fourier Transforms of Functions by Differential Calculus............. 619 16.1. Introduction .................................................................................................................. 619 16.2. Differential Operators Realizing Fourier Transform .................................................... 620 16.2.1. Introduction to Operator Calculus.................................................................................... 620 16.2.2. Operators Realizing the Fourier Transform ..................................................................... 622 16.2.3. The Fourier Transform of f ( x)  1 .................................................................................. 625 16.2.4. Geometric Signification of a Convolution Product ........................................................... 626 16.2.5. Fourier Transform of a Convolution Product ................................................................... 627 16.3. General Properties of the Fourier Transform ............................................................... 627 16.3.1. The Primordial Property ................................................................................................... 628 16.3.2. The Convolution Product .................................................................................................. 628 16.3.3. The double Fourier Transform.......................................................................................... 628 16.3.4. The Inverse Transform ...................................................................................................... 628 16.3.5. Transform of Complex Conjugate ..................................................................................... 628 16.3.6. Transform of Translated Functions Following (16.11) ..................................................... 628 16.3.7. Transform of a Dilated Function ...................................................................................... 629 16.3.8. Transform of a Sum of Two Functions .............................................................................. 629 16.4. Table of Fourier Transform of Differential Operators ................................................. 629 16.5. Table of Fourier Transform of Functions ..................................................................... 630 16.6. Remarks and Conclusion .............................................................................................. 633 Acknowledgments ................................................................................................................. 633 References ............................................................................................................................. 633 Index ............................................................................................................................. 635 15 Contributors Contributors Nikolai Andreev Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St, Nizhny Novgorod, 603950, Russia Alpan Bek Department of Physics, Middle East Technical University, 06800 Ankara, Turkey E-mail: bek@metu.edu.tr Edwin Caballero-Agosto Department of Chemistry, University of Puerto Rico, Mayagüez John R. Castro-Suárez Department of Chemistry, University of Puerto Rico, Mayagüez Shubo Cheng School of Physics and Electronics, Central South University, Changsha 410083, China Charles Ciret Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France Francheska M. Colón-González Department of Chemistry, University of Puerto Rico, Mayagüez Annette M. Colón-Mercado Department of Chemistry, University of Puerto Rico, Mayagüez Ling Fan School of science, Beijing Jiaotong University, Beijing, China Email: lfan@bjtu.edu.cn Nataly J. Galán-Freyle Department of Chemistry, University of Puerto Rico, Mayagüez Shuhong Gong Xidian University, Xi’an 710071, China Juan C. González de Sande Department of Materials Science, Universidad Politécnica de Madrid, Spain Kazuo Hasegawa The Graduate School for the Creation of New Photonics Industries (GPI), Hamamatsu, Shizuoka, Japan Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan Samuel P. Hernandez-Rivera Department of Chemistry, University of Puerto Rico, Mayagüez 17 Advances in Optics: Reviews. Book Series, Vol. 5 Boian A. Hristov Institute of Optical Materials and Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria Fırat İdikut Department of Physics, Middle East Technical University, 06800 Ankara, Turkey Paul Ikechukwu Iroegbu School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, 400044, China Key Laboratory of Optoelectronic Technology and Systems (Education Ministry of China), Chongqing University, Chongqing, 400044, China Email: paul.iroegbu@gmail.com Hiroshi Ito Institute of Materials Innovation, Institutes of Innovation for Future Society, Nagoya University, Japan Burcu Karagöz Department of Physics, Middle East Technical University, 06800 Ankara, Turkey Meriem Kemel Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France Renxian Li School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China Min Liu School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, 400044, China Key Laboratory of Optoelectronic Technology and Systems (Education Ministry of China), Chongqing University, Chongqing, 400044, China Alexandre Makarov Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St, Nizhny Novgorod, 603950, Russia Shintaro Mizuno Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan Tomoyoshi Motohiro Institute of Materials Innovation, Institutes of Innovation for Future Society, Nagoya University, Japan Ahmed Nady Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France 18 Contributors Salam Nazhan Department of Communication, College of Engineering, University of Diyala, Iraq Leonardo C. Pacheco-Londoño Department of Chemistry, University of Puerto Rico, Mayagüez Sahnggi Park Oprocessor Inc., Boston, MA, USA, E-mail: sahnggi@gmail.com Gemma Piquero Department of Materials Science, Universidad Politécnica de Madrid, Spain Antonio Parretta Physics and Earth Science Department, University of Ferrara, Academy of Sciences of Ferrara, Italy Mohamed Salhi Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France François Sanchez Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France Massimo Santarsiero Department of Materials Science, Universidad Politécnica de Madrid, Spain Georges Semaan Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045, Angers cedex 01, France Do Tan Si HoChiMinh-city Physical Association, Vietnam E-mail: tansi_do@yahoo.com Ningning Song Xidian University, Xi’an 710071, China Juan C. Suárez-Bermejo Department of Materials Science, Universidad Politécnica de Madrid, Spain Han Sun Xidian University, Xi’an 710071, China Yasuhiko Takeda Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan Huan Tang Xidian University, Xi’an 710071, China Shaohua Tao School of Physics and Electronics, Central South University, Changsha 410083, China 19 Advances in Optics: Reviews. Book Series, Vol. 5 Karla M. Vázquez-Vélez Department of Chemistry, University of Puerto Rico, Mayagüez Vladimir Villanueva-López Department of Chemistry, University of Puerto Rico, Mayagüez Ksenia Vlasova Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St, Nizhny Novgorod, 603950, Russia Bing Wei Xidian University, Xi’an 710071, China Bojian Wei Xidian University, Xi’an 710071, China Qun Wei Xidian University, Xi’an 710071, China Tian Xia School of Physics and Electronics, Central South University, Changsha 410083, China Liu Yang Xidian University, Xi’an 710071, China Shu Zhang Xidian University, Xi’an 710071, China Sijiong Zhang Nanjing Institute of Astronomical Optics and Technology, Nanjing, China University of Chinese Academy of Sciences, Beijing, China E-mail: sjzhang@niaot.ac.cn Tao Zhu Key Laboratory of Optoelectronic Technology and Systems (Education Ministry of China), Chongqing University, Chongqing, 400044, China 20 Preface Preface It is my great pleasure to introduce the fifth volume from our popular open access Book Series ‘Advances in Optics: Reviews’ started by the IFSA Publishing in 2018. The Vol. 5 of this Book Series is also published as an Open Access Book in order to significantly increase the reach and impact of this volume, which also published in two formats: electronic (pdf) with full-color illustrations and print (paperback). ‘Advances in Optics: Reviews’ Book Series is a comprehensive study of the field of optics, which provides readers with the most up-to-date coverage of optics, photonics and lasers with a good balance of practical and theoretical aspects. Directed towards both physicists and engineers this Book Series is also suitable for audiences focusing on applications of optics. A clear comprehensive presentation makes these books work well as both a teaching resources and a reference books. The book is intended for researchers and scientists in physics and optics, in academia and industry, as well as postgraduate students Like the first four volumes of this Book Series, the fifth volume also has been organized by topics of high interest to offer a fast and easy reading of each topic, every chapter in this book is independent and self-contained. All chapters have the same structure: first an introduction to specific topic under study; second particular field description including sensing or/and measuring applications. Each of chapter is ending by well selected list of references with books, journals, conference proceedings and web sites. The fifth volume is devoted to optics, lasers, optical communication and networks, and written by 53 authors from academia and industry from 12 countries: Belgium, Bulgaria, China, France, Italy, Iraq, Japan, Russia, Spain, Turkey, USA and Vietnam. But it is not a simple set of reviews. As usually, each chapter contains the extended state-of-the-art followed by new, unpublished before, obtained by the authors results. This book ensures that our readers will stay at the cutting edge of the field and get the right and effective start point and road map for the further researches and developments. By this way, they will be able to save more time for productive research activity and eliminate routine work. I shall gratefully receive any advices, comments, suggestions and notes from readers to make the next volumes of ‘Advances in Optics: Reviews’ Book Series very interesting and useful. Dr. Sergey Y. Yurish Editor IFSA Publishing Barcelona, Spain 21 Chapter 1. Physical Optics Chapter 1 Physical Optics Sijiong Zhang and Ling Fan1 1.1. Wave Equations of the Electromagnetic Field 1.1.1. The Maxwell’s Equations In this section, we will summarize the theory of the electromagnetic wave starting from Maxwell’s equations that lead to the equations of electromagnetic waves. The standard Maxwell’s macroscopic equations in a medium are written (1.1) 𝛻 · 𝑫 = 𝜌, 𝛻 · 𝑩 = 0, (1.2) 𝛻 × 𝑯 = 𝑱+ 𝜕𝑫 , 𝜕𝑡 𝜕𝑩 𝛻 × 𝑬 = − 𝜕𝑡 , (1.3) (1.4) where E and D are the electric field vector and the electric displacement vector respectively; B and H the magnetic induction vector and the magnetic field vector, respectively, and ρ and J the free charge density and the electric current density vector generated by free charges, respectively. And ∇·and ∇×, respectively, stand for the divergence and curl operators. The equation (1.1) is Gaussian Law for electric fields, the equation (1.2) Gaussian Law for magnetic fields, the equation (1.3) Ampere’s Law with 𝜕𝑫 the displacement current 𝜕𝑡 introduced by James C. Maxwell, and the equation (1.4) Faraday’s Law. Sijiong Zhang Nanjing Institute of Astronomical Optics and Technology, University of Chinese Academy of Sciences, China 23 Advances in Optics: Reviews. Book Series, Vol. 5 1.1.2. Waves in a Linear, Isotropic, and Homogeneous Medium In this chapter, we mainly consider about electromagnetic fields within a medium that is linear, isotropic, homogeneous, and non-conducting, in which free charge density ρ and free current density J are all zeros. The linearity of the medium is the property in which the response of the medium to an applied electromagnetic field is proportional to the field. An isotropic medium refers to one in which the properties of the medium in all directions are the same. A homogeneous medium is the one in which its index of refraction is identical at every point. Such a medium with the relative permittivity 𝜖 and the relative permeability 𝜇 can be described by material equations (1.5) 𝑫 = 𝜖𝜖0 𝑬, 𝑩 = 𝜇𝜇0 𝑯, where 𝜖0 and 𝜇0 are the vacuum permittivity and the vacuum permeability respectively; 𝜖 and 𝜇 are generally dependent on the frequency of the electromagnetic wave, which is the dispersive property of the medium. On inserting the equation (1.5) into the equations (1.1) to (1.4), and considering ρ = 0 and J = 0, we get the following equations 𝛻 · 𝑬 = 0, (1.6) 𝛻 · 𝑯 = 0, (1.7) 𝜕𝑬 𝛻 × 𝑯 = 𝜖𝜖0 𝜕𝑡 , 𝛻 × 𝑬 = −𝜇𝜇0 (1.8) 𝜕𝑯 𝜕𝑡 (1.9) On taking the curl of both sides of the equation (1.9) and inserting the equation (1.8) into it, we obtain 𝜕2 𝑬 𝜕 𝛻 × (𝛻 × 𝑬) = −𝜇𝜇0 𝜕𝑡 (𝛻 × 𝑯) = −𝜖𝜇𝜖0 𝜇0 2 𝜕𝑡 (1.10) By invoking the formula 𝛻 × (𝛻 × 𝑬) = 𝛻 (𝛻 · 𝑬) − 𝛻 2 𝑬 , where 𝛻 2 is Laplace’s operator, and considering the equation (1.6), then the equation (1.10) becomes 𝜕2 𝑬 𝛻 2 𝑬 = 𝜖𝜇𝜖0 𝜇0 𝜕𝑡 2 (1.11) The above equation is a vector wave equation for the electric field. Furthermore, 1 let 𝑣 2 = 𝜖𝜇𝜖 𝜇 , which 𝑣 is the wave speed for the wave equation (1.11). Then we get a 0 0 standard electromagnetic wave equation as 24 𝛻 2𝑬 = 1 𝜕2 𝑬 𝑣 2 𝜕𝑡 2 (1.12) Chapter 1. Physical Optics This is a fundamental and starting formula for studying physical optics or wave optics in a linear, isotropic, and homogeneous dielectric medium. 1.1.3. Wave Speed of Light and the Index of Refraction From the formula (1.11), we know that wave speed in the medium is In free space, the wave speed 𝑐 = 𝑣 = 1 1 𝜖𝜇𝜖 √ 0 𝜇0 √𝜖0 𝜇0 (1.13) , which is just the light speed in vacuum. Based on this point, Maxwell thought of light as an electromagnetic wave. According to the 𝑐 definition of the index of refraction 𝑛 = 𝑣 and the expression (1.13), such that we can get that the index of refraction of the medium is 𝑛 = √𝜖𝜇. In optics, it is generally valid that 𝜇 ≈ 1 (the reason is to be explained in Section 1.1.4), so that the index of refraction of the medium can be written 𝑛 = √𝜖. 1.1.4. Scalar Time Harmonic Waves When the Cartesian components of the light field 𝐄, 𝐇 are not coupled, the equation (1.12) can be reduced to a scalar wave equation 𝛻 2𝑈 = 1 𝜕2 𝑈 , 𝑣 2 𝜕𝑡 2 (1.14) where U is a scalar light field that represents any Cartesian components of E and H. In optics the scalar U just stands for the electric field. The reason for U representing for the electric field is that in view of atomic physics the electric field of light influences the atoms much stronger than the magnetic field does. In the optical frequency domain, the power of magnetic diploe radiation is basically equivalent to that of electric quadrupole radiation which is much weaker than that of electric dipole radiation that is the dominated contribution to the radiation of atoms [12]. For most of our work on optics, the scalar wave equation (1.14) is accurate enough to describe optical phenomena provided that the diffracting structures and the observation distance are large compared with the wavelength of light. Attention is now focused on the case of a monochromatic wave. By means of the Fourier transform theory, the generalization to polychromatic or nonmonochromatic waves can be done by Fourier synthesis by means of the results of the monochromatic waves, which will be discussed in Section 1.2. For a monochromatic wave, its analytic expression can be written 𝑈(𝒓, 𝑡) = 𝑈(𝒓)𝑒 −𝑗𝜔𝑡 , (1.15) 25 Advances in Optics: Reviews. Book Series, Vol. 5 where r is the spatial coordinate in the three dimensional space, t time, 𝜔 the angular frequency of light, and U(r) the complex amplitude of the monochromatic wave. On substituting the equation (1.15) into (1.14), it follows that U(r) obeys the Helmholtz equation 𝛻 2 𝑈(𝒓) + 𝑘 2 𝑈(𝒓) = 0, (1.16) 2𝜋 where k is called the wave number, given by k = , and 𝜆 is the wavelength of light in 𝜆 the medium. Equation (1.16) is used as a starting point to study the diffraction theory of light. There are many different solutions of the equation of vector waves (1.12) for different situations. Particularly, we provide two useful solutions without proof, plane waves and spherical waves, which are often used in this chapter. The electric field of a plane wave can be expressed as an analytic function 𝑬 = 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , where 𝒌 is the wave-vector in the medium through which light is propagating, 𝒓 the spatial position, and 𝐸0 the amplitude vector of the wave. The plane waves are the basis for Fourier optics. We should emphasize that the phase of the plane wave is denoted as the form (𝒌 ∙ 𝒓 − 𝜔𝑡). As a result of this sign convention of the right-handed Cartesian coordinate adopted, the electric field 𝑬𝟎 𝑒 𝒋(𝒌∙𝒓−𝝎𝒕+𝜹) lags the electric field 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) by δ if the additional phase δ > 0; conversely, if the additional phase δ < 0 the electric field 𝑬𝟎 𝑒 𝒋(𝒌∙𝒓−𝝎𝒕+𝜹) leads the electric field 𝐸0 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) by |δ|. We should aware of that the formalism of representation for polarization of light is dependent on the notation of the phase, which will be discussed in the following Section 1.6. The electric field of a spherical wave can be expressed as an analytic function 𝑬 𝐸 = 𝑟𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , where 𝑟 = |𝒓| stands for the distance from the origin, and the meaning of the other quantities in the expression is same as those for the plane wave. This wave represents one propagating outwards from the origin, if wavenumber is positive. The spherical waves are building blocks for Huygens’ principle. 1.1.5. The Energy Law of the Electromagnetic Field: Pointing Vector An electromagnetic field is a wave, and so is light. The wave can transport energy. The energy flux density vector, representing the power passing through unit area, is called Poynting vector expressed as 𝑺 = 𝑬 × 𝑯 The light intensity, 𝐼, is defined as the time averaged power crossing a unit area. 26 𝐼 = 1 𝑇/2 𝑆(𝑡) 𝑑𝑡 ∫ 𝑇 −𝑇/2 ≡< 𝑆(𝑡) >, (1.17) Chapter 1. Physical Optics where T is the response time of an optical detector. In the following, we will give out the concrete expression of the light intensity of a plane wave. On inserting the expression 𝑬 = 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) into Maxwell’s equations (1.6) to 𝜕 (1.9), and considering that 𝛻 ≡ 𝑖𝒌, ≡ −𝑖𝜔, we get the following equations 𝜕𝑡 𝒌 · 𝑬 = 0, 𝒌 · 𝑯 = 0, 𝒌 × 𝑯 = −𝜔𝜖𝜖0 𝑬, 𝒌 × 𝑬 = 𝜔𝜇0 𝑯 (1.18) From the above equations, it is easy to know that 𝑫, 𝑯, and 𝒌 form a right-handed mutually orthogonal triad, so do 𝑬, 𝑩, and 𝒌. For an isotropic medium, 𝑬, 𝑯, and 𝒌 also form a right-handed mutually orthogonal triad, and the electromagnetic wave is transverse. Furthermore, the magnetic field for the plane wave can be derived as 𝑯 = 1 𝑬 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , 𝑍 𝟎 where that 𝑍 = |𝑬| |𝑯| = |𝒌| 𝜖𝜖0 𝜔 = 1 𝜇0 √ 𝜖 𝜖0 √ = 𝑍0 𝑛 is the impedance of the light field in the medium, further the index of refraction of the medium 𝑛 = √𝜖 and 𝜇 the impedance of the light field in vacuum 𝑍0 = √ 𝜖 0 . Then the intensity of light for a 0 plane wave is 𝐼 = 𝑇 1 2 ∫ 𝐸 𝑐𝑜𝑠(𝜔𝑡) 𝐻0 𝑐𝑜𝑠(𝜔𝑡) 𝑑𝑡 𝑇 −𝑇 0 2 = 1 2 𝐸 2𝑍 0 = 1 𝑛𝐸02 , 2𝑍0 (1.19) 1 where 𝐸0 is the magnitude of 𝑬𝟎 . To get the equation (1.19) the condition 𝑇 ≫ is 𝜔 required, which is always valid because of the very high frequency, 𝜔, of the light field. From the expression (1.19), we know that the light intensity is proportional to the electric field squared. An optical detector can just detect the energy of light, such its detected result is proportional to the electric field squared. Based on this reason, the optical detector is called a square-law detector. 1.1.6. Vectorial Nature of the Electromagnetic Field As stated previously, the electromagnetic fields are vectorial and transverse vectors, so that, at every point in the wave, the electric field E, the magnetic field H, and Poynting vector S forms a mutually perpendicular triad. The direction of the electric field E is known as the polarization of the wave. Depending on how the electric field is oriented, the polarized light can be classified into three types of polarization: elliptical polarization, and its degenerate cases, i.e., circular polarization and linear polarization. Linear polarization: the electric field of light keeps the fixed direction in a single plane being perpendicular to the direction of propagation, as shown in Fig. 1.1. Circular polarization: two orthogonal components of the electric field of light are equal in magnitude, but have a phase difference of π/2. The resulting electric field then rotates in 27 Advances in Optics: Reviews. Book Series, Vol. 5 a circle around the direction of propagation. Depending on the rotation sense, there exists left- or right-hand circularly polarized light, as shown in Fig. 1.2. Fig. 1.1. Schematic of electric vectors of the linearly polarized light with different orientations. The light propagates out of the diagram along z-axis. Fig. 1.2. Schematic of electric vectors of the left- and right-circularly polarized light. The light propagates out of the diagram along z-axis. Elliptical polarization: two orthogonal components of the electric field of light are not equal in magnitude, and a phase difference is different from zero. The resulting electric field then rotates in an ellipse around the direction of propagation. This is the most general description of polarized light, and circular and linear polarized light can be viewed as the special cases of elliptically polarized light, as shown in Fig. 1.3. Fig. 1.3. Schematic of electric vectors of the left- and right-elliptically polarized light. The light propagates out of the diagram along z-axis. 28 Chapter 1. Physical Optics General treatment for completely polarized, unpolarized and partially polarized light will be discussed in Section 1.6. 1.1.7. Electromagnetic Theory of Radiation A linearly accelerating charge and an oscillating dipole are two important types of sources for radiation in optics and we will briefly summarize them here. 1.1.7.1. Radiation from an Accelerating Charge A charged particle moving at uniform velocity, according to Ampere’s law, just produces a constant magnetic field. From the equation (1.9), a constant magnetic field cannot induce an electric field, which means that a moving particle with a constant velocity does not radiate electromagnetic waves. However, if the particle accelerates, the magnetic field has a time derivative that induces an electric field. As a result, the accelerated particle generates radiation. For a particle of charge q moving with velocity 𝒗(𝑡) and then acceleration 𝒗̇ (𝑡) (the top dot refers to derivative), the far field of radiation at the spatial position r is 𝑬 = 𝑞 [𝒓̂ × 4𝜋𝜖0 𝑟𝑐 2 (𝒓̂ × 𝒗̇ )], (1.20) where 𝒓̂ is the unit vector along the direction of r, 𝑟 = |𝒓|, representing the magnitude of r and its value being large comparing with the wavelength of light, and c the speed of light. From the equation (1.20), the electric field of radiation is always perpendicular to the direction of 𝒓̂, and E, 𝒓̂, and 𝒗̇ are coplanar, as shown in Fig. 1.4. Fig. 1.4. Schematic of spatial relationship of a triad of E, 𝒓̂, and 𝒗̇ . From the Fig. 1.4 and the formula (1.20), we know that the magnitude of the electric field 𝜋 is maximum when 𝜃 = 2 ; and the magnitude of the electric field is minimum when 𝜃 = 0. 29 Advances in Optics: Reviews. Book Series, Vol. 5 1.1.7.2. Radiation from an Oscillating Dipole The radiative source most frequently encountered in optics is a periodically oscillating dipole that can radiate light, for example, in the scattering theory. Let a point charge q has a position vector with time, 𝐴𝑐𝑜𝑠(𝜔𝑡)𝒛̂, A being oscillating amplitude of the dipole, 𝜔 representing oscillatory frequency of a dipole with the direction 𝒛̂. The moment of the dipole is 𝒑 = 𝑞𝐴 𝑐𝑜𝑠(𝜔𝑡)𝒛̂. The acceleration of the dipole is 𝒑̈ 𝒗̇ (𝑡) = −𝐴𝜔2 𝑐𝑜𝑠(𝜔𝑡)𝒛̂ = 𝑞, where the double dot on the top of 𝒑 represents the second derivative with respect to time. From the equation (1.20), we obtain that the far field of radiation at the spatial position r is 𝑬 = 1 [𝒓̂ × 4𝜋𝜖0 𝑟𝑐 2 (𝒓̂ × 𝒑̈ )] (1.21) From the above equation, we can see that the electric field of radiation is always perpendicular to the direction of 𝒓̂, and 𝑬, 𝒓̂, and 𝒑̈ are coplanar, as shown in Fig. 1.5 (a). And the same as the case of radiation from an accelerating charge, the magnitude of position vector 𝒓 is also large comparing to the wavelength of light. Fig. 1.5. The spatial relationship of a triad of E, 𝒓̂, and 𝒑̈ (a), and the intensity radiation pattern of an oscillating dipole as a function of angle 𝜃 (b). As the case of the radiation from an accelerating charge, for the radiation from an oscillating dipole the magnitude of the electric field is maximum when 𝜃 = 𝜋/2; and the magnitude of the electric field is minimum when 𝜃 = 0, shown as in Fig. 1.5(b). For these two cases, radiation field E is always perpendicular to 𝒓̂ . This property of perpendicularity means that the light wave is a transverse wave. 1.1.8. Evanescent Waves When light in a dielectric medium with the index of refraction 𝑛1 is incident onto an interface of a different dielectric material with a lower index of refraction 𝑛2 , at an angle 30 Chapter 1. Physical Optics 𝜃1 , as shown in Fig. 1.6. The propagation of light, in view of geometrical optics, obeys Snell’s law, 𝑛1 𝑠𝑖𝑛 𝜃1 = 𝑛2 𝑠𝑖𝑛 𝜃2 , where 𝜃2 is the refracted angle. For an incident angle 𝜃1 greater than the critical angle defined as the following 𝑛 𝜃𝑐 = 𝑠𝑖𝑛−1 (𝑛 2 ), 1 (1.22) the total internal reflection occurs. Although all of the incident energy is reflected, Maxwell’s equations predict the existence of an electromagnetic field in the less dense medium 𝑛2 with the intensity decaying exponentially away from the interface. This field is known as the evanescent wave. Fig. 1.6. Schematic of the light refraction on the interface. The existence of the evanescent wave can be appreciated by considering the transmitted plane wave of light: 𝑬𝒕 = 𝑬𝒕𝟎 𝑒 𝑗(𝒌𝑡 ∙𝒓−𝜔𝑡) , (1.23) 𝒌𝑡 ∙ 𝒓 = 𝑘𝑡 𝑥 𝑠𝑖𝑛 𝜃2 + 𝑘𝑡 𝑧𝑐𝑜𝑠𝜃2 (1.24) where 𝑬𝒕 and 𝑬𝒕0 are the electric vector of the transmitted light wave and its amplitude vector respectively; 𝒌𝑡 is the wave-vector of the transmitted light wave, 𝒓 is the position vector, and ω is the angular frequency of light. According to the coordinate axes shown in Fig. 1.6, we have Combining Snell’s law with (1.24), we have 𝑐𝑜𝑠𝜃2 = √1 − 𝑛12 𝑠𝑖𝑛2 𝜃1 . 𝑛22 When incident angle 𝜃1 is greater than the critical angle 𝜃𝑐 , 𝑐𝑜𝑠𝜃2 becomes pure imaginary and we can write 𝑐𝑜𝑠𝜃2 = 𝑗√ 𝑛12 𝑠𝑖𝑛2 𝜃1 𝑛22 − 1. Thus the term (𝒌𝑡 ∙ 𝒓) can be written. 31 Advances in Optics: Reviews. Book Series, Vol. 5 2𝜋 𝒌𝑡 ∙ 𝒓 = 𝑘𝑡 𝑥 𝑠𝑖𝑛 𝜃2 + 𝑗𝑘𝑡 𝑧 √ 𝑛12 𝑠𝑖𝑛2 𝜃1 𝑛22 − 1, (1.25) where 𝑘𝑡 = 𝜆 𝑛2 is the magnitude of 𝒌𝑡 , 𝜆 is the wavelength of light in the vacuum space. On inserting the equation (1.25) into the equation (1.23) and omitting the time factor, we have 𝑬𝒕 = 𝑬𝒕𝟎 𝑒 𝑗𝑘𝑡𝑥 𝑠𝑖𝑛 𝜃2 𝑒 𝑛2 𝑠𝑖𝑛2 𝜃 −𝑘𝑡 𝑧√ 1 2 1 −1 𝑛 2 (1.26) From the expressions (1.19) and (1.26), the intensity of light field, 𝐼 ∝ |𝑬𝒕 |2 , can be written in the second medium 𝑧 𝐼 = 𝐼0 𝑒 −𝑑 , (1.27) where 𝐼0 ∝ |𝑬𝒕𝟎 |2 is the intensity of light field of the incident on the interface, and a penetration depth given by 𝑑 = 𝜆 4𝜋√𝑛12 𝑠𝑖𝑛2 𝜃1 −𝑛22 (1.28) This evanescent wave propagates parallel to the interface and has a penetration depth, 𝑑, on the order of the wavelength of the illuminating light, λ. However, light is not actually reflected at the exact position of the interface, but rather penetrates a distance 𝑑 into the medium with the lower index of refraction by optical tunneling. As a result, the reflected beam of light is shifted along the interface by a small amount 𝑥 = 2𝑑𝑡𝑎𝑛𝜃1 , which is known as the Goos-Hänchen Shift as shown in Fig. 1.7. Fig. 1.7. Schematic of the Goos-Hänchen Shift. The evanescent wave in the second medium, or optical tunnelling, can be detected. As shown in Fig. 1.8, two same right angle prisms are placed with their long faces close 32 Chapter 1. Physical Optics tighter but not in actual contact. Light from a source S is found to be partially transmitted into the second right-angle prism, the amount of light transmitted depending on the separation of the prism faces. Fig. 1.8. Schematic of the effect of optical tunnelling. The total internal reflection and the evanescent wave have been exploited to apply to many fields, for example, optical fibers and the evanescent wave microscopy. Optical fibers use the total internal reflection at the interface between core and cladding media to transfer energy along to the fiber. The evanescent wave microscopy or total internal reflection microscopy (TIRM) can increase the resolution of images. At last, we will discuss the laws of reflection and refraction, encountered in geometrical optics, on a plane boundary between two homogeneous dielectric media using Maxwell’s equations. 1.1.9. Laws of Reflection and Refraction The phenomena of reflection and refraction are very important in optics, especially they are the physical basis for instrumental optics. When a plane wave is incident onto a plane boundary between two homogeneous dielectric media with different indices, this wave is partly transmitted into the second medium and partly reflected back into the first medium. Here we are going to derive, by using macroscopic Maxwell’s equations, the laws of reflection and refraction on an abrupt plane boundary between two homogeneous dielectric media. 1.1.9.1. Laws of Reflection and Refraction for Homogeneous Dielectric Media As it is shown in Fig. 1.9, a plane wave, 𝑬𝐼 = 𝑬0𝐼 𝑒 𝑗(𝒌𝐼∙𝒓−𝜔𝑡) , at the incident angle 𝜃𝐼 , is incident on the interface between the medium with the index of refraction 𝑛1 and the one with the index of refraction 𝑛2 . The reflected wave, 𝑬𝑅 = 𝑬0𝑅 𝑒 𝑗(𝒌𝑅∙𝒓−𝜔𝑡) , is reflected back into the first medium at the reflected angle 𝜃𝑅 . And the refracted wave, 𝑬𝑇 = 𝑬0𝑇 𝑒 𝑗(𝒌𝑇 ∙𝒓−𝜔𝑡) , is transmitted into the second medium at the refracted angle is 𝜃𝑇 . The plane specified by the incident wave vector 𝒌𝐼 and the normal z-axis of the boundary is called the plane of incidence. 33 Advances in Optics: Reviews. Book Series, Vol. 5 At any point on the boundary between the two media, for example, the meeting point of three light rays at the boundary, as shown in Fig. 1.9, the phases of three light waves are the same at any time instant. We therefore get the following expression. (1.29) 𝒌𝐼 ∙ 𝒓 = 𝒌𝑅 ∙ 𝒓 = 𝒌𝑇 ∙ 𝒓, where 𝒓 = (𝑥, 𝑦, 0) represents any point on the boundary, 𝑘𝐼 = |𝒌𝐼 | = 2𝜋 𝜆 2𝜋 𝜆 2𝜋 𝜆 𝑛1 , 𝑛1 , and 𝑘 𝑇 = |𝒌𝑇 | = 𝑛2 . We write three wave vectors in the 𝑘𝑅 = |𝒌𝑅 | = form of components, respectively, as 𝒌𝐼 = (𝑘𝐼𝑥 , 𝑘𝐼𝑦 , 𝑘𝐼𝑧 ), 𝒌𝑅 = (𝑘𝑅𝑥 , 𝑘𝑅𝑦 , 𝑘𝑅𝑧 ), and 𝒌𝑻 = (𝑘 𝑇𝑥 , 𝑘 𝑇𝑦 , 𝑘 𝑇𝑧 ). On inserting them into the equation (1.29), we have 𝑥𝑘𝐼𝑥 + 𝑦𝑘𝐼𝑦 = 𝑥𝑘𝑅𝑥 + 𝑦𝑘𝑅𝑦 = 𝑥𝑘 𝑇𝑥 + 𝑦𝑘 𝑇𝑦 (1.30) Because the equations (1.30) are valid for all values 𝑥 and 𝑦, the following equations must be valid. Fig. 1.9. Reflection and refraction of a plane wave. 𝑘𝐼𝑥 = 𝑘𝑅𝑥 = 𝑘 𝑇𝑥 , 𝑘𝐼𝑦 = 𝑘𝑅𝑦 = 𝑘 𝑇𝑦 (1.31) The equations (1.31) shows that both 𝒌𝑅 and 𝒌𝑇 lie the plane of incidence. Based on this 𝟐𝝅 result and Fig. 1.9, we can, by trigonometry, obtain that 𝒌𝐼 = 𝝀 𝑛1 (𝑠𝑖𝑛𝜃𝐼 , 0, −𝑐𝑜𝑠𝜃𝐼 ), 𝟐𝝅 𝟐𝝅 then 𝒌𝑅 = 𝝀 𝑛1 (𝑠𝑖𝑛𝜃𝑅 , 0, 𝑐𝑜𝑠𝜃𝑅 ) , and 𝒌𝑇 = 𝝀 𝑛2 (𝑠𝑖𝑛𝜃𝑇 , 0, −𝑐𝑜𝑠𝜃𝑇 ) . On combining them with the equation (1.31), such that the equation (1.31) can be explicitly written as follows. 2𝜋 𝜆 𝑛1 𝑠𝑖𝑛𝜃𝐼 = 2𝜋 𝜆 𝑛1 𝑠𝑖𝑛𝜃𝑅 , To simplify the above equations, we have 34 2𝜋 𝜆 𝑛1 𝑠𝑖𝑛𝜃𝐼 = 2𝜋 𝜆 𝑛2 𝑠𝑖𝑛𝜃𝑇 Chapter 1. Physical Optics 𝑠𝑖𝑛𝜃𝐼 = 𝑠𝑖𝑛𝜃𝑅 or 𝜃𝐼 = 𝜃𝑅 , 𝑛1 𝑠𝑖𝑛𝜃𝐼 = 𝑛2 𝑠𝑖𝑛𝜃𝑇 (1.32) (1.33) The equation (1.32), together with the statement that the reflected wave vector 𝒌𝑅 lies in the plane of incidence is just the law of reflection; the equation (1.33), together with the statement that the refracted wave vector 𝒌𝑇 lies in the plane of incidence is just the law of refraction. 1.1.9.2. Fresnel’s Formulas We now consider the amplitudes of the reflected and the refracted waves. And assume that the two media are homogeneous, isotropic, of zero conductivity, and non-magnetic (𝜇 = 1). At an abrupt boundary between two media, the boundary conditions must be obeyed between the fields on the two sides. In words, the parallel components of the fields 𝑬 and 𝑯 to the boundary are equal on the two sides, whereas the normal components of 𝑫 and 𝑩 to the boundary are also continuous. Detailed derivation of the boundary conditions, readers can refer to the reference [1]. We resolve each field vector into the parallel components (𝑝) and the normal components (𝑠) to the plane of incidence. Then the fields 𝑬 and 𝑯 can be expressed as 𝑬 = 𝑬𝒑 + 𝑬𝒔 , and 𝑯 = 𝑯𝒑 + 𝑯𝒔 , as shown in Fig. 1.10. By means of the equation (1.18) and the expression of the plane wave, we have 𝑯𝒑 = 1 𝒌 𝜔𝜇0 × 𝑬𝒔 , 𝑯𝒔 = 𝟏 𝒌 𝝎𝝁𝟎 × 𝑬𝒑 , (1.34) where 𝒌 is the wave vector of the plane wave in the medium that light is traveling. Its 2𝜋 2𝜋 magnitude in the medium 𝑛1 is 𝜆 𝑛1 , and 𝜆 𝑛2 in the medium 𝑛2 . Fig. 1.10. Conventions for directions of field vectors, black dots stand for the normal components of field vectors pointing out the diagram. 35 Advances in Optics: Reviews. Book Series, Vol. 5 The components of the electric vector of the incident field then are 𝐸𝐼𝑥 = |𝑬𝐼𝑝 |𝑐𝑜𝑠𝜃𝐼 , 𝐸𝐼𝑦 = −|𝑬𝐼𝒔 |, 𝐻𝐼𝑥 = − 𝑘𝐼 |𝑬𝐼𝑠 |𝑐𝑜𝑠𝜃𝐼 , 𝜔𝜇0 𝐻𝐼𝑦 = − (1.35) 𝑘𝐼 |𝑬𝐼𝑝 | 𝜔𝜇0 (1.36) The components of the electric vector of the refracted field then are 𝐸𝑇𝑥 = |𝑬𝑇𝑝 |𝑐𝑜𝑠𝜃𝑇 , 𝐸𝑇𝑦 = −|𝑬𝑇𝒔 |, 𝑘 (1.37) 𝑘 𝐻𝑇𝑥 = − 𝜔𝜇𝑇 |𝑬𝑇𝑠 |𝑐𝑜𝑠𝜃𝑇 , 𝐻𝑇𝑦 = − 𝜔𝜇𝑇 |𝑬𝑇𝑝 | 0 0 (1.38) The components of the electric vector of the reflected field then are 𝐸𝑅𝑥 = |𝑬𝑅𝑝 |𝑐𝑜𝑠𝜃𝑅 , 𝐸𝑅𝑦 = −|𝑬𝑅𝒔 |, 𝐻𝑅𝑥 = 𝑘𝑅 |𝑬𝑅𝑠 |𝑐𝑜𝑠𝜃𝑅 , 𝝎𝝁𝟎 𝐻𝑅𝑦 = 𝑘𝑅 |𝑬𝑅𝒑 | 𝝎𝝁𝟎 (1.39) (1.40) Application of boundary conditions to the equation (1.34), we can obtain the following expressions. 𝐸𝐼𝑥 + 𝐸𝑅𝑥 = 𝐸𝑇𝑥 , 𝐸𝐼𝑦 + 𝐸𝑅𝑦 = 𝐸𝑇𝑦 , 𝐻𝐼𝑥 + 𝐻𝑅𝑥 = 𝐻𝑇𝑥 , 𝐻𝐼𝑦 + 𝐻𝑅𝑦 = 𝐻𝑇𝑦 (1.41) (1.42) Combining the equations (1.35) to (1.42) with algebraic manipulations and the law of reflection (1.32) and of refraction (1.33), we can get Fresnel’s formulas. 𝑟𝒔 = 𝑟𝑝 = 𝑡𝒔 = 𝑡𝑝 = |𝑬𝑅𝒔 | |𝑬𝑰𝒔 | |𝑬𝑅𝒑 | |𝑬𝐼𝒑 | |𝑬𝑇𝒔 | |𝑬𝑰𝒔 | |𝑬𝑇𝒑 | |𝑬𝐼𝒑 | sin(𝜃 −𝜃 ) = − 𝑠𝑖𝑛(𝜃𝐼 +𝜃𝑇 ), 𝐼 = = = 𝑇 (1.43) 𝑡𝑎𝑛(𝜃𝐼 −𝜃𝑇 ) , 𝑡𝑎𝑛(𝜃𝐼 +𝜃𝑇 ) (1.44) 2𝑐𝑜𝑠𝜃𝐼 𝑠𝑖𝑛𝜃𝑇 𝑠𝑖𝑛(𝜃𝐼 +𝜃𝑇 )𝑐𝑜𝑠(𝜃𝐼 −𝜃𝑇 ) (1.46) 2𝑐𝑜𝑠𝜃𝐼 𝑠𝑖𝑛𝜃𝑇 , 𝑠𝑖𝑛 (𝜃𝐼 +𝜃𝑇 ) (1.45) We exclude the cases of total external and internal reflections, then 𝜃𝐼 and 𝜃𝑇 are real. The transmitted coefficients 𝑡𝑠 in the equation (1.45) and 𝑡𝑝 in the equation (1.46) are also real and positive. Consequently, the phase of the refracted wave is equal to that of the incident wave, i. e., they are in phase. For the reflected wave, if the second medium is optically denser than the first (𝑛2 > 𝑛1 ), then 𝜃𝑇 < 𝜃𝐼 . According to the equations (1.43), 𝑟𝒔 < 0. For the normal component of the electric vector, the phases of the reflected wave 36 Chapter 1. Physical Optics and the incident wave therefore differ by 𝜋, which is always valid. Under the same 𝜋 circumstances if 𝜃𝐼 + 𝜃𝑇 > , then 𝑟𝑝 < 0. In this case the phases of the reflected wave 2 𝜋 and the incident wave also differ by 𝜋 . However, when 𝜃𝐼 + 𝜃𝑇 < 2 , the previous conclusion is not valid. Similar analyses apply when the second medium is optically rarer than the first (𝑛2 < 𝑛1 ). Especially, if we change the order of 𝜃𝐼 and 𝜃𝑇 in the equations (1.43) and (1.44), we know that the sign of 𝑟𝑠 and of 𝑟𝑝 will be changed to the opposite one. This can be also thought of as the 𝜋 phase shift of 𝑟𝑠 and of 𝑟𝑝 relative to each own original phase, respectively. As shown in Fig. 1.11, we now consider a situation that a parallel plane plate is put in a medium with the uniform index of refraction, for example, in the air, encountered in interferometry. The angles of incidence and refraction of ray 1 are, respectively, 𝜃𝐼 and 𝜃𝑇 at the first surface of the plate whereas the angles of incidence and refraction of ray 2 are, respectively, 𝜃𝑇 and 𝜃𝐼 at the second surface of the plate. The second reflection with respect to the first one is equivalent to swap over the order of 𝜃𝐼 and 𝜃𝑇 in the equations (1.43) and (1.44). Such that the relative phase shift between the ray 1 and the ray 2 generated by just reflections is 𝜋. Fig. 1.11. Light reflections from a parallel plane plate for the same incident ray. Based on the above discussion, we know that it is a misleading concept for the conventional statement that a light wave reflected from an optically denser medium undergoes a phase shift of half of a wavelength. 1.2. Scalar Theory of Diffraction Diffraction is any deviation of the light rays from rectilinear paths, predicted by geometrical optics, as they pass through apertures or around obstacles in a uniform medium. It is easy to observe fringes caused by diffraction when the size of the aperture or obstacle is of the same order of magnitude as the wavelength of the incident light wave. The essential features of diffraction phenomena are diffraction fringes which can be qualitatively explained with help of Huygens-Fresnel’s principle. Based on this principle, 37 Advances in Optics: Reviews. Book Series, Vol. 5 the diffraction fringes, in fact, result from the interference of all waves originating from a continuous wavefront. The distinction between diffraction and interference is somewhat vague. In general, the term diffraction refers to the case where all the superposed waves come from a continuous wavefront, and the mathematical formulation of the light field superposed is integral; the term interference refers to the case where separable waves are superposed, and the mathematical formulation of the light field superposed is summation. In this section we try to treat the scalar theory of diffraction in a more quantitative way. Rigorous solutions of diffraction problems encountered in optics are very rare. Because of mathematical difficulties, approximate approaches must be invoked in most of practical problems. Of these the Rayleigh-Sommerfeld approximating formula is adequate for the treatment of the majority of problems encountered in instrumental optics. We will, starting from Helmholtz equation (1.16), obtain the Rayleigh-Sommerfeld formula that casts Huygens’ principle onto a firm mathematical basis. Furthermore, the paraxial formula of the Fresnel diffraction and of the Fraunhofer diffraction is derived from this formula depending on the different propagation distances of light. 1.2.1. The Rayleigh-Sommerfeld Formulation of Diffraction Typically, the optical layout of a plane screen that is opaque except for the open aperture S in a uniform medium is shown in Fig. 1.12. Suppose that the light field 𝑈(𝑃1 ) with wavelength 𝜆 in the aperture S of the plane z1 is known，we want to seek the diffracted light field 𝑈(𝑃) on the right-hand plane z2 with some distance from the plane z1. Fig. 1.12. Optical layout of a plane screen for diffraction. Starting from the Helmholtz equation (1.16) for a monochromatic electromagnetic wave, we can solve the scalar wave equation by applying Green’s theorem and boundary conditions, provided that: a) The aperture S is large in comparison to the light wavelength considered; 38 Chapter 1. Physical Optics b) The distance of the plane for observing the diffracted field from the aperture S is large comparing to the size of the aperture. The detailed derivation of the following formula (1.47) is given in the appendix at the end of this chapter. Here we just give out its final expression of Rayleigh-Sommerfeld diffraction formula as follows. where 𝑘 = 2𝜋 𝜆 1 𝑈(𝑃) = 𝑗𝜆 ∬𝑆 𝑈(𝑃1 ) 𝑒 𝑗𝑘𝑟 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆, 𝑟 (1.47) is the wavenumber of light in the medium, 𝑟 the magnitude of the position vector 𝒓 from the point 𝑃 to the point 𝑃1 ， 𝒏 is the outward normal direction at each point on the aperture S， and 𝑐𝑜𝑠(𝒏, 𝒓) the cosine value of the counter clockwise angle from 𝒓 to 𝒏. The integral area is the light-transmission portion of the aperture represented by formula (1.47) is known as Rayleigh-Sommerfeld formula for diffraction. The formula (1.47) can be pictorially explained as follows. It expresses the observed light field 𝑈(𝑃) as a superposition of diverging spherical waves 𝑒 𝑗𝑘𝑟 𝑟 with the amplitude 1 𝑈(𝑃1 ) 𝜆 originating from secondary source located at every point 𝑃1 within the aperture S. 𝑗 in the 𝜋 formula cancels out the Gouy’s phase shift 2 gained by each secondary wavelet in the propagation of the wavelet. The 𝑐𝑜𝑠(𝒏, 𝒓) indicates the pattern of directivity for each secondary wavelet. Base on this explanation, we can know that the formula (1.47) is just the mathematical expression of Huygens-Fresnel’s principle. Let ℎ(𝑃, 𝑃1 ) = 1 𝑒 𝑗𝑘𝑟 𝑐𝑜𝑠(𝒏, 𝒓), 𝑗𝜆 𝑟 then the formula (1.47) can be written 𝑈(𝑃) = ∬𝑆 ℎ(𝑃, 𝑃1 )𝑈(𝑃1 )𝑑𝑆 (1.48) The above formula says the light field 𝑈(𝑃) on the plane z2 is a linear superposition of light field 𝑈(𝑃1 ) on the plane z1. In the point of view for a linear system, ℎ(𝑃, 𝑃1 ) is called the impulse response for the light propagation (or diffraction) from the plane z1 to the plane z2. The occurrence of this linear superposition is not surprise. It is a result of linearity of the Helmholtz equation (1.16). 1.2.2. Paraxial Approximations for Diffraction Although the Rayleigh-Sommerfeld formula gives a solution to the diffraction problem in general form, it is still difficult to calculate diffraction fields with this formula in practical problems. Attention is now turned to further approximations to the Rayleigh-Sommerfeld formula. The approximations commonly made are Fresnel and Fraunhofer ones that have comparatively simple mathematical manipulations. These two types of approximations can be obtained depending on different approximating conditions that will be, in detail, elaborated as follows. 39 Advances in Optics: Reviews. Book Series, Vol. 5 Before performing further approximations to the Rayleigh-Sommerfeld formula, it is convenient for setting up the Cartesian coordinates. As shown in Fig. 1.13, the diffracting aperture lies in the (x1, y1) plane and is illuminated in the +z direction. We are going to calculate the light field on the (x, y) plane that is parallel to the (x 1, y1) plane and at distance z from it along z-axis. Let the light field of point 𝑃1 on the (x1, y1) plane be 𝑈(𝑥1 , 𝑦1 , 0). While the light wave propagates a certain distance, the light field at point 𝑃 on the observation plane (x, y) is 𝑈(𝑥, 𝑦, 𝑧). Fig. 1.13. Diffraction geometry. For the above Cartesian coordinates the Rayleigh-Sommerfeld diffraction formula (1.47) can be written as 𝑈(𝑥, 𝑦, 𝑧) = 𝑒 𝑗𝑘𝑟 1 𝑈(𝑥 , 𝑦 , 0) 𝑐𝑜𝑠𝜃𝑑𝑥1 𝑑𝑦1 , ∬ 1 1 𝑟 𝑗𝜆 𝑆 (1.49) 𝑧 where 𝑟 = √(𝑥 − 𝑥1 )2 + (𝑦 − 𝑦1 )2 + 𝑧 2 , and 𝑐𝑜𝑠𝜃 = . 𝑟 1.2.2.1. Fresnel Diffraction When the distance z is much greater than the size of the diffracting aperture and the light wavelength, the 𝑟 appearing at the denominators in the formula (1.49) can be replaced by z and the result is accurate enough. The distance 𝑟 appearing in 𝑒 𝑗𝑘𝑟 for the formula (1.49) cannot be approximated to just the first order term because the wavenumber 𝑘 is very large. It can be approximated to the second order terms according to Taylor’s expansion as follows. 𝑟 = √(𝑥 − 𝑥1 )2 + (𝑦 − 𝑦1 1 𝑥−𝑥1 2 ) 𝑧 ≈ 𝑧 [1 + 2 ( 40 )2 + 𝑧2 = 𝑧 [1 + 1 𝑦−𝑦1 2 ) 𝑧 + 2( 1 − 8 [( 𝑥−𝑥 2 ( 𝑧 1) 𝑥−𝑥1 2 ) 𝑧 + 2 𝑦−𝑦1 2 ] ) ] 𝑧 +( 1 𝑦−𝑦 2 2 ( 𝑧 1) ] ≈ (1.50) Chapter 1. Physical Optics On inserting the above different approximations to 𝑟′𝑠 in the formula (1.49), its approximating formulation for the light field at the point P on the observation surface therefore becomes 𝑈(𝑥, 𝑦, 𝑧) = 𝑘 𝑒 𝑗𝑘𝑧 𝑗 [(𝑥−𝑥1 )2 +(𝑦−𝑦1 )2 ] 2𝑧 𝑑𝑥1 𝑑𝑦1 𝑈(𝑥 , 𝑦 , 0)𝑒 ∬ 1 1 𝑗𝜆𝑧 (1.51) The above formula (1.51) is called the Fresnel diffraction approximation, or the Fresnel diffraction integral. To get the formula (1.51), the condition 1 𝑥−𝑥1 2 𝑘𝑧 ) [( 8 𝑧 2 𝑦−𝑦1 2 ) ] 𝑧 +( ≪ 1 should be satisfied. However, the validity of the Fresnel diffraction approximation does not need so strict condition. When the diffraction distance is greater than about 50 , fairly accurate results can be obtained by the Fresnel diffraction approximation. In what follows we will qualitatively justify its legitimacy [16]. For the convenience of statement, we let 𝜀 = 1 𝑥−𝑥1 2 𝑘𝑧 ) [( 8 𝑧 2 𝑦−𝑦1 2 ) ] . When the point 𝑧 √(𝑥1 − 𝑥)2 + (𝑦1 − 𝑦)2 is +( (𝑥1 , 𝑦1 ) explores the diffracting aperture and the distance greater than some value, the function 𝜀 then changes by many wavelengths due to the large number 𝑘. Such that the exponential 𝑒 −𝑗𝜀 in the integral oscillates many times with alternatively positive and negative signs, and therefore the contributions to the integral from this region of the diffraction aperture will be negligible. However, when the point (𝑥1 , 𝑦1 ) on the diffracting aperture is the critical one [1], i.e., 𝜀 is stationary, then the integrand varies much more slowly, so that it significantly contributes the integral of diffraction. Based on the previous analysis, we know that the integral value is substantially contributed by a small region around the critical point (𝑥1 , 𝑦1 ), which is equivalent to the original diffracting aperture becoming the very small region around the critical point. Therefore, the condition 𝜀 ≪ 1 can be satisfied even if the diffraction distance z is just few of tens wavelengths. Equations (1.51) can also be re-written in an equivalent form as the following. = 𝑈(𝑥, 𝑦, 𝑧) = 𝑘 𝑘 𝑒 𝑗𝑘𝑧 𝑗 𝑘 (𝑥 2 +𝑦 2 ) 𝑗 (𝑥1 2 +𝑦1 2 ) −𝑗 (𝑥𝑥1 +𝑦𝑦1 ) 2𝑧 2𝑧 𝑧 𝑒 𝑈(𝑥 , 𝑦 , 0)] 𝑒 𝑑𝑥1 𝑑𝑦1 [𝑒 ∬ 1 1 𝑗𝜆𝑧 (1.52) We can see that, apart from the complex term outside the integration sign, the field 𝑘 2 2 𝑈(𝑥, 𝑦, 𝑧) is the Fourier transform of the term 𝑒 𝑗2𝑧(𝑥1 +𝑦1 ) 𝑈(𝑥1 , 𝑦1 , 0), evaluated at 𝑥 𝑦 two-dimensional spatial frequencies (𝜆𝑧 , 𝜆𝑧). Therefore, Fresnel diffraction integral can be calculated using the fast Fourier transform (FFT). 1.2.2.2. Fraunhofer Diffraction In the condition of the Fresnel diffraction, 𝑟 in phase factor 𝑒 𝑗𝑘𝑟 was approximated to the second order terms as 41 Advances in Optics: Reviews. Book Series, Vol. 5 1 𝑥−𝑥1 2 ) 𝑧 𝑟 = 𝑧 {1 + [( 2 That is 𝑟 = 𝑧+ (𝑥−𝑥1 )2 +(𝑦−𝑦1 )2 2𝑧 = 𝑧+ +( 𝑦−𝑦1 2 ) ]} 𝑧 𝑥 2 +𝑦 2 2𝑧 + 𝑥12 +𝑦12 2𝑧 − 𝑥𝑥1 +𝑦𝑦1 𝑧 If the propagation distance is further increased, and the third term of the above expression 𝑥 2 +𝑦 2 can be ignored, i.e., 𝑘 12𝑧 1 ≪ 1 , which means that the wavefront of light on the diffracting aperture is a plane. Then the formula (1.49) can be further approximated as 𝑈(𝑥, 𝑦, 𝑧) = 2 2 𝑒 𝑗𝑘𝑧 𝑗𝑘𝑥 +𝑦 2𝑧 𝑒 𝑗𝜆𝑧 ∬𝑆 𝑈(𝑥1 , 𝑦1 , 0)𝑒 −𝑗𝑘 𝑥𝑥1 +𝑦𝑦1 𝑧 𝑑𝑥1 𝑑𝑦1 (1.53) The above formula (1.53) is called the Fraunhofer diffraction approximation, or the Fraunhofer diffraction integral. The Fraunhofer diffraction approximation condition is 𝑥 2 +𝑦 2 that 𝑘 12𝑧 1 ≪ 1 . This condition required z be extremely large, so that Fraunhofer diffraction is also called the far field diffraction that is just a special case of Fresnel diffraction. The Fraunhofer diffraction integral (1.53) shows that the diffraction field on the plane (x, y) is the Fourier transformation of the input light field on the plane (x1, y1), except for one complex factor associated with (x, y). The Fraunhofer diffraction field is very far away from the diffracting aperture. For easy observations, we can put a converging lens near to the diffracting aperture between it and the diffraction field, such that the Fraunhofer diffraction pattern is formed onto the focal plane of the converging lens thanks to its imaging property. This is just about the starting point of Fourier optics. The Fraunhofer diffraction is, therefore, the physical basis for Fourier optics of a positive lens. Relying on different approximations taken for the distance 𝑟 , we can delimit the diffraction zone into three regions. The first is the very near zone to the aperture, where the sharp image of the aperture, predicted by geometrical optics, can be formed [16]. The second is the very far zone, where the diffraction of the aperture is governed by Fraunhofer diffraction pattern. The third is in between zone, where the diffraction pattern of the aperture can be predicted by Fresnel diffraction. Therefore, Fresnel diffraction patterns is a transition one between the patterns governed by geometrical optics at one extreme and Fraunhofer diffraction patterns at the other. Fresnel and Fraunhofer diffraction formulas derived from the Rayleigh-Sommerfeld integral through the paraxial approximations at different distances are valid only in certain regions, not very close to the input aperture plane. The Rayleigh-Sommerfeld integral is valid very close to the aperture, but not valid to the distance of the order of one wavelength. The valid regions for the Rayleigh-Sommerfeld integral, Fresnel, and Fraunhofer approximations are shown in Fig. 1.14. 42 Chapter 1. Physical Optics Fig. 1.14. Three diffraction regions. 1.2.3. Diffraction with Nonmonochromatic Light In above, we just considered the Rayleigh-Sommerfeld diffraction formula (1.47) which will serve a fundamental physical law governing the propagation of monochromatic light. In addition, it can be used to find a similar relation for the propagation of nonmonochromatic light. Let us now turn to investigate the diffraction case in which the incident light wave is nonmonochromatic. As shown in Fig. 1.13, we denote the nonmonochromatic light field as an analytic function 𝑈(𝑃, 𝑡) at point 𝑃, and express it in a non-rigorous way using Fourier synthesis as follows ∞ ̃(𝑃, 𝑓) 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑓, 𝑈(𝑃, 𝑡) = ∫−∞ 𝑈 (1.54) ̃(𝑃, 𝑓) the where 𝑓 is the temporal frequency of a monochromatic light wave, and 𝑈 Fourier transform of the light field 𝑈(𝒓, 𝑡) into the frequency domain. Let the ̃(𝑃1 , 𝑓), then we have the following equation monochromatic light field at point 𝑃1 be 𝑈 ̃(𝑃, 𝑓). according to Rayleigh-Sommerfeld diffraction formula (1.47) for 𝑈 ̃(𝑃, 𝑓) = 𝑈 𝑗𝑘𝑟 1 ̃(𝑃1 , 𝑓) 𝑒 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆 ∬𝑈 𝑟 𝑗𝜆 𝑆 (1.55) On inserting the equation (1.55) into the equation (1.54), we have 𝑈(𝑃, 𝑡) = = 1 ∞ 𝑓 ̃(𝑃1 , 𝑓) 𝑒 [∬𝑆 𝑈 ∫ 𝑗 −∞ 𝑣 𝑟 𝑗2𝜋𝑓( −𝑡) 𝑣 𝑟 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆] 𝑑𝑓 = 𝑟 ∞ 1 ̃(𝑃1 , 𝑓)𝑒 −𝑗2𝜋𝑓(𝑡−𝑣) 𝑑𝑓 ] 1 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆 (−𝑗2𝜋𝑓)𝑈 [∫ ∬ 2𝜋𝑣 𝑆 −∞ 𝑟 1 (1.56) 𝑓 In the equation (1.56), we have used the relation = , where 𝑣 is the speed of light in 𝜆 𝑣 the medium. We differentiate the formula (1.54), the following expression can be obtained 𝜕 𝑈(𝑃1 , 𝑡) 𝜕𝑡 ∞ ̃(𝑃1 , 𝑓) 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑓 = ∫−∞(−𝑗2𝜋𝑓)𝑈 (1.57) 43 Advances in Optics: Reviews. Book Series, Vol. 5 We further have 𝜕 𝑈 (𝑃1 , 𝑡 𝜕𝑡 𝑟 ∞ 𝑟 ̃(𝑃1 , 𝑓) 𝑒 −𝑗2𝜋𝑓(𝑡−𝑣) 𝑑𝑓 − ) = ∫−∞(−𝑗2𝜋𝑓)𝑈 𝑣 (1.58) Combining the equation (1.56) and (1.58), we have the diffraction formula for quasi-monochromatic light as follows. 𝑈(𝑃, 𝑡) = 𝑟 1 𝜕 1 ∬ [ 𝑈 (𝑃1 , 𝑡 2𝜋𝑣 𝑆 𝑟 𝜕𝑡 𝑟 (1.59) − 𝑣)] 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆 In the above expression, 𝑣 is the time of light propagation from the point 𝑃1 to the point 𝑃 , i.e., the delayed time. This means that the light field of the point 𝑃 at time 𝑡 is 𝑟 contributed by the light field of the point 𝑃1 at the previous time 𝑡 − . 𝑣 1.3. Coherence In the previous sections of this chapter, we discussed light that behaves in a totally predictable way for all time instants 𝑡 and positions 𝒓 , i.e., the light fields are deterministic. However, perfectly deterministic light never exists in actual physical realities. For actual light there is a range of frequencies 𝛥𝑓. If the center frequency of light 𝛥𝑓 ≪ 1, the light with this feature is called quasi-monochromatic. is 𝑓, and 𝑓 The bandwidth of a light source is mainly caused by the following several physical mechanisms: 1) A finite lifetime of the higher energy level leads to this state not just an exact energy level but with a range of energies according to the uncertainty principle. This energy uncertainty results in the natural linewidth of light which is a fundamental linewidth; 2) High temperature involved in a light source results in a largely thermal motion of the atoms. The light emitted from atoms moving towards or away from the observer will produce a change in frequency due to the Doppler effect. The Maxwell’s distribution of the speed of the thermal motion of atoms, therefore, broadens the spectral linewidth; 3) In a light source, especially for a gas source, there exist collisions between the various molecules. The collision will interrupt the wave train so that the phase correlation between the emitted light before and after the collision is destroyed. In Fourier theory, the collision then broadens the spectral linewidths. In practice, light with a finite bandwidth about the average (or center) optical frequency is, therefore, not fully predictable, such the real light has certain randomness. In addition, the phases of the light field (even if it is monochromatic) emitting from different parts of an extended light source are not in unison, so that the field of light still has a certain randomness. The phase randomness of the light field results in the different degrees of coherence for light fields. 44 Chapter 1. Physical Optics To deal, quantitatively and clearly, with the randomness of a light field, the statistical approach must be invoked. The statistical approach needed for describing coherence is the average intensity and the correlation functions. 1.3.1. Correlation Function: Definition We take the complex functions of space and time to define the correlation function. Let 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) be randomly complex functions at two different positions of space 𝒓 and time 𝑡, for example, and suppose that the observation time for these two functions is very long comparing to the varying time of two functions. Furthermore, we suppose that 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) are stationary, at least wide-sense stationary, and ergodic processes, and let 𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏. The correlation function of 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) is defined as 〈𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)〉, (1.60) where 〈∙〉 stands for the ensemble average, * the conjugate complex, and 𝜏 is the time shift. Based on the assumption of ergodicity, the definition can be further written as 1 𝑇/2 〈𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)〉 = lim ∫−𝑇/2 𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)𝑑𝑡 𝑇 𝑇→∞ (1.61) For convenience of statement, we simply denote two functions 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ), respectively, as 𝑈1 and 𝑈2 . If 𝑈1 and 𝑈2 are different functions, the correlation function is called cross-correlation one; if 𝑈1 with 𝑈2 are the same functions, the correlation function is called autocorrelation one. For most of time, if 𝑈1 and 𝑈2 tend to take both positive values together or negative values together, its correlation function will take on a positive value, being positively correlated. If 𝑈1 and 𝑈2 tend to take the values of opposite signs together for most of time, its correlation function will take on a negative value, being negatively correlated. For all time if 𝑈1 and 𝑈2 randomly take both positive and negative values, such that their product will be as often positive as negative, and its correlation function will be negligible, being uncorrelated. The correlation function, therefore, is a good way to describe the degree of resemblance of the two functions in a single number. In terminology when 𝑈1 and 𝑈2 are correlated, we say that 𝑈1 and 𝑈2 are statistically similar. Particularly, the autocorrelation of a function 𝑈(𝒓, 𝑡) is mathematically expressed as 1 𝑇/2 Γ(𝒓, 𝜏) = 〈𝑈 ∗ (𝒓, 𝑡)𝑈(𝒓, 𝑡 + 𝜏)〉 = 𝑙𝑖𝑚 𝑇 ∫−𝑇/2 𝑈 ∗ (𝒓, 𝑡)𝑈(𝒓, 𝑡 + 𝜏)𝑑𝑡 𝑇→∞ (1.62) 𝛤(𝜏) describes the resemblance of 𝑈(𝒓, 𝑡 + 𝜏) to 𝑈(𝒓, 𝑡) both in magnitude and in phase. 1.3.2. The Concept of Coherence of Light In the previous subsection, we defined the correlation of two general functions varying with time. We now turn to the correlation properties of a light field. In optics, we refer to 45 Advances in Optics: Reviews. Book Series, Vol. 5 the correlation between two values of a light field with different temporal and spatial coordinates as the coherence of the light field. Let functions 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) particularly represent two values of a light field, as shown in Fig. 1.15, then the coherence between 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) is 〈𝑈 ∗ (𝒓1 , 𝑡1 )𝑈(𝒓2 , 𝑡2 〉 which means that, given 𝑈(𝒓1 , 𝑡1 ) , what extent of information for 𝑈(𝒓2 , 𝑡2 ) can be obtained. The better the coherence is, much information for 𝑈(𝒓2 , 𝑡2 ) can be retrieved. Fig. 1.15. Two values, 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ), of a light field. If we confine the light field to be stationary, at least wide sense, then we can write that 𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏, we define the complex degree of coherence of a stationary light field as 𝛾(𝒓1 , 𝒓2 , 𝜏) = √〈𝑈 ∗ (𝒓 〈𝑈 ∗ (𝒓1 ,𝑡)𝑈(𝒓2 ,𝑡+𝜏)〉 1 ,𝑡)𝑈(𝒓1 ,𝑡)〉√〈𝑈 ∗ (𝒓 2 ,𝑡)𝑈(𝒓2 ,𝑡)〉 (1.63) Invoking the Schwarz inequality, we know that 0 ≤ |𝛾(𝒓1 , 𝒓2 , 𝜏)| ≤ 1. According to the properties of the correlation function, we can get the following conclusions. If |𝛾(𝒓1 , 𝒓2 , 𝜏)| = 0, the light field is completely uncorrelated, i.e. incoherent light; if |𝛾(𝒓1 , 𝒓2 , 𝜏)| = 1 , the light field is completely correlated, i.e. coherent light; if 0 < |𝛾(𝒓1 , 𝒓2 , 𝜏)| < 1, the light field is partially correlated, i.e. partially coherent light. In the real world, there only exists partially coherent light, and the other two extreme cases are just ideal situations. a) Temporal coherence Temporal coherence measures the coherence between 𝑈(𝒓, 𝑡1 ) and 𝑈(𝒓, 𝑡2 ), i. e. between two field values of light at the same spatial point r but different times, 〈𝑈 ∗ (𝒓, 𝑡1 )𝑈(𝒓, 𝑡2 〉. Temporal coherence can be used to define a coherence time 𝜏𝑐 , in which time duration we can think that 𝑈(𝒓, 𝑡1 ) and 𝑈(𝒓, 𝑡2 ) have nearly the same behaviors. Its reciprocal is the 1 bandwidth Δ𝑓 of a quasi-monochromatic light wave, i.e. 𝛥𝑓 = 𝜏 . The Wiener-Khintchine theorem 𝑐 A major practical importance of the autocorrelation function defined by the equation (1.62) is that there exists a very useful relationship associated with power spectral density. 46 Chapter 1. Physical Optics The Wiener-Khintchine theorem says that the autocorrelation function,Γ(𝒓, 𝜏), and the corresponding power spectral density, 𝑆(𝒓, 𝑓), form a Fourier transform pair between time domain 𝜏 and frequency domain 𝑓 ∞ (1.64) ∞ (1.65) 𝑆(𝒓, 𝑓) = ∫−∞ 𝛤(𝒓, 𝜏)𝑒 𝑗2𝜋𝑓𝜏 𝑑𝜏, 𝛤(𝒓, 𝜏) = ∫−∞ 𝑆(𝑟, 𝑓)𝑒 −𝑗2𝜋𝑓𝜏 𝑑𝑓 We follow the Goodman’s approach [8] to prove the above relations. Because of the stationarity of random light field 𝑈(𝒓, 𝑡), its Fourier transform does not exist. For defining its Fourier transform, i.e. the spectrum in frequency domain 𝑓, we perform its Fourier transform as ∞ 𝑡 𝑢 𝑇 (𝒓, 𝑓) = ∫−∞ 𝑟𝑒𝑐𝑡( 𝑇1 )𝑈(𝒓, 𝑡1 )𝑒 −𝑗2𝜋𝑓𝑡1 𝑑𝑡1 , ∞ 𝑡 𝑢 𝑇 (𝒓, 𝑓)∗ = ∫−∞ 𝑟𝑒𝑐𝑡( 𝑇2 )𝑈(𝒓, 𝑡2 )∗ 𝑒 𝑗2𝜋𝑓𝑡2 𝑑𝑡2 , 𝑡 (1.66) where 𝑟𝑒𝑐𝑡(𝑇) is a temporal rectangle function with its length 𝑇, * stands for the complex conjugate. The power spectral density can be defined as <𝑢𝑇 ∗ (𝒓,𝑓)𝑢𝑇 (𝒓,𝑓)> 𝑇 𝑇→∞ 𝑆(𝒓, 𝑓) = 𝑙𝑖𝑚 (1.67) On substituting (1.66) into (1.67), we get ∞ 1 <𝑢𝑇 ∗ (𝒓,𝑓)𝑢𝑇 (𝒓,𝑓)> 𝑇 𝑇→∞ 𝑆(𝒓, 𝑓) = 𝑙𝑖𝑚 𝑡 𝑡 = = 𝑙𝑖𝑚 𝑇 ∬−∞ 𝑟𝑒𝑐𝑡( 𝑇1 ) 𝑟𝑒𝑐𝑡( 𝑇2 ) < 𝑈 ∗ (𝒓, 𝑡2 )𝑈(𝒓, 𝑡1 ) > 𝑒 𝑗2𝜋𝑓(𝑡2 −𝑡1) 𝑑𝑡1 𝑑𝑡2 𝑇→∞ Let 𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏, the above expression becomes 𝑆(𝒓, 𝑓) = 1 ∞ 𝑡 𝑡+𝜏 ) 𝑇 = 𝑙𝑖𝑚 𝑇 ∬−∞ 𝑟𝑒𝑐𝑡(𝑇) 𝑟𝑒𝑐𝑡( 𝑇→∞ < 𝑈 ∗ (𝒓, 𝑡 + 𝜏)𝑈(𝒓, 𝑡) > 𝑒 𝑗2𝜋𝑓𝜏 𝑑𝑡 𝑑𝜏 (1.68) We now interchange orders of integration with respect to 𝑑𝜏 and the limit, and have 1 ∞ 𝑡 𝑡+𝜏 ) 𝑇 𝛤(𝒓, 𝜏) = 𝑙𝑖𝑚 𝑇 ∫−∞ 𝑟𝑒𝑐𝑡( )𝑟𝑒𝑐𝑡( 𝑇 𝑇→∞ < 𝑈 ∗ (𝒓, 𝑡 + 𝜏)𝑈(𝒓, 𝑡) > 𝑑𝑡 (1.69) On inserting the expression (1.69) into (1.68), we have the Wiener-Khintchine theorem (1.64). This theorem is the theoretical basis for the Michelson Fourier transform spectroscopy. 47 Advances in Optics: Reviews. Book Series, Vol. 5 b) Spatial coherence Spatial coherence measures the coherence between 𝑈(𝒓1 , 𝑡) and 𝑈(𝒓2 , 𝑡), i. e. between two field values of light at the different spatial points with the same times, 〈𝑈 ∗ (𝒓1 , 𝑡) 𝑈(𝒓2 , 𝑡)〉. By means of the spatial coherence, one can define a coherent area (in two-dimensional space) or a coherent volume (in three-dimensional space). In these coherent regions, 𝑈(𝒓1 , 𝑡) and 𝑈(𝒓2 , 𝑡) can be roughly thought of as having the same behaviors. The van Cittert-Zernike theorem This theorem reveals that when light propagates in space even a spatially incoherent source will generate a field that can become spatially coherent over a large region of space. This theorem is the spatial equivalent of the Wiener-Khintchine theorem (1.64). The van Cittert-Zernike theorem is as follows 𝛾(𝑃1 , 𝑃2 ) = 𝑒 −𝑗𝜑 ∬ 𝐼(𝑥1 ,𝑦1 )𝑑𝑥1 𝑑𝑦1 ∬ 𝐼(𝑥1 , 𝑦1 )𝑒 −𝑗𝑘(𝑝𝑥1 +𝑞𝑦1 ) 𝑑𝑥1 𝑑 𝑦1 (1.70) In what follows we will demonstrate for what every quantity in (1.70) stands. As shown in Fig. 1.16, in the plane denoted by (𝑥, 𝑦) parallel to the source plane S and far from it, there are the field points 𝑃1 and 𝑃2 , the van Cittert-Zernike theorem, expressed by the equation (1.70), reveals the relationship between the intensity distribution of the light source and the complex degree of coherence of the light field in the far field. In the source plane we choose a Cartesian coordinate system with its own origin and denote by (𝑥1 , 𝑦1 ) the coordinates of a point in the source plane; in the plane (𝑥, 𝑦) of the far region of the source we choose a Cartesian coordinate system with its own origin, and with 𝑥 and 𝑦-axes parallel to the 𝑥1 and 𝑦1 -axes. Let us denote that the Cartesian coordinates of the field points 𝑃1 and 𝑃2 , respectively, (𝑋1 𝑌1 ) and (𝑋2 𝑌2 ), and the distance between the origin of the source and the origin of the plane (𝑥, 𝑦) is 𝑅. The distance between the point 𝑆 and the point 𝑃1 is 𝑅1 ; the distance between the point 𝑆 and the point 𝑃2 is 𝑅2 . According to the Rayleigh-Sommerfeld diffraction formula, the light field at the point 𝑃1 from the field 𝑈(𝑆) of the source point 𝑆 is Similar to 𝑈(𝑃2 ). 𝑈(𝑃1 ) = 𝑒 𝑗𝑘𝑅1 1 𝑈(𝑆) 𝑅 ∫ 𝑗𝜆 1 𝑑𝑆 (1.71) 𝑈(𝑃2 ) = 𝑒 𝑗𝑘𝑅2 1 𝑈(𝑆) ∫ 𝑅2 𝑗𝜆 𝑑𝑆 (1.72) In above two expressions, 𝜆 is the center wavelength of the quasi-monochromatic light 2𝜋 wave, 𝑘 = 𝜆 , and the direction factors have been omitted because 𝑅1 and 𝑅2 are very large in comparison with the size of the source 𝑆. 48 Chapter 1. Physical Optics According to the definition of the complex degree of coherence of the expression (1.63), we obtain the following formula for the equal-time complex degree of coherence of light between two field points 𝑃1 and 𝑃2 . 𝛾(𝑃1 , 𝑃2 ) = 1 √𝐼(𝑃1 )𝐼(𝑃2 ∫ 𝐼(𝑆) ) 𝑒 𝑗𝑘(𝑅2 −𝑅1 ) 𝑑𝑆, 𝑅1 𝑅2 (1.73), where 𝐼(𝑆) is the intensity of the point 𝑆 in the source. Fig. 1.16. Notation for the van Cittert-Zernike theorem. And we can further get, from the equations (1.71) and (1.72), that 𝐼(𝑃𝑖 ) = ∫ (𝑖 = 1,2) which are, respectively, light intensities at two field points 𝑃1 and 𝑃2 . 𝐼(𝑆) 𝑑𝑆 𝑅𝑖2 From Fig. 1.16, we have 𝑅12 = (𝑋1 − 𝑥1 )2 + (𝑌1 − 𝑦1 )2 + 𝑅 2, 𝑅22 = (𝑋2 − 𝑥1 )2 + (𝑌2 − 𝑦1 )2 + 𝑅 2 (1.74) The linear dimensions of the light source and the distance between 𝑃1 and 𝑃2 are small comparing to the distance R, so that 𝑅1 = 𝑅 + (𝑋1 −𝑥1 )2 +(𝑌1 −𝑦1 )2 , 2𝑅 𝑅2 = 𝑅 + (𝑋2 −𝑥1 )2 +(𝑌2 −𝑦1 )2 2𝑅 (1.75) In above expressions, we just retained the leading terms of the power-series expansions. It follows from equations (1.75) that We set 𝑅2 − 𝑅1 = (𝑋2 2 +𝑌2 2 )−(𝑋1 2 +𝑌1 2 ) 2𝑅 − 𝑥1 (𝑋2 −𝑋1 )+𝑦1 (𝑋2 −𝑋1 ) 𝑅 (1.76) 49 Advances in Optics: Reviews. Book Series, Vol. 5 𝑝 = 𝜑 = 𝑋2 −𝑋1 ,𝑞 𝑅 = 𝑌2 −𝑌1 , 𝑅 𝑘[(𝑋22 +𝑌22 )−(𝑋12 +𝑌12 )] 2𝑅 (1.77) (1.78) On inserting the formulas (1.76), (1.77), and (1.78) into the formula (1.73), we get the van Cittert-Zernike theorem as the formula (1.70). The van Cittert-Zernike theorem says the equal-time degree of coherence 𝛾(𝑃1 , 𝑃2 ), excepting a phase factor, is equal to the normalized Fourier transform of the intensity distribution across the light source. This theorem provides a basis for the technique of aperture synthesis. In summary, Coherence of light is classified into temporal and spatial coherence for convenience, not for the deep physical difference. Temporal coherence relates directly to the bandwidth of the light source; Spatial coherence to its finite extent of light source. Their applications will be discussed in Section 1.5. 1.4. Fourier Theory and Fourier Optics Fourier Optics is a branch of optical physics that Fourier analysis methods were applied into light and optical systems to calculate the propagation of light and analyze the properties of optical systems. Under certain conditions, optical systems are linear and invariant ones that can easily be processed by Fourier analysis methods. Although the targets processed in Fourier optics are still the propagation of light, interference, imaging, and information processing etc., application of Fourier analysis methods into optics has enabled us to address optical issues in an easy and physically insightful way in the spatial frequency domain. This section will first describe the concept of the Fourier transform. Second, the concept of the angular spectrum, a positive lens, and grating diffraction are discussed by Fourier analysis method. Third, the imaging of coherent and incoherent illumination through the optical system is discussed in the approach of Fourier optics. Finally, a typical optical system for application of Fourier optics, the Fourier Transform Spectrometer, will be studied. 1.4.1. Fourier Theory 1.4.1.1. Analysis of Periodic Functions: Fourier Series The idea of Fourier series is to express a periodic function by the linear combination of a series of orthogonal functions that form a set of basis in a function space. For Fourier series, its basis is a set of trigonometric functions or complex exponential functions relying on the representation formalism of Fourier series. 50 Chapter 1. Physical Optics Suppose that the period of a periodic function 𝑔(𝑥) is , and furthermore 𝑔(𝑥) has limited extreme points and discontinuity points and is absolutely integrable. Then 𝑔(𝑥) can be expanded into Fourier series. 𝑔(𝑥) = 1 𝑎0 2 + ∑∞ 𝑛 = 1[𝑎𝑛 𝑐𝑜𝑠(2𝜋𝑛𝑓0 𝑥) + 𝑏𝑛 𝑠𝑖𝑛(2𝜋𝑛𝑓0 𝑥)], (1.79) where 𝑓0 = 𝜏 , being called the fundamental frequency of 𝑔(𝑥), and n’s are called the orders of the terms that are harmonics. 𝑎𝑛 = 2 𝜏 ∫ 𝑔(𝑥) 𝑐𝑜𝑠(2𝜋𝑛𝑓0 𝑥) 𝑑𝑥 𝜏 0 𝑎𝑛 , 𝑏𝑛 are called Fourier coefficients. , 𝑏𝑛 = 2 𝜏 ∫ 𝑔(𝑥) 𝑠𝑖𝑛(2𝜋𝑛𝑓0 𝑥) 𝑑𝑥, 𝜏 0 The expression (1.79) shows that the periodic function 𝑔(𝑥) can be represented as a linear combination of infinite cosine and sine functions that have different frequencies. According to the Euler’s formula 𝑒 𝑗𝑥 = 𝑐𝑜𝑠𝑥 + 𝑗𝑠𝑖𝑛𝑥 , the Fourier series in the trigonometric form can be rewritten as the exponential series. where 𝑐𝑛 = 𝑗2𝜋𝑛𝑓0 𝑥 𝑔(𝑥) = ∑∞ , 𝑛 = −∞ 𝑐𝑛 𝑒 (1.80) 1 𝜏 ∫ 𝑔(𝑥)𝑒 −𝑗2𝜋𝑛𝑓0 𝑥 𝑑𝑥. 𝜏 0 The expression (1.80) shows that a periodic function 𝑔(𝑥) can be represented as a linear combination of infinite complex exponential functions that have different frequencies. The Fourier coefficients 𝑐𝑛 are the weight coefficients for each complex exponential function. The above two kinds of series expansions of a periodic function are called Fourier series. A function with a period of  can be expanded into a Fourier series. Its fundamental 1 𝑛 frequency 𝑓0 = , the harmonic frequencies 𝑛𝑓0 = , and the frequency interval of the 𝜏 adjacent frequencies is ∆𝑓 = 𝑓0 = 1 . 𝜏 𝜏 1.4.1.2. Analysis of Non-periodic Functions: Fourier Transform If a function 𝑔(𝑥) is a non-periodic one, it can be regarded as a periodic function with an infinite period. When 𝜏 → ∞，∆𝑓 can be expressed as 𝑑𝑓. Discrete frequencies 𝑛𝑓0 turn to a continuous frequency 𝑓. The sum of polynomials in (1.80) can be rewritten into an integral as the below. ∞ where 𝑔(𝑥) = ∫−∞ 𝐺(𝑓) 𝑒 𝑗2𝜋𝑓𝑥 𝑑𝑓, (1.81) 51 Advances in Optics: Reviews. Book Series, Vol. 5 ∞ 𝐺(𝑓) = ∫−∞ 𝑔(𝑥) 𝑒 −𝑗2𝜋𝑓𝑥 𝑑𝑥 (1.82) The expressions (1.81) and (1.82) are called a Fourier transform pair. The formula (1.82) of 𝐺(𝑓) obtained by 𝑔(𝑥) is called the Fourier transform or Fourier analysis and can be, in shorthand, denoted as 𝐺(𝑓) = 𝐹{𝑔(𝑥)}. The formula (1.81) of 𝑔(𝑥) obtained by 𝐺(𝑓) is called the Inverse Fourier transform or Fourier synthesis and can be, in shorthand, denoted as 𝑔(𝑥) = 𝐹 −1 {𝐺(𝑓)}. Furthermore 𝐺(𝑓) is also called the spectrum of 𝑔(𝑥), and 𝑔(𝑥) the original function of 𝐺(𝑓). Because two-dimensional spatial functions are common in optics ， the definition of two-dimensional Fourier transform is given below. Let 𝑔(𝑥, 𝑦) be the original function and its spectrum function 𝐺(𝑓𝑥 , 𝑓𝑦 ). The Fourier transform of the function 𝑔(𝑥, 𝑦) is ∞ 𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔(𝑥, 𝑦)} = ∫ ∫−∞ 𝑔(𝑥, 𝑦)𝑒 −𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) 𝑑𝑥𝑑𝑦, (1.83) where 𝑓𝑥 , 𝑓𝑦 are spatial frequencies. The Inverse Fourier transform of the function 𝐺(𝑓𝑥 , 𝑓𝑦 ) is ∞ 𝑔(𝑥, 𝑦) = 𝐹 −1 {𝐺(𝑓𝑥 , 𝑓𝑦 )} = ∫ ∫−∞ 𝐺(𝑓𝑥 , 𝑓𝑦 )𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) 𝑑𝑓𝑥 𝑑𝑓𝑦 (1.84) The mutual Fourier transform between the function 𝑔(𝑥, 𝑦) and the function 𝐺(𝑓𝑥 , 𝑓𝑦 ) is usually denoted as 𝑔(𝑥, 𝑦) ⇔ 𝐺(𝑓𝑥 , 𝑓𝑦 ). Fourier transform has some basic mathematical properties that are described by the theorem of linearity, of similarity, of phase shift, of the convolution, and of correlation. Readers can consult reference [6]. These theorems can greatly simplify the manipulations of the Fourier transform. 1.4.2. Fourier Optics 1.4.2.1. The Concept of Angular Spectrum The spatial distribution of light field obeys the Helmholtz equation (1.16). Plane waves are the simplest solution to this equation. Plane waves have the same amplitude and phase in the plane perpendicular to the propagation direction of the light wave. The scalar light field of a monochromatic plane wave with the amplitude 𝐴 can be mathematically denoted as 𝑢(𝒓, 𝑡) = 𝐴𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) . The factor related to its spatial position is 𝑢(𝒓) = 𝐴𝑒 𝑗𝒌∙𝒓 , where 𝒓 = (𝑥, 𝑦, 𝑧) is the spatial coordinate in the three-dimensional space, 𝒌 the 2𝜋 ̂ wave-vector in the medium through which light is propagating, given by 𝒌 = 𝜆 𝒌 , 𝜆 the ̂ ̂ as wavelength of the light wave, and 𝒌 the unit vector of the wave-vector 𝒌. We denote 𝒌 52 Chapter 1. Physical Optics ̂ = (𝑐𝑜𝑠𝛼, 𝑐𝑜𝑠𝛽, 𝑐𝑜𝑠𝛾). Then the expression 𝑢(𝒓) = 𝐴𝑒 𝑗𝒌∙𝒓 can its direction cosines 𝒌 be rewritten as 𝑈(𝑥, 𝑦, 𝑧) = 𝐴𝑒 𝑗𝒌∙𝒓 = 𝐴𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽+𝑧 𝑐𝑜𝑠 𝛾) (1.85) For most of optical systems, the object and image planes are perpendicular to the optical axis. We are concerned with the distribution of light fields in these planes. Let the optical axis as z-axis, we discuss the distribution of light field in the 𝑧 = 𝑧1 plane. Inserting 𝑧 = 𝑧1 into formula (1.85), we have 𝑈(𝑥, 𝑦) = 𝐴𝑒 𝑗𝑘𝑧1 √1−𝑐𝑜𝑠 2 𝛼−𝑐𝑜𝑠 2 𝛽 𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽) (1.86) For the above formula, we have used the relation 𝑐𝑜𝑠 2 𝛼 + 𝑐𝑜𝑠 2 𝛽 + 𝑐𝑜𝑠 2 𝛾 = 1. For the 2 2 plane 𝑧1 , 𝐴𝑒 𝑗𝑘𝑧1 √1−𝑐𝑜𝑠 𝛼−𝑐𝑜𝑠 𝛽 is a complex constant just depending on 𝑧1 but irrelevant of 𝑥 and 𝑦, and we denote it as 𝑈0 , so that the expression (1.86) becomes 𝑈(𝑥, 𝑦) = 𝑈0 𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽) (1.87) The isophase line equation of the plane wave in the 𝑧 = 𝑧1 plane is 𝑘(𝑥 𝑐𝑜𝑠 𝛼 + 𝑦 𝑐𝑜𝑠 𝛽) = 𝐶, where 𝐶 is a constant. Each 𝐶 value corresponds to a contour line representing an isophase one. These isophase lines are parallel to each other. From the equation (1.87), it is obvious that the period of the isophase lines is 2𝜋. For a group of isophase lines whose 𝐶 values differ by 2𝜋 , the spatial period between them in the direction of light wave propagation is wavelength λ, and then the spatial periods in 𝑥 and 𝑦 directions are, respectively, as shown in Fig. 1.17. 𝑑𝑥 = 2𝜋 𝑘𝑐𝑜𝑠𝛼 = 𝜆 , 𝑐𝑜𝑠𝛼 𝑑𝑦 = 2𝜋 𝑘𝑐𝑜𝑠𝛽 = 𝜆 𝑐𝑜𝑠𝛽 Fig. 1.17. Spatial periods in 𝑥 and 𝑦 directions. Then the spatial frequencies corresponding to 𝑥 and 𝑦 directions are, respectively 53 Advances in Optics: Reviews. Book Series, Vol. 5 𝑓𝑥 = 1 𝑑𝑥 = 𝑐𝑜𝑠𝛼 , 𝑓𝑦 𝜆 = 1 𝑑𝑦 = 𝑐𝑜𝑠𝛽 𝜆 (1.88) On substituting the equation (1.88) into the equation (1.87), we can get the complex amplitude distribution of a plane wave expressed by spatial frequency 𝑈(𝑥, 𝑦) = 𝑈0 𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) (1.89) Comparing the kernel, 𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) , of the inverse Fourier transform (1.84) with the expression of the plane wave (1.89), we can see that the kernel has the same form as the expression of plane wave (1.89), i.e., the kernel in the inverse Fourier transform is a plane wave with a unit amplitude in the direction of 𝑐𝑜𝑠 𝛼 = 𝜆𝑓𝑥 ， 𝑐𝑜𝑠 𝛽 = 𝜆𝑓𝑦 . If 𝑔(𝑥, 𝑦) is a light field distribution on some plane, it can be regarded as the weighted superposition of numerous plane waves in different directions. In other words, Fourier transform is a process of decomposing a complex light wave into a series of plane waves with different spatial frequencies. The weight factor 𝐺(𝑓𝑥 , 𝑓𝑦 ) is the spectrum of the light field distribution 𝑔(𝑥, 𝑦). Since 𝑓𝑥 = 𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛽 𝑐𝑜𝑠𝛼 ，𝑓𝑦 𝜆 = 𝑐𝑜𝑠𝛽 𝜆 , 𝐺(𝑓𝑥 , 𝑓𝑦 ) can be written as 𝐺 ( 𝜆 , 𝜆 ), which means that the propagation direction of one plane wave component is proportional to its spatial frequency, therefore it is called the angular spectrum of the light field distribution 𝑔(𝑥, 𝑦). 1.4.2.2. Fourier Transform of a Positive Lens Lenses are the most common elements in optical systems. Here we will in detail study a positive lens that plays the role in Fourier optics. In Section 1.2, we qualitatively explained that a positive lens could perform Fourier transform due to the property of Fraunhofer diffraction. In the following, we will rigorously prove in the case for the thin lens that the positive lens can perform Fourier transform under certain conditions. We will firstly discuss the phase transformation of the lens, and then the property of the Fourier transform of the lens. We suppose that there is no energy loss when light passes through the lens, such that the lens only changes the phase distribution of the incident light wave. Fig. 1.18 shows the imaging of a point light source by a positive lens. Without loss of generality, we just consider the case that the lens is in air. If the diffraction and aberrations of the lens with a limited aperture are not considered, the object point 𝑃 is ideally imaged at the point 𝑃′ through a positive thin lens. Therefore, the optical path lengths of all light rays from the object point 𝑃 to the image point 𝑃′ are the same. Furthermore, δ0 and δ are optical path lengths of the on-axis and off-axis light ray from the plane 𝑈𝑙 , lying on the front surface of the lens, to the plane 𝑈𝑙′ , lying on the behind surface of the lens, and the geometric meaning of all other quantities is shown in Fig. 1.18. Thus we have the following equation 54 𝑑0 + 𝛿0 + 𝑑𝑖 = √𝑑02 + (𝑥 2 + 𝑦 2 ) + 𝛿 + √𝑑𝑖2 + (𝑥 2 + 𝑦 2 ) (1.90) Chapter 1. Physical Optics Further manipulations we get the expression for 𝛿 − 𝛿0 as follows 𝛿 − 𝛿0 = 𝑑0 + 𝑑𝑖 − √𝑑02 + (𝑥 2 + 𝑦 2 ) − √𝑑𝑖2 + (𝑥 2 + 𝑦 2 ) (1.91) In the above formula, the term, 𝛿 − 𝛿0 , stands for the variation of the optical path length of the off-axis light ray with respect to the on-axis light ray. Fig. 1.18. Imaging scheme of a lens. Expanding the terms in the square root sign of the formula (1.91) into Taylor’s series up to the second order terms in the paraxial approximation, the formula (1.91) can be simplified as the following. 𝛿 − 𝛿0 = − (𝑥 2 +𝑦2 ) 2𝑑0 − (𝑥 2 +𝑦 2 ) 2𝑑𝑖 1 𝑑𝑖 1 2 = − (𝑥 2 + 𝑦 2 ) ( + 1 ) 𝑑0 (1.92) The sign convention of the right-handed Cartesian coordinate system has been applied in the equations (1.90) to (1.92). Therefore, under the condition of the paraxial approximation the variation of the optical path length caused by the lens is given by the formula (1.92). Invoking the imaging property of the lens, we know that the object distance 𝑑0 and the image distance 𝑑𝑖 follow 1 1 1 + 𝑑 = 𝑓 , where 𝑓 is the focal length of the lens, so that the the imaging formula 𝑑𝑖 0 transmittance function of the lens can be expressed as where 𝑘 is the wavenumber. 𝑡𝑙 (𝑥, 𝑦) = 𝑒 −𝑗𝑘 𝑥2 +𝑦2 2𝑓 , (1.93) Although the above formula is obtained in the case of a positive lens, it is also applicable to a negative lens. Under the condition of the paraxial approximation, the lens without loss is just a phase converter in the optical system. We will demonstrate that a lens performs Fourier transform in the following two cases: the object is in front of the lens, and the object is behind the lens. 55 Advances in Optics: Reviews. Book Series, Vol. 5 As shown in Fig. 1.19, the focal length of a positive lens in air is f. The object with the amplitude transmissivity, 𝑡(𝑥0 , 𝑦0 ), is placed at a finite distance 𝑑 in front of the lens and normally illuminated by a plane wave with the amplitude 𝐴. The light field just behind the object is then 𝑈0 (𝑥0 , 𝑦0 ) = 𝐴𝑡(𝑥0 , 𝑦0 ). Fig. 1.19. Fourier transform of a lens for 𝑈1 . According to the Fresnel diffraction formula (1.52), the light field distribution at the front surface, 𝑈1 , of the lens is 𝑈1 (𝑥1 , 𝑦1 ) = 𝑘 𝐴𝑒 𝑗𝑘𝑑 𝑗 [(𝑥1 −𝑥0 )2 +(𝑦1 −𝑦0 )2 ] 2𝑑 𝑡(𝑥 , 𝑦 )𝑒 𝑑𝑥0 𝑑𝑦0 , ∬ 0 0 𝑗𝜆𝑑 (1.94) where (𝑥1 , 𝑦1 ) is the coordinate in the plane, 𝑈1 , of the lens. By means of the transmittance function (1.94) of the lens, the light field distribution just behind the lens, 𝑈2 , can be obtained as 𝑈2 (𝑥1 , 𝑦1 ) = 𝑈1 (𝑥1 , 𝑦1 )𝑒 2 2 𝑥 +𝑦 −𝑗𝑘 1 1 2𝑓 (1.95) Due to the thin lens, the light ray coordinates at the front and back surfaces of the lens are both (𝑥1 , 𝑦1 ). From the back surface, 𝑈2 , of the lens to the back focal plane, with help of the Fresnel diffraction formula again, then the complex amplitude distribution of the light wave on the back focal plane is 𝑈𝑓 (𝑥, 𝑦) = 𝑘 𝑗 [(𝑥−𝑥1 )2 +(𝑦−𝑦1 )2 ] 𝑒 𝑗𝑘𝑓 𝑈2 (𝑥1 , 𝑦1 )𝑒 2𝑓 𝑑𝑥1 𝑑𝑦1 , ∬ 𝑗𝜆𝑓 (1.96) where (𝑥, 𝑦) are the coordinates of the back focal plane. By taking (1.95) into (1.96), we can get 𝑈𝑓 (𝑥, 𝑦) = 𝑘 2𝜋 −𝑗 (𝑥1 𝑥+𝑦1 𝑦) ∞ 𝑒 𝑗𝑘𝑓 𝑗2𝑓(𝑥 2 +𝑦2 ) 𝑑𝑥1 𝑑𝑦1 𝑒 ∫ ∫−∞ 𝑈1 (𝑥1 , 𝑦1 ) 𝑒 𝜆𝑓 𝑗𝜆𝑓 (1.97) Obviously, the integral in the above formula is the Fourier transform of 𝑈1 (𝑥1 , 𝑦1 ). By means of the shorthand notation of the Fourier transform, the expression (1.97) can be written 56 Chapter 1. Physical Optics 𝑈𝑓 (𝑥, 𝑦) = = 𝑘 𝑒 𝑗𝑘𝑓 𝑗2𝑓(𝑥 2 +𝑦 2 ) 𝐹{𝑈1 (𝑥1 , 𝑦1 )}, 𝑒 𝑗𝜆𝑓 (1.98) According to the convolution theorem, from the expression (1.94) we have 2 2 𝐹{𝑈1 (𝑥, 𝑦)} = 𝐴𝑇(𝑓𝑥 , 𝑓𝑦 )𝑒 𝑗𝑘𝑑 𝑒 −𝑗𝜋𝜆𝑑(𝑓𝑥 +𝑓𝑦 ) , where 𝑇(𝑓𝑥 , 𝑓𝑦 ) is the Fourier transform of the transmittance function 𝑡(𝑥0 , 𝑦0 ) , and 𝑦 𝑥 𝑓𝑥 = 𝜆𝑓, 𝑓𝑦 = 𝜆𝑓. On inserting the above expression into the equation (1.98), and considering 𝑥 𝑦 𝑓𝑥 = 𝜆𝑓 , 𝑓𝑦 = , then we have the following expression. 𝜆𝑓 𝑈𝑓 (𝑥, 𝑦) = 𝑘 𝑑 𝐴𝑒 𝑗𝑘(𝑑+𝑓) 𝑗2𝑓(1−𝑓)(𝑥 2 +𝑦 2 ) 𝑥 𝑦 𝑇 (𝜆𝑓 , ) 𝑒 𝑗𝜆𝑓 𝜆𝑓 (1.99) From this formula, we can get the following conclusions. If a lens is placed at a finite distance behind the object, the product of the quadratic phase factor and the Fourier transform of the object's transmittance function can be obtained on the back focal plane of the lens, such that it is an inexact Fourier transform. If the object is just lying on the front focal plane of the lens, i.e. 𝑑 = 𝑓，and the quadratic phase factor disappears, the formula (1.99) can be simplified as 𝑈𝑓 (𝑥, 𝑦) = 𝑥 𝑦 𝐴𝑒 𝑗2𝑘𝑓 𝑇 (𝜆𝑓 , 𝜆𝑓) 𝑗𝜆𝑓 (1.100) The above formula means that if the object is placed on the front focal plane of the lens, and the light field on the back focal plane is the exact Fourier transform of the light field transmitted through the object. If the object is placed behind the lens, it is still illuminated normally by a plane wave with amplitude 𝐴, as shown in Fig. 1.20. Under a paraxial approximation, the light wave 𝐴𝑓 𝑘 2 2 𝐴𝑓 illuminating the object is 𝑑 𝑒 −𝑗2𝑑(𝑥0 +𝑦0 ) ., in which the factor 𝑑 is the amplitude of the spherical wave impinging on the object, due to the fact that the linear dimension of the circular converging bundle of rays has been reduced by this factor. Then the amplitude of the wave transmitted by the input could be written 𝐴𝑓 −𝑗 𝑘 (𝑥0 2 +𝑦02 ) 𝑒 2𝑑 𝑡(𝑥0 , 𝑦0 ). 𝑑 The propagation of the light wave from the object to the back focal plane can be calculated with help of the Fresnel diffraction formula (1.52). Thus, the field distribution on the focal plane behind the lens can be obtained. 𝑈𝑓 (𝑥, 𝑦) = 𝑘 2 2 𝑥 𝑦 𝐴𝑓 𝑒 𝑗2𝑑(𝑥 +𝑦 ) 𝑇 (𝜆𝑑 , 𝜆𝑑), 2 𝑗𝜆𝑑 (1.101) 57 Advances in Optics: Reviews. Book Series, Vol. 5 𝑥 𝑦 where 𝑇 (𝜆𝑑 , 𝜆𝑑) is the spectrum of the object's transmittance function 𝑡(𝑥0 , 𝑦0 ). The expression (1.101) shows that when the object is placed behind the lens, up to the quadratic phase factor, the light field on the back focal plane of the lens is the Fourier 𝑥 𝑦 transform of the light field from the object. In this case, 𝑓𝑥 = , 𝑓𝑦 = the size of the 𝜆𝑑 𝜆𝑑 light spot on the back focal plane can be changed by changing d. In the optical information processing, this flexibility will bring a lot of convenience. Fig. 1.20. Fourier transform of a lens for 𝑈0 . In summary, no matter is the object in front of, against, or behind the lens at a finite distance from it, the light field distribution on the back focal plane of the lens is, up to a quadratic phase factor, the Fourier transform of the field distribution transmitted through the object. If the intensity distribution of the light field on the back focal plane of the lens is just concerned, the quadratic phase factor disappears for the intensity. Therefore, the power spectrum of the light field from the object can be obtained on the back focal plane of the lens. Furthermore, with help of the concept of the angular spectrum in the formula (1.88), the Fourier transform of the light field transmitted through the object is, in essence, to decompose the light field into plane waves with different directions. Then these plane waves with different directions are converged to different positions to form diffraction spots on the back focal plane of the lens. The higher the frequency is, the larger the diffraction angle with respect to the optical axis of the lens, as shown in Fig. 1.21. The intensity of each diffraction spot is proportional to the square of its own Fourier coefficient. The Fourier spectrum of the light field from the object appears directly on the back focal plane of the lens, so that the back focal plane of the lens is often called Fourier spectrum plane. In the above discussion, the limitation of lens aperture is ignored. This limited aperture of the lens may cause the distortion of the object spectrum, because the lens is actually a low-pass filter: the low frequency signal can pass through, the cut-off frequency one can partially pass through, and the high frequency one cannot pass through. Therefore, due to the influence of the limited aperture of the lens, the object spectra cannot be fully obtained 58 Chapter 1. Physical Optics on the back focal plane. The higher the frequency is, the greater the spectrum loss. This phenomenon is called vignetting effect. Fig. 1.21. Optical system for Fourier transform. 1.4.2.3. Fourier Optics for Analyzing Diffraction Gratings A diffraction grating is an important optical element in optical instruments or optical information processing systems. Any object with the spatial periodic structure can be used as a grating. Gratings can modulate the amplitude or phase of a light wave periodically. Next, we will discuss two kinds of plane diffraction gratings: cosine and Ronchi’s gratings. As shown in Fig. 1.22, the transmittance function in the plane (𝑥0 , 𝑦0 ) for a cosine 1 amplitude grating with an infinite length and the grating constant is mathematically 𝑓0 expressed as 𝑡(𝑥0 , 𝑦0 ) = 1 1 + 2 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 ) 2 (1.102) Fig. 1.22. A cosine amplitude grating. If the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens, a plane light beam with the amplitude 𝐴 is normally incident on it, then the diffraction light field on the back 59 Advances in Optics: Reviews. Book Series, Vol. 5 focal plane, (𝑥, 𝑦),of the lens with the focal length 𝑓 can be calculated by the equation (1.100). On inserting the transmittance function (1.102) into (1.100), then we have 𝑈𝑓 (𝑥, 𝑦) = = 𝐴𝑒 𝑗2𝑘𝑓 𝐴𝑒 𝑗2𝑘𝑓 1 1 𝐹{𝑡(𝑥0 , 𝑦0 )} = 𝐹 { + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )} = 𝑗𝜆𝑓 2 2 𝑗𝜆𝑓 𝑦 𝑥 1 𝑥 1 𝐴𝑒 𝑗2𝑘𝑓 𝛿 (𝜆𝑓) [2 𝛿 (𝜆𝑓) + 4 𝛿 (𝜆𝑓 𝑗𝜆𝑓 = 1 𝑥 − 𝑓0 ) + 4 𝛿 (𝜆𝑓 + 𝑓0 )] = 𝐴𝑒 𝑗2𝑘𝑓 1 1 1 𝛿(𝑓𝑦 ) [ 𝛿(𝑓𝑥 ) + 𝛿(𝑓𝑥 − 𝑓0 ) + 𝛿(𝑓𝑥 + 𝑓0 )] 𝑗𝜆𝑓 2 4 4 The above formula indicates that a cosine amplitude grating with an infinite length illuminated by a plane wave can obtain three δ functions at different positions on the focal plane behind the lens, as shown in Fig. 1.23. The first term in the square bracket corresponds to the zero order component of diffraction at x = 0; the second term to the +1 order of diffraction; the third term to -1 order of diffraction. The zero order component has no dispersion, and the other two components have dispersion. Fig. 1.23. Diffraction distribution of a cosine amplitude grating. The transmittance function of the grating can be rewritten as the following form by Euler’s 1 1 1 1 formula 𝑡(𝑥0 , 𝑦0 ) = [1 + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥1 )] = + 𝑒 𝑗2𝜋𝑓0 𝑥1 + 𝑒 −𝑗2𝜋𝑓0 𝑥1 . 2 2 4 4 It can be seen that the light field behind the grating contains three plane wave components, 1 1 1 represents the plane wave propagating along the optical axis, and 4 𝑒 𝑗2𝜋𝑓0 𝑥1 , 4 𝑒 −𝑗2𝜋𝑓0 𝑥1 2 a pair of inclined plane waves with a certain angle between the propagation direction and the optical axis and symmetrical about the optical axis. The three plane waves correspond to the frequency components of 0, 𝑓0, −𝑓0 in the complex amplitude distribution. The lens converges the three plane waves on the back focal plane to form three focal points at different positions, namely the above δ functions, which are called 0-order, 1-order and -1-order components, respectively, as shown in Fig. 1.24. 60 Chapter 1. Physical Optics If the size of the grating discussed above is limited, and assume that the grating is a square with the side length L, as shown in Fig. 1.25, then the transmittance function can be expressed as 1 1 𝑥 𝑦 𝑡(𝑥0 , 𝑦0 ) = [2 + 2 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )] 𝑟𝑒𝑐𝑡( 𝐿0 )𝑟𝑒𝑐𝑡( 𝐿0 ) (1.103) If the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens with the focal length 𝑓 and illuminated normally by a plane wave with the amplitude 𝐴, the light field on the back focal plane, (𝑥, 𝑦), of the lens is = 𝑈𝑓 (𝑥, 𝑦) = 𝐴𝑒 𝑗2𝑘𝑓 𝐹{𝑡(𝑥0 , 𝑦0 )} = 𝑗𝜆𝑓 𝑥0 𝑦0 1 1 𝐴𝑒 𝑗2𝑘𝑓 𝐹 {[ + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )] 𝑟𝑒𝑐𝑡( )𝑟𝑒𝑐𝑡( )} 2 2 𝐿 𝐿 𝑗𝜆𝑓 Fig. 1.24. Grating diffraction. Fig. 1.25. Cosine amplitude grating with the finite aperture. By substituting the Fourier transforms of 𝑐𝑜𝑠( ∗), and 𝑟𝑒𝑐𝑡(∗), furthermore applying the convolution theorem and the scaling transform theorem of the Fourier transform, we can get 61 Advances in Optics: Reviews. Book Series, Vol. 5 𝑈𝑓 (𝑥, 𝑦) = = = = 1 1 𝐴𝑒 𝑗2𝑘𝑓 1 {[2 𝛿(𝑓𝑥 ) + 4 𝛿(𝑓𝑥 − 𝑓0 ) + 4 𝛿(𝑓𝑥 + 𝑓0 )] ∗∗ 𝐿2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 )} 𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) = 𝑗𝜆𝑓 1 1 𝐴𝑒 𝑗2𝑘𝑓 𝐿2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) {𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 ) + 2 𝑠𝑖𝑛𝑐[𝐿(𝑓𝑥 − 𝑓0 )] + 2 𝑠𝑖𝑛𝑐[𝐿(𝑓𝑥 + 𝑓0 )]} = 𝑗𝜆𝑓 2 𝐴𝑒 𝑗2𝑘𝑓 𝐿2 𝑦 𝑥 1 𝑥 1 𝑥 𝑠𝑖𝑛𝑐(𝐿 ) {𝑠𝑖𝑛𝑐(𝐿 ) + 𝑠𝑖𝑛𝑐 [𝐿( − 𝑓0 )] + 𝑠𝑖𝑛𝑐 [𝐿( + 𝑓0 )]} 𝑗𝜆𝑓 2 𝜆𝑓 𝜆𝑓 2 𝜆𝑓 2 𝜆𝑓 In the above formula the notation, ∗∗, represents the operation of convolution. This formula shows that due to the limitation of diffraction grating size, the complex amplitude of the light field on the back focal plane with the focal length 𝑓 is no longer the superposition of three δ functions, but of three 𝑠𝑖𝑛𝑐 functions with the same width and different position. The intensity distribution of the light field on the back focal plane is 𝐼(𝑥, 𝑦) = 𝑈𝑓 (𝑥, 𝑦)𝑈𝑓 ∗ (𝑥, 𝑦) = 𝐴𝐿2 2 𝐿𝑦 𝐿𝑥 1 1 𝐿 𝐿 2 = (2𝜆𝑓) 𝑠𝑖𝑛𝑐 2 (𝜆𝑓) {𝑠𝑖𝑛𝑐(𝜆𝑓) + 2 𝑠𝑖𝑛𝑐 [𝜆𝑓 (𝑥 − 𝑓0 𝜆𝑓)] + 2 𝑠𝑖𝑛𝑐 [𝜆𝑧 (𝑥 + 𝑓0 𝜆𝑓)]} For general gratings, the size of a grating is much larger than the grating constant, i.e. 2𝜆𝑓 2 𝐿 ≫ 𝑓 , then 𝑓0 𝜆𝑓 >> 𝐿 . Here we should note that 𝑓0 represents frequency and 𝑓 the 0 focal length of the lens. In this case, the value of overlapped part of 𝑠𝑖𝑛𝑐 functions is small, so the cross multiplication terms in the above formula can be ignored, then we have the following formula. 𝐴𝐿2 2 𝐿𝑦 𝐼(𝑥, 𝑦) = (2𝜆𝑓) 𝑠𝑖𝑛𝑐 2 (𝜆𝑓)⋅ 𝐿𝑥 1 𝐿 1 𝐿 ⋅ {𝑠𝑖𝑛𝑐 2 (𝜆𝑓) + 4 𝑠𝑖𝑛𝑐 2 [𝜆𝑓 (𝑥 − 𝑓0 𝜆𝑓)] + 4 𝑠𝑖𝑛𝑐 2 [𝜆𝑓 (𝑥 + 𝑓0 𝜆𝑓)]} (1.104) And the diffraction pattern is as shown in Fig. 1.26. Fig. 1.26. Light intensity distribution from a cosine grating. Next we discuss Ronchi amplitude grating. The transmittance function for a Ronchi grating with grating constant 𝑑 is shown in Fig. 1.27. 62 Chapter 1. Physical Optics If its size is a square with side length L, the transmittance function can be written as 𝑥 𝑥 1 𝑥 𝑦 (1.105) 𝑡(𝑥0 , 𝑦0 ) = {𝑟𝑒𝑐𝑡 ( 𝑎0 ) ∗∗ [𝑑 𝑐𝑜𝑚𝑏 ( 𝑑0 )]} 𝑟𝑒𝑐𝑡 ( 𝐿0 ) 𝑟𝑒𝑐𝑡 ( 𝐿0 ) In the above formula comb(∗) is called the comb or shah function [7]. Its concrete 𝑥 expression is that 𝑐𝑜𝑚𝑏 ( 0 ) = 𝑑∑𝛿(𝑥0 − 𝑛𝑑) , 𝑛 = 0,1,2, … ∞ . And the Fourier 𝑑 1 𝑥 transform of 𝑑 𝑐𝑜𝑚𝑏 ( 𝑑0 ) is 𝐹[∑𝛿(𝑥0 − 𝑛𝑑)] = 1 ∑𝛿 (𝑓𝑥 𝑑 𝑛 − 𝑑) = ∑𝛿(𝑑𝑓𝑥 − 𝑛) = 𝑐𝑜𝑚𝑏(𝑑𝑓𝑥 ) Fig. 1.27. Ronchi’s grating. In the formula (1.105), and 𝑟𝑒𝑐𝑡(∗) is called the rectangular function or the normalized boxcar function. When the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens with the focal length 𝑓 and illuminated normally by a plane wave with the amplitude 𝐴, the light field on the back focal plane, (𝑥, 𝑦), of the lens is 𝑈𝑓 (𝑥, 𝑦) = = = 𝐴𝑒 𝑗2𝑘𝑓 𝐹{𝑡(𝑥0 , 𝑦0 )} 𝑗𝜆𝑓 = 𝐴𝑒 𝑗2𝑘𝑓 [{𝑎[𝑠𝑖𝑛𝑐(𝑎𝑓𝑥 )][𝑐𝑜𝑚𝑏(𝑑𝑓𝑥 )]} ∗∗ 𝐿2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 )]𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) 𝑗𝜆𝑓 𝐴𝑒 𝑗2𝑘𝑓 𝑎𝐿2 𝑦 𝑛𝑎 𝑥 𝑠𝑖𝑛𝑐 (𝐿 𝜆𝑓) ∑∞ 𝑛 = −∞ 𝑠𝑖𝑛𝑐 ( 𝑑 ) 𝑠𝑖𝑛𝑐 [𝐿 (𝜆𝑓 𝑗𝜆𝑓 𝑑 𝑛 − 𝑑)] = (1.106) In the above formula the notation, ∗∗ , represents the operation of convolution. The complex amplitude of the light field on the focal plane behind Ronchi’s grating is the 𝑛 superposition of infinite sinc(x) functions with the same width and the translation, 𝑑 , respectively, where 𝑛 is an integer called the diffraction order. Similar to the cosine amplitude grating, the cross multiplication terms can be ignored, and the intensity distribution of the light field on the back focal plane is 63 Advances in Optics: Reviews. Book Series, Vol. 5 𝐼(𝑥, 𝑦) = 𝑈𝑓 (𝑥, 𝑦)𝑈𝑓 ∗ (𝑥, 𝑦) = 𝑎𝐴𝐿2 2 𝑦 𝑛𝑎 𝑥 𝑛 2 2 = ( 𝜆𝑓𝑑 ) 𝑠𝑖𝑛𝑐 2 (𝐿 𝜆𝑓) ∑∞ 𝑛 = −∞ 𝑠𝑖𝑛𝑐 ( 𝑑 ) 𝑠𝑖𝑛𝑐 [𝐿 (𝜆𝑓 − 𝑑 )] (1.107). The light intensity distribution of plane(𝑥, 0) is shown in Fig. 1.28, only limited number of diffraction orders are shown. Fig. 1.28. Light intensity distribution from a Ronchi’s grating. 1.4.2.4. Fourier Optics for Imaging under Coherent Illumination In the following two sections, Fourier optics is used to analyze the imaging properties of optical systems illuminated, respectively, under coherent and incoherent light. Most optical systems can be regarded as linear and spatially invariant systems. We can describe the imaging characteristics of the system not only through its point spread function (the impulse response) in the space domain but also through its transfer function in the frequency domain. If the transfer function of the optical system is known, the spectrum of the image can be obtained by multiplying the transfer function with the spectrum of the object or its ideal image. Under illumination of coherent light, the imaging system is linear to the complex amplitude. In the actual imaging systems, the size of the incident beam is limited by the entrance pupil of the optical system, and the size of the outgoing beam is restricted by the exit pupil of the optical system. The imaging process can be described as follows: the light wave passes from the object surface to the entrance pupil, from the entrance pupil to the exit pupil through the optical system, and then from the exit pupil to the image plane, as shown in Fig. 1.29. 64 Chapter 1. Physical Optics In the ideal imaging case, the spherical wavefront emitted by the point light source will become a convergent spherical wavefront centered on its ideal image point after passing through the system. This convergent spherical wavefront can be taken as a reference one for evaluating wave aberrations for a practical system. The difference between the actual wavefront from a practical system and the reference wavefront is called the wave aberration that can assess the imaging quality of the optical system. Fig. 1.29. An imaging system. The general exit pupil function is defined as 𝐻(𝑥1 , 𝑦1 ) = 𝑐𝑖𝑟𝑐 ( where 𝑐𝑖𝑟𝑐 ( √𝑥12 +𝑦12 𝜌 √𝑥12 +𝑦12 𝜌 ) 𝑒 𝑗𝜑(𝑥1 ,𝑦1 ), (1.108) ) is the circular function with radius ρ, and 𝜑(𝑥1 , 𝑦1 ) is the wave aberration. The distribution of light field on the exit pupil can be expressed as the product of exit pupil function (1.108) and the ideal convergent spherical wave (1.93) 𝐻′(𝑥1 , 𝑦1 ) = 𝐻(𝑥1 , 𝑦1 )𝑒 −𝑗 𝑘 (𝑥 2 +𝑦12 ) 2𝑧0 1 , (1.109) where 𝑧0 is the distance from the exit pupil to the image plane. The propagation of light wave from the exit pupil to the image plane generally meets the Fresnel diffraction condition. On inserting the formula (1.109) into the Fresnel diffraction formula (1.52), we can get the light field distribution of the image plane as follows 𝑝(𝑥, 𝑦) = 𝑒 𝑗𝑘𝑧0 𝑗𝜆𝑧0 𝑒 𝑘 𝑗 (𝑥 2 +𝑦 2 ) 𝑧 0 ∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑘 𝑧0 𝑗 (𝑥𝑥1 +𝑦𝑦1 ) 𝑑𝑥1 𝑑𝑦1 In the above formula, the coefficient in front of the integral does not affect the relative intensity distribution of the image plane and can be ignored. Then we have 65 Advances in Optics: Reviews. Book Series, Vol. 5 𝑝(𝑥, 𝑦) = ∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑘 𝑧0 𝑗 (𝑥𝑥1 +𝑦𝑦1 ) (1.110) 𝑑𝑥1 𝑑𝑦1 The formula (1.110) represents the complex amplitude distribution of the light field on the image plane after the spherical wave emitted by a point object passes through the optical system. 𝑝(𝑥, 𝑦) is called the Amplitude Point Spread Function (APSF) of the optical system. It is the Fourier transform of the exit pupil function 𝐻(𝑥1 , 𝑦1 ), which only depends on the properties of the optical system. Assuming that the object distribution is 𝑜(𝑥0 , 𝑦0 ), the complex amplitude distribution of the geometrical optical ideal image on the image plane is where 𝑀 = 𝑥 𝑥0 = 𝑦 𝑦0 𝑔𝑖 (𝑥, 𝑦) = 1 𝑜(𝑥, 𝑦), 𝑀 (1.111) is the absolute value of the magnification of the image. The ideal image is the same as the object, but the coordinates are scaled. The imaging system can be regarded as a linear and spatially invariant system with diffraction, and the form of APSF does not change with its position in the paraxial regime. By means of the properties of the linear system, the image can be written as the convolution between the ideal image and APSF as follows. ∞ 𝑔(𝑥, 𝑦) = ∬−∞ 𝑔𝑖 (𝜉, 𝜂)𝑝(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝜉𝑑𝜂 , (1.112) 𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐺𝑖 (𝑓𝑥 , 𝑓𝑦 )𝑇(𝑓𝑥 , 𝑓𝑦 ), (1.113) Performing Fourier transform on the both sides of the above formula and invoking the convolution theorem of Fourier transform, we have where 𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔(𝑥, 𝑦)} , 𝐺𝑖 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔𝑖 (𝑥, 𝑦)} , 𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} , y 𝑥 and 𝑓𝑥 = 𝜆𝑧 ，𝑓𝑦 = 𝜆𝑧 . 0 0 𝑇(𝑓𝑥 , 𝑓𝑦 ) is the Fourier transform of APSF 𝑝(𝑥, 𝑦). By using the expression (1.107) we can get 𝑇(𝑓𝑥 , 𝑓𝑦 ) as follows 𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} = 𝐹 {∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑘 𝑗 (𝑥𝑥1 +𝑦𝑦1 ) 𝑧0 = 𝐻(−𝜆𝑧0 𝑓𝑥 , −𝜆𝑧0 𝑓𝑦 ) 𝑑𝑥1 𝑑𝑦1 } If we ignore the immaterial negative sign in the above expression, then we have 𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} = 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ), (1.114) 𝑇(𝑓𝑥 , 𝑓𝑦 ) is called the Amplitude Transfer Function (ATF) of the optical system, which has the same form as the exit pupil function of the optical system. The imaging system generally acts as a low-pass filter due to the Amplitude Transfer Function. 66 Chapter 1. Physical Optics 1.4.2.5. Fourier Optics for Imaging under Incoherent Illumination Under the illumination of the coherent light, the imaging process is the superposition of complex amplitudes. There is a very simple and direct relationship between the exit pupil and the amplitude transfer function (ATF). When the incoherent light is used for illumination, the imaging process is the superposition of intensity. Although the transfer function of the imaging system is still determined by the exit pupil function, the relationship between them is not straightforward. The intensity distribution of the image is the time average of the instantaneous intensity in detection duration. The intensity distribution can be obtained by squaring the coherent complex amplitude distribution of the optical system (1.112), and then by time average, the incoherent intensity distribution on the image plane of the optical system can be obtained ∞ 2 𝐼(𝑥, 𝑦) = 〈|∬−∞ 𝑔𝑖 (𝜉, 𝜂)𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝜉𝑑𝜂 | 〉, (1.115) where 〈∙〉 is the time average or ensemble average. By exchanging the order of the integral and the time average in the above formula, we can get ∞ 𝐼(𝑥, 𝑦) = ∬ ∬−∞〈𝑔𝑖 (𝜉, 𝜂)𝑔𝑖 ∗ (𝜉′, 𝜂′)〉 𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)𝑃∗ (𝑥 − 𝜉′, 𝑦 − 𝜂′)𝑑𝜉𝑑𝜂𝑑𝜉′𝑑𝜂 ′ (1.116) With help of the properties of completely incoherent light, we have 〈𝑔𝑖 (𝜉, 𝜂)𝑔𝑖 ∗ (𝜉′, 𝜂′)〉 = 𝐼𝑖 (𝜉, 𝜂)δ(𝜉 − 𝜉′, 𝜂 − 𝜂′), where 𝐼𝑖 (𝜉, 𝜂) is the intensity distribution of the geometrical optical ideal image. The point spread function p (x, y) is only related to the properties of the system, so that the average does not affect its value. On inserting the above formula into the formula (1.116), we have the imaging formula for an inherent optical system. ∞ 𝐼(𝑥, 𝑦) = ∬−∞ 𝐼𝑖 (𝜉, 𝜂)|𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)|2 𝑑𝜉𝑑𝜂 = = 𝐼𝑖 (𝑥, 𝑦) ∗ |𝑃(𝑥, 𝑦)|2 (1.117) The above formula shows that the incoherent intensity distribution of image can be written as the convolution of the ideal image intensity distribution 𝐼𝑖 (𝑥, 𝑦) with incoherent point spread function |𝑃(𝑥, 𝑦)|2 . Performing Fourier transform on both sides of the above formula, we have where Π(𝑓𝑥 , 𝑓𝑦 ) = Π𝑖 (𝑓𝑥 , 𝑓𝑦 )𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ), Π(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝐼(𝑥, 𝑦)}, Π𝑖 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝐼𝑖 (𝑥, 𝑦)}, (1.118) 67 Advances in Optics: Reviews. Book Series, Vol. 5 and 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{|𝑃(𝑥, 𝑦)|2 }. 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) is called the non-normalized Optical Transfer Function (OTF) of the incoherent illumination that is a comprehensive parameter for evaluating the quality of the optical system. According to the correlation theorem of Fourier transform and the formula (1.114), we can obtain the relationship between the optical transfer function and the exit pupil function as follows 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{|𝑃(𝑥, 𝑦)|2 } = 𝐹{𝑃(𝑥, 𝑦)𝑃∗ (𝑥, 𝑦)} = 𝐹{𝑃(𝑥, 𝑦)} ⊗ 𝐹{𝑃∗ (𝑥, 𝑦)} = = 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ) ⊗ 𝐻 ∗ (𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ), (1.119) where ⊗ stands for the correlation operation. The above formula shows that the optical transfer function 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) is the autocorrelation of the exit pupil function 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ). If the exit pupil function is known, the optical transfer function can be calculated by the expression (1.119). In general, OTF is a complex, and its modulus is called the Modulation Transfer Function (MTF), which represents the ratio of the contrast of the actual image to the contrast of the ideal image at a particular frequency. The argument of the OTF is called the Phase Transfer Function (PTF), which represents the phase shift of each frequency component of the actual image distribution relative to the ideal image distribution. There is a very important concept, resolution, in an optically imaging system. The resolution of the optical system is dependent on the size of its aperture stop and the coherence of the illumination. There exist the simple expressions of the resolution for the perfect coherence and perfect incoherence of the illumination. However, the partially coherent illumination is complicated and the results can only be given in rather general terms. Here we do not discuss them further. Interested readers can consult references [8] and [10]. 1.4.2.6. Fourier Transform Spectroscopy Fourier transform spectroscopy is a technique for the measurement of the its spectrum of a light source, which is based on the temporal coherence of light and the tool of the Fourier transform. The schematic of the layout of the Fourier transform spectrometer is shown in Fig. 1.30, and its core part is the Michelson’s interferometer with a movable mirror in one arm. The basic principle is as follows. The movable mirror is scanned a series of distances, and the corresponding intensity at each scanned position is recorded by a point detector. Then the spectrum of the light source can be obtained through the Fourier transform of a series of recorded intensities. For simplicity, suppose that light emitted from a source is monochromatic, and the intensities of two light waves arriving at the point detector are the same, denoted as 𝐼0 (𝑘). From the following Section 1.5 for interferometry to be discussed, the interferometric 2𝜋 intensity on the point detector can be expressed as 2𝐼0 (𝑘)[1 + cos(𝑘𝑙)], where 𝑘 = 𝜆 , 68 Chapter 1. Physical Optics 𝜆 is the wavelength of light, and 𝑙 is the difference between optical paths of two light waves passing through the two arms of the Michelson interferometer. Since light emitted from an actual source includes light with different wavelengths, the intensity on the point detector will be the superposition of intensities for different wavelengths, and let 1 wavenumber 𝑝 = , such that the total intensity on the point detector is as follows 𝜆 ∞ 𝐼(𝑙) = ∫0 2𝐼0 (𝑝)[1 + cos(2𝜋𝑝𝑙)] 𝑑𝑝 = ∞ ∞ = 2𝐼 + ∫0 𝐼0 (𝑝)(𝑒 𝑗2𝜋𝑝𝑙 + 𝑒 −𝑗2𝜋𝑝𝑙 )𝑑𝑘, (1.120) where 𝐼 = ∫0 𝐼0 (𝑝) 𝑑𝑝. Fig. 1.30. Illustration of Fourier transform spectrometer. If let 𝐼𝑡 (𝑙) = 𝐼(𝑙) − 2𝐼, then we have from (1.117) ∞ ∞ 𝐼𝑡 (𝑙) = ∫0 𝐼0 (𝑝)(𝑒 𝑗2𝜋𝑝𝑙 + 𝑒 −𝑗2𝜋𝑝𝑙 )𝑑𝑝 = ∫−∞ 𝐼0 (𝑝)𝑒 𝑗2𝜋𝑝𝑙 𝑑𝑝 (1.121) This equation indicates that 𝐼𝑡 (𝑙) and 𝐼0 (𝑝) are a Fourier transform pair. Thus, the power spectrum of the light source 𝐼0 (𝑝) can be got by the Fourier transform of 𝐼𝑡 (𝑙), and ∞ 𝐼0 (𝑝) = ∫−∞ 𝐼𝑡 (𝑙)𝑒 −𝑗2𝜋𝑝𝑙 𝑑𝑙 (1.122) The difference, 𝑙, of the optical paths is proportional to the displacement of the movable mirror in Fig. 1.30 he mirror, the power spectrum of the light source under measurement can be determined. 1.5. Optical Interferometry In Section 1.3 we have discussed the coherence of light, which is manifested by interferometric phenomena. We know that that when light waves are superimposed, the field vibration is strengthened at some positions, while weakened at some other positions. 69 Advances in Optics: Reviews. Book Series, Vol. 5 This phenomenon is called the interference of light waves, which obeys the superposition principle of light waves. When light beams superposed satisfy certain conditions, interferometric fringes are stable over certain time duration. In this section, the superposition principle for any types of waves is first introduced, then the interferometric phenomenon of light waves is explained, and at final, some examples for light interference are given. 1.5.1. Superposition of Waves 1.5.1.1. General Principle The principle of superposition is that when two or more waves meet at a point, the field vibration at that point is equal to the sum of the field vibrations of the individual waves. Furthermore, the propagation of waves at the meet point is independent of each other, and the characteristics of a series of waves will not be changed by the existence of other waves. After two (or more) waves meet in space, each wave still keeps its own characteristics of frequency, wavelength and vibration direction, and continues to propagate along the original direction. This is called the principle of independent propagation of waves. The above two principles are not universal. Generally, for a physical system, if the differential equation describing the change of physical quantity is linear, the corresponding physical quantity satisfies the principle of superposition. However, for very strong waves, the principle of superposition fails. When two waves meet and superimpose in space, the composite waves is often very complicated. For simplicity, we just consider the superposition of two simple harmonic waves with the same vibration direction, and consider the superposition of two harmonic waves with the same frequency and different frequencies, respectively. 1.5.1.2. Superposition of Waves with Same Frequencies As shown in Fig. 1.31, two plane waves with the same vibration direction are emitted from wave sources 𝑆1 and 𝑆2 , both of which have the same frequency 𝜔. They propagate in the same medium and have the same wavenumber 𝑘. Let the initial phase of the wave source be 𝜑1 and 𝜑2 , and meet at a point P after propagating the distance of 𝑟1 and 𝑟2 respectively. Fig. 1.31. Supposition of two waves. 70 Chapter 1. Physical Optics The amplitudes of the two waves at the point 𝑃 are 𝐴1 and 𝐴2 . The vibration expressions are 𝑈1 (𝑡) = 𝐴1 cos( 𝑘𝑟1 − 𝜔𝑡 + 𝜑1 ), 𝑈2 (𝑡) = 𝐴2 cos( 𝑘𝑟2 − 𝜔𝑡 + 𝜑2 ) (1.123) 𝑈 = 𝑈1 + 𝑈2 = 𝐴 cos( 𝜔𝑡 + 𝜑) (1.124) 𝐴 = √𝐴1 2 + 𝐴2 2 + 2𝐴1 𝐴2 𝑐𝑜𝑠∆𝜑 (1.125) 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼11 𝐼2 𝑐𝑜𝑠∆𝜑 (1.126) ∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ), (1.127) Because of the same frequency, the composite vibration at the point 𝑃 is still a simple harmonic vibration The amplitude 𝐴 of the composite vibration is determined by the following equation. On taking the formula (1.125) squared, we have following expression. In formulas (1.125) and (1.126), ∆𝜑 is the phase difference of two waves at the point 𝑃 where 𝐼1 = 𝐴1 2 and 𝐼2 = 𝐴2 2 are, respectively, the intensities of waves 𝑈1 (𝑡) and 𝑈1 (𝑡) at the point 𝑃 , (𝜑2 − 𝜑1 ) is the initial phase difference of two waves, and 𝑘(𝑟2 − 𝑟1 ) is the phase difference caused by different wave paths from two wave sources to the point 𝑃. For a given point 𝑃, the wave path difference 𝑟2 − 𝑟1 is certain. If the changing time of the initial phase difference (𝜑2 − 𝜑1 ) is much longer than the detection time of the detector, stable fringes can be formed at the point 𝑃. When the phase difference at a space point satisfies ∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ) = ±2𝑚𝜋, 𝑚 = 0,1,2 ⋯ , (1.128) 𝐴𝑚𝑎𝑥 = 𝐴1 + 𝐴2 , 𝐼𝑚𝑎𝑥 = 𝐼1 + 𝐼2 + 2√𝐼11 𝐼2 (1.129) ∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ) = ±(2𝑚 + 1)𝜋, 𝑚 = 0,1,2 ⋯ , (1.130) 𝐴𝑚𝑖𝑛 = |𝐴1 − 𝐴2 |, 𝐼𝑚𝑖𝑛 = 𝐼1 + 𝐼2 − 2√𝐼11 𝐼2 (1.131) the amplitude and intensity at a point have a maximum value of When the phase difference at a point satisfies the amplitude and intensity at a point have a minimum value of 71 Advances in Optics: Reviews. Book Series, Vol. 5 1.5.1.3. Standing Waves When two coherent waves with the same amplitude propagate in opposite directions, a standing wave is formed. Suppose that two coherent plane waves with the same amplitude propagating along the positive and negative directions of z-axis as the following expressions 𝑈1 (𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 − 𝜔𝑡), 𝑈2 (𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 + 𝜔𝑡). The superposition wave function of these two waves is (1.132) 𝑈 = 𝑈1 + 𝑈2 = 2𝐴 cos(𝑘𝑧) cos(𝜔𝑡) This is the expression of a standing wave, where the space variable z and the time variable t are separated. The time function cos 𝜔𝑡 shows that all points vibrate harmonically at the same frequency. The space function 2𝐴 cos(𝑘𝑧) shows that the amplitudes of each point are different and distributed according to the cosine function. At the position of |cos(𝑘𝑧)| = 1, the maximum amplitude is 2𝐴, and its coordinate is 𝜆 2 𝑧 = 𝑚 , 𝑚 = 0, ±1, ±2, ⋯ These positions of the maximum amplitude are called antinodes. At the position of |cos(𝑘𝑧)| = 0, the minimum amplitude is zero, and its coordinate is 𝜆 4 𝑧 = ±(2𝑚 + 1) ， 𝑚 = 0, ±1, ±2, ⋯ These positions with the smallest amplitude are called nodes. The positions of antinodes and nodes of the standing wave are fixed, and the distance between adjacent antinodes or 𝜆 nodes is . 2 At every moment, a standing wave has a certain waveform, but this waveform does not move with time like a traveling wave. The standing wave propagates neither the vibration state nor the energy. This is because the average energy flux density of two traveling waves forming the standing wave is equal and opposite, so the average energy flux density of the composite wave is zero. The standing wave has stable energy state, called the energy steady state. 1.5.1.4. Superposition of Waves with Diffident Frequencies: Group Velocity, Phase Velocity The plane wave with the frequency of 𝜔 propagates along z-axis, and the wave function can be written 𝑧 72 𝑈(𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 − 𝜔𝑡)] = 𝐴𝑐𝑜𝑠𝜔 (𝑡 − 𝑣), (1.133) Chapter 1. Physical Optics where 𝐴 is the amplitude, 𝜔 is the angular frequency, and 𝑣 is the speed of wave, 2𝜋 𝜔 𝑘 = = wavenumber or spatial frequency. For the constant phase (𝑘𝑧 − 𝜔𝑡), its 𝜆 𝑣 𝑑𝑧 moving speed is 𝑑𝑡 = 𝑣, which is called phase velocity. For the superposition of two simple harmonic waves with the same frequency, the same vibration direction and the same wave velocity 𝑣, the composite wave is still the simple harmonic wave with the same frequency and propagates with the phase velocity 𝑣 . However, the superposition of simple harmonic waves with different frequencies, the composite wave is no longer simple harmonic, which is generally a more complex wave. Next, we will discuss this situation. Suppose that the two plane waves propagate along the z-axis and the wave function is as follows. 𝑈1 (𝑧, 𝑡) = 𝐴 cos( 𝑘1 𝑧−𝜔1 𝑡), 𝑈2 (𝑧, 𝑡) = 𝐴 cos(𝑘2 𝑧 − 𝜔2 𝑡), where 𝑘1 = 2𝜋 𝜆1 = 𝜔1 ，𝑘2 𝑣1 = 2𝜋 𝜆2 = 𝜔2 ，𝜔1 𝑣2 and 𝜔2 are the frequencies of the wave, 𝜆1 and 𝜆2 the wavelengths of two waves, which are very close, and 𝑣1 and 𝑣2 the phase velocities of two waves. The composite wave of these two waves is 𝑈(𝑧, 𝑡) = 𝑈1 (𝑡) + 𝑈2 (𝑡) = 𝐴[cos(𝑘1 𝑧 −𝜔1 𝑡) + cos( 𝑘2 𝑧 − 𝜔2 𝑡)] With help of trigonometric formulas, the above formula can be expressed as 𝑈(𝑧, 𝑡) = 2𝐴 cos( 𝑘𝑚 𝑧−𝜔𝑚 𝑡) cos( 𝑘̅ 𝑧 − 𝜔 ̅𝑡), (1.134) 𝜔 −𝜔 𝑘 −𝑘 𝜔 +𝜔 𝑘 +𝑘 where 𝜔𝑚 = 1 2 2 , 𝑘𝑚 = 1 2 2, 𝜔 ̅ = 1 2 2，and 𝑘̅ = 1 2 2. Because that 𝜔1 and 𝜔2 are near the same, such that 𝜔𝑚 ≪ 𝜔1 or 𝜔2 and 𝜔 ̅ ≈ 𝜔1 or 𝜔2 . The composite wave expression (1.134) can be regarded as a plane wavecos( 𝑘̅ 𝑧 − 𝜔 ̅𝑡) with a slowly changing amplitude 2𝐴 cos( 𝑘𝑚 𝑧−𝜔𝑚 𝑡) . The waveform of the composite wave is shown in Fig. 1.32. In the figure above, two simple harmonics with similar frequency are represented by solid line and dotted line respectively. The figure below shows the superposition of the two waves. The solid line represents the waveform with the frequency of 𝜔 ̅, and the dotted line represents the waveform with the slowly varying amplitude. Due to the change of amplitude, the composite wave forms wave packet in shape. There are two propagation velocities in the wave packet. One is the phase propagation velocity of the high frequency wave in the composite wave. From (1.134), let 𝜔 ̅𝑡 − 𝑘̅ 𝑧 = constant, the phase velocity of the wave packet can be obtained 𝑣𝑝 = 𝑑𝑧 𝑑𝑡 = ̅ 𝜔 ̅ 𝑘 (1.135) Since 𝜔 ̅ ≈ 𝜔1 or 𝜔2 and 𝑘̅ ≈ 𝑘1 or 𝑘2 , this phase velocity is approximately equal to the phase velocities of the two plane waves that consist of the wave packet. 73 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 1.32. Wave packet. The other velocity is the group velocity of the wave packet, that is, the moving velocity of the point with the maximum amplitude of the composite wave. According to (1.134), the change of amplitude is 2𝐴 cos( 𝜔𝑚 𝑡 − 𝑘𝑚 𝑧), let 𝜔𝑚 𝑡 − 𝑘𝑚 𝑧 = constant, the moving velocity of the maximum amplitude point of the wave packet can be obtained 𝑑𝑧 𝑑𝑡 𝑣𝑔 = = 𝜔𝑚 , 𝑘𝑚 (1.136) The formula (1.136) is the group velocity of the wave packet formed by the superposition of two plane waves with different frequencies. The wave energy is proportional to the square of the amplitude, so that the group velocity represents the propagation velocity of the wave energy. If the wave packet is composed of multiple frequency components, the group velocity should be the moving velocity of the point with the maximum amplitude, and the maximum amplitude appears at the same phase position of all plane waves. Let the phase of any plane wave be 𝜙 = 𝜔𝑡 − 𝑘𝑧, and the extremum point of the same phase can be expressed as 𝑑𝜙 𝑑𝑘 𝑑𝜔 = 𝑡 𝑑𝑘 − 𝑧 = 0 (1.137) Then the moving velocity (group velocity) of the point with the maximum amplitude of the wave packet is 𝑣𝑔 = 𝑧 𝑡 = 𝑑𝜔 𝑑𝑘 (1.138) For a plane wave, there only exists the phase velocity not the group velocity that propagates the signal. In order to transmit the signal, the wave must be modulated, which involves the complex wave composed of more than one frequency wave, that is, the signal can only propagate with the complex wave, and its propagation velocity is that of wave packet movement. If we take 𝜔 = 𝑘𝑣, where 𝑣 is phase velocity, into (1.138), we get 74 Chapter 1. Physical Optics and considering 𝑘 = 𝑣𝑔 = 2𝜋 ，and 𝜆 𝑑𝜔 𝑑𝑘 𝑑𝑘 = − 𝑑𝑣 = 𝑣 + 𝑘 𝑑𝑘, 2𝜋 𝑑𝜆, 𝜆2 we furthermore have 𝑑𝑣 (1.139) 𝑣𝑔 = 𝑣 − 𝜆 𝑑𝜆 The formula (1.139) indicates that the relationship between group velocity and phase velocity is related to the dispersion of the medium. When the medium dispersion is small, the group velocity and phase velocity are almost the same, and the wave packet has a stable shape. When the dispersion is large, the group velocity and phase velocity may be very different. If the complex wave propagates in the medium with large dispersion, the wave packet will gradually flatten and eventually disappear due to the significant difference of phase velocity of each frequency. At this scenario, the concept of group velocity is meaningless. 1.5.2. Superposition of Light Waves: Optical Interferometry In Section 1.5.1.3, we discussed that if the two waves satisfy the coherent conditions, as discussed in Section 1.3, they will interfere when they are superposed, and the composite amplitude and its intensity will form a stable distribution in space, with the light field vibration strengthened in some locations and weakened in some locations. The reason can be understood from the luminous characteristics of ordinary light source. The emission mechanism of an ordinary light source is due to the spontaneous emission of atoms. The atoms in the high energy state will spontaneously transit to the low energy state and emit light waves. The duration of each emission is about 10−8 s, and the length of the wave train is less than 3 m. There are a large number of atoms in the light source, and the transitions occur randomly. The wave trains emitted by one atom successively and those emitted by different atoms are completely independent. Even if the frequencies of the wave trains emitted by ordinary light sources are the same, their initial phase difference is changing rapidly, so that the composite vibration distribution is also changing rapidly. The response speed of common detectors including human eyes is limited. As a result, the light intensity detected is the average value in the response time of the detector. The intensity expression (1.126) for the composite light field is averaged in the response time interval τ of the detector. 𝐼̄ = 1 𝜏 ∫ 𝐼𝑑𝑡 𝜏 0 1 𝜏 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝜏 ∫0 𝑐𝑜𝑠∆𝜑𝑑𝑡 (1.140) Because the initial phase difference of the two wave trains changes rapidly and irregularly, it experiences all possible values from 0 to 2π for many times in the time interval τ. The integral of the third interference term in the above formula is equal to zero, that is, the instantaneous interference term is averaged out by the average effect of the detector. The average intensity of the composite vibration in the time interval τ is 75 Advances in Optics: Reviews. Book Series, Vol. 5 𝐼̄ = 𝐼1 + 𝐼2 (1.141) In this case, the average intensity of the composite light field is equal to the sum of the intensities of the individual fields, and there is no interference phenomenon. Usually the light waves from two independent light sources or different parts of the same light source do not satisfy the coherent condition. 1.5.2.1. Interference Phenomena of Monochromatic Light In order to produce observable interference pattern by using an ordinary light source (suppose that the light source was monochromatic), we can use some optical methods to divide a light wave train emitted by one luminescent atom into two beams. Such two divided light beams have the same initial phases. Then the phase difference between them is only determined by the difference of two different optical paths through which light beams propagate. The methods splitting a light wave from an ordinary source includes wavefront division and amplitude division. Wavefront division The wavefront of the same light beam is divided into two parts, and the two divided beams are overlapped after passing through different paths to produce interference. Young's experiment is the most famous example of wavefront division interferometry. The Young's experimental setup is shown in Fig. 1.33. After the light from the point light source S0 on the optical axis passes through two pinholes S1 and S2 with equal sizes and symmetry about the optical axis, the interference fringes can be seen on the screen with the distance D from the two pinholes. Because the two light waves emitted from S1 and S2 are separated from the same wavefront, and S1 and S2 are equidistant from S0, they can be used as two coherent light sources with the same phases. The light intensity distribution of a point P near the optical axis on the observation screen is (1.142) 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 𝛿, where 𝛿 is the phase difference between S1 to P and S2 to P, and 𝛿 = 𝑘(𝑟2 − 𝑟1 ). Taking it into the above formula, we have 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 [ 2𝜋(𝑟2 −𝑟1 ) ] 𝜆 (1.143) The light intensity of the point P on the surface depends on the optical path difference of two light waves at this point. Let the coordinates of point P be (x, y, D), as shown in the Fig. 1.33, the distances from S1 and S2 to P are, respectively 2 76 2 𝑑 𝑑 𝑟1 = √(𝑥 + 2 ) + 𝑦 2 + 𝐷 2, 𝑟2 = √(𝑥 − 2 ) + 𝑦 2 + 𝐷 2, Chapter 1. Physical Optics where 𝑑 is the distance between S1 and S2. From the above two formulas, we can get 2𝑥𝑑 𝑟12 − 𝑟22 = 2𝑥𝑑. Such the optical path difference is |𝑟1 − 𝑟2 | = 𝑟 +𝑟 . If the distance of 1 2 the screen is far enough to make D»d, and the observation range is small, i.e. D»x,y, there will be 𝑟1 + 𝑟2 ≈ 2𝐷, then |𝑟1 − 𝑟2 | = 𝑥𝑑 . 𝐷 Bringing it into (1.143), we obtain 2𝜋𝑥𝑑 ) 𝜆𝐷 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 ( (1.144) Fig. 1.33. Young's experimental setup. The formula (1.144) indicates that the interference pattern of Young's experiment near the z-axis on the screen is a series of parallel fringes. The direction of fringes is perpendicular to the line connecting S1 and S2. The fringes are distributed according to the cosine 𝑚𝜆𝐷 (𝑚 = 0, ±1, ±2, ⋯ ) on the screen, function, as shown in Fig. 1.34. When 𝑥 = 𝑑 there is the maximum light intensity 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 , the bright fringe. When 1 𝜆𝐷 𝑑 𝑥 = (𝑚 + 2) (𝑚 = 0, ±1, ±2, ⋯ ) , there is the minimum light intensity 𝐼 = 𝐼1 + 𝐼2 − 2√𝐼1 𝐼2, the dark fringe. Fig. 1.34. Fringes from Young's experiment. The wavefront division method requires the size of the source be small enough. A point or line light source is usually used. Because the light wave emitted by each point on the light source will form its own set of interference fringes, the positions of fringes 77 Advances in Optics: Reviews. Book Series, Vol. 5 corresponding to different points will shift, and the larger the light source is, the greater the movement will be, which will cause the fringes to blur until they are invisible. Other devices for generating coherent light by wavefront division include Fresnel biprism, Fresnel double-sided mirror, and Lloyd's mirror, etc. Fresnel biprism is composed of two thin prisms with a very small vertex angle with the common bottom, as shown in Fig. 1.35. A beam of light from the point source is refracted by the biprism and divided into two beams, which is equivalent to the two beams from the two virtual images S1 and S2 formed by the prism. Therefore, they can be regarded as the coherent light source and produce interference fringes in the overlapping area. Fig. 1.35. Fresnel biprism. Fresnel double-sided mirror is composed of two mirrors with a small angle, as shown in Fig. 1.36. The light wave emitted by the point light source S is reflected by the double-sided mirror M1 and M2 to form two beams, which irradiate the screen to produce interference fringes. It can be regarded as the two beams of light emitted by the two virtual images S1 and S2 formed by S in the double-sided mirror, so that S1 and S2 are equivalent to a pair of coherent light sources. Fig. 1.36. Fresnel double-sided mirror. 78 Chapter 1. Physical Optics In the Lloyd’s mirror experiment as shown in Fig. 1.37, the point source S1 is placed quite far away from the mirror MM and close to the mirror plane. One part of the light wave emitted by S1 directly irradiates the screen E, and the other part of the light wave is incident on the mirror MM at nearly 90 degrees (grazing incidence), and then it is reflected by the mirror MM to reach the screen E. The two beams come from the same light wave by dividing its wavefront, so that they are coherent. The corresponding coherent light source is S1 and its virtual image S2 in the mirror. Fig. 1.37. Lloyd’s mirror. In the experiment, when the screen E is moved to contact with one end M of the mirror, the distance of the two beams from S1 and S2 to the point M is the same, so that the point M seems to be the center of the bright fringe, but it is actually the center of the dark fringe. The reason for this phenomenon, readers can consult to the discussion about Fresnel’s formulas in Section 1.2. Amplitude division A beam of light incident on a transparent dielectric film is successively reflected and refracted by the upper and lower surfaces to form the multiple beams of reflected light (or refracted light). They can be regarded as coherent light beams formed by dividing the amplitude of the incident light. The basic principle of the amplitude division method will be discussed by taking the interference of parallel thin films as an example. As shown in Fig. 1.38, there is a uniform transparent parallel film with refractive index 𝑛 and thickness ℎ, which is placed in a transparent medium with refractive index 𝑛′ . The monochromatic light with wavelength 𝜆 is incident on the upper surface of the film, and the incident angle is 𝑖 and the refraction angle 𝛾. A pair of coherent parallel beams 1 and 2 is produced after the incident light is reflected by the upper and lower surfaces of the film. The two reflected beams can only overlap at infinity. If a convergent lens is used, they can be superimposed on the focal plane to produce interference fringes. For an arbitrary point P on the focal plane I  I1  I 2  2 I1 I 2 cos  , 79 Advances in Optics: Reviews. Book Series, Vol. 5 where 𝛿 = 𝑘(𝑟2 − 𝑟1 ) is the phase difference between the two beams from point A to point P，which depends on the optical path difference 𝑟2 − 𝑟1 . The paths of the two reflected beams are ADP and ACBP respectively, and BD is the vertical line perpendicular to beam 1. According to the equal optical path property of the lens, 𝐷𝑃 = 𝐵𝑃. The optical path difference between the two beams is 𝑟2 − 𝑟1 = (𝐴𝐶 + 𝐶𝐵)𝑛 − 𝐴𝐷𝑛 . From the geometric relationship in Fig. 1.38, we have 𝐴𝐶 = 𝐶𝐵 = ℎ , 𝐴𝐷 𝑐𝑜𝑠 𝛾 = 𝐴𝐵 𝑠𝑖𝑛 𝑖 = 2ℎ 𝑡𝑎𝑛 𝛾 𝑠𝑖𝑛 𝑖 Fig. 1.38. Parallel thin films. Because two beams of light are reflected by the film surfaces, and the refractive index of the film is different from that of the surrounding medium, the additional optical path 𝜆 difference 2 between two beams should also be considered (see discussion about Fresnel’s formulas in Section 1.1). By means of the refraction law, the optical path difference can be expressed as 𝜆 𝜆 𝑟2 − 𝑟1 = 2ℎ𝑛 𝑐𝑜𝑠 𝛾 + 2 = 2ℎ√𝑛2 − 𝑛′2 𝑠𝑖𝑛2 𝑖 + 2 (1.145) The distribution of interference fringes is determined by the optical path difference, and then by 𝑛, ℎ, and 𝑖 in thin film interference. If 𝑛 and ℎ are fixed, i.e., for a uniform parallel film, the optical path difference is only determined by the incident angle. The incident light with the same incident angle 𝑖 forms interference fringes of the same order. The interference fringes of different orders will be formed when the reflected light interferes with the incident light at different incident angles. The interference formed by the parallel film with the same thickness is called equal inclination interference. If 𝑛 and 𝑖 are fixed, i.e., a beam of parallel light is incident on the film with varying thickness, the optical path difference is only related to the film thickness ℎ. When the film thickness is the same, the same order of interference fringes is formed; when the film thickness is different, the different orders of interference fringes are formed. This kind of interference is called equal thickness interference. 80 Chapter 1. Physical Optics It can be seen from the above discussions that the formation of interference fringes in the amplitude division method is not related to the size of the light source, which is differ from wavefront division method. Different points in the source emit light waves in all directions, and the light beams with the same angle have the same optical path difference, which will form interference fringes of the same order and the same position on the screen. After superposition, the contrast of the fringes can be increased. Therefore, this method can use the extended light source to produce clearer and brighter interference fringes. However, the fringes formed from an extended light source are localized. The above two methods of forming stable fringes with coherent light generated by ordinary light source have been introduced. Next we will define the contrast, or visibility, of interference fringes as follows. 𝑉 = 𝐼𝑚𝑎𝑥 −𝐼𝑚𝑖𝑛 , 𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛 (1.146) where 𝐼𝑚𝑎𝑥 and 𝐼𝑚𝑖𝑛 are the maximum intensity and the minimum intensity of the fringe, which can be obtained from the equations (1.129) and (1.131). When 𝐼𝑚𝑖𝑛 = 0, visibility 𝑉 = 1, the contrast is the biggest, and the fringe is clearest. When 𝐼𝑚𝑖𝑛 = 𝐼𝑚𝑎𝑥 , 𝑉 ≈ 0, the fringes are indistinct or even illegible. If the equations (1.129) and (1.131) are brought into the equation (1.146), we can also get 𝑉 = 2𝐴1 𝐴2 𝐴21 +𝐴22 = 𝐴 2 1 𝐴2 𝐴 1+( 1 ) 𝐴2 2 (1.147) The above formula shows that the amplitude ratio will affect the visibility of interference fringes. For an ideal monochromatic coherent light source, when the amplitudes of two beams are equal, the visibility of interference fringes is the largest. 1.5.2.2. Interference Phenomena of Partial Coherent Light In Section 1.5.2.1, we analyzed the interference of completely coherent light, however in practice light from the source is partially coherent. The light emitted by the source is not an ideal single wavelength, but has a certain bandwidth; the luminous area of the light source has a certain size, rather than an ideal point light source. This section will discuss how these two factors affect the visibility of interference fringes. As mentioned above, the interference generated by an ordinary light source is actually the result of superposition of wavelet trains separated from the same wave train emitted by atoms. Due to the short duration of the atomic luminescence, which is the time about ∆𝜏 = 10−8 s, the length of the wave train 𝐿 = 𝑐∆𝜏 is less than 3 m (c is the light speed in vacuum). Taking Young's experiment as an example, two wave trains are divided by double slits. When their optical path difference reaches the point P satisfies |𝑟2 − 𝑟1 | < 𝐿, i.e. the time difference satisfies ∆𝑡 < ∆𝜏 , they can meet and interfere, as shown in Fig. 1.39 (a). If the optical path difference is too large at P, then |𝑟2 − 𝑟1 | > 𝐿, two wavelets from the same wave train can not meet at this point and will not interfere, as 81 Advances in Optics: Reviews. Book Series, Vol. 5 shown in Fig. 1.39 (b). The duration, ∆𝜏, of the wave train can be called coherent time, further denoted as 𝜏𝑐 . If the two beams of light interfere, the maximum optical path difference should be less than the wave train length. The length of wave train can be called coherent length 𝐿𝑐 = 𝑐𝜏𝑐 . The coherence length 𝐿𝑐 and coherence time 𝜏𝑐 measure the temporal coherence of the light source from the different dimensional quantities and they are in fact equivalent. a b Fig. 1.39. Two wavelets from the same wave train. The coherence length is related to the monochromaticity of light. The plane wave is an infinite wave train with a single frequency, while the actual light source emits a wave train with a limited length. According to Fourier transform, it is impossible to have a single wavelength (or frequency) for a finite length wave. A wave train with a finite length always contains a certain range of wave length ∆𝜆, or frequency range ∆𝜈，as shown in Fig. 1.40. The intensity at the center wavelength is 𝐼0 , and the wavelength range ∆𝜆 where 𝐼 the intensity is reduced to 20 is usually taken as the line width. The smaller the line width of the light source, the better the monochromaticity is. 82 Fig. 1.40. Actual light source with the wavelength range ∆𝜆. Chapter 1. Physical Optics Obviously, light with different wavelengths in the line width will produce their own interference fringes due to different optical path differences. Because of incoherence of light with different wavelengths, the total light intensity overlapped by different fringes with different wavelengths is just addition of the light intensities of fringes of various wavelengths, which will affect the contrast of fringes. As shown in Fig. 1.41, the lower Δ𝜆 curve shows the distribution of interference fringes with wavelengths of 𝜆 − , 𝜆, 2 Δ𝜆 𝜆 − 2 . Because the width of the fringes is proportional to the wavelength, the same order interference fringes with different wavelengths have different positions. With the increase of the order, the misplacement of different wavelength fringes becomes more and more serious, and these fringes will overlap to reduce the contrast. The upper curve is the total intensity of incoherent superposition. It can be seen that with the increase of the order, the contrast of interference fringes decreases gradually, and finally the fringes disappear. Fig. 1.41. Light intensity of fringes of various wavelengths. According to the condition of bright fringes, the optical path difference of the m-order 𝑥𝑑 bright fringe is |𝑟1 − 𝑟2 | = 𝐷 = 𝑚𝜆 (𝑚 = 0, ±1, ±2, ⋯ ). If the line width of the light is ∆𝜆, the position where the interference fringes disappear should be the position where Δ𝜆 the m + 1 order bright fringe of the light with the wavelength of 𝜆 − coincide with the Δ𝜆 . 2 2 The reason of fringe m order bright fringe of the light with the wavelength of 𝜆 + disappearance under this condition can be appreciated as the follows. If this condition is satisfied, it is equivalent to that the phases of light fields from all wavelengths of light experience all possible values from 0 to 2π, so that the fringes disappear due to the average effect of the detector. Then this condition can be expressed in the mathematical form as Δ𝜆 Δ𝜆 follows: (𝜆 − 2 ) (𝑚 + 1) = (𝜆 + ) 𝑚. 2 From the above formula, we have 𝑚Δ𝜆 = 𝜆 − Δ𝜆 . 2 Since Δ𝜆 ≪ 𝜆, in the above formula we can ignore Δ𝜆 and obtain that 𝑚 = 𝜆 . Δ𝜆 83 Advances in Optics: Reviews. Book Series, Vol. 5 The maximum optical path difference, i.e. the coherence length, should meet that 𝐿𝑐 = 𝜆𝑚 = 𝜆2 Δ𝜆 (1.148) From the above formula, we can know the coherence length 𝐿𝑐 of the light source is inversely proportional to the line width Δ𝜆. The smaller the Δ𝜆 is, the larger 𝐿𝑐 , and the better the monochromaticity, that is, the easier interference occurs. Even if the ideal monochromatic light source has the infinite time coherence, however, for an extended light source the interference fringes generated by different points on the source are still not completely coherent and their superposition will also affect the fringe contrast. This is illustrated by Young's experiment. As shown in Fig. 1.42, if the point light source S is on the optical axis and the distances from 𝑆 to the double slit 𝑆1 and 𝑆2 (the distance between 𝑆1 and 𝑆2 is 𝑑) are equal, the light waves emitted from the double slits have no initial phase difference, and the 𝑏 zero-order bright fringe is at point 𝑂 on the optical axis. When 𝑆 moves down 2, there is a certain initial phase difference in the two beams emitted from the double slit. At this time, the zero-order bright fringe will move up to 𝑂′ above the axis. The distance between 𝑂 and 𝑂′ is 𝑥 = 𝑏 𝐷 2𝑅 (1.149) The width of fringes produced by different parts of the light source with the wavelength of λ is the same. The width of fringes can be calculated by the formula of bright fringes ∆𝑥 = 𝐷𝜆 𝑑 𝑏 𝐷 2𝑅 = If the width of the light source is b, the fringes generated by each part of the extended source are displaced, and the dislocation position is determined by the formula (1.149). Each part of the light source is incoherent, and the total intensity of the fringes is the result of incoherent superposition of several displaced fringes. This causes the contrast of the fringes to decrease. When the fringe generated by the edge point moves half of the fringe width relative to the fringe generated by the center point, i.e. 𝑥 = 𝐷𝜆 2𝑑 (1.150) The bright fringes of the interference fringes generated by the center of the light source coincide with the dark fringes generated by the edge of the light source, and the fringes will be blurred. If we denote this particular width of the light source as 𝑏0 , according to equation (1.150), the width, 𝑏0, of the light source S is 𝑏0 = 84 𝑅 𝜆 𝑑 (1.151) Chapter 1. Physical Optics When the size if the light source is 𝑏0 , fringes will disappear. It shows that only when the width of the light source satisfies 𝑏 < 𝑏0 , or when the distance of double slit satisfies 𝑅𝜆 𝑑 < 𝑏 , the interference fringes can be observed. The influence of the width of the light 0 source or the distance between two slits on the interference fringes is called spatial 𝑅𝜆 coherence, and the critical value of the distance between two slits 𝑑0 = 𝑏 is called the 0 spatial coherence length of light. Fig. 1.42. Young's interference experiment. From the formula (1.151) the angular diameter β of the light source can be expressed in the spatial coherence length 𝑑0 and the wavelength of light 𝜆 as follows. 𝛽 = 𝜆 𝑑0 (1.152) 1.5.2.3. Michelson Stellar Interferometer As shown in the equation (1.152), the angular diameter of the light source can be measured by the interferometric method. In 1920s, Michelson and Pease demonstrated experimentally that the angular diameter of a star might be obtained with help of the interferometer as shown schematically in Fig. 1.43. The basic idea for measuring the angular diameter of a star can be understood as follows. Light from the star illuminates the outer mirrors M1 and M2, and then is reflected onto the two inner mirrors M3 and M4. Furthermore, light is further reflected from the mirror M3 and M4 and brought to the focal plane of the telescope that is attached to the interferometer. A detector is placed at the back focal plane of the telescope. The distance between the mirror M1 and the mirror M2 in the interferometer could be symmetrically changed freely in the direction joining M1 and M2, and the inner mirrors M3 and M4 are fixed. On the detector a diffraction image of the star, crossed by fringes generated by the two interfering beams, could be obtained. For a certain size of a star, suppose that its angular diameter is 𝛽. According to the equation (1.152), when the distance between the mirror M1 and the mirror M2 is some value, suppose 𝑑0 , the visibility of the fringes on the detector is just zero. Such the angular 85 Advances in Optics: Reviews. Book Series, Vol. 5 diameter of the star can be approximately calculated according to the formula (1.152). For the circular aperture of the optical system, the angular diameter of the star can be accurately calculated by means of the following formula. 𝜆 𝛽 = 1.22 𝑑 0 (1.153) In essence, Michelson’s stellar interferometer is just Young’s double slit setup. Fig. 1.43. Michelson’s stellar interferometer. 1.6. Polarization of Light The concept of polarization of an electromagnetic wave was introduced in Section 1.1.5, and furthermore the various types of polarized plane waves were briefly described. In the current section, the several methods for producing polarized light are going to be described, the mathematical representation of polarized light and of polarizers will be explained, and finally the basic theory of the polarization of stochastic light fields is introduced. There are various applications of polarized light for different fields. For example, communication and detection systems, liquid crystal display technology, optical microscopy, optical spectroscopy, geology material science, astronomy, and so on. 1.6.1. Generation of Polarization Light When light acts on any matter whose optical properties are asymmetric along transverse directions to the light propagation vector, the polarized light will be produced. In Section 1.1, we know that light is a transverse wave. Based on this property of light waves, one of asymmetrical examples along transverse directions to the light propagation vector is light traveling through anisotropic matters, which can provide a means for generating polarized light. In essence, the process of polarizing light is taking unpolarized light and 86 Chapter 1. Physical Optics extracting from it a polarized light by means of asymmetric properties of matter due to the transverse property of light. In the following, several most important approaches for generating polarized light are discussed in detail. 1.6.1.1. Polarization by Reflection from Dielectric Surfaces Light reflected from a dielectric interface becomes partially polarized if the angle of incidence is nonzero, which is asymmetric with respect to the local normal of the dielectric interface in the nonzero angle of incidence. When the angle of incidence is at Brewster’s angle, the reflected light becomes the completely polarized one being perpendicular to the plane of incidence (S-polarization, here S is the initial of German word ‘Senkrecht’, meaning perpendicular). And the refracted light becomes partially polarized light that is polarized more in the plane of incidence than in the perpendicular plane to the plane of incidence (i.e. more P-polarization than S-polarization, here P is the initial of German word ‘parallele’). In Fig. 1.44, the dielectric interface lies in the x-z plane, and z-axis is also the normal of the interface surface. 𝑛1 is the index of refraction of the first medium through which the incident light propagates, and 𝑛2 the index of refraction for the second medium through which the refracted light propagates. When the reflected and refracted light rays are perpendicular to each other, the reflected light becomes S-polarization, and the refracted light becomes partially polarized. As shown in Fig. 1.44, 𝜃𝐵𝑖 and 𝜃𝐵𝑟 are the angle of incidence and of reflection, respectively, and 𝜃𝐵𝑡 is the refracted angle. At Brewster’s angle of incidence, or polarization angle, we have the following condition 𝜃𝐵𝑟 + 𝜃𝐵𝑡 = 𝜋 2 (1.154) Fig. 1.44. Polarization by reflection and refraction through an interface. Snell’s law says that 𝑠𝑖𝑛𝜃𝐵𝑖 𝑠𝑖𝑛𝜃𝐵𝑡 = 𝑛2 , 𝑛1 (1.155) 87 Advances in Optics: Reviews. Book Series, Vol. 5 and further the reflection law gives the expression that 𝜃𝐵𝑟 = 𝜃𝐵𝑖 . Taken all expressions together, we get Brewster’s law as 𝑡𝑎𝑛𝜃𝐵𝑖 = 𝑛2 𝑛1 (1.156) For unpolarized light being incident onto the interface between two different media at Brewster’s angle 𝜃𝐵𝑖 , the reflected light has pure S-polarization, and the refracted light is partially polarized with more P-polarization than S-polarization. The physical mechanism for S-polarization of the reflected light can be qualitatively explained as follows. When light is incident onto the interface between two media, it is absorbed by atoms in the second medium, and due to the transverse property of light waves, dipoles in these atoms will execute the forced oscillation in the direction perpendicular to the traveling direction of the refracted light. These oscillating dipoles re-emit light to form both the reflected and refracted light waves. From radiation formula (1.21), we know that the P-polarization in the reflected light just stems from the dipole oscillations caused by the P-polarization of the refracted light and the electric field does not exist in the direction of the dipole oscillations. When light strikes onto the interface at the Brewster’s angle, the reflected and refracted light rays are mutually perpendicular. As a result, the direction of the dipole oscillations of the atoms in the second medium is the same direction as the reflected light, so that there is no P-polarization in the reflected light, just S-polarization left. The light polarization for Brewster reflection is the same as the light polarizations along y- and z-axes for light scattering as shown in Fig. 1.46, to be discussed in Section 1.6.1.4. Polarization by reflection at Brewster’s angle is very sensitive to surface quality and structure, which is exploited in ellipsometry for investigating surfaces. 1.6.1.2. Polarization through Diattenuation Some materials, natural or synthetic, absorb light with different polarizations by different proportions. This asymmetric property of selective absorption of the material is referred to as diattenuation that can be used to polarize light. The most common material, Polaroid, uses this property to polarize light and make a polarizer. The polarizer easily transmits light with electric vibrations along the direction perpendicular to the direction of absorption. This preferred transmission direction is called the transmission axis (TA) of the polarizer. For an ideal polarizer, the transmitted light is linearly polarized in the direction of transmission axis. When a polarizer is used to test the state of polarization of light, the polarizer with this purpose is called an analyzer. Malus’ law says that the light intensity transmitted through a polarizer and an analyzer with an angle θ between their transmission axes is given by 𝐼 = 𝐼0 cos 2 𝜃, (1.157) where 𝐼0 ∝ 𝐸02 (𝐸0 is the amplitude of the incident field) is the light intensity incident on the polarizer, and I is the light intensity transmitted through the analyzer. 88 Chapter 1. Physical Optics Malus’ law is easily understood through Fig. 1.45 with the help of the relationship between the light intensity and the electric field defined by the formula (1.19). Fig. 1.45. Schematic of Malus’ law. 1.6.1.3. Polarization by Scattering Light scatterings occur whenever there is an irregular distribution of minute particles or the fluctuations of the density of medium through which light is traveling. There are many ways for light scattering, to mention a few, for example, dust particles whose size is compared with the light wavelength, the particles in a colloidal, and air molecules. We take light scattering from one air molecule, as shown in Fig. 1.46, to elaborate the mechanism for the production of polarized light from unpolarized light by means of radiation theory in Section 1.1. According to formula (1.21), we know that the orientation of the electric vector of the scattered light follows the dipole radiation pattern, and the electric vectors of the scattered light are perpendicular to the propagation direction, as shown in Fig. 1.46. If the incident wave is unpolarized, the scattered light in the forward x-axis direction is completely unpolarized; off that x-axis it is partially polarized, becoming increasingly more polarized as the angle increases away from x-axis. For y-axis direction or z-axis direction of observation that is normal to x-axis of the incident beam, the light is completely linearly polarized. In y-axis and z-axis directions, if light were not completely linearly polarized, as shown in Fig. 1.46, the radiation formula (1.21) would not be satisfied. 1.6.1.4. Polarization by Birefringence Crystalline substances whose lattice atoms were not completely symmetrically arrayed are optically anisotropic. Their optical properties are not the same in all directions within a given crystal. Birefringence is associated with anisotropy in the binding forces between the atoms forming a crystal. Mechanically, we can visualize that the atoms are bounded by springs with different stiffness in different crystalline directions. As a result, the different polarizations of the electric field lead to the different indices of refraction in anisotropic materials, which, therefore, results in the phenomenon of double refraction or birefringence in optically anisotropic media. 89 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 1.46. Schematic of polarization by scattering. For uniaxial crystals (hexagonal, tetragonal, and trigonal crystal), there is one direction so that any light in that direction in the crystal has the same index of refraction, regardless of its state of polarization. This direction is called the optic axis (OA) that is also the symmetric axis of the crystal. The remaining crystal systems (orthorhombic, monoclinic, and triclinic) have two optic axes and are said to be biaxial. Any plane that contains the optic axis is referred to as the principal plane, and the principal plane being normal to one of the cleavage surfaces of the crystal is called the principal section. In the following, we focus our discussion on the phenomena of light traveling through uniaxial crystals. When there exists an angle between the propagation direction and the optic axis, for one part of the light wave, its electric field is normal to the optic axis, and for the other part of the light wave, its electric field is parallel to the optic axis. The light wave with its electric field being everywhere perpendicular to the optic axis will propagate in the crystal in all directions with the same speed, as they would be in an isotropic medium. This wave is called the ordinary- or o-ray. The light wave with its electric field parallel to the optic axis will propagate in the crystal in a speed different from the speed of o-ray. This wave is called the extraordinary- or e-ray. Thanks to the anisotropy or asymmetry of the crystal, the propagation speed of e-ray and of o-ray in the crystal is different. Therefore, the index of refraction 𝑛𝑜 of the ordinary ray is different from 𝑛𝑒 of the extraordinary ray. The ordinary and extraordinary rays traveling along the optic axis have the same indices of refraction no = ne because the electric field is always perpendicular to the optic axis in such a case. The difference of the indices of refraction ∆𝑛 = 𝑛𝑒 − 𝑛𝑜 between the ordinary and the extraordinary rays is maximized in the direction of propagation perpendicular to the optic axis. This difference ∆𝑛 is a measure of the birefringence, If ∆𝑛 < 0, the crystal is negative uniaxial; If ∆𝑛 > 0, the crystal is positive uniaxial. Then the speed of the extraordinary ray and of the 𝑐 𝑐 ordinary ray are 𝑣𝑒 = 𝑛 and 𝑣𝑜 = , respectively, here 𝑐 is the speed of light in 𝑛 𝑒 0 vacuum. As shown in Fig. 1.47, if we choose Cartesian coordinate system having its 90 Chapter 1. Physical Optics z-axis along the optic axis, after time 𝑡 the equation for the spheroid of the e-ray wavelet 𝑥 2 +𝑦 2 𝑣𝑒2 𝑥 2 +𝑦 2 +𝑧 2 𝑣𝑜2 can be written be written + 𝑧2 𝑣𝑜2 = 𝑡 2 , and the equation for the sphere of the o-ray wavelet can = 𝑡 2 . Drawings (a) and (b) in Fig. 1.47 schematically demonstrate the Huygens’ wavelets for e-ray and o-ray in the negative and positive uniaxial media, respectively. In Fig. 1.48, birefringence is shown for a uniaxial crystal in drawing (a). The black dots in the drawing stand for the polarization perpendicularly pointing out the principal section (o-ray), and the arrow lines represent the polarization in the principal section (e-ray). Based on Huygens’ wavelets for e-ray and o-ray, in the drawing (b) we can qualitatively explains the formation of o-ray and e-ray in a negative uniaxial crystal using Huygens’ principle (1.47) that was discussed in Section 1.2. Fig. 1.47. Huygens’ wavelets in uniaxial media: negative crystal (a), positive crystal (b). Fig. 1.48. Polarization by birefringence in a principal section (a); Huygens principle of birefringence (b). It will now be a simple matter, in principle, to make some sort of linear birefringent polarizers. Any scheme that can get the aim for angular separation between the e-ray and the o-ray can be taken as a polarizer, all schemes exploiting the fact that 𝑛𝑜 ≠ 𝑛𝑒 . There are various devices that can accomplish the physical separation of e-ray from o-ray, for 91 Advances in Optics: Reviews. Book Series, Vol. 5 example, Wollaston prism, Glan-Foucault prism. These prisms are also named as polarizing beam-splitting prisms. Wollaston prism (Fig. 1.49) Fig. 1.49. Wollaston prism. A Wollaston prism is an optical device, invented by William H. Wollaston that separates unpolarized light into two linearly polarized outgoing beams with mutually perpendicular polarization directions. The Wollaston prism consists of two orthogonal prisms of birefringent material – typically a uniaxial material such as calcite. These prisms are cemented together on their base (traditionally with Canada balsam) to form two right triangle prisms with perpendicular optic axes. The o-ray in the first part of the prism vibrating in the direction perpendicular to the diagram, as shown in Fig. 1.49, becomes the e-ray in the second part, and the e-ray in the first part of the prism vibrating in the plane of the diagram becomes the o-ray in the second part of the prism. In this way, the two further spatially separated beams with mutually orthogonal polarization emerge on the right side of the prism. Glan-Foucault prism (Fig. 1.50) Fig. 1.50. The Glan-Foucault prism. The Glan-Foucault prism consists of two right-angled calcite prisms spaced with an air gap and can produce and analyze a polarized plane wave, as shown in Fig. 1.50. The Glan-Foucault prism is suitable for a broad spectral range. The incoming unpolarized ray is incident on the surface normally, and the electric field of light travelling into the prism can be resolved into components that are either completely parallel or perpendicular to the optic axis. The two orthogonally polarized components of light are angularly separated by 92 Chapter 1. Physical Optics the birefringence property of the calcite. The angle-of-incidence on the calcite-air interface is needed to arrange in such a way that for the o-ray totally internal refection occurs, but not for the e-ray. 1.6.1.5. Phase Shift by Birefringence The characteristics of optically anisotropic media can be used to manipulate the phase shift between the o-ray and the e-ray so as to produce the other types of polarized light beam from an incident linearly polarized beam. A device called the waveplate, or the phase shifter, can shift the phase between two perpendicular polarization components of a light wave. A typical waveplate is a birefringent crystal cut into a plate with a carefully chosen orientation and thickness. For convenience, we define the fast and slow axis for a waveplate. When the speed of the e-ray is faster than that of the o-ray, we call the direction of the optic axis the fast axis, and the direction perpendicular to it is the slow axis, i.e., the polarization direction of the e-ray is the fast axis, and the polarization direction of the o-ray is the slow axis. When the speed of the e-ray is slower than that of the o-ray, the situation is reversed. If the optic axis is parallel to the surface of the waveplate and light is incident on the plate at normal to the surface, as shown in Fig. 1.51, both o- and e- rays are going to propagate in the same direction, but with the different indices of refraction. A phase shift ∆𝜙 between o- and e- rays is ∆𝜙 = 2𝜋 (|𝑛0 𝜆 (1.158) − 𝑛𝑒 |)𝑑, where 𝜆 is the wavelength of the incident light, 𝑛0 and 𝑛𝑒 are, respectively, the indices of refraction of o- and e- rays in the waveplate, and d is the thickness of the plate. Fig. 1.51. Schematic of phase shift by birefringence. 1) Quarter-wave plate 𝜋 If the thickness d of the plate is such as to make ∆𝜙 = 2 , it is a quarter-wave plate (QWP). This plate is referred to as the zero-order plate. For the manufacturing engineering 93 Advances in Optics: Reviews. Book Series, Vol. 5 𝜋 reason the plate is made thicker for the higher order m, satisfying ∆𝜙 = (4𝑚 + 1) 2 , where m is integers. A quarter-wave plate could convert linearly polarized light into elliptically polarized light when the incident light polarized a certain angle 𝜃 with respect to the fast axis of the quarter-wave plate. Specially, the polarization conversion from linear to circular occurs on passage through a quarter-wave plate when the polarization direction of the incident light makes an angle of 45𝑜 with respect to the fast axis of the quarterwave plate. Conversely, when the fast and slow axes of a quarter-wave plate and the axes of the ellipse coincide, elliptically polarized light will be converted linearly on light passing through a quarter-wave plate. 2) Half-wave plate If the thickness d of the plate is such as to make ∆𝜙 = 𝜋, it is a half-wave plate (HWP). These are called the zero-order plates. Because such plates are extremely thin, its thickness being of order of one wavelength, in practice the plate is made thicker for the higher order m, satisfying ∆𝜙 = (2𝑚 + 1)𝜋 , where m is integers. Suppose that the polarization direction of an incident beam of linearly polarized light makes some angle 𝜃 with respect to the fast axis, as shown in Fig. 1.52. When the light emerges from the plate, the polarization direction of light will have rotated through 2𝜃, but the light emerged from the plate is still linear polarized. When the polarization direction of the incident light makes an angle of 45𝑜 with respect to the fast axis of the half-wave plate, the linearly polarized light coming out from the plate is perpendicularly polarized to the polarization direction of the incident light. Fig. 1.52. Schematic of a half-wave plate. 1.6.1.6. Optical Activity Optical activity is the capability of a material to rotate the plane of polarization of light passing through it. If we are looking towards the light source, the plane of polarization appears to have revolved clockwise, the material is referred to dextrorotatory; if looking in the direction of the light source the plane of polarization appears to have revolved counterclockwise, the material is levorotatory. 94 Chapter 1. Physical Optics 1.6.2. Matrix Treatment for Completely Polarized Light In above, we have discussed the physical mechanism for the generation of polarized light and several corresponding devices. In what follows, we will elaborate mathematical representation of completely polarized light and of polarizers using Jones formalism. Jones formalism is the mathematical method of describing polarized light in terms of amplitudes and phases. Here we focus on Jones algebra of polarization for a monochromatic light wave. For nonmonochromatic light, its Jones vector is the same as that of monochromatic light, which is also the starting point for the other statistical methods for describing polarization of light and of polarizers. We will not discuss this subject further. The interested readers can consult reference [13]. 1.6.2.1. Mathematical Representation of Completely Polarized Light: Jones Vector As shown in Fig. 1.53, the x and y complex field components 𝐸𝑥 and 𝐸𝑦 for a monochromatic light wave traveling in the z-direction with amplitudes 𝐸𝑥0 and 𝐸𝑦0 , and initial phases 𝜑𝑥 and 𝜑𝑦 , according to the convention introduced in Section 1.1, can be written 𝐸𝑥 = 𝐸𝑥0 𝑒 𝑗(𝑘𝑧−𝜔𝑡+𝜑𝑥 ), 𝐸𝑦 = 𝐸𝑦0 𝑒 𝑗(𝑘𝑧−𝜔𝑡+𝜑𝑦 ) (1.159) The electric field denoted by the above expressions can represent the different polarization states of light. Fig. 1.53. Cartesian coordinate system. If we ignore the common factors in the expressions (1.159), and write them in a vector form, we can get the following form. 𝐸𝑥0 𝑒 𝑗𝜑𝑥 [ ] 𝐸𝑦0 𝑒 𝑗𝜑𝑦 (1.160) This vector is also called Jones vector that describes the polarization state of light. 95 Advances in Optics: Reviews. Book Series, Vol. 5 Let us now give out the particular forms of Jones vectors for the linear, circular, and elliptical polarizations. As stated in Section 1.1, there are three cases (a), (b), and (c) for linear polarizations. The corresponding normalized Jones vector is as the following, respectively: Case (a) 1 [ ] linear polarization along x-axis; 0 Case (b) 0 [ ] linear polarization along y-axis; 1 Case (c) 𝑐𝑜𝑠𝛼 [ ] linear polarization along at angle 𝛼 with respective x-axis. 𝑠𝑖𝑛𝛼 Suppose that normalized Jones vector taking the following form 1 √𝐸𝑥0 2 +𝐸𝑦0 2 [ 𝐸𝑥0 𝜋 𝐸𝑦0 𝑒 𝑗 2 (1.161) ], where 𝐸𝑥0 and 𝐸𝑦0 represent the amplitudes of x and y components of the electric vibration. This Jones vector refers to elliptically polarized light with counterclockwise rotation. We will demonstrate why the expression (1.161) stands for counterclockwise rotation of polarization. According to the expressions (1.159) and (1.161), the real x and y components of the electric field can be written 𝜋 𝐸𝑥 = 𝐸𝑥0 𝑐𝑜𝑠(𝑘𝑧 − 𝜔𝑡), 𝐸𝑦 = 𝐸𝑦0 𝑐𝑜𝑠 (𝑘𝑧 − 𝜔𝑡 + 2 ) = −𝐸𝑦0 𝑠𝑖𝑛(𝑘𝑧 − 𝜔𝑡) First, let 𝑧 = 0, 𝑡 = 0, then 𝐸𝑥 = 𝐸𝑥0, 𝐸𝑦 = 0; second, take a little bit later time Δ𝑡 𝜋 (suppose that 𝜔𝛥𝑡 < 2 ) and at the same spatial point 𝑧 = 0 , then we get 𝐸𝑥 = 𝐸𝑥0 𝑐𝑜𝑠(𝜔𝛥𝑡) > 0, and 𝐸𝑦 = 𝐸𝑦0 𝑠𝑖𝑛(𝜔𝛥𝑡) > 0. According to the right-handed Cartesian coordinate system shown in Fig. 1.53, we know that electric field of light is rotating counterclockwise at an angular frequency of 𝜔. The same analysis can be applied to the elliptically polarized light with clockwise rotation. If Jones vector takes the following form 1 √𝐸𝑥0 2 +𝐸𝑦0 2 [ 𝐸𝑥0 𝜋 𝐸𝑦0 𝑒 −𝑗 2 ] (1.162) This means that the state of polarization for light is elliptically polarized light with clockwise rotation. 96 Chapter 1. Physical Optics If 𝐸𝑥0 = 𝐸𝑦0 , the cases represented by (1.161) and (1.162) are, respectively, circularly polarized light with counterclockwise and clockwise rotations. For these cases, we write their Jones vectors in a normalized form as follows. The normalized form of the Jones vector for counterclockwise rotation, viewed head-on, is 1 1 [ ] √2 𝑗 (1.163) This mode is called left-circularly polarized (LCP) light. The normalized form of the Jones vector for clockwise rotation, viewed head-on, is 1 1 [ ] √2 −𝑗 (1.164) This mode is called right-circularly polarized (RCP) light. 0 1 For the linear polarization along x-axis [ ] and the linear polarization along y-axis [ ], 1 0 they are orthogonal to each other and therefore can be used as a basis for the Jones vector of any polarization state. And they are also orthogonal to each other between the 1 1 left-circularly polarized (LCP) light [ ]and the right-circularly polarized (RCP) light √2 𝑗 1 1 [ ] and likewise they can be used as a new basis for the Jones vector of any √2 −𝑗 polarization state. Suppose that in the linear polarization basis the x and y components of 𝑎𝑥 a polarization state are 𝑎𝑥 and 𝑎𝑦 and denote as [𝑎 ], and in the circular polarization basis 𝑦 the LCP and RCP components of a polarization state are, respectively, 𝑎𝐿 and 𝑎𝑅 and 𝑎𝐿 denote as [𝑎 ], such that the transformation from the linear polarization basis to the 𝑅 𝑎𝐿 1 1 −𝑗 𝑎𝑥 circular polarization basis is as follows: [𝑎 ] = 2 [ ] [ ]. 1 𝑗 𝑎𝑦 √ 𝑅 To get the above expression, a polarization vector is decomposed onto the basis of the linear polarization and of the circular polarization, respectively, such that an equation relating the components in the different base is obtained. Solving the equation, the above transformation expression can be obtained. It should be emphasized that Jones vectors representing different polarized states are not unique. It is obvious that any Jones vector can be multiplied a real or complex prefactor the polarization mode represented by this vector is not changed, just changing amplitude or/and promoting the phase of each polarization component by the same value. 97 Advances in Optics: Reviews. Book Series, Vol. 5 1.6.2.2. Mathematical Representation of Polarizers: Jones Matrices Any optical device that can transmit light and modify the state of polarization can serve as a polarizer. The action of the polarizer on light can generally described by a 2×2 Jones matrix, 𝑎 [ 𝑐 𝑏 ], 𝑑 (1.165) where the matrix entries a, b, c, and d indicate the effects of polarizers on the state of polarization of the light traveling through them. In what follows, we give the Jones matrices of linear polarizers, waveplates, and rotators. In order to study the matrix forms for polarizers we have the following assumption that is generally legitimate for the common scenarios encountered if the intensity of light is not very high. Assumption: The components of the electric field, 𝐸𝑥′ and 𝐸𝑦′ , of a light beam emerging from a polarizer are linearly related to the components of the electric field, 𝐸𝑥 and 𝐸𝑦 , of the incident light beam, as shown in Fig. 1.54. This relationship can be expressed in the matrix form as: [ 𝐸𝑥′ 𝑎 ] = [ 𝐸𝑦′ 𝑐 𝑏 𝐸𝑥 ][ ] 𝑑 𝐸𝑦 (1.166) Fig. 1.54. Schematic of a polarizing element. a) Ideal linear polarizers A linear polarizer doesn't affect the vibration direction of either 𝐸𝑥 or 𝐸𝑦 . What this means is that 𝐸𝑥′ is just related to 𝐸𝑥 , and has nothing to do with 𝐸𝑦 , and the same as y component. For the linear polarizer, the Jones matrices are: for the transmission axis (TA) of a linear polarizer along x-axis 98 1 [ 0 0 ], 0 (1.167) Chapter 1. Physical Optics for the transmission axis (TA) of a linear polarizer along y-axis 0 0 (1.168) ], [ 0 1 for the transmission axis (TA) of a linear polarizer at angle 𝜃 with respect to x-axis 2 [ 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 sin 𝜃 cos 𝜃 ] 𝑠𝑖𝑛2 𝜃 (1.169) Next we will in detail illustrate how to get the above Jones matrix for the situation that the transmission axis of a linear polarizer makes an angle 𝜃 with respect to x-axis. First, let light polarized along the same direction as the TA go through the polarizer, then its 𝑐𝑜𝑠𝜃 Jones vector is [ ]. Second, let light with the polarization direction perpendicular to 𝑠𝑖𝑛𝜃 −𝑠𝑖𝑛𝜃 the TA go through the polarizer, then its Jones vector is [ ]. Then, we can get the 𝑐𝑜𝑠𝜃 following expressions 𝑐𝑜𝑠𝜃 𝑎 [ ] = [ 𝑐 𝑠𝑖𝑛𝜃 𝑎 𝑏 𝑐𝑜𝑠𝜃 0 ][ ], [ ] = [ 𝑐 𝑑 𝑠𝑖𝑛𝜃 0 𝑏 −𝑠𝑖𝑛𝜃 ][ ] 𝑑 𝑐𝑜𝑠𝜃 (1.170) Solving the above equations with four unknowns a, b, c, and d, we can get the Jones matrix (1.169) for the transmission axis (TA) of a linear polarizer making an angle 𝜃 with respect to x-axis. For the special case TA at 𝜃 = 45𝑜 , the Jones matrix is 1 1 [ 2 1 b) Waveplates 1 ] 1 (1.171) 𝐸𝑥0 𝑒 𝑗𝜑𝑥 Suppose that the light field incident on the waveplate is [ ]. According to the 𝐸𝑦0 𝑒 𝑗𝜑𝑦 property of the waveplate, the emerging light field behind it can be expressed as 𝐸𝑥0 𝑒 𝑗(𝜑𝑥 +𝛿𝑥 ) ] , where 𝛿𝑥 and 𝛿𝑦 are, respectively, the phase delays of x and y [ 𝐸𝑦0 𝑒 𝑗(𝜑𝑦 + 𝛿𝑦 ) components of the light field. Then we have 𝐸𝑥0 𝑒 𝑗(𝜑𝑥 +𝛿𝑥 ) 𝑎 [ 𝑗(𝜑𝑦 + 𝛿𝑦 ) ] = [ 𝑐 𝐸𝑦0 𝑒 𝑗𝜑 𝑏 𝐸𝑥0 𝑒 𝑥 ] ][ 𝑑 𝐸𝑦0 𝑒 𝑗𝜑𝑦 (1.172) Then the matrix form representing a waveplate is 𝑎 [ 𝑐 𝑗𝛿𝑥 𝑏 ] = [𝑒 𝑑 0 0 ] 𝑒 𝑗𝛿𝑦 (1.173) From this general form of Jones matrix for a phase retarder, we can get the Jones matrix of a quarter-wave plate and of a half-wave plate as followings. 99 Advances in Optics: Reviews. Book Series, Vol. 5 1) Quarter-wave plates 𝑒 −𝑗𝜋⁄4 [ 1 0 1 0 ], fast axis is in x direction, 𝑒 𝑗𝜋⁄4 [ 0 𝑗 2) Half-wave plates 𝑒 −𝑗𝜋⁄2 [ 0 ], fast axis is in y direction. −𝑗 1 1 0 ], fast axis is in x direction, 𝑒 𝑗𝜋⁄2 [ 0 0 −1 c) Rotators 0 ], fast axis is in y direction. −1 𝑐𝑜𝑠𝛼 For a linearly polarized light with the normalized field [ ] incident on a rotator of 𝑠𝑖𝑛𝛼 cos(𝛼 + 𝛽) angle 𝛽, the emerging field is [ ]. Moreover, the emerging and incident fields sin(𝛼 + 𝛽) satisfy the following relationship, [ cos(𝛼 + 𝛽) 𝑎 ] = [ sin(𝛼 + 𝛽) 𝑐 𝑏 𝑐𝑜𝑠𝛼 ][ ] 𝑑 𝑠𝑖𝑛𝛼 (1.174) We can get that 𝑎 = 𝑐𝑜𝑠𝛽, 𝑏 = −𝑠𝑖𝑛𝛽, 𝑐 = 𝑠𝑖𝑛𝛽, and 𝑑 = 𝑐𝑜𝑠𝛽, so that the Jones 𝑐𝑜𝑠𝛽 −𝑠𝑖𝑛𝛽 matrix of the rotator is [ ]. 𝑠𝑖𝑛𝛽 𝑐𝑜𝑠𝛽 We have got all Jones matrices for three types of polarizers. If the light with the electric 𝐸𝑥 field [𝐸 ] is sent through a train of polarizers or optical systems, the output light field 𝑦 𝐸𝑥′ [ ′ ] is given by matrix multiplication: 𝐸𝑦 [ 𝐸𝑥′ 𝑎𝑛 ′] = [ 𝐸𝑦 𝑐𝑛 𝑏𝑛 𝑎 ]…[ 2 𝑑𝑛 𝑐2 𝑏2 𝑎1 ][ 𝑑2 𝑐1 𝑏1 𝐸𝑥 ] [ ], 𝑑1 𝐸𝑦 (1.175) where every 2×2 matrix represents the corresponding polarizer. 1.6.3. Basic Theory of Polarization of Stochastic Light Fields In the Subsection 1.6.2 we just described the cases of monochromatically polarized light by Jone’s method. This subsection deals with, in a general method, the problem of characterizing the polarization of stochastic light fields that exist in practice. The reason for the generation of stochastic light fields in nature has been explained in Section 1.3. Here we will generally describe the completely polarized, unpolarized, and partially polarized light by means of the degree of polarization. 100 Chapter 1. Physical Optics 1.6.3.1. Correlation Matrix of a Light Field Let us consider a light beam that propagates along z direction, as shown in Fig. 1.53, and suppose that the light fields are statistically stationary, at least at wide sense. Let 𝐸𝑥 (𝑡) and 𝐸𝑦 (𝑡) be the components of the electric field of light at the point O in the beam. The second-order correlation properties of the electric field at the point O can be characterized by a 2×2 polarization matrix as the following. 𝐽𝑥𝑥 < 𝐸𝑥∗ (𝑡)𝐸𝑥 (𝑡) > < 𝐸𝑥∗ (𝑡)𝐸𝑦 (𝑡) > 𝐽 = [ ] = [ ∗ ∗ 𝐽𝑦𝑥 < 𝐸𝑦 (𝑡)𝐸𝑥 (𝑡) > < 𝐸𝑦 (𝑡)𝐸𝑦 (𝑡) > 𝐽𝑥𝑦 ], 𝐽𝑦𝑦 (1.176) where the asterisk denotes the complex conjugate, and < ∗ > ensemble average. The matrix J is an equal-time correlation one whose elements can be used to describe the state of polarization of a light field. It is clear that 𝐽𝑥𝑥 and 𝐽𝑦𝑦 are always real and nonnegative, and 𝐽𝑥𝑦 and 𝐽𝑦𝑥 mutually complex conjugate. Thus 𝐽 is a Hermitian matrix and can be written in the form 𝐽𝑥𝑥 𝐽 = [ 𝐽𝑦𝑥 𝐽𝑥𝑦 ] 𝐽𝑦𝑦 (1.177) Furthermore, with help of Schwarz’s inequality, we get that ｜𝐽𝑥𝑦 ｜ ≤ √𝐽𝑥𝑥 𝐽𝑦𝑦 (1.178) Therefore the determinant of 𝐽 is nonnegative det(𝐽) = 𝐽𝑥𝑥 𝐽𝑦𝑦 − ｜𝐽𝑥𝑦 ｜2 ≥ 0 (1.179) The trace of 𝐽 is equal to the average intensity, I, of the light wave. tr(𝐽) = 𝐽𝑥𝑥 + 𝐽𝑦𝑦 = 𝐼 (1.180) We can characterize the correlation between the x and y components of the electric field by the correlation coefficient 𝑗𝑥𝑦 = 𝐽𝑥𝑦 √𝐽𝑥𝑥 𝐽𝑦𝑦 (1.181) According to the inequality (1.178), we have 0 ≤ 𝑗𝑥𝑥 ≤ 1. 1.6.3.2. Completely Polarized, Unpolarized, and Partially Polarized Light 1) For completely polarized light, |𝑗𝑥𝑦 | = 1. This condition means that the statistical dependence between x and y polarization components is completely correlated, which is 101 Advances in Optics: Reviews. Book Series, Vol. 5 referred to as a completely polarized light. For this case, the correlation matrix J can be written 𝐽𝑥𝑥 𝐽 = [ √𝐽𝑥𝑥 𝐽𝑦𝑦 𝑒 −𝑗𝜑 √𝐽𝑥𝑥 𝐽𝑦𝑦 𝑒 𝑗𝜑 𝐽𝑦𝑦 ], (1.182a) where 𝜑 is the phase difference between x and y components and a real number. We express the formula (1.182a) as a general form as follows. 𝐴 𝐽 = [ ∗ 𝐶 𝐶 ], 𝐵 (1.182b) where * is the complex conjugate. From the matrix (1.182a) for a completely polarized light we know that 𝐴 ≥ 0, 𝐵 ≥ 0, and 𝐴𝐵 − 𝐶𝐶 ∗ = 0. 2) For unpolarized light, |𝑗𝑥𝑦 | = 0. This condition means that the statistical dependence between x and y polarization components is completely uncorrelated, therefore, the light field is called unpolarized because no any direction of the polarization component being dominant. For this case, the correlation matrix J can be written 1 𝐽 = 𝐽𝑥𝑥 [ 0 0 ] 1 (1.183) 3) Partially polarized light and the degree of polarization. In the above, we just described two extreme cases of the completely polarized and unpolarized light, expressed by |𝑗𝑥𝑦 | = 1 and |𝑗𝑥𝑦 | = 0, respectively. In what follows, it can be shown that any light beam, at each spatial point, can be expressed uniquely as the sum of the completely polarized and unpolarized light beams. First we introduce the concept of the degree of polarization. It is the ratio of the intensity of the polarized part to the total intensity of a light beam, denoted as ℘. The polarization matrix J is a Hermitian matrix and semidefinite positive. From matrix theory, a unitary matrix transformation can transform J into a diagonal matrix as 𝛴 = [ 𝜎1 0 0 ], 𝜎2 (1.184) where 𝜎1 and 𝜎2 (𝜎1 ≥ 𝜎2 ) are the nonnegative real-valued eigenvalues of J. We further write the above diagonal matrix as [ 𝜎1 0 0 𝜎 ] = [ 2 𝜎2 0 0 𝜎 − 𝜎2 ]+[ 1 𝜎2 0 0 ] 0 (1.185) From the formulas (1.182b) and (1.183), we know that the first matrix on the right hand side represents unpolarized light with average intensity 2𝜎2 , whereas the second one represents the linearly polarized light of intensity 𝜎1 − 𝜎2 . Thus the light beam with 102 Chapter 1. Physical Optics arbitrary polarization can be expressed as a sum of completely polarized and unpolarized components. According to the definition of the degree of polarization, it is ℘ = 𝜎1 −𝜎2 𝜎1 +𝜎2 (1.186) The degree of polarization can be expressed more explicitly in terms of the elements of the polarization matrix J. From the matrix theory, 𝜎1 and 𝜎2 can be gotten by solving the following characteristic equation. (1.187) det[𝐽 − 𝛴𝕀] = 0, where 𝕀 represents a 2×2 unit matrix. The solution of the above quadratic in 𝛴 yields 𝜎1,2 = 1 𝑡𝑟(𝐽) [1 ± 2 𝑑𝑒𝑡(𝐽) (1.188) √1 − 4 (𝑡𝑟(𝐽))2 ] Thus the degree of polarization, in terms of elements of the matrix J, can be expressed as 𝑑𝑒𝑡(𝐽) (1.189) ℘ = √1 − 4 (𝑡𝑟(𝐽))2 That ℘ = 0 stands for unpolarized light, that ℘ = 1 for completely polarized light and that 0 < ℘ < 1 for partially polarized light. The monochromatic plane wave expressed by the formula (1.159) is a completely polarized light one. For a quasi-monochromatic beam, if the ratio of the amplitudes of the two components of 𝐸𝑥 (𝑡) and 𝐸𝑦 (𝑡) and the 𝐸 (𝑡) 𝐸 (𝑡) difference between their phases are all constants, i.e. |𝐸𝑦(𝑡)| = 𝐶1 and arg(𝐸𝑦(𝑡)) = 𝐶2 , 𝑥 𝑥 where 𝐶1 and 𝐶2 are constants, and arg() represents the phase difference, the degree of polarization ℘ of the quasi-monochromatic beam is still one. This result implies that for this situation the conventional experiment, employing only a compensator and a polarizer for determining the elements of the polarization matrix, cannot distinguish between a random quasi-monochromatic light beam and a strictly monochromatic beam. Representation of the Jones’s vector and of the polarization matrix for the light field is used to process interference effects, i.e. the combination of coherent light beams. For incoherent light, the Stokes vector and Mueller matrices are used to describe it. Here we will not discuss it further. Interested readers can consult the reference [5] In most of literatures, the subjects of coherence and polarization of light have been generally treated separately. However, both of them are manifestations of the correlations between fluctuations in light beams. Coherence of light describes the correlation between fluctuations at two or more points in space; Polarization of light is a description of correlation between fluctuating components of the electric field at a single point in space. The unified theory of coherence and polarization is, in detail, explained in the reference [15]. 103 Advances in Optics: Reviews. Book Series, Vol. 5 References [1]. M. Born, E. Wolf, Principles of Optics, Cambridge University Press, 1999. [2]. G. Brooker, Modern Classical Optics, Oxford University Press, 2003. [3]. G. Fowles, Introduction to Modern Optics, Dover Publications, 1989. [4]. O. K. Ersoy, Diffraction, Fourier Optics, and Imaging, John Wiley and Sons Inc., 2007. [5]. D. Goldstein, Polarized Light, 2nd Ed., CRC Press, 2011. [6]. G. Giusfredi, Physical Optics: Concepts, Optical Elements, and Techniques, Springer, 2019. [7]. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996. [8]. J. W. Goodman, Statistical Optics, John Wiley and Sons Inc., 2015. [9]. E. Hecht, Optics, Pearson, 2017. [10]. A. Lipson, S. Lipson, L. Lipson, Optical Physics, Cambridge University Press, 2011. [11]. M. V. Klein, Optics, John Wiley and Sons Inc., New York, 1970. [12]. L. D. Landau, E. M. Lifshits, The Classical Theory of Fields, Pergamon, 1975. [13]. E. L. O'Neill, Introduction to Statistical Optics, Addison-Wesley, Reading, MA, 1963. [14]. F. Pedrotti, L. Pedrotti, Introduction to Optics, Cambridge University Press, 2018. [15]. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, 2007. [16]. S. Zhang, C. Li, S. Li, Understanding Optical Systems through Theory and Case Studies, SPIE, 2017. Appendix. Derivation of the Rayleigh-Sommerfeld Formula This appendix tries to derive the Rayleigh-Sommerfeld formula (1.47) in Section 1.2 in the vein of Chapter 3 of the reference [7]. Any monochromatic optical field 𝑈(𝒓) propagating in a uniform medium obeys the Helmholtz equation (1.16), rewritten as follows. 𝛻 2 𝑈(𝒓) + 𝑘 2 𝑈(𝒓) = 0 (1.190) Solving this partial differential equation with certain boundary conditions, we can get the optical field 𝑈(𝒓) at a spatial position 𝒓 of observation. In what follows, we will elucidate how to get Rayleigh-Sommerfeld formula (1.47) from the Helmholtz equation (1.190). A1. Green's Theorem Calculation of the light field 𝑈(𝒓) at a spatial position 𝒓 of interest can be accomplished with the help of Green’s theorem. Let the light field 𝑈(𝒓) and an auxiliary function 𝐺(𝒓) be any two complex-valued functions of position 𝒓, and 𝑆 be a closed surface surrounding a volume V. If 𝑈(𝒓), 𝐺(𝒓), and their first and second partial derivatives are single-valued and continuous in the volume V and on surface S, Green’s theorem can be stated as follows 𝜕𝐺 104 𝜕𝑈 ∭𝑉 (𝑈𝛻 2 𝐺 − 𝐺𝛻 2 𝑈)𝑑𝑉 = ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 ) 𝑑𝑆, 𝜕𝑛 (1.191) Chapter 1. Physical Optics 𝜕 where 𝜕𝑛 represents a partial derivative in the outward normal direction 𝒏 at a point on surface S, and for simplicity we omit the spatial position vector 𝒓 in functions U and G. This theorem is the starting point for solving the Helmholtz equation of scalar diffraction theory. In what follows we will address how to choose the function G and a closed surface S for applying this theorem to diffraction problems. A2. The Integral Theorem of Helmholtz and Kirchhoff As shown in Fig. 1.55, let 𝑃 be the point of observation, 𝑆 an arbitrary closed surface enclosing 𝑃, and 𝒏 the outward normal direction of the surface 𝑆. Our goal is to express the optical field at 𝑃 in terms of the values and derivatives of the optical field on the surface 𝑆. We now apply Green’s theorem and choose an auxiliary function G as a unit-amplitude spherical wave from the point 𝑃 to an arbitrary point 𝑃1 in the volume V, i.e., 𝐺(𝑃) = 𝑒 𝑗𝑘𝑟 , 𝑟 (1.192) where 𝑟 is the length of the 𝒓 pointing from 𝑃 to 𝑃1 . As stated above, the function 𝐺 must be continuous within the enclosed volume V. For this reason, a spherical surface 𝑆𝜀 with small radius 𝜖 is inserted about the point 𝑃. In this way, Green’s theorem can be applied in which the volume of integration for 𝑉′ lies between 𝑆 and 𝑆𝜀 , and the surface of integration is on the composite surface 𝑆 ′ = 𝑆𝜀 + 𝑆. Within the volume 𝑉′, The Green function 𝐺 also satisfies the Helmholtz equation 𝛻 2 𝐺 + 𝑘 2𝐺 = 0 (1.193) Fig. 1.55. Surface of integration. On inserting the two the Helmholtz equations (1.190) and (1.193) into Green’s theorem 𝜕𝐺 𝜕𝑈 (1.191), we get ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 ) 𝑑𝑆 = 0, or 𝜕𝑛 105 Advances in Optics: Reviews. Book Series, Vol. 5 𝜕𝐺 𝜕𝑈 𝜕𝑈 𝜕𝐺 (1.194) − ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜕𝑛 ) 𝑑𝑆 = ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜕𝑛 ) 𝑑𝑆 𝜀 For a point 𝑃1 on the composite surface 𝑆 ′ , according to the expression of the Green function, we have its derivative as follows 1 𝑒 𝑗𝑘𝑟 , 𝑟 𝜕𝐺(𝑃1 ) 𝜕𝑛 (1.195) = cos(𝒏, 𝒓 ) (𝑗𝑘 − 𝑟) where cos(𝒏, 𝒓 ) stands for the cosine of the angle between the outward normal 𝒏 and 𝒓 joining 𝑃 to 𝑃1 . For the case of 𝑃1 on 𝑆𝜀 , cos(𝒏, 𝒓 ) = −1 , and 𝐺(𝑃1 ) = 1 𝑒 𝑗𝑘𝜖 𝜕𝐺 𝜕𝑈 ) 𝑑𝑆 𝜕𝑛 𝜕𝐺(𝑃1 ) 𝜕𝑛 = (𝜖 − 𝑗𝑘) 𝜖→0 𝜀 lim ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜖 . Let 𝜖 become arbitrarily small, then 1 = 𝑙𝑖𝑚4𝜋𝜖 2 [𝑈(𝑃) (𝜖 − 𝑗𝑘) 𝜖→0 Substituting this result into (1.192), we have 𝑈(𝑃) = 𝑒 𝑗𝑘𝑟 𝜕𝑈 1 [ ∬ 4𝜋 𝑆 𝑟 𝜕𝑛 𝜕 𝑒 𝑗𝑘𝜖 𝜖 − 𝜕𝑈(𝑃) 𝑒 𝑗𝑘𝜖 ] 𝜕𝑛 𝜖 𝑒 𝑗𝑘𝜖 𝜖 and = 4𝜋𝑈(𝑃) 𝑒 𝑗𝑘𝑟 )] 𝑑𝑆 𝑟 (1.196) − 𝑈 𝜕𝑛 ( The formula (1.196) is called the integral theorem of Helmholtz and Kirchhoff, which states that the light field at any point 𝑃 can be expressed in terms of the boundary values of the optical field on any closed surface surrounding the point 𝑃. A3. The Kirchhoff Formulation of Diffraction by a Planar Screen Consider now the problem of diffraction of light by an aperture in an infinite opaque with an open aperture Σ as shown in Fig. 1.56. A monochromatic light wave is assumed to being incident on the screen and the aperture from the left. We try to seek the light field at the point 𝑃 behind the aperture 𝑆 through applying the integral theorem of Helmholtz and Kirchhoff (1.196). In this case, the closed surface 𝑆 is a composite one with 𝑆1 + 𝑆2 , as shown in Fig. 1.56. The plane surface 𝑆1 lies directly behind the diffracting screen, and it connects to a large spherical cap, 𝑆2 , of radius 𝑅 and centered at the observation point 𝑃. Thus applying the formula (1.196) to this case, we have 𝑈(𝑃) = 𝑒 𝑗𝑘𝑟 𝜕𝑈 1 ) 𝜕𝑛 [( ∬ 𝑟 4𝜋 𝑆1 +𝑆2 𝜕 𝑒 𝑗𝑘𝑟 )] 𝑑𝑆 𝑟 (1.197) − 𝑈 𝜕𝑛 ( We further examine the integration on the surface 𝑆2 , i.e. the following integral 𝑒 𝑗𝑘𝑅 𝜕𝑈 𝜕 𝑒 𝑗𝑘𝑅 𝜕 𝑒 𝑗𝑘𝑅 1 𝑒 𝑗𝑘𝑅 𝑒 𝑗𝑘𝑅 ∬𝑆 [( 𝑅 ) 𝜕𝑛 − 𝑈 𝜕𝑛 ( 𝑅 )] 𝑑𝑆 , and 𝜕𝑛 ( 𝑅 ) = (𝑗𝑘 − 𝑅) 𝑅 ≈ 𝑗𝑘 𝑅 , where the 2 last approximation is valid provided very large 𝑅. The above integral can therefore be reduced to 2 106 𝑒 𝑗𝑘𝑅 𝜕𝑈 ) 𝜕𝑛 𝑅 ∬𝑆 [( 𝜕 𝑒 𝑗𝑘𝑅 )] 𝑑𝑆 𝑅 − 𝑈 𝜕𝑛 ( = ∬𝛺 𝑒 𝑗𝑘𝑅 𝜕𝑈 (𝜕𝑛 𝑅 − 𝑗𝑘𝑈) 𝑅 2 𝑑𝛺, (1.198) Chapter 1. Physical Optics 𝑒 𝑗𝑘𝑅 where Ω is the solid angle subtended 𝑆2 at 𝑃. The quantity | 𝑈| is uniformly bounded 𝑅 on 𝑆2 . Furthermore, let the light field have the property, known as the Sommerfeld 𝜕𝑈 radiation condition, lim 𝑅 ( − 𝑗𝑘𝑈) = 0. Then the integral (1.198) will vanish. We 𝑅→∞ 𝜕𝑛 can get that the light field at the point 𝑃 from the expression (1.197) is as follows 𝑈(𝑃) = 𝜕𝑈 1 ∬ [𝐺 𝜕𝑛 4𝜋 𝑆1 𝜕𝐺 − 𝑈 𝜕𝑛] 𝑑𝑆 (1.199) Fig. 1.56. Surface of integration. In the above expression, we have used the formula (1.192). As shown in Fig. 1.56, the screen 𝑆1 is opaque, except for the open aperture denoted as Σ. A4. The Rayleigh-Sommerfeld Formulation of Diffraction The conditions for validity of the equation (1.199) are: 1. The scalar theory of diffraction holds; 2. Both 𝑈 and 𝐺 satisfy the Helmholtz equation (1.190); 𝜕𝑈 3. The Sommerfeld radiation condition, which 𝑙𝑖𝑚 𝑅 (𝜕𝑛 − 𝑗𝑘𝑈) = 0, is satisfied. 𝑅→∞ The light field 𝑈 satisfies the Helmholtz equation (1.190), and we choose the Green’s function expressed by (1.200) in the following, which Green's function 𝐺 is generated together by a point source located at 𝑃, and a second point source 𝑃̃ at a mirror position of 𝑃 relative to the screen as shown in Fig. 1.57. 𝐺 = 𝑒 𝑗𝑘𝑟 𝑟 − 𝑒 𝑗𝑘𝑟̃ 𝑟 (1.200) Obviously such a function 𝐺 vanishes on the aperture Σ, and satisfies the Helmholtz equation. Furthermore Kirchhoff boundary conditions are modified as follows: 1. Across the aperture Σ, the field distribution 𝑈 is exactly the same as it would be in the absence of the screen; 107 Advances in Optics: Reviews. Book Series, Vol. 5 2. Over the portion of 𝑆1 that lies in the geometrical shadow of the screen, the field 𝜕𝑈 distribution 𝑈 is identically zero, but 𝜕𝑛 ≠ 0 that is one of Kirchhoff boundary conditions. Fig. 1.57. Geometry of a point source 𝑃 and its mirror point source 𝑃̃ . Such the light field at observation point 𝑃, according to the formula (1.199), is reduced to 1 𝑈(𝑃) = − 4𝜋 ∬𝛴 [𝑈 𝜕𝐺 ] 𝑑𝑆 𝜕𝑛 (1.201) This solution is called the first Rayleigh-Sommerfeld solution. 𝜕𝐺 Using (1.200), we can get 𝜕𝑛 on the aperture Σ 𝜕𝐺 𝜕𝑛 1 𝑒 𝑗𝑘𝑟 𝑟 𝑟 = 𝑐𝑜𝑠(𝒏, 𝒓) (𝑗𝑘 − ) 1 𝑒 𝑗𝑘𝑟̃ 𝑟̃ 𝑟̃ − 𝑐𝑜𝑠(𝒏, 𝒓̃) (𝑗𝑘 − ) (1.202) Further we have 𝑟 = 𝑟̃ , and 𝑐𝑜𝑠(𝒏, 𝒓) = −𝑐𝑜𝑠(𝒏, 𝒓̃). Therefore the expression can be 1 𝑒 𝑗𝑘𝑟 . 𝑟 𝜕𝐺 written 𝜕𝑛 = 2𝑐𝑜𝑠(𝒏, 𝒓) (𝑗𝑘 − 𝑟) 1 For 𝑘 ≫ 𝑟, the above expression can be approximated as follows 𝜕𝐺 𝜕𝑛 = 2𝑗𝑘𝑐𝑜𝑠(𝒏, 𝒓) 𝑒 𝑗𝑘𝑟 𝑟 (1.203) On substituting (1.203) into the equation (1.201), we get the Rayleigh-Sommerfeld formula as follows 𝑈(𝑃) = 1 𝑒 𝑗𝑘𝑟 𝑈(𝑃 ) 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆 ∬ 1 𝑗𝜆 𝛴 𝑟 This is just the formula (1.47) in Section 1.2. 108 (1.204) Chapter 2. CPU Technology Trend and Optical Interconnects Chapter 2 CPU Technology Trend and Optical Interconnects Sahnggi Park1 2.1. Introduction Computer CPUs are a measure that reveals the highest level of technology that humans can attain. Any type of intelligent models and programs that can be imagined by humans are affected by the CPU performance. This article describes the technology, characteristics and market trend of the latest CPUs used in smartphones, computers, and supercomputers released in 2020 as well as the optical network-on-chip architectures proposed by researchers of major corporations and universities. A CPU is the device that executes programs that list a large number of instructions. The Section 2.2 describes CPU design and fabrication. The CPU design corresponds to creating an instruction set and converting all logic functions contained in the instruction to transistors. CPU fabrication is the work of drawing all the transistors and communication lines in photomasks and carrying out the CMOS production process according to the drawing [1-3]. The system on chip (SoC) technology, the internal configuration of SoCs, and the types of ARM company licenses are described in the Section 2.3, and the latest 2020 SoC performances and characteristics, manufacturers, market shares, and the smartphone disassembly layouts are also described [4-7]. The internal architecture of Intel CPUs, the communication line width and communication speeds between registers and caches, and between the L1, L2, and L3 caches, are described in the Section 2.4. The ring bus and mesh network which are the communication systems inside Intel CPUs are also described in the same chapter. Supercomputer server configuration and 3D torus and fat-tree optical network architectures are described in the Section 2.5. Furthermore, the technology and characteristics of x86 CPU and Sahnggi Park Oprocessor Inc., Boston, MA, USA 109 Advances in Optics: Reviews. Book Series, Vol. 5 supercomputers developed by China for IT technology independence are also described [8-15]. Major corporations and universities have proposed the network architectures that can be adopted when optical communication technology would be implemented to the CPU chip. Among them, the frequently cited ATAC architecture of MIT and the Corona Architecture of HP are outlined in the Section 2.6. In addition, the main reasons why these architectures have not been realized are described [16-20]. It would be the most realizable to adopt the technology and components utilized in supercomputers, the vertical cavity surface emitting laser (VCSEL) and photodiode (PD) and the fat-tree optical network, while replacing only the optical fiber with SiON/SiN optical waveguides, to implement optical network-on-chip to the Intel server CPUs. For this, three key technologies and details proven through experiments are described in the Section 2.7 [21-24]. 2.2. CPU Design and Fabrication For CPU design, it is essential to understand the structures of programming languages. As shown in Fig. 2.1, C++, Python, and Java are used in coding. Once the compilation is finished, it is converted into assembly language. The assembly language is a language that represents an instruction set architecture (ISA) such as ARM, x86, MIPS and RISC-V, which is converted into a machine language such as 10101 by an assembler. Although CPU recognizes machine language when programs are executed, instruction sets expressed in the assembly language play an important role in actual CPU design. Fig. 2.1. Programming language level. The x86 belongs to the complex instruction set computer (CISC), which has been extensively developed by Intel and is frequently called the x86 instruction set. While each instruction in the reduced instruction set computer (RISC) has a fixed-length (typically 4 bytes), which limits the expression of instructions, and there is not a large number of 110 Chapter 2. CPU Technology Trend and Optical Interconnects instructions, the x86 instruction set has a variable length (typically 1-16 bytes), allowing for a free expression of instructions, and has a much larger number of total instructions. Although it is confidential to find out the instruction set of ARM or x86, the similar RISC-V details are freely available and has a total number of about 100 instructions, and the x86 instruction set is known to have thousands of instructions. As shown in Fig. 2.2, a processor is largely composed of the logic part, register part and cache part. The logic part performs arithmetic calculations using the given function and data, and the register and cache parts store the instructions and data. The register part stores instructions and data that are currently undergoing calculations, and the cache part stores instructions and data that are finished with the calculation or are waiting to the L1, L2, L3 caches. The functions to be processed and the necessary data are designated in the instructions received by the register, and after the calculation is done in the logic part corresponding to each function, the processed result goes to the register for the next calculation or are stored in the cache, and if the order of priority is very low, it is sent to the main memory (DRAM). Fig. 2.2. Schematic diagram of the processor. The logic part is composed of transistors of logic functions such as AND, OR, NOT, NAND, NOR. A transistor, nMOS or pMOS, is a switch to turn the current on/off between the source and drain according to the voltage applied to the gate. Fig. 2.3 is a conceptual diagram of 1 bit NOT function formed on a silicon wafer. Each instruction contains functions, each of which contains many logic functions, and it performs 64-bit calculations through complex connections and arrangements and then sends the results to the register. For an x86 instruction set, due to the thousands of instructions, the logic part has an assigned area for transistors corresponding to each instruction, and the register and input/output lines are connected through a complex connecting structure. To arrange the entire instruction set, a CPU contains about 1~10 billion transistors. Between the register, the logic and cache parts is the control unit (CU) that controls the transmission of instructions and data, and all transmission signals go through it. 111 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 2.3. Schematic diagram of NOT Function. The register and cache parts are composed of SRAM transistors to store instructions and data. Fig. 2.4 shows the basic structure of an SRAM storing a 1-bit signal. The two NOT functions are connected to each other to form a structure that connects output to input and input to output. When the word line (WL) signal turns off after a 1-bit signal is input to the bit line (BL), the signal goes to the input of the NOT function on the right and comes out reversed as output. The signal goes back to the input of the NOT function on the left and reverses back to its original state and goes back to the input of the NOT function on the right, maintaining the state of storage through permanent repeating cycles. The register and the L1, L2, and L3 caches are assigned to their corresponding locations according to the CPU design with the determined number of bytes and are connected to the 64-bit bus line in the CPU. Fig. 2.4. 1-bit storage of SRAM circuit. The description of CPU design and manufacturing is outlined as follows. As technology advances, instructions are changed or instructions with new functionalities are added to the existing instruction set. Once the final instruction set is determined, logic functions that calculate the function of each instruction are placed and connecting lines are drawn to the transistor. Since the logic part of the CPU must be able to execute all of the instructions, for x86, logic functions and transistors corresponding to the thousands of instructions need to be placed and connected. Registers, caches, control units, and bus lines must also be drawn according to the determined specifications. For this work, each of the components is drawn using the high-level synthesis tools owned by each company 112 Chapter 2. CPU Technology Trend and Optical Interconnects and the work is divided and executed by many engineers. A lot of manpower and time are required especially for simulation and verification due to the loss of billions of dollars that occurs when a bug is confirmed after production has begun. Once this work is finished, the work of drawing the mask layer for the actual chip manufacturing begins. At this time, if a manufacturing request is made to a foundry company, such as TSMC, the mask layer must be drawn according to the technical specifications provided by the foundry company. For example, when using the 7 nm technology of a foundry company, the gate width of the corresponding transistor must meet this specification. Once the basic specifications are established, the mask layer work is automatically transferred using a computer program. When the mask layer drawing is finished, it is sent to the foundry company to manufacture the chip. Fig. 2.5 is a cross-sectional view of a Broadwell CPU chip manufactured by Intel in 2013. It used the 14 nm process and applied the finFET transistor technology. The finFET transistor is formed on the bottommost silicon layer. Although the gate width is 14 nm, it is known that the gate pitch is 70 nm, the fin pitch is 42 nm, and the minimum pitch of the metal line is 52 nm; these are the parts that directly affect the number of transistors per area. The number of transistors integrated per area increases gradually according to the technology node but the width of the metal line, in order to maintain the electrical characteristics of the signal transmission line, cannot decrease in proportion. Therefore, the number of metal layers has been increasing three-dimensionally, and Fig. 2.5 shows the cross-sectional view of the first chip with 13 layers. As of 2019, Intel is applying a 10 nm technology to manufacture the latest CPU chips, and the metal layer is known to be 13 layers. TSMC, which manufactures the latest CPU chips for AMD, Apple, Huawei and Qualcomm, announced that they will introduce the 5 nm process and use 15 metal layers. Samsung is known to have used the 8 nm process for the production of Exynos chips that were sold in 2019 and introduced the 7 nm process in the second half of 2019. Fig. 2.5. Cross section of Broadwell CPU chip (Source: https://www.chipworks.co.uk). When a user runs a program through Windows or Android OS, the program stored on the hard disk is brought to the main memory (DRAM). A program is a structure in which many instructions are sequentially arranged and machine signals such as 101010 are 113 Advances in Optics: Reviews. Book Series, Vol. 5 transmitted. Each instruction is brought to the CPU sequentially and stored in the L1 cache in a waiting state. When it is their turn, it is sent to the decoder and after the instruction content is analyzed, it is transferred to the register. In addition, if the function included in the instruction needs data, data is transferred from the hard disk to the CPU register or cache. When both instruction and data are prepared, they are sent to the corresponding logic part to undergo calculations. Then, the results are stored in the register or the cache. Processes like this can be seen as a diagram in Fig. 2.6. Fig. 2.6. Pipeline diagram. Instead of executing one instruction and storing the result data and then executing the next instruction, the process is divided into stages: the fetch stage that receives the instructions, the decode stage, the execute stage where data is sent to the logic part to calculate the function, and the writeback stage where the calculated results are stored. It is more like a machine that is assembled as it moves along a conveyor belt. As multiple instructions are in the waiting state, when the first instruction enters the decode stage after the fetch stage, the second instruction enters the fetch stage. When the first instruction enters the execute stage, then in sequential order, the next instructions enter their respective next stages. Processes like this are called pipelines. The technology in which one core in the CPU operates two or more pipelines is called the superscalar technology, and such CPU is called a superscalar processor. The components that operate a pipeline are called threads. In order to operate two or more threads, the register, logic part and cache are shared and as many decoders as the threads are placed and operated individually. When one core executes two instructions on two treads at the same time, since they share the logic part, if one instruction is a real number calculation, then the other instruction should be an integer calculation. All recent Intel and AMD CPUs have two threads per core and are called hyper threads. For Quad-core CPUs, there are four physical cores, but OS recognizes it as eight. One of the key technologies in CPU design is parallel computing technology. Parallel computing has a variety of parallel computing methods and is normally divided into bit-level, instruction-level, task-level, data-level parallel computing. All CPUs used in the latest smartphones and PCs support 64 bit. In CPU, 64 bit means that there are at least 114 Chapter 2. CPU Technology Trend and Optical Interconnects 64 signal lines between the register and logic or cache, and that the 64 bit signals, 1 bit per line, are simultaneously transmitted and calculated or stored during one clock cycle. That is, bit-level parallel computing indicates that 64 bit of data is processed in parallel at the same time. For a 32 bit computer, it calculates and stores by dividing the 64 bit data into two. Therefore, a 64 bit computer has a processing speed that is twice as fast as a 32 bit computer through bit-level parallel computing. If parallel execution occurs in a program with many instructions sequentially listed, each core executes multiple instructions simultaneously, this is called instruction-level parallel computing. Large programs consist of many tasks and each task has a large number of instructions. The CPU distributes each task to each core or thread for parallel execution and this is called task-level parallel computing. In special programs, many data can be processed by one instruction. Let's say there is a program that calculates the sum (A + B = C) of Vector A = [1, 2, … 10000] and B = [10001, 1002, …, 20000]. Suppose that 10,000 components are divided into 1,000 pieces and distributed over 10 cores to calculate the final sum. The ten cores execute the same one instruction and all data is input regularly in a sequence (SIMD, single instruction multiple data). This type of calculation is called data-level parallel computing, the instruction is called vector instruction and the processor is called vector processor [1, 2]. An ARM CPU or x86 CPU is a processor for general function calculation, with logic functions corresponding to hundreds or thousands of instructions arranged in the logic part. When a large data is processed with a small number of instructions with general-purpose CPU, such as data-level parallel processing, the logic functions that correspond to other remaining instructions will be inactive and the CPU performance won't be utilized properly. In order to solve this problem, various accelerator processors are being developed. Typical examples are a graphics processing unit (GPU), an AI processor or a neural processing unit (NPU), an image processing unit (IPU), an image signal processor (ISP) or a digital signal processor (DSP). Using GPU as an example, there are hundreds of cores in the GPU and each core consists of independent logic parts and resisters that can process a small number of instructions. By properly distributing the large data, hundreds of cores process the instructions in parallel at the same time. Since the design and manufacturing process of the accelerator processors is the same as that of general-purpose CPUs, the CPUs for smartphones and PCs are in the form of a system on chip (SoC) in which the CPU, GPU, IPU, and NPU are manufactured as one die and integrated on the same chip. While the clock frequency of CPU is fixed in the range of 1.5 ~ 4.0 GHz, CPU's processing speed of instructions is continuously increasing due to the advancement of parallel computing technology. In other words, because instructions are executed in parallel at the same time, instructions per cycle (IPC) continues to increase. While cycles per instructions (CPI) or IPC is used as an indicator of CPU speed, for x86 CPU that has different lengths of instructions, floating point operations per second (FLOPS), which is used for supercomputers, are recently used by ARM CPU and x86 CPU to express the speed. Floating point represents real number values in a format such as +1.12345×108 and the name was given because the decimal point can move freely. If this number is expressed as a binary number, it will be in a format such as +10101010×210101 (not the same numeric 115 Advances in Optics: Reviews. Book Series, Vol. 5 value as the former). 64 bit data is called double precision (precision below the decimal point) and 1 bit is assigned to the +/- position, 52 bit to the 10101010 position and 11 bit to the 10101 quotient position to express the number. 32 bit data is called single precision and 1 bit, 23 bit, and 8 bit are assigned to each respective position. The operation used to calculate FLOPS is called fused multiply-add (FMA) in which two numbers are multiplied and another number is added to the sum, like (A × B) + C, then finally, the number is rounded up to determine the last digit. It includes 2 operations of multiplication and addition. During one clock cycle, 64 bit data A and B are sent to the logic function from the cache to be multiplied, then value C that was stored in the register and the result of the former are sent to the addition function to find the sum. When one core completes this operation, it becomes two FLOPS and when the clock frequency is 3 GHz, then the core speed becomes 6 Giga FLOPS. If the CPU has four cores, it will have a speed indicator of approximately 24 Giga FLOPS. In addition, it will have a speed indicator of 48 Giga FLOPS based on single precision data calculation for the same CPU. In reality, the FLOPS indicator is not directly proportional to the number of cores but depends on the CPU design. It is obtained by executing a benchmark program that measures FLOPS. In order to increase the processing speed of CPU, technology is developing to allow many of the hardware elements in the CPU to participate as much as possible in the execution of work without being idle each hour. If the order of instructions written in the program is A-B-C, then B uses the result from A after A is executed, but on the other hand, if C can be executed through cache data regardless of the result of A, then the CPU order of process becomes A-C-B, which is different from the program order. That is, if the optimal order for the whole program is found and executed, CPUs with this capability are marked out-of-order in its performance indication. In some cases, it proceeds by selecting either task 1 or task 2 depending on a condition like the if function in the program. This type of function's instruction is called branch instruction and it takes up about 20 % of the program. In order to increase the processing speed, CPU starts operating by determining the thread or core that is expected to run the task by predicting it before the function's selection result is provided. If the prediction is correct, the process continues and if it is wrong, the work is discarded and it starts the selected task again. This type of function is called branch prediction and it requires a hardware installation, which is called a branch predictor. Prediction programs like this are designed to increase its accuracy by learning through the program and they are said to have a rate of accuracy over 90 %. When the program starts running, it conducts speculation and scheduling to utilize all the necessary elements in the optimal condition, so the necessary scheduler has to be installed on the CPU. One of the most essential technology in enabling parallel computing, out-of-order, branch prediction, and scheduling is cache coherence. Coherence is a perfect coordination state, with no dissonance or inconsistency between the components that are connected and move together for each process. Take a CPU with four cores as an example. Each core owns L1 and L2 caches privately while L3 cache is shared. From the L1 and L2 caches belonging to core 1, only core 1 can read to and write from L1. Although core 1 has priority over L2, it is possible for cores 2~4 to read from and also write to L2. L3 can be read from and written to with the same qualification without special priority. Table 2.1 shows the storage 116 Chapter 2. CPU Technology Trend and Optical Interconnects capacity and latency, which is the time it takes to read and write, for each storage location. Let's say that data A is stored at a specific address in L3 and core 1, core 2, and core 3 read value A and store it in their respective L1 cache. If core 1 changes data A from its L1 cache data into B, it must notify the change to all the other cores and share the fact that data A is no longer valid. Data A located at the L3 address must also change from A to B. If cache data is not correctly coordinated between the cores, some cores will input A in the formula where B should be entered or vice versa, and the entire program will be in error. Cache coherence is a state in which cache data is quickly and accurately coordinated to prevent errors, even when all the cores or threads read and write numerous times. For cache coherence, communication speed between cores plays a very important role. As the number of cores increases, the communication in the CPU using electrical signals is expected to reach its limit. Therefore, researches to introduce optical signals, under the name of silicon photonics, began in the mid-2000s with the participation of world's leading corporations and research institutes, but it has not been successful yet. Table 2.1. Latency and capacity by data storage location. Component L1 cache L2 cache L3 cache Memory SSD storage HDD storage Example latency 1 ns 3 ns 10 ns 100 ns 30,000 ns 15,000,000 ns Example capacity 64 kB per core 256 kB per core 20,480 kB 64 GB 100 GB 1 TB Various protocols have been developed and are used for cache coherence and a representative protocol would be MESI (modified, exclusive, shared, invalid) protocol used by Intel, shown in Table 2.2. When core 1 reads or writes data A from L3, core 1 goes into an exclusive state and uses the corresponding address in L3 exclusively, and all existing data in the remaining cores becomes invalid and cannot be used. When core 2 and core 3 send a signal (called snoopy signal) to read data A over the snoopy bus, the cache data of core 1 is transmitted through the data bus. At this time, all cores are in a shared state. After that, if core 3 changes data A to B, core 3 goes into a modified state and the other cores become invalid and the existing data A cannot be used. If core 3 writes data B on the corresponding address of L3, core 3 becomes exclusive and the corresponding address of L3 can only be used by core 3. Then the remaining cores will go into an invalid state. If core 4 sends the read signal over snoopy bus, it receives data B of core 3 through the data bus, and at this time all cores are in a shared state. Data on the data bus is sent to all the cores, so core 1 and core 2 can each store data B in their cache. When a perfect cache coherence is accomplished, CPUs with multiple cores run the program as if it had only one large core [1-3]. The market size and share of ARM CPU and x86 CPU according to 2020 statistics are shown in Fig. 2.7. Although parts of the Tablet, Cell-phone App and Embedded areas use x86 CPU, most of them use ARM CPU. So, the sum of the three areas can be estimated 117 Advances in Optics: Reviews. Book Series, Vol. 5 to be the share of ARM CPU. It can be seen that the share of ARM CPU is approximately 50 %. According to the 2012 statistics by the same agency, the share of pure x86 CPU decreased from 64 % to 50 % in 2020. The growth of ARM CPU market share is expected to accelerate even more considering that Intel has stopped the production of Atom CPU and Apple has announced that they will be using ARM CPU instead of Intel CPU for their Mac-Book computers from 2020. The total market size is about 79.3 billion dollars. Table 2.2. MESI protocol. Core1 Core2 Core3 Core4 E I I I Read/Write A S S S S Read A A A I I M I Modified A A B I I E I Write A A B S S S S Read B B Fig. 2.7. Market Share of ARM CPU and x86 CPU (source: 2020 edition of IC Insights’ McClean Report). 2.3. Smartphone SoC Trend The latest processor models released on ARM's homepage include the Cortex-A78, Cortex-A78AE, and Cortex-X1. Cortex-A78 was released and adopted in 2020 by the smartphones with the best specifications. Cortex-A78AE has functions improved in the automotive enhancement area from Cortex-A78. It is known that Cortex-X1 was focused on maximizing performance with compromising on higher power usage and a larger die area. The features of Cortex-A78 published on the homepage are shown in Table 2.3. Depending on the manufacturing process and technology node, it can have a clock frequency of up to 3.0 ~ 3.3 GHz. In addition to the general register and logic parts, each 118 Chapter 2. CPU Technology Trend and Optical Interconnects core has a cryptography unit that enhances the security function and a neon/floating point unit for vector computation. Cores consisting of big-littles can have up to 4 bigs and 4 littles [4-6]. The ARM processor has a different characteristic than x86. In the case of octa-core CPU with 8 cores, 4 big cores of the latest specification with high computing speed and high-power consumption and 4 little cores with relatively slower computing speed but high-power efficiency are arranged in 1:1 pairs. Big cores are used when a high performance of CPU is required, and the small cores are used when a lower performance can be used and when it is trying to reduce power consumption. Therefore, the actual number of cores used corresponds to a quad-core. In 2017, ARM announced the DynamIQ technology that they developed as a technology to efficiently operate the workload and data sharing between the accelerator processors inside a SoC. Depending on the type of the license that was signed, this technology has been applied to the latest SoC. DynamIQ technology allows for various configurations such as big-little as well as big-middle-little. For Qualcomm's Snapdragon 888, it is configured as 1-3-4. The ARM processor models compatible with DynamIQ technology are Cortex-A55, Cortex-A75~A78 and Cortex-X1. Cortex-A55 is configured as little and the rest are used as middle or big [6]. Table 2.3. Features of Cortex-A78. ARMv8.2-A (cryptography extensions, RAS extensions), ARMv8.3-A Max. CPU clock 3.0 GHz in phones and 3.3 GHz in tablets/laptops Out of order pipeline, Superscalar, Neon/Floating Point Unit, Microarchitecture Cryptography Unit 1. 64 bit program registers 64 bit registers 31, 128 bit registers 32 Registers 2. 32 bit program registers 32 bit 13, 64 bit 32 L1 cache: I and D 64 KB, L2 cache: 256~512 KB, Memory system L3 cache: optional, 512 KB~4 MB ISA version ARM licenses apply various options depending on the company they are signing with, and though it is difficult for a third party to understand in detail, the licenses are roughly classified into three types (see Table 2.4). In a cortex license, ARM provides the instruction set and CPU core that completed the circuit design, simulation, and verification work. The partner company takes on the mask work necessary for manufacture, which includes the design of the above CPU and of the other components (e.g. GPU, NPU, etc.,) included in the SoC. Huawei CPUs are known to belong to this type. In an architecture license, ARM provides only the instruction set while the core design, verification and other follow-up work for manufacture are done by the partner company. Since 2016, the 'Built on ARM Cortex' license became available in which the partner company can request their requirements to reflect in the instruction set. It is said that the customized ISA cannot be provided to other companies. The range of technology provision of ARM goes up to 119 Advances in Optics: Reviews. Book Series, Vol. 5 the stage of completed circuit design and verification, like the cortex license. Other than this simple classification, there are many details per each type or partner. Apple, Samsung, and Qualcomm belong to the architecture license type, and Qualcomm has since been known to have changed to the 'Built on ARM Cortex' license. In addition, the architecture license cannot use the DynamIQ technology provided by ARM. For the two licenses other than the cortex license, the partner's brand name may be used for the processor. Table 2.4. License types of ARM. Cortex License Architecture License ‘Built on ARM Cortex’ license Partner licenses complete microarchitecture design. CPU differentiation through - flexible configuration options - wide implementation envelope with different process technologies Partner designs complete CPU microarchitecture from scratch. Freedom to develop any design - must conform to the rules and programmers’ model of a given architecture - Must pass ARM architecture validation to preserve software compatibility A new engagement model for Cortex-A partners, announced in Feb. 2016. Partners work with ARM to develop a custom RTL database. Currently, in 2020, the four companies with the highest technology and market share among SoC suppliers, including smartphone CPUs that purchased the IP license from ARM CPU are Qualcomm, Samsung, Apple, and Huawei. The market share of smartphone SoCs in the second quarter of 2020 according to the data from a statistics company (Statista) is shown in Fig. 2.8. Qualcomm has 32 %, Huawei 22 %, Apple 19 %, and others 27 %. From the percentage of others, Samsung and MediaTek have the largest share. HiSilicon is a subsidiary of Huawei specializing in design, and MediaTek is a SoC design specialist company in Taiwan and supplies SoC for mid to low priced smartphones. In order to identify the highest technology as of 2020, it is necessary to analyze the SoCs of the four companies above. Table 2.5 shows the SoC model names and features of the four companies that were released in 2020, and the 2020 smartphone models that applied SoCs. Each SoC has CPUs consisting of six to eight cores, GPUs, AI engines, ISPs, 4G or 5G modems, and memory controllers integrated on a chip about 1 cm2 in size, with the total number of 8~10 billion transistors. Qualcomm is the company with the highest level of technology among the four companies. Although it does not produce its own smartphone models, many companies including Samsung have released their latest smartphones in 2021 using Qualcomm's SoC. Qualcomm designed all the devices with its own technology except for the CPU. For the CPU, the company brand Kryo 680 was used under ARM's 'Built on ARM Cortex' license. Qualcomm Kryo 680 SoCs were manufactured by using the Samsung foundry with 5 nm technology node in 2020. 120 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.8. ARM CPU market share by company (source: Statista 2020). Table 2.5. Features of 2020-2021 smartphone SoC. Qualcomm 2021 SoC 2021 Top device Features 2021 SoC 2021 Top device Features Snapdragon 888 Galaxy S21, S21+, S21 Ultra, Mi 11, iQOO 7, OnePlus 9 CPU 1 × Kryo 680 Prime (Cortex X1-based), 2.84 GHz, L2 cache 1024 KB 3 × Kryo 680 Gold (Cortex-A78-based), 2.42 GHz, L2 cache 3×512 KB 4 × Kryo 680 Silver (Cortex-A55-based), 1.8 GHz, L2 cache 4×128 KB L3 cache 4 MB GPU: Adreno 660 (840 MHz) DSP: Hexagon 780 (AI engine) ISP: Triple 14-bit Spectra 580 ISP Modem: Internal LTE Modem (4G): DL 2500 Mbit/s, UL 316 Mbit/s Internal 5G Modem: DL 7500 Mbit/s, UL 3000 Mbit/s Memory controller: LPDDR4X 4-channel 16-bit (64-bit) 3200 MHz (51.2 GB/s) Samsung 5 nm Technology node Samsung Exynos 2100 Galaxy S21, S21+, S21 Ultra CPU 1 × Cortex X1, 2.91 GHz, L2 cache 512 KB 3 × Cortex-A78, 2.81 GHz, L2 cache 3×512 KB 4 × Cortex-A55, 2.20 GHz, L2 cache 4×64 KB L3 cache 4 MB GPU: Mali G78MP14, 854 MHz NPU: Triple-core 26 TOPS ISP: 8 K@30 fps & 4 K@150 fps Modem: Internal LTE(4G), DL 3000 Mbit/s, UL = 422 Mbit/s, Internal 5G Exynos DL 7350 Mbit/s, UL 3670 Mbit/s. Memory controller: LPDDR4X Quad-channel 16-bit (64-bit) 3200 MHz (51.2 GB/s) Samsung 5 nm Technology node 121 Advances in Optics: Reviews. Book Series, Vol. 5 Table 2.5. Features of 2020-2021 smartphone SoC (continued). Huawei 2020 SoC 2020 Top device Features 2020 SoC 2020 Top device Features Kirin 9000 Mate 40 Pro, Mate 40 RS, Mate X2 CPU 1 × Cortex-A77, 3.13 GHz, L2 cache N/A 3 × Cortex-A77, 2.54 GHz, L2 cache N/A 4 × Cortex A55, 2.04 GHz, L2 cache N/A L3 cache: No information available GPU: Mali G78MP24, 759 MHz, 2331 GFLOPs NPU: 2 + 1 Da Vinci NPU ISP: 8 K @30 fps, 4 K @60 fps Modem: Integrated Balong 5G Modem, DL 7500 Mbit/s, Upload 3500 Mbit/s Memory controller: LPDDR4X Quad-channel 16-bit 2133 MHz (34.13 GB/s) TSMC 5 nm Technology node Apple A14 Bionic iPhone 12, iPhone 12 Pro, iPhone 12 Pro Max CPU 2 Firestorm, 2.99 GHz 4 Icestorm, 1.82 GHz L1, L2 cache: No information available, L3 8 MB GPU: Apple Custom GPU (quad-core) NPU: Neural Engine (octa-core) + AMX Blocks (dual-core)) ISP: 4 K @60 fps Modem: Intel Gigabit-Class LTE (4G), Qualcomm X55 (5G) Memory controller: LPDDR5 Dual-channel 64-bit (42.7 GB/s) TSMC 5 nm, Die size: 88 mm² OS: iOS 14.4.1 Samsung established Samsung’s Austin R&D Center (SARC) in Austin, Texas in 2010 and recruited former AMD, Intel, and ARM engineers to design SoCs. In addition, Samsung established Advanced Computing Lab (ACL) in San Jose, California in 2017 and is developing GPUs by introducing MediaTeks' GPU technology. Exynos 9820 and 9825 in 2019, predecessor of Exynos 2100 have used the M4 core which has its own brand name. Exynos cores and GPUs use ARM designs, but other devices, such as NPU, ISP, and 5G modems, have been developed using the company's own technology. Fig. 2.9 shows the market share statistics of smartphone manufacturers. Samsung has been the world's largest smartphone manufacturer from 2012. Huawei surpassed Apple in 2018 and became the second-largest smartphone manufacturer. The Da Vinci NPU and 5G modem were considered to be superior to those of Qualcomm and Samsung. The 5G modem of Kirin990 is the first to be integrated on the SoC. Although having made the boldest investments, Huawei’s technology development may be in difficulty due to the trade conflict between US and China, of which effect starts to appear in the second quarter of 2020. 122 Chapter 2. CPU Technology Trend and Optical Interconnects While Android OS smartphones account for about 90 % of the market share, the market share of Apple, which solely uses iOS, has been decreasing rapidly. It is difficult to maintain a competitive edge as a single company that is responsible for iOS development, SoC design, and smartphone production. The company's technology of CPU, GPU (PowerVR company license) and modem which are the main devices of SoC are relatively low and smartphone production is done by consignment assembly. Therefore, other than iOS, it lacks competitiveness in the remaining areas. For 5G modem, while the three competing companies finished development in the first half of 2019 and released smartphones with the 5G modem attached in the second half of 2019, Apple, which acquired 4G modem supplies from Intel, terminated its supply contract with Intel when Intel failed to keep the promise to finish developing the 5G chip by the second half of 2019. As Apple was looking for a third-party 5G chip supplier and Intel had given up on the 5G chip development, in July 2019, Apple ultimately bought Intel's modem business for about 1 billion dollars and started its own development in the second half of the year. As of September 2019, the companies that have completed the development of the 5G modem chip are the three companies mentioned above and MediaTek. It is known that Spreadtrum, now renamed Unisoc, is preparing to integrate 5G chips on low-end phones. Huawei and Unisoc are virtually owned by China and their technology can be seen as shared. As of March 2021, Apple’s latest device, iPhone 12 Pro Max has a 5G Modem provided by Qualcomm. Fig. 2.9. Quarterly smartphone market share (source: Statista 2021). Fig. 2.10 shows the image of the main PCB of Samsung Galaxy S10 5G released in 2019. Major technology analysis companies have uploaded on the internet the teardown photo, along with the cost analysis of the major components. This image is one of those photos taken from a company, Techinsights. As a package on package (PoP), there is the Samsung SoC Exynos9829 and the main memory, 8 GB SDRAM. There are also the Samsung Exynos 5100 external 5G modem, 256 GB flash memory, and WiFi/Bluetooth module. The PCB consists of two parts, one large and one small. The large main PCB has 123 Advances in Optics: Reviews. Book Series, Vol. 5 devices attached to the front and back, and only the front of the main PCB is shown in the image. Fig. 2.10. Samsung Galaxy S10 5G (Reprinted with permission from Techinsights (https://www.techinsights.com/ko)). Fig. 2.11 shows the front and back of the main PCB of the Apple iPhone Xs Max, released in 2018. Like the Samsung product, the PCB consists of two parts and Intel's 4G LTE modem is on the smaller part, which is not shown in the figure. Apple SoC A12 Bionic and 4 GB SDRAM of Micron are integrated by the PoP technology and there's Sandisk's 256 GB flash memory. Fig. 2.12 is the image without the SDRAM on an Apple SoC A12 Bionic released in 2018. The locations of the big cores, little cores, caches, GPUs, and neural engines are displayed. Fig. 2.11. Apple iPhone Xs Max (Reprinted with permission from Techinsights (https://www.techinsights.com/ko)). 124 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.12. Apple SoC A12 Bionic Image (Reprinted with permission from Techinsights (https://www.techinsights.com/ko)). Fig. 2.13 shows the front view of the Huawei Mate 20X (5G)'s main PCB, which was released in 2019. The Mate 30Pro (5G) product released in the second half of 2019 has the 5G modem integrated on the SoC, but this product has an external 5G. Huawei SoC Kirin 980 and SK Hynix 8 GB SDRAM are integrated through the PoP technology and there's Samsung's 256 GB flash memory. Fig. 2.13. Huawei Mate 20X (5G) (Reprinted with permission from Techinsights (https://www.techinsights.com/ko)). 2.4. x86 CPU Technology and Trend Intel, AMD, and VIA are the three companies that currently design and manufacture x86 CPUs. There were about 10 companies in the past but by the 2000s, the companies either 125 Advances in Optics: Reviews. Book Series, Vol. 5 merged with the above three companies or lost their market competitiveness and stopped production. VIA was founded in Taiwan in 1987, and mainly designed and manufactured chipsets for motherboards. From the late 1990s, VIA merged with US processor companies such as IDT and Cyrix and started the x86 CPU business. In cooperation with the municipal government of Shanghai, China, VIA established a venture company, Zaoxin, in 2013. Starting with ZX-A in 2014, it produced ZX-E and KX-6000 in 2019. The performance of KX-6000 is close to that of the 7th generation Intel i5 produced by Intel in 2016 and it is mainly sold in mainland China. In order to end the dependency on the US for IT technology, China is pursuing strategic projects at the national level. VIA developed the x86 instruction set through the merger of American companies and production is seen to be done by TSMC. According to Statista's 2019 statistics, Intel and AMD's market share was 77 % and 23 %, respectively, in the 1st quarter of 2019. It is difficult to find the market share of VIA's CPU. The CPUs for laptops, desktops and servers by performance are on the homepages of Intel and AMD. Since there is not a big difference between the CPUs for a desktop and a server other than the number of cores, the CPU features and internal communication structures of a laptop and server is analyzed here. The general features are shown in Table 2.6. Intel's mobile CPU i7 is a SoC with multiple devices integrated on one chip, as shown in Fig. 2.14. Four cores, GPU and system agent are connected by ring bus. The system agents include IPU, NPU (GNA), integrated memory controller (IMC), display and thunderbolt ports. Fig. 2.15 shows the overlay of colors on a real die image to show the placement of each device. The die size is 122.52 mm2. Table 2.6. x86 CPU features. Features Core (Thread) Clock (overclock) L1/L2/L3 GPU IPU/NPU TDP Tech. node (Die. mm2) Launched 126 Intel i7(Laptop) Xeon Platinum 9282 1065G7 AMD Ryzen7 3780U EPYC 7742 48 56 112 48 64128 1.3 GHz (3.9 GHz) 2.6 GHz (3.8 GHz) 2.3 GHz (4.0 GHz) I32 KB, D48 KB/ 512 KB/8 MB 32 KB/1 MB/ 77 MB I64 KB, D32 KB/ 512 KB/4 MB Intel Gen11 4th Gen/ GNAv1.0 15 W 10 nm (122.52) 2019. 8. external AMD Vega11 2.25 GHz (3.4 GHz) I64 KB, D32 KB/ 512 KB/ 256 MB external external external external 400 W 14 nm (2 X 694) 2019. Q2 15 W Global Found. 12 nm 2019. 10. 225 W TSMC 7 nm 2019. 8. Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.14. Internal architecture diagram of Intel mobile CPU i7 (source: https://en.wikichip.org/wiki/WikiChip). Fig. 2.15. Actual image of Intel mobile CPU i7 (source: https://en.wikichip.org/wiki/WikiChip). Fig. 2.16 is an architecture diagram illustrating the ring bus. The ring bus is a two-ring architecture that runs in both directions. Each ring has a data bus consisting of 256 wires and sub communication lines of snoop bus, request bus, and acknowledgment bus. There are approximately 1,000 total communication lines going in both directions. The ring bus is placed on the top layer of the 13 metal layers in the chip manufacturing process. The core and ring bus use the same clock frequency and the data bus transmits 32 B (256 bit) per cycle, which is a 96 GB/s transmission speed per ring for 3.0 GHz CPU. As 64 bytes of cache is stored per address block and data is transmitted in block units, the minimum transmission unit is 64 bytes of data transmitted over two cycles. If there are four cores, there are six access points on the ring and it moves by one access point per cycle. When core 1 transmits data to L3 of core 4, it has to pass over three transmission sections and will have a transmission latency of 3 cycles. For ring bus, the latency not only continues to increase as the number of cores increases but it also has the disadvantage of dependency on transmission location. All of Intel's mobile CPU released after 2015 transmits at a speed of 32 B/cycle through the 256 wires in the ring bus at all times when data is transmitted from the L2 of a core to L3, or from L3 to L2, or from GPU to CPU. L3 has a total of 127 Advances in Optics: Reviews. Book Series, Vol. 5 8 MB of storage and 2 MB is allocated per core. Instruction cache L1 has a storage capacity of 32 KB, data cache L1 has 48 KB, and L2 cache has 512 KB. Fig. 2.16. Ring bus architecture diagram. The internal architecture of each core is shown in Fig. 2.17. Since Intel uses the same core architecture from mobile to server, the core's internal architecture is nearly identical within the product family of the same period, regardless of CPU performance. When a program is executed, the program is transferred from the hard disk to main memory (DRAM), and the instructions pass through L2 or L3 according to the order written in the program, and then arrives at the L1 instruction cache. There are hundreds of instructions stored on L1 on standby. The communication line from L1 to the predecoder consists of 128 wires and transmits at a speed of 16 B (128 bit)/cycle. As x86 instruction lengths vary from 1 byte to 16 bytes, at least 1 ~ 16 instructions are sent in one cycle. The predecoder fetches instructions, predecodes the boundaries between multiple instructions and carries out fusion of instructions, where it fuses two instructions that can be combined to run at one time in succession. The completed 16 B signal is divided into several macrooperations (MOPs) and is transmitted to the decoder. The following step by step processes are divided into several branches for transmission and each step occurs in 1 cycle. The decoder reads and translates MOPs and converts them into 5 micro-operations (μOP) and sends them to the next step. All μOPs have the same length and a regular signalling system. While ARM instruction sets have a uniform length of 4 bytes, x86 varies in length and goes through a complicated process, omitted in Fig. 2.17, in order to reach the decode step. There are also many fused-μOPs of two instructions. Although there were 4 μOPs in Broadwell products, from Skylake products, there are 5 μOPs and they are transmitted to the ReOrder Buffer. In addition, until the decode step, two threads operate independently in rotation by cycle. That is, there exist two predecoders and decoders and if the instruction of thread 1 goes through the first predecoder and decoder in one cycle, then the instruction of thread 2 goes through the second predecoder and decoder in the next cycle. From the ReOrder buffer, 128 Chapter 2. CPU Technology Trend and Optical Interconnects the two threads use the same hardware for each step. Some of the μOps that have finished the decoding process are stored in the L0 cache and join the ReOrder buffer in the order determined by the branch predictor unit (BPU). The communication line between L0 cache and ReOrder buffer has a high priority of 64 B/cycle. Fig. 2.17. Internal core architecture diagram. From the L1 cache to the decode step, each instruction is entered in the order written in the program, but the ReOrder buffer collects 224 μOps and proceeds out-of-order by reassigning priority. When five μOPs enter the ReOrder buffer, they are renamed in the entry, allocated their order and direction and the completed μOps go into retirement. New μOps can enter only when retirement occurs. When μOPs are sent in fives to the scheduler in the new order, the scheduler saves up to 97 μOPs and performs the calculation by sending it to the arithmetic logic unit (ALU) through each register according to the time allocation and order. If data is needed, it is retrieved from the L1 data cache. When calculation finishes, the outcome is either stored in the register of the scheduler or in the L1 data cache based on the need for reuse. The completed μOPs remain in the ReOrder buffer if they need to be reused but otherwise go into retirement. There is a total of 512 communication lines that transmits data from L1 data cache to the logic part and the speed is 64 B/cycle (1,028 lines and a speed of 128 GB/cycle for 2017 server CPU). There are 256 communication lines for storing data and the communication speed is 32 B/cycle (512 lines and a speed of 64 GB/cycle for 2017 server CPU). There are 512 communication lines from the L2 cache to each L1 cache and the speed is 64 B/cycle. The process of starting at the L1 instruction cache to storing the ALU calculation in the L1 data cache is done in a pipeline format. The steps shown in Fig. 2.17 are pipeline steps and the total number of steps is 14 for the fastest route and 19 for the slowest route. Fig. 2.17 shows multiple steps condensed into each step. 129 Advances in Optics: Reviews. Book Series, Vol. 5 Two methods are typically used to speed up CPU computation. The first is to increase the IPC by increasing the number of parallel lines shown in Fig. 2.17. The second is to increase the clock frequency of the CPU. In 14~19 pipeline steps, the slowest step determines the highest CPU clock frequency, so the execution speed of all steps should increase evenly. To achieve this, the communication line speed per step and the switching speed of the transistors need to increase. The switching speed of transistors is inversely proportional to the area of the gate since it depends on the speed that the gate is charged. As the technology node gets smaller, the switching speed increases. As of 2020, Intel produces the latest products using the 10 nm process at the company's fab and AMD produces its latest products using the TSMC 7 nm process. There used to be a north bridge that included the CPU to DRAM and display interface and a south bridge that connected the other peripheral devices. But the north bridge has been integrated into the CPU chip and only the south bridge (PCH, platform control hub) remains. For mobile products, the CPU and PCH are mounted on the same board and packaged as seen in Fig. 2.18. The signal between CPU and PCH is transferred by an on-package interconnect (OPI) on the board. An OPI has the same transmission speed as the four lanes of PCIe 3.0. Each lane has a transmission speed of 8 Gb/s. The maximum transmission speed for each direction is 32 Gb/s, four lanes from CPU to PCH and 4 lanes in the opposite direction. PCH connects to the Internet, audio, SATA, USB, printer, etc. As can be seen in the image, the size is 53.76 mm2. (a) (b) Fig. 2.18. Intel i7 SoC interface (a) architecture diagram, (b) actual image of package (source: https://en.wikichip.org/wiki/WikiChip). The layout of Intel's latest server CPU, Xeon Platinum 9282, is shown in Fig. 2.19. 28 cores, 2 DDR4s (3 channels each), 2 IMCs, 4 PCIe3.0 (16 lanes each), 3 UPIs (20 lanes each), and 1 PCIe3.0 (4 lanes, used for OPI) are arranged on one die as shown in the figure. It is a CPU with 56 cores, a combination of two dies in one package. Intel uses a mesh interconnect architecture as shown in Fig. 2.19(a) for desktop and server products with more than 10 cores instead of a ring bus. There are bi-directional half rings arranged along the boundaries of each device in x and y directions. The mesh lines are on the top of the 13 metal layers and the devices are on the layer below. Therefore, the mesh lines actually pass over the top of each device and they do not overlap. As shown in Fig. 2.19(b), every device has a converged mesh stop (CMS) at its access point that manages signal 130 Chapter 2. CPU Technology Trend and Optical Interconnects connections and routing. Each core has an L3 cache of 1.375 MB allocation and a caching/home agent (CHA). When its core requests data from the L3, it is delivered to the core by sending it to the CHA. If it's requested by a different device, it is delivered to the CMS through the CHA. Through the cache coherency control and routing function, CMS sends the data to the address of the destination within the specified time slot by selecting the fastest route. For example, when core 05 sends data to core 51, it moves 4 points in the y-direction, then it switches paths and moves 5 points in the x-direction. Except for the cores, there are 7 access points connected to the mesh. In this case, there are no L3 and CHA, and the signal is sent directly to the CMS to go on the mesh. In 2017, Intel introduced the mesh architecture to its products, but the details such as the width of the communication line (number of wires) and communication speed have not been disclosed yet. Fig. 2.20 is a die image and its size is 694 mm2. (a) (b) Fig. 2.19. Intel server CPU Xeon Platinum, (a) Layout, (b) Core Detail Diagram (source: https://en.wikichip.org/wiki/WikiChip). Fig. 2.20. Actual image of Intel server CPU Xeon Platinum (source: https://en.wikichip.org/wiki/WikiChip). 131 Advances in Optics: Reviews. Book Series, Vol. 5 Table 2.8 shows the core communication speeds of 2019 CPUs of AMD and ARM in comparison to Intel CPUs. While the communication speed from the instruction cache IL1 to the predecoder of Intel and ARM core is 16 GB/s, 128 lines, that of AMD core is twice as fast. The communication that stores data from the scheduler register to data cache 1 (DL1) is 64 GB/s, 512 lines for Intel's server CPU, but that of AMD and ARM core is half as much. The communication of retrieving data from DL1 to ALU is 128 GB/s, 1024 lines for Intel while that of AMD and ARM core is ¼ as much. The speed and communication architecture of other communication lines are shown in Table 2.7. Compared to Intel, the microarchitecture and interconnect architecture of AMD and ARM cores are not disclosed in detail. Table 2.7. 2019 CPU core internal communication. Core internal Intel (Server) AMD (Server) ARM communication IL1 to Predecode 16 GB/s 32 GB/s 16 GB/s DL1 to ALU Store 64 GB/s 32 GB/s 32 GB/s or Register Read 128 GB/s 32 GB/s 32 GB/s IL1 to L2 64 GB/s 32 GB/s 32 GB/s DL1 to L2 64 GB/s 32 GB/s 64 GB/s L2 to L3 32 GB/s 32 GB/s 64 GB/s Ling Bus Interconnect Infinity Fabric DynamIQ Or Mesh Number of lines 16 GB/s: 128, 32 GB/s: 256, 64 GB/s: 512 There are three types of server CPU dies for Intel: 10 cores (LCC, low count cores), 18 cores (HCC, high count cores), and 28 cores (XCC, extreme count cores). One of the three types of CPU dies are used to place two dies on one package board and then a socket is attached. In Fig. 2.21, four dies are tied together in pairs to create two sockets, and the four dies are connected through UPI links. In Fig. 2.19, one die has three UPI links (20 lanes each) and six DDR4 channels (two 3-channels). Each of the three UPI links of the dies is used to connect the total of four dies in Fig. 2.21. Also, the 6 channels of DDR4 are connected to a DRAM through the mother board. Fig. 2.21(b) shows the actual packaged image. Fig. 2.22 shows the method of connecting a total of 8 dies using the same method. On a server computer or a Supercomputer, up to 28×8 = 224 Intel cores can be loaded on one board. Like this, the mesh interconnect and UPI links are currently connected by electric communication but optical communication is the most effective way. This has been the goal for related fields for the last 15 years. The ultra-path interconnect (UPI), an upgrade of the quick path interconnect (QPI) developed by Intel, is a point-to-point communication architecture that has been applied to chips since 2017. The communication characteristics of QPI are as follows. In one QPI link are 42 lanes and each lane consists of two communication lines. The same signal is sent to the two lines and the differential signal is found to remove noise. Therefore, there are a total of 84 signal lines in one QPI link. Of the 42 lanes, 21 are used for transmission and the other 21 are used for reception. From the 21 unidirectional lanes, 1 lane is used 132 Chapter 2. CPU Technology Trend and Optical Interconnects for the clock signal, 4 lanes are used for the control signal and 16 lanes are used for the actual data transmission. The UPI in Fig. 2.19 has 20 unidirectional lanes, but it can be assumed that the characteristics are almost the same as QPI. When the clock signal changes from 0 to 1 and from 1 to 0, 1 bit is sent each, and so 2 bits (double data rate) are sent per 1 cycle. The maximum transmission speed for one lane is 10.4 GT/s (giga transfer per second). It means that since there are 2 transfers per one cycle, the maximum link frequency can be up to 5.2 GHz. If the CPU frequency is 3.2 GHz and the link is synchronized to the CPU frequency, then the transmission speed per lane is 6.4 Gb/s. The unidirectional transmission speed of 16 lanes is 102.4 Gb/s and it can send and receive simultaneously. The CPU die has four PCIe 3.0 and each PCIe has 16 unidirectional lanes (32 lanes in both directions). The transmission method is the same as QPI and the maximum transmission speed per lane is 8 GT/s. That is, the maximum link frequency is up to 4.0 GHz, and when the link frequency is synchronized to the CPU 3.2 GHz, it has the same transmission speed as QPI. VCSEL, which is commercially available for sale currently in 2019, has a direct modulation speed of 50 GHz. It can transmit 120 Gb/s when it uses differential signals, which is 12 times faster than QPI. Even when 25 GHz VCSEL is used, which has higher stability, the transmission speed is six times faster. The heat generation rate is typically 2 mW per Gb/s for electric signals, but it is about 0.08 mW for optical signals, which is about 20 times less. (a) (b) Fig. 2.21. 2 Socket connection of server CPU (a) Architecture, (b) Image (source: https://en.wikichip.org/wiki/WikiChip). Fig. 2.22. 8 connected server CPU die architecture (source: https://en.wikichip.org/wiki/WikiChip). 133 Advances in Optics: Reviews. Book Series, Vol. 5 The structure diagram of the socket (socket model: LGA2011-3) used in Intel CPU i7 is shown in Fig. 2.23. There are 2,011 solder balls on the bottom of the socket and they electrically contact the PCB which has the same number of contact arrangements. On top of the socket are 2,011 contact arrangements that connect to the bottom side of processor substrate. The socket is bonded to the PCB through the surface mount technology (SMT) process. A solder ball is composed of 3 % silver (Ag), 0.5 % copper (Cu) and 96.5 % tin (Sn), and has a melting point of 217 °C. It adheres at a higher temperature than the melting point through thermal reflow. Normally, the packaging process refers to the process of electrically contacting the CPU die to the processor substrate and coating a protective film. In the case of Intel, the surface of the CPU die is turned upside down to face the substrate surface, and the substrate electrodes that has the solder pad and the CPU electrodes are bonded by a heat-treatment process (flip-chip bond). The substrate has the necessary electrical circuit and electrode pad at the bottom side where it corresponds to the contact points of the socket. The CPU die is in contact with the lid and a thermal interface material (TIM). The lid is designed to function as an integrated heat spreader (IHS) as well as a protective case. The top of the lid is in contact with a heat sink block and TIM and there is usually a fan over it. The lower electrodes of the substrate and the upper electrodes of the socket come in contact with a mechanical force applied by an independent loading mechanism (ILM), a type of a clamp. In addition to applying force, ILM has the function of aligning the substrate to the correct position of the socket. The processor substrate is made out of fiber-reinforced resin and the lid is nickel-plated copper, including TIM, and each has standards for area and thermal properties. (a) (b) Fig. 2.23. Intel CPU package structure, (a) Structure diagram, (b) Cross-section diagram. One of the biggest reasons for slow CPU performance or failures is the temperature of the CPU. Several techniques are used to maintain the temperature properly. The CPU die contains at least four digital thermal sensors (DTS). The case temperature (Tcase) is monitored by connecting a thermocouple (metal temperature sensor) to the center of the CPU's case lid (IHS). The experimental values of the power used by the CPU, the DTS temperature (TDTS), and the case temperature for the Intel i7 products are as follows. Tcase = 0.17×P + 43.3 and TDTS = 0.398×P + 43.3. The thermal design power (TDP) of the CPU is the power consumption amount that is set at the time of CPU design, and it is the amount of power used when all the cores of the CPU execute at the clock frequency 134 Chapter 2. CPU Technology Trend and Optical Interconnects set at the time of design. This is the power that is consumed when CPU is used at peak performance under normal conditions without using turbo boost. The TDP of Intel i7-5960X Extreme Edition is 140 W. In this case, the die temperature is T DTS = 99.02 °C and the case lid is Tcase = 67.1 °C. In other words, when the die temperature is about 100 °C, the case lid is 67.1 °C and CPU should always be below this temperature to ensure a stable lifespan and no calculation errors. Intel CPUs have a built-in thermal control circuit (TCC) and a register for program execution. If the temperature gets higher than the set value, besides maximally rotating the fan, a program that forcibly lowers the temperature is executed using two methods. The first method drops the clock frequency of the CPU to slow down the execution speed. The second method repeats the on and off states of the CPU clock at about 32 ms. If this fails, the CPU is forcibly stopped [8-12]. Recently, compared to computer CPUs, smartphone SoC package technology has been improving greatly and there are many investments being made. Examples include Package on Package (PoP) technology and 2.5D and 3D package technology (see Fig. 2.24). PoP is the technology currently used by most smartphone SoC manufacturers, as shown in Fig. 2.24(a). The SoC die and the memory (SDRAM) die are electrically connected to each package substrate and processed with a protective film. Then the two package substrates are stacked above and below to connect the electrodes using a ball grid array (BGA) method. In general, SoC connects to the package substrate by flip-chip bond and the memory die connects to the package substrate by the wire-bonding method. In the 2.5D package technology, there is a silicon interposer plate between the package substrate and the die, and the SoC die and memory die are connected to the top of the plate by flip-chip bond. The interposer plate is also connected to the package substrate by the BGA method. Unlike plastic substrates, silicon plates use a CMOS process that allows much finer wires to be stacked. The bottom electrodes are connected to the top electrodes using a through silicon via (TSV) process, and this is the same for a 3D package. The 3D package connects die 1 and die 2 by the flip-chip bond process, forms a TSV on die 1 and connects to the lower electrodes of die 1. Then, the die 1 is connected to the package substrate by the BGA method. Though there is an advantage in the communication speed due to the higher package density and shorter connecting lines, the disadvantage is that the metal line design has to be shared between different manufacturers. The TSV process involves not only simply forming holes but also the processes that increase the manufacturing cost, such as thinning the wafer (approximately 650→100 μm) and applying thin wafers to the CMOS process line. Therefore, 2.5D and 3D packages are not yet commonly used. (a) (b) (c) Fig. 2.24. SoC package structure, (a) POP, (b) 2.5D, (c) 3D. 135 Advances in Optics: Reviews. Book Series, Vol. 5 2.5. Supercomputer Architecture As of November 2020, the top performance supercomputer rankings 1, 2, and 3 were Fujitsu’s Fugaku (442 PFlops), IBM's Summit (148.6 PFlops) and Sierra (94.64 PFlops), and in fourth place was Sunway TaihuLight (93.01 PFlops) of China. Out of the total 500 rankings, 59.8 % or 299 units were manufactured by the three Chinese companies Lenovo, Inspur, and Sugon. American companies HP, Cray, etc. manufactured 21.6 % or 108 units. China invested a large budget ($200B) for IT technology independence. Based on the five-year unit step-by-step technology development strategy that started in 2001, China holds considerable technology for designing and manufacturing their own CPUs, along with supercomputers. In particular, the processor mounted on the Sunway TaihuLight uses the product (Sunway SW26010 1.45 GHz) developed by the Chinese researchers through technology that they accumulated for approximately 15 years. Its detailed performance and architecture are discussed in the next section. IBM was the world's largest supplier of supercomputers, but the company's overall share decreased significantly since selling its business division that used Intel processors in its server manufacturing business to Lenovo, a Chinese company, in 2014, and only using its own brand IBM Power9 line processors. Among the top 500 supercomputers, Intel processors (Xeon E5, Xeon Gold, etc.) account for about 91.8 %, which shows that a vast majority are using Intel products. In the 500 top supercomputers, China has 214 units, the US has 113 units, Japan has 34 units, France has 18 units, and Germany has 17 units. There are about 100,000 ~ 1,000,000 computer servers in a supercomputer that are connected by an optical communication network and a CPU in the server consists of typically 10~100 cores. Therefore, one supercomputer has about 1~10 million cores. Each server CPU is designed to enable its own cache coherence and each server memory (DRAM) is accessible to all other servers and is designed in a non-uniform memory access (NUMA) network to share data with each other. The network topology used in supercomputers mostly adopts the fat-tree network architecture or 3D torus network (a type of mesh network) architecture. Most of the top-ranking supercomputers have adopted the fat-tree architecture which is advantageous in connecting more servers. The communication path between any two servers is the same, so the latency to retrieve data from another server is uniform. But there is difficulty in adding servers later to upgrade the supercomputer. And if one of the switching devices on the upper level fails, it has a large effect on the whole. The 3D torus architecture, other than the advantages over the fat-tree architecture, is beneficial for programs where data locality between adjacent servers is important, and because of the efficient communication architecture, the computation speed is faster with a small number of cores. The supercomputer “Gordon,” installed at the San Diego Supercomputer Center (SDSC) with a $200M grant from the National Science Foundation (NFS), began service in late 2011 and was ranked 48th (285.8 TFlops) at the time. Cray manufactured it by upgrading the previous model (CS300-AC) and adopted the 3D torus network architecture. Gordon has 1,024 compute nodes (servers primarily used for program executions) and 64 I/O nodes (servers primarily used for data input and output), and each node is connected to 64 Infiniband network switches. 136 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.25 shows the internal architecture of a compute node and I/O node. The compute node has two CPU sockets with 8 cores connected to QPI 2 links on top of one motherboard as shown in the figure. Each CPU is connected to a 64 GB DRAM by 4 channels and to a chipset by OPI. The 80 GB SSD (hard disk memory) is connected to the chipset by SATA. The host channel adapter (HCA) is a card type that slides into the motherboard slot. It is connected to the CPU through PCle3.0 and has two optical communication access terminals. The I/O also has a similar architecture and the detailed features are shown in the figure. Table 2.8 shows the number of communication lines between each device, the communication speed, and detailed characteristics. (a) (b) Fig. 2.25. Gordon's server, (a) Compute node, (b) I/O node. Table 2.8. Internal communication width and speed of server. Devices CPU Compute node I/O node Intel Xeon E5-2670, 2.6 GHz Intel Xeon X5650, 2.67 GHz DDR3 1600, DDR3 1333, DRAM 8 channels, 64 GB 6 channels, 48 GB Hard disk SSD 80 GB SSD 4.8 TB CPU to CPU QPIx2, 20.8 GB/s QPI×2, 21.36 GB/s 32 lines/ch, 32 lines/ch, CPU to DRAM 8 ch 102.4 GB/s 6 ch 64 GB/s CPU to HCA PCIe3 16 ch 10.4 GB/s PCIe3 16 ch 10.68 GB/s CPU to chipset OPI 4 ch, 2.6 GB/s OPI 4 ch, 2.67 GB/s The network switch uses a Mellanox product with 36 ports as shown in Fig. 2.26(b). It can switch freely between 36 optical communication terminals and has 90 ns switching latency. It is powered by the built-in dual-core x86 CPU and switch chip. For the active optical cable (AOC), 200 Gb/s is currently the highest communication speed product (see Table 2.9). Four VCSELs with a 56 Gb/s modulation rate connects to four multi-mode fibers to send signals. Because it is a duplex, it is actually composed of 8 optical fibers, 8 VCSELs, and 8 PDs, and the maximum speed is 200 Gb/s in each direction. These optical cables follow the standard specification for package size and pin architecture. The 137 Advances in Optics: Reviews. Book Series, Vol. 5 standard specification is named Small Form-factor Pluggable (SFP) for 1 lane or Quad Small Form-factor Pluggable (QSFP) for four lanes. The Gordon system used the QDR 40 Gb/s QSFP, the latest product at the time of manufacture. The optical connection connects through the electric signal pin on the other side of the optical fiber when it is plugged into the HCA card and Infiniband switch port. As of November 2020, 154 units from the top 500 supercomputers use the Infiniband switch while 251 units use the Gigabit Ethernet switch. Percentage by performance shows 40 % for Infiniband and 19.2 % for Gigabit Ethernet, showing that Infiniband is advantageous for higher performances. For Mellanox products, the switching latency for Gigabit Ethernet is 300 ns, which is disadvantageous compared to Infiniband. However, the advantage is that connection between the internal and external networks is easy. The optical cables are able to connect with the same QSFP standard and instead of HCA, it uses an Ethernet card. (a) (c) (b) (d) Fig. 2.26. (a) HCA, (b) Infiniband 36-port switch, (c) Gigabit Ethernet 32-port switch, (d) HDR 200 Gb/s AOC. Table 2.9. AOC Communication Speed. AOC SFP QSFP nm/distance SDR 2.5 Gb/s 10 Gb/s 850/500 m 850/400 m DDR 5 Gb/s 20 Gb/s 1310/10 km 850/300 m QDR 10 Gb/s 40 Gb/s 1310/10 km 850/300 m FDR 14 Gb/s 56 Gb/s 1310/10 km 850/100 m EDR 25 Gb/s 100 Gb/s 1310/10 km HDR 50 Gb/s 200 Gb/s 850/100 m 138 Chapter 2. CPU Technology Trend and Optical Interconnects Gordon's 3D torus network is shown in Figs. 2.27. 18 out of 36 Infiniband ports connect to 16 compute nodes and 2 I/O nodes, and the remaining 18 ports connect to other Infinibands. From the 18 ports, 3 ports are allocated to each of the 6 directions of +x, -x, +y, -y, +z, and -z. Since each port is connected through QDR 40 Gb/s QSFP, there is a communication speed of 40 Gb/s between each node and the Infiniband, and a communication speed of 120 Gb/s between the Infinibands. The +x at the edge of 3D torus connects to the -x at the opposite edge, and the y and z directions connect in the same way as well so that all of the Infinibands have the same connecting architecture. Therefore, all the nodes have the same connecting architecture regardless of location. When this type of connection occurs in 2D, it forms a torus structure (a hollow donut structure) and this is where the name comes from. (a) (b) Fig. 2.27. (a) Gordon 3D torus network, (b) Gordon network architecture. 139 Advances in Optics: Reviews. Book Series, Vol. 5 In the Gordon supercomputer, HCA1 of each node forms one 3D torus and HCA2 forms another 3D torus, so it has two independent 3D torus architectures. This not only doubles the communication speed between nodes but also increases safety against accidents by operating a different torus network when one torus network fails. The 3D torus architecture is a mesh network architecture and it communicates between two nodes by skipping over each of the multiple switching points along the shortest path. Therefore, communication time is advantageous when two nodes are adjacent, but it is disadvantageous when they are far apart. Either one supernode, consisting of 32 adjacent compute nodes and 2 I/O nodes, executes the program alone or the 32 total supernodes carry out the computation in parallel. As shown in Fig. 2.27(b), 64 I/O nodes are connected to the system storage of 4 PB and each node is connected to an external Ethernet network through multiple Ethernet servers. The theoretical maximum computational speed of the Gordon supercomputer is 336.1 TFLOPs and the actual measurement is 285.8 TFLOPs. In addition to China's smartphone SoC (Kirin990) with the ARM license and x86 CPU(ZX-E, KX-6000) produced in collaboration with a Taiwanese company, the representative CPUs developed using pure Chinese technology are Loongson and Sunway. From the 10th five years (2001~2005) to the 12th five years (2011~2015) of CPU development using China's own technology, towards the end in 2015, the performance of the above two processors reached the level of application to commercial products. Loongson was developed as a national project (863 programs) by the Institute of Computing Technology (ICT) under the Chinese Academy of Sciences and is now produced and sold by Loongson Technology, a company established through public-private collaboration. The latest products released in 2019 is the Loongson 3B4000, 2.0 GHz, 64 bit (4 cores). Its reported theoretical maximum performance (peak FLOPS) is 128 GFLOPS. For comparison, the maximum performance for the 2011 Intel Xeon E5-2670 (8 cores) used by Gordon, after conversion to 4 cores, is 83.25 GFLOPS. The Loongson processor used LoongISA that they developed based on the MIPS instruction set, and for the four instructions they lacked, they purchased the license from MIPS Technologies. Since it is based on MIPS, it cannot use Windows OS of x86 CPU or Android OS of ARM CPU but has to use Linux. To compensate for this disadvantage, multiple translation instructions were added to LoongISA so that it can retrieve Windows and Android OS programs on the Linux OS. Since 2015, it is known to be used in national facilities of China where security is important such as national defense, communications, e-government, transportation, etc. In addition, it developed a product (Loongson X-CPU) that applied radiation hardening by design (RHBD) to use for satellites and it has been mounted on the BeiDou satellite, which was launched on March 30, 2015. The Sunway processor was developed primarily for national defense use at the National High Performance IC (Shanghai) Design Center, and the latest product, 2016 Sunway SW26010, is used by the processor of supercomputer 'Sunway TaihuLight'. Sunway TaihuLight was manufactured by a Chinese research institute, National Research Center of Parallel Computer Engineering & Technology (NRCPC) and is installed at the National Supercomputing Center in Shanghai. It has 40,960 nodes (10,649,600 cores) based on a three-layer fat-tree network architecture. Each node consists of a Sunway SW26010 processor with four core groups (CG) and 260 cores, as shown in Fig. 2.28. 140 Chapter 2. CPU Technology Trend and Optical Interconnects Each core group has a management processing element (MPE) of one big core and computing processing elements (CPE) of 64 small cores and a memory controller (MC) as shown in Fig. 2.28(b). The MPE core each has 32 KB of L1 data cache and instruction cache and 256 KB of L2 cache. For the CPE, there is a common L1 data cache of 64 KB and an L1 instruction cache of 16 KB for the 64 cores. Each core group has 8 GB SDRAM each. The four core groups are connected through a network-on-chip bus (128 bit) and they connect to the fiber optic terminals by a system interface (SI). The SI uses PCIe 3.0 and its maximum transmission speed is 16 GB/s. The transmission speed for Sunway's frequency of 1.45 GHz is 5.8 GB/s. It claims that, though the instruction set is based on RISC, it uses the ISA and architecture of its own development and has not disclosed the details. The MPE core has two pipelines and the CPE core has a single pipeline. Both have the 64-bit, SIMD, out-of-order features. When comparing the performance and architecture with an Intel processor, it can be assumed that it combined one quad-core CPU and a GPU with 256 processing elements and that instruction sets necessary for the changed architecture based on RISC ISA were added. Each pipeline has a computation speed of 8 FLOPS/cycle. Since the core frequency is 1.45 GHz, the theoretical maximum computation speed per node is (16 FLOPS ×4 + 8 FLOPS ×256) ×1.45 = 3.0624 TFLOPS. The value of 125.4 PFLOPS is for all 40,960 nodes and the value measured by Linpack Benchmark is 93.01 PFLOPS. It was ranked number one in 2018 and currently ranks fourth. (a) (b) Fig. 2.28. Sunway SW26010 processor composition, (a) 250 cores in 1 die, (b) Individual group architecture. A fat-tree network architecture has a tree-shaped structure as shown in Fig. 2.29(a) and the name comes from the increased connection lines as the level gets higher. A common tree structure has the same number of communication lines regardless of the level, so the higher levels don't have enough communication lines. A non-blocking network has the same number of lines that go up and down at the switch. In this case, all the signal input at the switch are output simultaneously without waiting time in the buffer. Most supercomputers and data centers have the non-blocking architecture. 141 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (b) (c) Fig. 2.29. Fat tree network architecture. A three-layer fat-tree architecture with switches that have four ports each is shown in Fig. 2.29(b). Normally, the upper layer is called the core, the middle layer is called the aggregation, the lower layer is called the edge, and the server is called the leaf. The left two nodes can reach each other only through the edge switch between them, the four nodes can reach one another only through the aggregation switch and the rest needs to go through the core switch. The characteristic of the fat-tree is that when a path goes through the core switch, all of the paths have the same length. If one path fails, there are many bypass paths and the path lengths are all the same. When configuring a three-layer fat-tree with the same switches and n number of ports, typically the core switch can connect up to (n/2)2, the aggregation and edge switches can connect up to n2/2 and the server can connect up to 2(n/2)3. If a switch with 64 ports is used, more than 130,000 servers can be used. Fat-trees are advantageous in connecting large numbers of servers but is very disadvantageous in adding new servers after it has been completed. For 3D torus, there are no upper and lower levels, so a new server can be added by opening the last line and it can be connected in the same method as before. By internally connecting the four switches inside the dotted boxes to create one 16-port switch and four 8-port switches, a two-layer fat-tree network can be created as shown in Fig. 2.29(c). Although it looks different externally, it is the same network architecture. Network device companies internally connect multiple 16-port, 32-port or 36-port switches and sell switch devices with a large number of ports such as 64 ports, 128 ports, and 256 ports. Sunway has purchased switch chips and HCAs from Mellanox. It is assumed that the custom switch devices were manufactured using imported switch chip and HCA. Sunway TaihuLight has a fat-tree architecture as shown in Fig. 2.30. The upper core layer has 40 cabinets and each cabinet has 4 supernodes. Each supernode has 32 boards and each board has eight connected nodes. To have a non-blocking architecture, each board has switches for 16 ports and 8 of these are connected to the nodes and the other 8 are connected to the supernode switches. The supernode switch has 512 ports and 256 of them are connected downwards and the other 256 are connected upwards to the cabinet switch. The cabinet switch has 1,024 ports and is dispersed and connected downward to 160 supernodes. The 256 downward ports of a supernode are dispersed and connected to 128 boards. The entire network is broken down into a central switch network, management network, and storage network and the switches of each layer are connected to either the management server or the storage system (see reference [15] for more details). In addition, they connect to the external Ethernet through the management server. EDR 100 Gb/s QSFP is used for the optical cable. It took $270M USD to build this supercomputer, which 142 Chapter 2. CPU Technology Trend and Optical Interconnects was divided between the Central Chinese government, the province of Jiangsu and the city of Wuxi [13-15]. Fig. 2.30. Sunway TaihuLight network architecture. 2.6. Optical Network-on-chip Architecture Since the early 2000s, leading semiconductor companies such as Intel, IBM, HP, Sun Microsystems, Samsung, and others have started conducting silicon photonics research in order to implement optical communication technology to core-to-core or CPU-to-memory communication on semiconductor chips. In particular, companies that focus on designing and manufacturing CPUs have conducted competitive large-scale research projects. Universities and companies have also proposed the network architecture for the optical communication technology to be applied for. Among them, the frequently cited ATAC architecture of MIT and the Corona Architecture of HP are outlined [16-19]. ATAC Architecture: This is the optical communication network architecture for CPU chip application proposed by MIT silicon photonics research team in 2010. Optical communication technology was applied to a CPU chip consisting of 1,024 cores and its cache coherence and CPU performance were compared to those of electrical communication. It targeted relatively low-performance cores with a clock frequency of 1 GHz, in-order instruction, and 16 nm technology application. This was to minimize the footprint of a single core in order to integrate 1,024 cores on a chip with an area of about 400 mm2. Chip composition: As shown in Fig. 2.31, there are 64 clusters in total and each cluster has 16 cores. There is a memory controller in each cluster and they are connected to the DRAM at 40 Gb/s. Network: The 16 cores of each cluster are connected to an electrical signal-based mesh network (EMesh) like Intel's mesh network. The 64 clusters are connected to the ring optical communication network (ONet) as shown in Fig. 2.31(a). Each cluster has a hub that connects electrical signals to optical signals and optical signals to electrical signals. 143 Advances in Optics: Reviews. Book Series, Vol. 5 For example, when core 1 transmits data to core 4 within the cluster, the signal goes through the EMesh. When core 1 transmits data to core 1024 in a different cluster, it first sends the signal to the hub of cluster 1 through the EMesh, then it converts to an optical signal and gets sent to the hub of cluster 64. Bnet is used to convert the optical signal into an electrical signal and transmits it from the hub to core 1024. ATAC designers introduced Bnet to simplify the transmission process by omitting the routing function for the signal going from the hub to the core. Bnet only has a broadcasting function that delivers signals to all the cores. The 16 cores are divided into two groups and when the specific digit of the signal is an odd number, it gets sent to group 1 and when it is an even number, it gets sent to group 2. (a) (b) Fig. 2.31. ATAC network architecture, (a) 64 clusters, (b) Internal architecture of cluster (Source: George Kurian et al., “ATAC: A 1000-Core Cache-Coherent Processor with On-Chip Optical Network,” PACT’10, Sept.11–15, 2010). Communication line composition: The number of data signal communication lines of EMesh and Bnet is 128 and 128-bit signals are transmitted in parallel per clock cycle. ONet has more than 129 optical waveguides in total including 128 waveguides for data, one optical waveguide for the control signal and many waveguides for meta-signal. These waveguides pass through each cluster in a ring shape as shown in Fig. 2.31(a). One optical waveguide has 64 wavelengths and one wavelength is allocated to each cluster. Therefore, each cluster transmits 128-bit data signal in parallel one wavelength at a time to 128 optical waveguides. For example, when data is transmitted from cluster 1 to cluster 64, the signal is sent on wavelength 1 of 128 optical waveguides. Clusters 2 ~ 64 receive the signal of wavelength 1 each by 1/63 of the strength and all clusters except cluster 64 ignore the signal and only cluster 64 takes in the transmitted data. Multiple clusters can send signals to cluster 64 since they use different wavelengths from one another. Latency: The time it takes to transmit a signal for each of the communication line is shown in Table 2.10. In the case of EMesh, after 1 cycle at the router, 1 cycle is added for each hopping. 6 hopping is needed to start at core 1 and to arrive at core 16. In comparison, Onet requires 3 cycles across the board. In other words, 1 cycle for E/O conversion, 1 cycle for O/E conversion and 1 cycle for waveguide signal transmission regardless of distance. BNet requires 1 cycle for O/E conversion and 1 cycle is added for every hopping. 144 Chapter 2. CPU Technology Trend and Optical Interconnects Since the CPU clock is 1 GHz, the communication speed of the electric cable is 1 Gb/s, and in the case of the optical waveguides, the total communication speed per optical waveguide is 64 Gb/s since there are 64 wavelengths. Table 2.10. Communication line width and latency of ATAC. Core model EMesh Hop Latency ONet Hop Latency BNet Hop Latency EMesh ONet BNet Memory Bandwidth In-order, 1 GHz, 1024 cores, 16 nm technology 2 cycles (router delay – 1, link traversal – 1) 3 cycles(E/O + O/E conversion – 2, link traversal – 1) 2 cycles(O/E conversion – 1, link traversal – 1) 128-bit wide 128-bit wide 128-bit wide 64 memory controllers, 5 GB/s per controller Optical device composition: The light source is outside of the CPU chip and shoots a continuous wave (cw) signal through the optical fiber into the power optical waveguide inside the CPU chip. Then, it connects from the power optical waveguide to the ONet optical waveguide through the ring resonators. Excluding the meta signal, there are at least 129 optical waveguides with 64 wavelengths each, requiring a minimum of 64 external light sources and a maximum of 8,192 external light sources. Each cluster has one modulator per waveguide to send signals on the allocated wavelengths and a Ge-based Electro-Absorption (EA) modulator. In this case, there are at least 129 modulators per cluster and more than 8,256 modulators on the chip as a whole. For detectors, Ge-based ones are used and at least 8,127 detectors per cluster and 520,000 detectors on the chip as a whole need to be integrated. The ring resonator has the function of connecting modulated signals to optical waveguides and the function of dividing the optical signals of each wavelength into 1/63 strengths. Therefore, as many ring resonators are integrated as the modulators and detectors. In addition, more ring resonators are added according to the number needed to connect the cw signals from the power optical waveguide to the ONet optical waveguide. ATAC designers assumed the ONet optical device to be integrated on a SOI substrate optical network chip separate from the CPU chip and joined to the CPU chip using 3D package technology. The CPU performance difference of connecting 64 clusters by ONet optical communication and by Emesh electric communication of 256-bit width was compared using 3 benchmark application programs of Splash2, Parsec, and Synthetic. The cache coherence protocol for this calculation used the ACKwise protocol proposed by ATAC designers and two other existing protocols. In conclusion, ACKwise protocol based on optical communication showed 60 % to 2.5 times better performance than any other protocol combinations based on electric communication. Since this difference in performance is based on the same core performance, it is a significant difference as it shows the difference in the cache coherence based on optical communication. There is also a greater significance in that it notably reduces heat generation and enables 145 Advances in Optics: Reviews. Book Series, Vol. 5 communication in a range that is nearly impossible with Emesh through optical communication [16]. Corona Architecture: This is the optical communication network architecture for CPU chip proposed by the HP research team and participants from two universities in 2008. The details of the CPU chip composition and network architecture are as follows [17]. CPU Chip Composition: Each cluster has four cores and a total of 64 clusters are connected by optical communication. The clock frequency of the core is 5 GHz with two L1 caches allocated per core and one L2 cache per four cores. Each cluster has a memory controller and is connected to the DRAM by optical communication. Assuming 4 threads per core, the CPU is set to run 256 cores and 1,024 threads. Optical Communication Network: The 64 clusters are connected by the Corona network optical communication. There isn't an explanation for the communication between the four cores inside the cluster but it can be regarded as the same bus type electric communication as a commercial CPU. There is a total of 260 optical waveguides including 256 optical waveguides for data transmission, 1 broadcast optical waveguide, 2 arbitration optical waveguides, and 1 clock signal optical waveguide. One optical waveguide carries independent signals of 64 wavelengths. Each cluster is allocated four optical waveguides (256 transmission channels) for data transmission. The four optical waveguides start at the corresponding cluster and pass through 63 clusters to return to the original position and takes on a broken ring architecture that does not continue. All clusters start and end with four optical waveguides of this type. For example, when cluster 1 transmits data to cluster 60, the cw signal is divided from the power optical waveguide to the optical waveguide allocated to cluster 1 to transmit the cw signal to cluster 60. It has already informed all clusters that cluster 1 will transmit data to cluster 60 and so all the remaining clusters except for cluster 60 turns their ring resonators off. The cw signal that started at cluster 1 arrives straight at cluster 60. Then cluster 60 uses the ring resonator modulator on the cw that was sent to generate data and sends it back to cluster 1. Arbitration is the act of creating an order to prevent multiple clusters from sending data at the same time to the same cluster. From the two arbitration waveguides, 64 wavelengths are allocated to waveguide 1 and 1 wavelength is allocated to waveguide 2. The two waveguides have an unbroken ring-type architecture. 1 wavelength of the 64 are allocated per cluster. Arbitration is executed before cluster 1 sends data to cluster 60, and at this time, cluster 1 sends a token signal to the wavelength of cluster 60. When cluster 60 receives the token signal, token issuance of all other clusters stops. At this time, necessary notification proceeds in the form of broadcast through waveguide 2 (one wavelength). To give equal opportunities, arbitration gives the opportunity to cluster 1 and once cluster 1's data transmission ends or it gives up, the next opportunity is given to cluster 2, and then 3, 4,...., until 64 in sequential order (Round Robin Method). In Corona networks, one wavelength is allocated to one waveguide for clock signal transmission and 64 wavelengths are allocated to one waveguide for the broadcast signal. This is separate from the arbitration broadcast and it used for general-purpose broadcast signalling. 146 Chapter 2. CPU Technology Trend and Optical Interconnects Optical Device Composition: Corona designers use external light sources like the ATAC and in this case, it would require a minimum of 64 and a maximum of 16,514 lasers. A ring resonator modulator is used for signal generation and each cluster needs 16,257 modulators that include 256×63 for data transmission, 65 for arbitration and 64 for broadcast, which is a total of 1.04 million ring resonator modulators. For detectors, a total of 20,000 are needed for all the clusters including 256 for data transmission, 65 for arbitration, 1 for broadcast and 1 for clock per cluster. Ring resonator filters for wavelength division multiplexing (WDM) require a total of 1.1 million filters, including the combined number of modulators and detectors and 386 for power divergence. Corona Performance: The target operating speed of the CPU is 10 TFLOPS, the number of data communication lines between clusters is 256 (256 bit-wide), and the communication speed per line is 10 Gb/s. The optical communication in the CPU has a total bandwidth of 20 TB/s and a total bandwidth of 10 TB/s between the CPU and the DRAM. The power consumption of the CPU was calculated as 82 W and the area as 423 mm2. From this, the power consumption for optical communication was calculated as 39 W. To compare the performance of optical communication and electric communication, simulations were performed using synthetic benchmark and SPLASH2 benchmark. In conclusion, it showed a performance difference of 2~6 times. Also, the paper reported that the Corona optical network architecture was the unique solution to not only the heat generation (power wall) but also the communication barrier (bandwidth wall) between the cores and between the CPU and memory. Causes of Failure: A number of optical communication-based network architectures for CPU application have been proposed, but they have not reached the stage of experimental integration and testing on actual chips. Rather, studies have been gradually fading out since 2015. This is because the factors that lower its feasibility have been highlighted more as research accumulated. First is the disadvantage of using SOI wafers. Currently, wafers used in CPU and memory chip production are ordinary bulk silicon. The fact that electronic devices and optical devices can be manufactured with the same CMOS process in the same wafer, which motivated the research of silicon photonics, did not have many benefits in reality. When using a silicon optical waveguide, the optical waveguide and the optical device should be arranged in the same silicon layer as the gates of the CPU [20]. This is not only disadvantageous in increasing the gate density of the CPU, but also makes it nearly impossible to design metal layers (13 layers) of electronic devices. Contrary to the motives, for most proposed architectures, it is assumed that optical devices and CPU were manufactured on different wafers and bonded in a 3D package. Second, the transmission loss of the silicon optical waveguide is large. Silicon has a high refractive index (n = 3.45), which is advantageous in making the size of the optical waveguide as small as possible, but disadvantageous in designing the low transmission loss of waveguide. Although ATAC assumed 0.3 dB/cm, the minimum value reported in the paper was 1.7 dB/cm [18], but in typical CMOS process, it is measured greater than 2 dB/cm. The optical waveguide length in ATAC is approximately 8 cm and 16 cm for 147 Advances in Optics: Reviews. Book Series, Vol. 5 Corona, of which the transmission loss will be 16~32 dB. It becomes impossible for 63 clusters to receive data at 1/63 strength. Third, the yield and performance of optical modulators, detectors, and ring resonators are poor. ATAC uses Ge-based optical modulators and detectors, while Corona uses ring resonator-based optical modulators and Ge-based detectors. For the network to function, the failure rate of each device needs to be lower than the surplus device rate (generally 10 %). The average yield has to be over 90 %. When manufacturing modulators and detectors by epitaxial growth of Ge materials on silicon wafers, it is possible for a few selected devices to reach their target performance, but it is very difficult to obtain a uniform yield by the necessary ratio. Particularly, it becomes nearly impossible when it is accompanied by fabrication-attributed statistical errors such as the resonance wavelength of the ring resonator. In this case, micro-heaters needs to be attached locally and heated for wavelength tuning. Heating 500,000 resonators for ATAC and 1.1 million resonators for Corona takes away the effective value of introducing optical signals. Also, the insertion loss of Ge-based optical modulator was assumed to be 1 dB for ATAC but the currently reported insertion loss of the device is about 3~6 dB, which is a significant difference. Fourth is the light source. Intel has developed in collaboration with a research team at UC Santa Barbara an on-chip laser by joining an InP wafer to a silicon wafer. However, it has yield problems like the Ge-based optical device. ATAC and Corona architectures were presumed to use external light sources, but it is difficult to integrate hundreds to thousands of lasers in the alignment of optical fiber arrays with optical waveguide arrays on a chip. It is also difficult to secure space to attach them on the side of the chip when considering the cross-sectional area of optical fibers. The coupling loss measured when aligning one optical fiber to a silicon optical waveguide is reported to be a minimum of 1~2 dB, but it is only accomplished by modifying or adding a special structure at the inlet of the optical waveguide. In general, coupling loss is measured 3~6 dB or more. When integrating tens of optical fibers in an array, it is nearly impossible to align all the optical fibers with a uniform coupling loss below 3 dB to single-mode silicon waveguide arrays. 2.7. Key Elements toward Optically Interconnected CPU In fact, the 4 causes of failure may all be attributed to the use of silicon as the optical waveguide. Among the materials used in the CMOS process, there are no materials other than silicon nitride (Si3N4), silicon oxynitride (SiOxNy), silicon dioxide (SiO2), and silicon (Si) that can be used to an optical waveguide. Since silicon dioxide (SiO2) is used for the cladding of the waveguide and if we don't use silicon, then the remaining candidate materials are only silicon nitride (Si3N4) or silicon oxynitride (SiOxNy). Although there are many papers that analyze optical devices using these two materials as optical waveguides, there are no papers that suggest or test a network architecture that can be integrated on a CPU chip using these two materials. Here, it is suggested that the 4 causes of failure can be overcome by the key elements, three technologies, and a possible network architecture based on these two materials. 148 Chapter 2. CPU Technology Trend and Optical Interconnects Silicon nitride or silicon oxynitride film deposition in the CMOS process is done by a low pressure chemical vapor deposition (LPCVD) or plasma enhanced CVD (PECVD) process. LPCVD can deposit films of silicon nitride with a precise stoichiometric ratio (3:4 of Si3N4). The deposition temperature ranges from 425-900 °C. The stoichiometric film used in the CMOS process is deposited most frequently around 770 °C and has a refractive index of 2.0. The transmission loss for the optical waveguide is also very low and a measurement < 0.1 dB/cm is reported. The disadvantage is that cracks occur in the film when the thickness is over 200 nm due to tensile stress. The melting point of aluminium and copper used for the metal layers in the CMOS process are 660 °C and 1,084 °C, respectively. At high temperatures, copper becomes an impurity source that penetrates into other thin films, and for aluminium, the deposition temperature is higher than its melting point. Therefore, LPCVD thin film is used only up to the first metal layer. The PECVD film can be deposited at 200~400 °C and fast at 780 nm/min (LPCVD 4.8 nm/min). Therefore, it is advantageous for the formation of thick films and is used after the first metal layer. Though PECVD silicon nitride film doesn't fit a stoichiometric ratio, it has relatively low stress and can be deposited up to several micrometers without cracks. In order to integrate the optical and electronic devices on the same wafer, it is most advantageous to form the optical waveguide on the top protective film layer shown in Fig. 2.5. Therefore, the first cause of failure, the same layer of optical devices with electric devices, can be overcome by using a PECVD silicon nitride or oxynitride film. There are many papers on the formation and characterization of optical waveguides using PECVD silicon nitride or oxynitride films. The ratio of the injection gas can be adjusted when depositing silicon oxynitride (SiOxNy) to have a specific refractive index of the film. As shown in Fig. 2.32, when the nitrogen component in the film is 0 %, the refractive index (1.45) is the same as that of PECVD oxide (SiO2) and as the nitrogen component increases, the refractive index increases linearly to 2.0 [21]. According to reference [22], the refractive index and transmission loss of PECVD SiON film are n = 1.68 and 0.3 ± 0.15 dB/cm, respectively, for the wavelength 1310 nm and the optical waveguide shown in Fig. 2.33(a). According to reference [23], the refractive index is n = 1.515 and transmission losses are 0.2 dB/cm in TE mode and 0.3 dB/cm in TM mode, for the wavelength 850 nm and the optical waveguide shown in Fig. 2.33(b). Therefore, the transmission loss of PECVD silicon oxynitride optical waveguide is about 0.3 dB/cm, which is the same as the transmission loss of the optical waveguide set in the ATAC architecture. The second cause of failure, large transmission loss of silicon waveguide, can be overcome. Although silicon has active optical properties, the performance and yield of the manufactured devices are far below the level for implementing a network-on-chip. The technology of VCSEL had developed fast and reached the level of 5 GHz direct modulation in the early 2000s, and from mid 2010s, 50 GHz products became commercialized, and as shown in Table 2.9, it came to a level where HDR products were already being used for supercomputer production. The performance is superior to optical modulators implemented by silicon optical waveguides. Therefore, it is more efficient in terms of area and power to construct the light source and the optical modulator with the same device using VCSEL rather than constructed with different devices. The third and 149 Advances in Optics: Reviews. Book Series, Vol. 5 fourth causes of failure, the low yield and poor performance of optical devices and external light sources, can be overcome by using VCSEL and the three technologies developed and verified experimentally by Oprocessor, Inc. (start-up company founded by the author). Fig. 2.32. Nitrogen component ratio and refractive index graph (source: M. I. Alayo et al., ‘Deposition and characterization of silicon oxynitride for integrated optical applications, Journal of Non-Crystalline Solids, 2004, pp. 76-80). (a) (b) Fig. 2.33. SiON optical waveguide, (a) n = 1.68, (b) n = 1.515 (Source: [a] Richard Jones et al., “Integration of SiON gratings with SOI,” Conference: Group IV, Photonics, 2005, [b] Tai Tsuchtza, ‘Low-loss Silicon Oxynitride Waveguides and Branches for the 850 nm Wavelength Region’, Japanese Journal of Applied Physics, vol. 47, no. 8, 2008, pp. 6739-6743). Among the methods that can inject light into PECVD silicon oxynitride optical waveguide, it is difficult to use the methods that were widely used in silicon photonics, butt coupling and grating coupling. For a direct modulated VCSEL, butt coupling can only generate signals at the edges of the chip, so it is impossible to generate signals from cores that are not located at the edge. It is also difficult to secure space to attach them on the side of the chip. For grating coupling, the difference of refractive index between silicon oxynitride (SiON) and silicon dioxide (SiO2) is not only small (Δn = 0.1 to 0.55), which leads to low coupling rate, it also creates statistical errors like the ring resonator. Therefore, it is difficult to overcome the yield problem. Excluding the two methods, the only possible method is to use the prism. 150 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.34 uses a prism method to measure the transmission loss of light passing through a silicon oxynitride material. The light was refracted by the prism with applying the index matching oil on the surface of the prism and silicon oxynitride material [24]. In order to use the prism method at the network architecture, the following three technologies must be achieved. Fig. 2.34. Transmission loss measurement of SiON material (Source: B. S. Sahu et al., ‘Influence of hydrogen on losses in silicon oxynitride planar optical waveguides’, Semicond. Sci. Technol., Vol. 15, 2000). First, an adhesive technology that is transparent, satisfies the refractive index matching condition, maintains its adhesive strength at 200-400 °C, and is not affected by solvents used in the CMOS cleansing process such as BOE, isopropanol, acetone, etc. needs to be achieved. There are no commercial adhesives that are transparent with a refractive index greater than 1.6. Adhesives that maintain its strength in cleansing solvents and at 400 °C have not been reported in papers. Therefore, it is necessary to develop a technology that can bond a prism to the surface of an optical waveguide without using a commercial adhesive. Second, microlenses that can produce collimated light under the conditions of Fig. 2.35 need to be achieved. Microlens consisting of quartz and silicon materials are commercially available, but their refractive indices are not suitable for the current purpose. Third, a special package technology for VCSEL and PD needs to be achieved. The current commercial package is a hermetically sealed structure that gets filled with nitrogen or argon gas after a specific distance between the laser light exit and the lens is maintained. This type of packaging is overly bulky, and dozens to hundreds of VCSELs and PDs cannot be integrated on the CPU chip. The area and height of a single optical device after packaging has to be at most <0.7 mm2 and <1.0 mm, respectively. Oprocessor, Inc. has developed an adhesive technology, a microlens, and a package technology that satisfies the requirements. 151 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 2.35. Conceptual diagram of optical signal connection on SiON optical waveguide. Fig. 2.36(a) is a micrograph of a GaP prism bonded to the surface of a SiON optical waveguide and (b) is an enlarged photograph of the sideview. The adhesion characteristics between the prism and wafer are shown in Table 2.11. It had a transmissivity of over 99 % at the thickness 2 μm required for adhesion, and the refractive index was measured up to 1.72 depending on the heat treatment conditions. When force was applied from the side to observe adhesion strength, the prism broke more times than it fell off. (a) (b) Fig. 2.36. Real Images of Prism Bonded to SiON Optical Waveguide. Table 2.11. Adhesion characteristics of prism. Refractive index Transmission High temperature resistance Solvent resistance adhesion strength 152 Specification 1.69~1.72 at 850 nm > 99 % for 2 μm thickness at 850 nm 400 C (compatible with CMOS metal process) BOE, Aceton, Isopropanol, etc. (unaffected by alcoholic cleaning) > 1 N/mm2 (not detachable before breaking GaP prism at 25 C) Chapter 2. CPU Technology Trend and Optical Interconnects The radiation angle of VCSEL in the air is typically in the range of 20~40°. The microlens must have the right value of refractive index to collimate light emitted by the median value of 30°. Fig. 2.37 shows a scanning electron microscope (SEM) image of the microlens array (a) and an enlarged image (b) of one. The diameter of the microlens is 30 μm. (a) (b) Fig. 2.37. SEM images of microlens. After attaching the microlens to the 25 GHz VCSEL chip, the shape of the output light was observed as shown in Fig. 2.38. The picture was taken by placing an infrared card 7 cm vertically from the VCSEL. Since the distance from the VCSEL to the optical waveguide is <1 mm, the collimated light proves a sufficient performance of the lens. (a) (b) Fig. 2.38. Output light images of VCSEL before and after microlens attachment. The test results of the microlenses can be used to complete a package. Fig. 2.39 (a) shows the package structure. The lens parts were made separately and then attached to the VCSEL in the final step. Fig. 2.39 (b) shows a micrograph of the VCSEL packaged with the structure. The base is 1.2×1.0 mm2 wide and 640 μm high. It does not show the minimum size because it was made for experimental verification of optical coupling. Since the size of VCSEL is 0.25×0.25 mm2, < 0.25 mm2 is possible to achieve. As shown in Fig. 2.40 (a), the prisms were attached to both ends of the SiON optical waveguide, and the VCSEL module was attached to a prism on one side. An experiment was conducted to measure the intensity of VCSEL light entering the optical waveguide and coming out of the opposite prism. Fig. 2.40 (b) is a micrograph showing the experimental laser light leaving the opposing prism after entering the optical waveguide. figure (c) is a micrograph showing the laser light leaving the opposite prism while the 153 Advances in Optics: Reviews. Book Series, Vol. 5 scattered laser light cannot be seen due to the higher ambient brightness. The laser light is strongly emitted from the center of the inclined side of the prism despite the partially broken state of the prism. The average value of the coupling loss measured from multiple modules and optical waveguides is 3.54 dB, which is the value without anti-reflection coating (AR) applied to the prism. The theoretical gain when AR coating is applied on both sides of the prism is 2.6 dB, which results in a coupling loss of about 0.94 dB. It is a much better value than the coupling loss of 3 dB set by ATAC. If the incident light is collimated and the refractive index matching condition is met correctly, the theoretical minimum value is close to 0 dB, excluding the reflection loss of prism. (a) (b) Fig. 2.39. A package module of VCSEL, (a) Package structure, (b) Actual image. (a) (b) (c) Fig. 2.40. Laser light transmission experiment, (a) Experimental scheme, (b) Image of input and output light, (c) Image of output light on the opposite prism. In the above experiment, the alignment tolerance that causes 1 dB loss was measured 18.9 μm parallel to the optical waveguide and 7.2 μm perpendicular to the optical 154 Chapter 2. CPU Technology Trend and Optical Interconnects waveguide. The alignment tolerance of the perpendicular direction can be adjusted freely by using an optical waveguide wider than the optical waveguide (30 μm wide) used in the measurement. Therefore, the alignment tolerance parallel to the optical waveguide is important and the measured value (18.9 μm) is within the range that enables automation in mass production. In the study of silicon photonics, ring resonators are assumed to be used for wavelength division multiplexing filter (WDM filter) to transmit multiple optical wavelengths in one optical waveguide. However, the ring resonator filter has not been used as a commercial product for on-chip nor off-chip due to its poor performance. Currently, the WDM filter with the most verified performance among the commercially available products is the thin film filter in which two thin films with different refractive indices are alternately stacked. Fig. 2.41(a) shows the schematic diagram of four optical wavelengths transmitted through a single optical waveguide by the coating of WDM thin film filters on the bottom sides of the prisms. For example, λ3 passes through the prism coated with the filter of the corresponding wavelength and enters into the waveguide, and it is reflected off the bottom sides of the prisms coated with other wavelength filters and proceeds along the waveguide and is emitted from the prism of the corresponding wavelength. Fig. 2.41(b) graphically shows the structure of a WDM filter and the light path. Fig. 2.42 shows the reflection and transmission graphs derived from theoretical calculation using TiO2 and Ta2O5 films, which are often used to design WDM filters. (a) (b) Fig. 2.41. 4-channel WDM, (a) Light input and output scheme, (b) Thin film filter and light path. (a) (b) Fig. 2.42. Theoretical graphs for (a) reflection, and (b) transmission. 155 Advances in Optics: Reviews. Book Series, Vol. 5 The refractive index of the prism is nprism = 3.16 for 850 nm, that of the TiO2 is nH = 2.5086 and that of Ta2O5 is nL = 2.0908. When the prism angle is θ, the light that enters perpendicularly to the inclined side enters the bottom side as θ and proceeds as ф, as shown in Fig. 2.41(b). The thickness of each film is determined so that the path at the corresponding angle is λ/4, and the TiO2 film is denoted as TH and the Ta2O5 film is denoted as TL. Fig. 2.42 (a) shows the reflection spectrum calculated when T H and TL films are repeated in alternation 8 times [(TH TL)8]. Fig. 2.42 (b) is the transmission spectrum calculated when a spacer with a width of 4λ is inserted between the two reflecting mirrors formed by repeating the TH and TL films 8 times [(TH TL)8(TH TH)8(TL TH)8]. Such calculations are technically very mature and used to produce commercial products. The thin film filters of commercial products meet the WDM performance required by ATAC and Corona architectures. As shown in Fig. 2.19, Intel's server CPU die is 694 mm2 with a maximum of 28 cores connected to the mesh network through electrical communication. Desktop CPU die is ~174 mm2 with a maximum of 8 cores connected by a ring bus. When optical communication is introduced to server CPU, 9 cores are connected by a ring bus and 36 cores are divided into 4 clusters that are connected by optical communication as shown in Fig. 2.43. The ring bus uses the electrical communication like Intel's desktop CPUs. Optical communication has a point-to-point crossbar network. That is, each cluster has a point-to-point communication with the remaining three clusters. Therefore, two or more clusters can send data to the same cluster at the same time and does not require arbitration. In addition, all optical communication lines have the same latency (3 cycles). If the CPU supports 3 GHz clock frequency and 64 bit process, the minimum bandwidth per communication line is 192 Gb/s. Each optical line in Fig. 2.43 can be assumed the bandwidth of 256 Gb/s unidirectionally and 512 Gb/s bidirectionally, which requires 10 optical waveguides and 10 VCSELs and 10 PDs of 50 Gb/s. This is twice as fast as that of ATAC. Each cluster connected with the other three clusters needs 30 optical waveguides and 15 VCSELs and 15 PDs of 50 Gb/s. Since the individual package area of VCSEL and PD is <1 mm2, the area per cluster for VCSELs and PDs is <30 mm2 which is much smaller than the Intel's 8 core CPU area 174 mm2. Fig. 2.43. A server CPU with 36 cores (Blue line: ring bus electrical communication, red line: point-to-point optical communication). 156 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.44 shows a candidate package structure to be adopted if optical communication lines are arranged on the top layer of CPU in Fig. 2.5. CPU is flip-chip bonded on the processor substrate, and the part corresponding to the optical device is cut out and the wires and electrodes are arranged appropriately at the substrate. Fig. 2.44. Flip-chip bond package after processor substrate processing. In Fig. 2.43, there are a total of six optical lines or 60 optical waveguides. The aggregate bandwidth is 3.072 Tb/s (6×512 Gb/s). Based on this, the power consumption between electrical and optical communications is compared. For electrical communication, it consumes about 2 mW per 1 Gb/s, and there is an average of 5 hops for 6×6 electric mesh network, which leads to 30 W total (3072 Gb/s × 5 × 2 mW/Gb/s). For optical communication, it consumes about 0.08 mW per 1 Gb/s, and there is 1 hopping per communication, which leads to 0.25 W total (3072 Gb/s × 1 × 0.08 mW/Gb/s). Considering TDP 200 W for Intel’s 28 core CPU, 30 W takes a significant portion that can limit CPU performance. Since optical communication has 3 cycles of latency per 50 Gb/s (2 cycles for O/E and E/O conversion and 1 cycle for traversal), a total of 184 cycles of latency occurs (3072 Gb/s × 3 / 50 Gb/s). For electrical communications, there is an average of 5 cycles of latency per 3 Gb/s (5 hops per communication and 3 GHz clock frequency), resulting in a total of 5,120 cycles of latency (3072 Gb/s × 5/3 Gb/s). In other words, a time gain of approximately 5,000 cycles is obtained, which is reflected in the cache coherence. Fig. 2.45 shows an example of a fat-tree network architecture connecting 1,024 cores by optical communication. Each part (P1, P2, etc.) represents a cluster consisting of eight cores which may include a memory controller or a GPU that connects to the optical communication. The controller unit (CU) is a 32-port switch (16 ports down, 16 ports up) with the same function as the InfiniBand of a supercomputer. There are 8 CUs, each CU having 16 parts, and the total number of cores is 1,024 (8×16×8). The eight cores in each part are connected by the same electrical ring bus as Intel's desktop CPUs. If the CPU supports 3 GHz and 64 bit, the optical lines are assumed a bandwidth of 512 Gb/s bidirectionally as before. Communication between all CUs is allocated two optical lines, and four optical lines are allocated between adjacent CUs. For example, the bandwidth of 2,048 Gb/s bidirectionally is allocated between CU1 and CU2, and bandwidth of 1,024 Gb/s bidirectionally is allocated between CU1 and CU3 ~ CU8. Only the optical lines starting from CU1 are shown in the Fig. 2.45, but the same optical lines repeat for 157 Advances in Optics: Reviews. Book Series, Vol. 5 CU2 to CU8. The CPU cores, the optical waveguides, and the VCSELs/PDs are arranged vertically on different layers and can overlap in the same area. Therefore, if the CPU is designed as the area of eight cores < 50 mm2, then each part having eight cores can contain its optical waveguides and optical devices within its area, and the total size of the wafer containing 1,024 cores is calculated 8 cm × 8 cm (50 mm2×1024/8 = 6400 mm2). The current supercomputer with the highest performance has a fat-tree architecture with bidirectional bandwidth 200 Gb/s. Our network architecture not only has a bandwidth that is at least twice as fast but also maintains a line width of 64 bit/cycle between all cores. The aggregate bandwidth for all the optical communication lines is 98 Tb/s (8×16×512+32×1024). The aggregate bandwidth of ATAC is 8.192 Tb/s and that of Corona is 160Tb/s. Since the bandwidth for individual communication lines is twice as fast, most of the remaining characteristics are similar to or superior to ATAC except for the latency that occurs from passing through the electrical switch, CU once (3 cycles) or twice (6 cycles). As before, based on the aggregate bandwidth, the power consumption and latency between optical and electrical communications are compared. For electrical communication, when 1,024 cores are connected by a 32×32 metric mesh network, an average of 32 hops/communication is assumed. Except for the average of 4 hops from movement within the cluster, about 28 hops correspond to the optical communication distance. For optical communication, there are 2 or 3 hops for traversals through optical lines. Since latency has a minimum of 7 cycles (2 cycles for 2 O/E, 2 cycles for 2 E/O, 2 cycles for 2 traversals, 1 cycle for 1 routing within a CU) and a maximum of 12 cycles (3 cycles for 3 O/E, 3 cycles for 3 E/O, 2 cycles for 2 traversals, 2 cycles for 2 routing, 2 cycles for a far traversal from CU to CU), an average of 9 cycles can be assumed. The power consumption for electrical communication is 5,488 W (98 Tb/s × 28 × 2 mW/Gb/s) and for optical communication 23.5 W (98 Tb/s × 3 × 0.08 mW/Gb/s). The aggregate latency for electrical communication is 915,000 cycles (98 Tb/s × 28 / 3 Gb/s) and for optical communication 18,000 cycles (98 Tb/s × 9 / 50 Gb/s). More than these numerical differences, the greater difference is that, while the heat generation for the optical communication is within the controllable range, it is out of range for the electrical communication as considering the wafer size and the heat dissipation rate, which was already experienced in supercomputers. Fig. 2.45. Fat-tree network with 1,024 cores. 158 Chapter 2. CPU Technology Trend and Optical Interconnects Fig. 2.46 shows a 2.5D package of CPU with the interposer chip replaced by an optical network chip (ONC), as compared with Fig. 2.24. The 2.5D interposer chip made of a silicon wafer is manufactured by the CMOS process. The optical waveguide and optical devices are formed on the bottom side of the interposer, and the flip-chip bond electrodes are formed on the top side, and the top and bottom sides are connected by TSVs. On the top, the necessary chips such as CPU, DRAM, GPU, and SoC are integrated by flip-chip bonding. For the bottom side, the proper heights and placements for the optical devices are designed on the package substrate. The interposer upgraded as an optical network chip is manufactured independently by a third party company. When the corresponding chips are supplied by CPU and memory companies, they can be integrated on the ONC at the package stage. In addition to x86 server CPUs, as technology and demand grow, smartphone SoCs can also be integrated on the ONC to communicate optically with each other. In case that the yield of individual core is 90 % in CPU manufacturing, the probability that all 28 cores will be successful drops to 5 %. Therefore, it is much more economical to manufacture and integrate CPU chips with 4~8 cores than to manufacture CPU chips with tens to hundreds of cores. Fig. 2.46. 2.5D package and optical network chip. As shown in Table 2.12, Intel's 28-core server CPU ranges from $10,000~$18,000 and 8-core CPU for desktop ranges from $440~$600. The production cost for optically connected 32-cores of CPU is estimated. For 2.2 GHz and 64 bit CPU, since the minimum bandwidth for an optical line is 140.8 Gb/s, 25 Gb/s per lane and 6 lanes for 150 Gb/s may be used for the configuration shown in Fig. 2.43. Based on the current prices in Table 2.12, the cost for optical devices with six lanes is $245 (duplex four lanes $162.9 × 6/4). The total cost of the optical devices for six lines is $1,470. Subtracting the cost of the AOC's package, the actual cost is assumed at most $1000. Excluding VCSELs, PDs, and prisms, the price of an ONC wafer is estimated to be less than $1,000. The optical waveguide, LD/PD driver, TIA, TSV and electrical lines are arranged on the ONC wafer by the CMOS process. Assuming that the 8 core CPU die before package costs $300, the price for 4 chips, 32 cores is $1200. As added together, the cost is at most $3,200 which is less than 1/3 of the 28-core server CPU that is currently on sale in 2021. Considering the power consumption and latency, it has superior performance and price competitiveness. For smartphones, numerous electric devices on the motherboard such as SoC, 5G modem, SDRAM, external GPU, and NPU can be placed on the ONC. This will not only be advantageous in the overall hardware size but also in price competitiveness. As you can see from ATAC, Corona, and the above calculations, 4~16 are suitable numbers for cores. Connecting multiple clusters through optical communication is 159 Advances in Optics: Reviews. Book Series, Vol. 5 beneficial in increasing CPU performance and lowering costs, and this is how supercomputers are already being designed. Table 2.12. Intel CPU and 100 Gb/s QSFP. Intel CPU 100 Gb/s QSFP CPU i9-9980HK Xeon Platinum 8276M Form factor QSFP28 AOC Duplex Number of cores 8 28 Wavelength 850 nm Frequency 2.4 GHz 2.2 GHz Data rate/lane 25 Gb/s Photolithography 14 nm 14 nm Length 25 m Price as of 2021 $ 583 $ 11,722 Price as of 2021 $ 162.9 2.8. Conclusion In order to improve the performance of CPUs, various technologies such as many cores, multi-threads, parallel computing, superscalar, vector, out-of-order, branch prediction, and scheduling are applied to the CPU design. In particular, one of the most essential technologies that enable parallel computing is cache coherence. Communication speeds between cores play an important role for cache coherence, and a lot of research projects to introduce optical signals into CPUs has been actively conducted, but there have not been successful results yet. Since the early 2000s, leading semiconductor companies such as Intel, IBM, HP, Sun Microsystems and Samsung have invested a large number of research funds and human resources to apply the optical communication technology to core-to-core or CPU-tomemory communication of semiconductor chips. Universities and corporations such as MIT, Columbia University, HP and Sun Microsystems have proposed the network architecture for which optical communication technology will be applied, but it has not reached a status of experimentally integrating and testing on actual chips. The main cause of failure is that a silicon material has been used as the optical waveguide. Among the materials used in the CMOS process, there are no materials other than silicon nitride (Si3N4), silicon oxynitride (SiOxNy), silicon dioxide (SiO2), and silicon (Si) that can be used as an optical waveguide. Since silicon dioxide (SiO2) is used for the cladding of the waveguide and if we don't use silicon, then the remaining candidate materials are only silicon nitride (Si3N4) or silicon oxynitride (SiOxNy). It is suggested that the causes of failure can be overcome by the three technologies and a possible network architecture based on these two materials. To implement the optical network architecture on CPU chip, it would be the most realizable to use VCSELs, PDs, and the fat-tree network architecture that are already utilized in supercomputers with replacing just the optical fiber with PECVD SiN/SiON optical waveguide. As this architecture is implemented in the 32 core server CPU with the 2.5D ONC package, it would allow a price that is 1/3 lower than the current price of Intel's server 28-core CPU. The three technologies which are necessary to realize the architecture 160 Chapter 2. CPU Technology Trend and Optical Interconnects have been developed and experimentally verified by Oprocessor Inc. If technology advances in the direction expected by optical network designers, then in addition to the CPU and memory, ONC will be the most important component of computers. References [1]. D. Page, A Practical Introduction to Computer Architecture, Springer, New York, NY, 2009. [2]. M. Brorsson, Multi-core and many-core processor architectures, Chapter 2, in Programming Many-Core Chips (A. Vajda, Ed.), Springer Science+Business Media, Berlin, Germany, 2011. [3]. J. Aad van der Stee, Overview of Recent Supercomputers, NCF/HPC Research, Netherlands, 2008. [4]. ARM Architecture Reference Manual; ARMv8, for ARMv8-A Architecture Profile, ARM Company, 2013-2019. [5]. ARM Compiler Version 6.6; User Guide, ARM Company, 2013-2016. [6]. E. Blemd, J. Menon, K. Sankaralingam, A detailed analysis of contemporary ARM and x86 architecture, in Proceedings of the 19th IEEE International Symposium on High Performance Computer Architecture (HPCA’13), 2013. [7]. A. La Manna, Outlook for 3D and 2.5D chips in smartphones, Chip Scale Review, Vol. 19, Issue 2, Mar. 2015, pp. 9-11. [8]. J. Turley, Introduction to Intel Architecture, White Paper, Intel Corporation, 2014. [9]. An Introduction to the Intel QuickPath Interconnect, White Paper, Intel Corporation, 2009. [10]. A. Kumar, New Intel Mesh Architecture: The ‘Superhighway’ of the Data Center, White Paper, Intel Corporation, 2017. [11]. M. Berktold, CPU Monitoring with DTS/PECI. White Paper, Intel Corporation, 2010. [12]. Thermal/Mechanical Specification and Design Guide (TMSDG); Intel Core i7 Processor Family for the LGA2011-3 Socket. White Paper, Intel Corporation, August 2014. [13]. S. Strande, M. Mclaughlin, Building a Data – Intensive Supercomputer Architecture for the National Research Community. White Paper, Cray Inc., 2013. [14]. W. Hu, Y. Zhang, J. Fu, An introduction to CPU and DSP design in China, Science China Information Sciences, Vol. 59, Jan. 2016, 012101. [15]. J. Dongarra, Report on the Sunway TaihuLight System, Tech Report UT-EECS-16-742, University of Tennessee, 2016. [16]. G. Kurian, J. E. Miller, J. Psota, J. Eastep, J. Liu, J. Michel, L. C. Kimerling, A. Agarwal, ATAC: A 1000-Core cache-coherent processor with on-chip optical network, in Proceedings of the 19th International Conference on Parallel Architecture and Compilation Techniques (PACT’10), September 11-15, 2010, pp. 477-488. [17]. D. Vantreas, R. Schreiber, M. Monchiero, M. McLaren, N. P. Jouppi, M. Fiorentino, A. Davis, N. Binkert, R. G. Beausoleil, J. Ho Ahn, Corona: System implications of emerging nanophotonic technology, in Proceedings of the 35th Annual International Symposium on Computer Architecture (ISCA’08), 2008, pp. 153-164. [18]. A. Biberman, K. Preston, G. Hendry, N. Sherwood-droz, J. Chan, J. S. Levy, M. Lipson, K. Bergman, Photonic network-on-chip architectures using multilayer deposited silicon materials for high-performance chip multiprocessors, ACM Journal on Emerging Technologies in Computing Systems, Vol. 7, Issue 2, 2011, 7. [19]. A. V. Krishnamoorthy, X. Zheng, J. Lexau, I. Shubin, Computer systems based on silicon photonic interconnects, Proceedings of the IEEE, Vol. 97, Issue 7, Jul. 2009, pp. 1337-1361. [20]. C. Sun, M. T. Wade, Y. Lee, J. S. Orcutt, L. Alloatti, M. S. Georgas, A. S. Waterman, J. M. Shainline, R. R. Avizienis, S. Lin, B. R. Moss, R. Kumar, F. Pavanello, A. H. Atabaki, 161 Advances in Optics: Reviews. Book Series, Vol. 5 H. M. Cook, A. J. Ou, J. C. Leu, Y.-H. Chen, K. Asanović, R. J. Ram, M. A. Popović, V. M. Stojanović, Single-chip microprocessor that communicates directly using light, Nature, Vol. 528, Issue 24, Dec. 2015, pp. 534-538. [21]. M. I. Alayo, D. Criado, L. C. D. Goncalves, I. Pereyra, Deposition and characterization of silicon oxynitride for integrated optical applications, Journal of Non-Crystalline Solids, Vols. 338-340, pp. 76-80. [22]. R. Jones, O. Cohen, H. Chan, D. Rubin, A. Fang, M. Paniccia, Integration of SiON gratings with SOI, in Proceedings of the IEEE International Conference on Group IV Photonics 2 nd (GFP’05), Sept. 2005, pp. 192-194. [23]. T. Tsuchizawa, T. Watanabe, K. Yamada, H. Fukuda, S. Itabashi, J. Fujikata, A. Gomyo, J. Ushida, D. Okamoto, K. Nishi, K. Ohashi, Low-loss silicon oxynitride waveguides and branches for the 850 nm wavelength region, Japanese Journal of Applied Physics, Vol. 47, Issue 8, Aug. 2008, pp. 6739-6743. [24]. B. S. Sahu, O. P. Agnihotri, S. C. Jain, R. Mertens, I. Katok, Influence of hydrogen on losses in silicon oxynitride planar optical waveguides, Semicond. Sci. Technol., Vol. 15, pp. L1-L11. 162 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera Chapter 3 Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera Juan C. Suárez-Bermejo, Juan C. González de Sande, Massimo Santarsiero and Gemma Piquero1 3.1. Introduction Polarimetry is a well-established technique that has a large number of applications. The primary goal of polarimetry is the characterization of the optical behavior of samples when light is transmitted through or reflected by the surface of the sample [1]. Different materials, for example, technological materials or biological tissues, can interact with light in different ways. From the changes produced in the polarization state of the beam, the actual optical properties of the samples can be obtained. In this way, it is possible to identify some characteristics or properties of the sample that may be related to the specific behavior of the technological material or to some pathologies in biological tissues [2, 3]. The state of polarization of a light beam can be described by its Stokes parameters that are usually arranged in a 4×1 column vector [1, 4]. These parameters are measurable quantities. If a polarized light beam passes through a given sample, the state of polarization changes depending on the sample characteristics. The linear optical sample ̂ , that relates the Stokes characteristics can be represented by a 4×4 Mueller matrix, 𝑀 vector of the input and of the output beam. The sample is usually tested with a uniformly and totally polarized input beam with a Stokes vector that is sequentially varied to reach at least four linearly independent states of polarization to measure the sample Mueller matrix [1, 5-9]. Many nonuniformly totally polarized beams contain four or more independent states of polarization [10, 11], so they can be used as parallel polarization state generator (PSG) in Juan C. Suárez-Bermejo, Department of Materials Science, Universidad Politécnica de Madrid, Spain 163 Advances in Optics: Reviews. Book Series, Vol. 5 a Mueller matrix polarimetric system [12-14]. In particular, the so-called full Poincaré beams (FPBs) present all possible states of polarization across their transverse section [15, 16]. A simple method to obtain a specific FPB has been recently described [17, 18] and this beam has been used to determine the Mueller matrix of a sample by measuring the change of the state of polarization at only four specific points of its transverse section by using a commercial Stokes polarimeter [14, 19]. This FPB has been proposed for Mueller matrix polarimetry by measuring the change of the state of polarization in the whole transverse section of the beam by means of a CCD camera instead of measuring it at only four points with a commercial polarimeter [20]. The states of polarization before and after the sample are measured by means of a polarization state analyzer (PSA) consisting of a quarter wave phase plate and a linear polarizer that are positioned before the CCD camera in six different configurations. An image of the transmitted intensity is recorded for each of the PSA configurations with and without sample. From these recorded images, many combinations of at least four different pixels can be used to determine the Mueller matrix of the sample [20]. The goal of this work is to present different strategies for recovering the Mueller matrix of a sample from simulated intensity images. A comparison among different strategies to process this set of data will be presented. This work will allow to determine the best strategy and to obtain an estimation of the errors in the measured Mueller matrix when only the inaccuracy of the intensity values at each pixel is considered as the main source of error. This section constitutes the Introduction. In Section 3.2 the FPB Polarimetry method is described. To study the errors of the intensity measurements, the characterization and different tests carried out for our CCD camera are described in Section 3.3. In Section 3.4 the different strategies with the considered errors in the measurements are presented. In Section 3.5 the main results and discussion is presented through the comparison among the previous forms of data treatment. Finally, the main conclusions are remarked in Section 3.6. 3.2. Polarimetry with a Full Poincaré Beam and a CCD Camera The Stokes vector at a given point in the transverse section of a light beam, 𝑆⃗𝑖𝑛 (𝑥, 𝑦), can be obtained by measuring the light intensity at such a point after a linear polarizer (P) with its transmission axis oriented at four different angles, namely, 0°, 45°, 90°, 135° relative to the x-axis of a suitable reference system (they will be denoted as I0, I45, I90, I135, respectively); and after the combination of a quarter wave phase plate (QWP) with its fast axis along the horizontal axis and the linear polarizer oriented at 45° and 135° (denoted as IQWP,45 and IQWP,135) [4, 21]. This system constitutes an optimum PSA when six intensity measurements are taken [1, 22] and is sketched in Fig. 3.1(a). By recording these intensities with a CCD camera, a map of the Stokes parameters across the section of a nonuniformly polarized beam can be obtained as 164 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera 𝑖𝑛 (𝑥, 𝑦) 𝐼0𝑖𝑛 (𝑥, 𝑦) + 𝐼90 𝑆0𝑖𝑛 (𝑥, 𝑦) 𝑖𝑛 (𝑥, 𝑖𝑛 (𝑥, 𝑖𝑛 (𝑥, 𝑦) − 𝐼90 𝑦) 𝐼0 𝑦) 𝑆 𝑆⃗𝑖𝑛 (𝑥, 𝑦) ≡ 1𝑖𝑛 = . 𝑖𝑛 (𝑥, 𝑖𝑛 (𝑥, 𝑦) 𝐼45 𝑦) − 𝐼135 𝑆2 (𝑥, 𝑦) 𝑖𝑛 𝑖𝑛 [𝑆3𝑖𝑛 (𝑥, 𝑦)] [𝐼QWP,45 (𝑥, 𝑦) − 𝐼QWP,135 (𝑥, 𝑦)] (3.1) Fig. 3.1. Scheme of Mueller matrix determination using a Full Poincaré beam and a CCD camera. (a) Parallel polarization state generator (PSG) by means of a specific FPB and polarimetric characterization through a PSA; (b) theoretical images for six specific configurations of the PSA; (c) the same images as in (b) with added Gaussian noise (AGN); (d) noisy Stokes parameter maps for the FPB; (e) polarimetric characterization of the generated FPB after passing through a sample ̂𝑇 ; (f) theoretical images for six configurations of the PSA; represented by a Mueller matrix 𝑀 (g) the same images as in (f) with AGN; (h) noisy Stokes parameter maps for the output beam. 165 Advances in Optics: Reviews. Book Series, Vol. 5 In the case of a FPB, the whole Poincaré sphere [1, 4], is mapped at least once across the beam section. If this light beam passes through a given sample, the state of polarization at each point changes depending on the sample characteristics. Then the sample can be tested for all polarization states at once when a FPB impinges on it. The Stokes parameters of the output beam, 𝑆⃗𝑜𝑢𝑡 (𝑥, 𝑦), can also be obtained by measuring six images as for the input beam [see Fig. 3.1 (e)]. The linear optical characteristics of the sample can be represented ̂ , that relates the Stokes vector of the input and output beam by a 4×4 Mueller matrix, 𝑀 as ̂ 𝑆⃗𝑖𝑛 (𝑥, 𝑦). 𝑆⃗𝑜𝑢𝑡 (𝑥, 𝑦) = 𝑀 (3.2) An advantage of using a FPB as input beam is that all input states of polarization can be tested at the same time. By repeating Eq. (3.2) at four points whose states of polarization are linearly independent, the Mueller matrix of the sample can be recovered [14, 19, 20]. However, if the Stokes parameters are measured with a CCD camera before and after the sample, many combinations of four different pixels can be used to determine the Mueller matrix of the sample. The theoretical Stokes vector at any point (𝑟, 𝜃) of the beam section at the exit of a uniaxial crystal illuminated with a divergent quasi-monochromatic beam [see Fig. 3.1(a)] can be approximated by [14, 19] 1 −[cos2 2𝜃 + sin2 2𝜃 cos 𝛿(𝑟)] 𝑆⃗(𝑟, 𝜃) = 𝑆0 (𝑟, 𝜃) × [ ], −sin2𝜃 cos2𝜃[1 − cos 𝛿(𝑟)] sin2𝜃 sin 𝛿(𝑟) being 𝛿(𝑟) a phase given by 𝑟2 𝑙 𝛿(𝑟) = 𝑘(𝑛𝑒 − 𝑛𝑜 ) , (3.3) (3.4) where 𝑘 is the vacuum wave number, 𝑙 is the crystal length, and 𝑛𝑜 and 𝑛𝑒 are the ordinary and extraordinary refractive indexes of the uniaxial crystal, respectively. It has been shown that for this particular beam, all possible totally polarized states of polarization can be found across the beam section inside a semicircle with radius [14, 17]. 𝑟𝑀 = √𝑘|𝑛 𝜋𝑙 . 𝑒 −𝑛𝑜 | (3.5) A possible way to recover the Mueller matrix consists in using the Stokes vector values at four different pixels (the same set for the input and output maps). The selected pixels must be chosen in such a way that the four corresponding input Stokes vectors be linearly independent (we recall that the coordinates of a point across the beam transverse section presenting a specific state of polarization can be obtained using specific analytical expressions [14, 17]). The optimum choice of the input states of polarization is such that 166 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera they form a regular tetrahedron inscribed in the Poincaré sphere [see Fig. 3.2(a)] [22]. The corresponding polarization states are found at the locations shown in Fig. 3.2(b) [14, 17]. (a) (b) Fig. 3.2. (a) Poincaré sphere with a set of optimum polarization states for polarimetric measurements; (b) Polarization map across the beam section and points where the polarization states represented in a) are found. Red (blue) ellipses represent right-handed (left-handed) polarization states. The semicircle has radius rM given in Eq. (3.4). All possible polarization states are found in this semicircle. A continuous rotation of the tetrahedron around the s1 axis yields a movement of the three points out of the center along the green ovoids in Fig. 3.2(b). The recovered Mueller matrix can be found by repeating the inversion of Eq. (3.2) for many sets of four selected ̂𝑆 , pixels and averaging results. The differences between the calculated Mueller matrix, 𝑀 ̂ and that corresponding to the ideal sample, 𝑀𝑇 , will be evaluated. For this approach, a critical point is the inaccuracy of the intensity measurement at each pixel of the CCD camera. In the next section, we analyze the noise introduced by the CCD camera and justify the Gaussian distribution considered for the added noise in the simulations. 3.3. Noise in CCD Images Noise in a camera arises from the aggregate spatial and temporal variations in the captured images. If a stabilized light source is used, the measured signals for every individual pixel vary according to different sources of noise. The main sources of noise due to the CCD camera are the dark shot noise, read noise, photon shot noise, and fixed pattern noise [23]. The shot noise is due to the current that appears even in the case of no photons reaching the camera, it has thermal origin and is uncorrelated to the signal level. The read noise is related to the production of the electronic signal in the CCD camera, depends on the sensor and electronics design, is more important for faster CCD pixel clock rates, but it is independent of the signal level and temperature of the sensor. The photon shot noise is 167 Advances in Optics: Reviews. Book Series, Vol. 5 associated with the statistical uncertainty due to the arrival of individual photons to the detector, it depends on the signal level for every individual pixel, but independent of the sensor temperature. Finally, the fixed pattern noise has its origin in the spatial nonuniformity of the pixel array in the CCD sensor, it is independent of the signal level or temperature of the sensor, and usually negligible for high quality sensors. The total effective noise per pixel could be estimated as the quadrature sum of the noise sources [24, 25]. When acquiring images with the CCD camera, several adjustments can be done to improve the quality of the images while they are being captured by the sensor and control software of the CCD camera, previously to other treatments of data. Here we analyze the noise in the acquisition of images by means of a Spiricom SP620U CCD camera RoHS, with 4.4 µm square pixels. It has an active area of 7.1 mm wide by 5.4 mm height, and a number of 1600 × 1200 active pixels, that is, 1.92 Mpx. The spectral response is in the range of 190 – 1100 nm. The minimum system dynamic range is 62 dB, and linearity with power is ±1 %. Saturation intensity in optimal conditions – camera is set to full resolution at maximum frame rate and exposure time, running CW at 632.8 nm wavelength – is 2.2 µW/cm2. Background noise due to the faint light in the room – coming from the computer screen and other dim sources but shutting down the laser output – has been recorded and subtracted from every recorded image for subsequent computations. The average power of the background noise is approximately 62 levels out of a maximum 4096 at saturation with a standard deviation of 2 levels. It can be observed that this background is less than 2 % of the maximum level and is quite uniform across the whole CCD. Afterwards, to estimate the image errors, the configuration shown in Fig. 3.1(a) without the QWP and with the transmission axis of the linear polarizer P oriented vertically (90º) is used to acquire a set of 25 images of the beam cross-transverse section. This configuration of the PSA is selected because is the one for which the light received by the CCD camera when measuring the Stokes parameters of the input FPB is maximum. Selected frame rate has been the lowest value allowed by the equipment (1.850 Hz) to minimize the errors due to the recording of the images. Exposure time of 40 ms has been used to get the maximum intensity level without reaching saturation in the sensor. All images show quite similar intensity patterns. The next step is to choose one out of the 25 stored images and compare it with the average of the remaining 24 images. This procedure is iterated 25 times, selecting a different image for every iteration, and comparing with the average of the remaining ones. One of these iterations is presented in Fig. 3.3, where the left-hand side shows one individual registered image and the right-hand side shows the average of the 24 remaining images. For both cases, the background has been subtracted. The accumulated noise at each pixel, computed as the difference between one of the images and the average of the 24 remaining images is shown in Fig. 3.4. Here we show a front view of these differences along either the 1200 rows or 1600 columns of the CCD 168 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera sensor, so the maximum and minimum difference of the pixels located on each column can be easily visualized. It can be observed that the computed noise level is higher for higher intensity values registered at every individual pixel (in the 400-800 row range and 700-1100 columns range). It should be noted that the computed noise includes not only the noise due to the CCD camera but also the possible temporal fluctuations of the light source intensity at each pixel. Fig. 3.3. Example of the procedure followed to estimate the image noise: (a) individual recorded image and (b) average image obtained from the 24 remaining images. Fig. 3.4. Difference at each individual pixel between the first recorded image and the average of the 24 remaining ones, (a) Row view, (b) Column view. A histogram of the differences shown in Fig. 3.4 has been done for the 1.92 Mpx present in each image and is shown in Fig. 3.5. The best fit to a single Gaussian distribution (blue solid line), has been obtained and some deviation from the experimental data can be observed, specially for the tails. The R-square parameter of this fit is 0.964. However, when a superposition of two Gaussian distributions is considered (red dashed line), the histogram is accurately fitted, with an adjusted R-square parameter of 0.998 for this example. Similar results are obtained for the rest of the images. The fact that a better fit is obtained with two Gaussian distributions could be due to a correlation between the differences and the intensity level measured at each pixel. Fig. 3.6(a) shows a representation of the difference level versus the intensity level measured at each pixel. That is, the average intensity level measured at each pixel (intensity value shown in the right-hand side of Fig. 3.3) is represented along the horizontal axis and the difference between the intensity at that pixel in the image of the 169 Advances in Optics: Reviews. Book Series, Vol. 5 left-hand side of Fig. 3.3 and the average intensity is represented along the vertical axis. It can be observed that the distribution of these differences broadens for higher and higher mean measured intensities, so the behavior is slightly different depending on such intensity level. There is a slight skew in the average towards negative values (respect zero difference level) for higher intensity levels. Fig. 3.5. Histogram of the number of pixels with a given level of difference between the first recorded image and the average of the 24 remaining ones. Black dots are the total number of pixels whose intensity difference is within a given range. Solid line: best fit to a Gaussian distribution. Dashed line: best fit to a superposition of two Gaussian distributions. From the graph in Fig. 3.6(a), a ±2 % wide band around a given intensity level has been selected for several measured intensity levels (500, 1000, 1500, 2000, 2500, 3000, and 3300). The vertical shaded bands indicate the one around 500 and 3300 intensity levels, respectively. A histogram of the differences has been built for each of these bands. Fig. 3.6(b) shows the obtained histograms for the bands corresponding to 500 and 3300 measured levels (both histograms include pixels where the measured intensity is within 2 % of the selected center value). The histogram for each band is fitted very well by a single Gaussian distribution. The R-values of the fits are always over 0.993. The standard deviations for the fits in the right-hand side of Fig. 3.6 are 6.5 (500-level band) and 21 (3300-level band), respectively. This process is repeated for each of the seven mentioned bands. The results of the fits are shown in Fig. 3.7. In the left side of Fig. 3.7, the standard deviation is represented versus the intensity level whereas in the right side, the standard deviation normalized to the intensity level is represented as a function of the intensity level. It is observed that the standard deviation grows steadily with the level of the measured intensity as it could be expected. The dashed line in Fig. 3.7(a) is the best linear fit to this data. Finally, the behavior followed by the normalized value of the standard deviation of the intensity level of the band for each band is depicted in Fig. 3.7(b). It can be observed that this relative standard deviation is around or below 0.01. 170 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera Fig. 3.6. (a) Difference between the first recorded image and the average of the 24 remaining ones versus the intensity level measured at each pixel; (b) Histogram of the number of pixels with a given level of difference between the first recorded image and the average of the 24 remaining ones for pixels whose measured intensity level is within 2 % of 500; (c) same as (b) for 3300 intensity level band. Black dots are the total number of pixels whose intensity difference is within a given range. Solid line is the best fit to a Gaussian distribution. Fig. 3.7. Absolute (a) and relative (b) standard deviation for each intensity level band. The blue line is (a) best linear fit to the data (b) a guide for the eye. 171 Advances in Optics: Reviews. Book Series, Vol. 5 This number will be used in the simulations presented in the following section. We will try to estimate the expected errors in the determination of the Mueller matrix of a sample, when the polarimetric measurements are carried out using a FPB as a parallel PSG, the PSA sketched in Fig. 3.1, and a CCD camera with similar characteristics than the studied in this work for. It should be noted that the simulation results that will be presented in the following section, including levels of noise above and below obtained for this camera. 3.4. Strategies for Obtaining the Mueller Matrix Besides the errors due to imperfections or errors in the positioning or the optical elements of the PSA (polarizer and quarter wave phase plate), an important source of error is the inaccuracy in the measurement of the light intensity at each pixel of the acquired images, [22, 26-28]. To estimate how this source of error affect to the Mueller matrix measurement, the following simulations will be carried out. The intensity maps that would be obtained when the considered FPB passes through the PSA for six different configurations are theoretically calculated [see Fig. 3.1 (b)]. This calculation is carried out by using Eq. (3.3) and the corresponding Mueller matrix of the specific PSA configuration, considering that 𝑆0 (𝑟, 𝜃) presents a Gaussian shape with a spot size twice the value of 𝑟𝑀 [see Eq. (3.5)]. Then, a random Gaussian noise is added to each of these calculated images [see Fig. 3.1(c)]. The amplitude of this added Gaussian noise (AGN) at each point presents a standard deviation that is proportional to the calculated intensity at such point. From these noisy images, the maps of Stokes parameters for the input beam are calculated [see Fig. 3.1 (d)]. The same procedure is followed when a sample (with a ̂𝑇 ) is inserted before the PSA [see Fig. 3.1 (e)-(h)]. Finally, from known Mueller matrix 𝑀 ̂𝑆 , is the input and output Stokes parameter maps, the Mueller matrix of the sample, 𝑀 evaluated. This simulation process has been repeated Ns = 5 times and the average of the final results has been taken. A quantitative estimate of the errors in the determination of Mueller matrix is given as the root mean square of the elements of the difference ̂𝑇 − 𝑀 ̂𝑆 . 𝑀 Two different approaches were considered for recovering the Mueller matrix of the sample from the simulated Stokes parameters: the center point and the nearest neighbour strategy. Let us briefly describe both approaches. 3.4.1. Center Point Strategy The first approach is an adaptation of the procedure followed in reference [14, 20]. Four pixels are chosen in such a way that the four corresponding input Stokes vectors correspond to the vertices of a tetrahedron inscribed in the Poincaré sphere [red dots in Fig. 3.2(a)]. The process is repeated for several orientations of the tetrahedron. Note that, for the chosen configuration of the FPB, the linear vertical polarization state, given by the Stokes vector 172 𝑆⃗𝑖𝑛 (𝑟0 , 𝜑0 ) = [1, −1, 0, 0]𝑇 , (3.6) Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera can be found at the FPB axis. This point is always taken because the intensity is maximum there. Several sets of the three remaining points across the transverse plane of the beam have been selected in such a way that their polarization states correspond to rotations of the tetrahedron inscribed in the Poincaré sphere around the axis s1 [see Fig. 3.2 (a)] [14, 17, 19, 22]. Due to its symmetry, the rotation of the tetrahedron can be limited to the 𝜋 𝜋 interval [− , ) . 6 6 The coordinates of the four considered points are mapped on the CCD images (red pixels in Fig. 3.8, for one of the choices), both for the input beam [without the sample and with AGN, Fig. 3.1 (d)], and for the output beam [after passing through the sample, and with a different pattern of AGN, Fig. 3.1 (h)]. Theoretical matrix for every ideal sample is compared to the matrix obtained from simulations and the differences are derived for ̂ . An average estimate of the error is obtained from 15 analyzed sets, every element of 𝑀 corresponding to 15 equally spaced tetrahedron orientations, and the uncertainties of these values are computed. Fig. 3.8. Red squares represent pixels that contain the point where the ideal FPB presents the states of polarization corresponding to the vertexes of the tetrahedron of Fig. 3.1a. They are the ones used for the center point strategy. Green squares together the center one represent the 3×3 nearest neighbor. Any combination of four of these pixels around the red ones is used for the nearest neighbor strategy. Shaded semicircle represents a possible area of the beam cross section where all possible states of polarization are found. 3.4.2. Nearest Neighbor Strategy The second strategy uses a wider set of points, all located around each of the center point considered in the previous case. A total of 𝑁 × 𝑁 nearest neighbor pixels are considered around every center point (see Fig. 3.8). Mueller matrices are obtained for all possible four-point sets (input and output) that can be chosen from the 𝑁 2 × 4 pixels included in the analysis. That means a total of (𝑁 2 )4 possible combinations. For all of them, the condition number for the equation system formed by evaluating Eq. (3.2) at four different locations is computed and the resulting Mueller matrix is discarded if such condition number [1] is over 20. An average value is then computed from all valid cases, for the 15 analysed orientations of the tetrahedron in the Poincaré sphere. 173 Advances in Optics: Reviews. Book Series, Vol. 5 3.5. Results from the Simulation Mueller matrices are obtained from the simulated Stokes input and output vectors for four different standard samples: air, a linear polarizer with horizontal transmission axis (LP0) and with its transmission axis at 60º with respect to the horizontal (LP60), and a quarter wave phase retarder with its fast axis along the horizontal direction. Fig. 3.9 shows the root mean square value of the difference between the mean matrix obtained for the Ns simulations at each orientation of the tetrahedron and the theoretical ̂𝑆 〉 − 𝑀 ̂𝑇 ), for each of the four samples considered. one, 𝑟𝑚𝑠(〈𝑀 Fig. 3.9. Individual and mean values of the rms of differences between the Mueller matrix obtained from numerical simulation and theoretical ones, for four different ideal samples, assuming a normalized standard deviation of the AGN amplitude  = 0.01. The four samples are air, a linear polarizer with horizontal transmission axis (LP0) and with its transmission axis at 60º with respect to the horizontal (LP60), and a QWP retarder with its fast axis along the horizontal direction. The value of the root mean square difference is calculated as ̂𝑆 〉 − 𝑀 ̂𝑇 ) = 𝑟𝑚𝑠(〈𝑀 1 𝑆 √∑3𝑖,𝑗 = 0(𝑚 ̅ 𝑖𝑗 4 2 𝑇 − 𝑚𝑖𝑗 ) , (3.7) 𝑆 where 𝑚 ̅ 𝑖𝑗 are the mean values of the Mueller matrix elements obtained from the 𝑇 simulation and 𝑚𝑖𝑗 are the theoretical ones. For these simulations, a Gaussian noise with amplitude  = 0.01 of the maximum intensity of the beam has been added to each pixel of the images in Fig. 3.1b and Fig. 3.1f. The 174 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera results for both the center point and nearest neighbor strategies are presented. In all cases, the nearest neighbor strategy renders a smaller value of the error compared to the center point strategy. It is worth noting that the orientation of the tetrahedron around the 𝑠1 axis has no influence on the obtained results. When the maximum amplitude of the added Gaussian noise (characterized by the parameter ) is varied, the uncertainty in the errors predicted by both strategies also changes. Fig. 3.10 shows the average values and the error bars for seven values of  when the sample is air: the beam does not go through any sample at all but two different Gaussian noises are added to the images in Fig. 3.1 (b). Fig. 3.10. Mean values of the rms of the differences between the Mueller matrix obtained from numerical simulation and the theoretical one, for both center point and nearest neighbor strategies when the considered sample is air. Uncertainties are indicated by vertical bars. A comparative graph for the four samples is included in Fig. 3.11(a), for the center point strategy. The general trend is the same for all of them: the error in the determination of the elements of the Mueller matrix grows linearly as the Gaussian noise is increased. However, quantitatively, the errors for the case of a linear polarizer with horizontal transmission axis are less than a half of the other three considered samples. Similarly, a comparative graph for the four samples following the nearest neighbor strategy is shown in Fig. 3.11(b). In this case, the error in the determination of the elements of the Mueller matrix does not grow linearly as the Gaussian noise is increased. Calculated errors are a minimum for the case of a linear polarizer (0º), followed by the linear polarizer (60º). Mean errors in this strategy are very similar for the cases of air (no sample) and quarter wave plate linear retarder. The final goal of these simulations is to assess the differences in the mean errors that are ̂ during experimental expected using the first or the second strategy for the evaluation of 𝑀 measurement using real samples. Comparison of the values in Fig. 3.11(a)-(b) for both strategies, gives a reduction in errors using the nearest neighbor strategy, compared to the center point strategy. For all cases, the nearest neighbor strategy is superior in terms of error reduction. A value of 80 % in error reduction can be noted for situations where the 175 Advances in Optics: Reviews. Book Series, Vol. 5 Gaussian noise parameter is  ≤ 0.02, for any of the samples. However, if higher levels of noise are present in the CCD images, the error reduction starts to drop steadily. (a) (b) Fig. 3.11. Mean values of the rms of the differences between the Mueller matrix obtained from numerical simulation and the theoretical ones, for (a) the center point strategy, and (b) nearest neighbor strategy. 3.6. Conclusions In the present work, it is proposed to use a full Poincaré beam together with a simple polarization state analyzer and a CCD camera for determining the Mueller matrix of an unknown sample. Two different strategies are tested, which we have called the center point strategy or nearest neighbor strategy, respectively. From the presented simulations, it can be deduced that the information collected from the images taken with a CCD camera for six configurations of the PSA could be useful to reduce the errors due to the inaccuracy of the intensity measurement. The results obtained from these simulations will be useful for deciding a measurement strategy during the experimental development of this polarimetric method. Acknowledgements This work has been partially supported by Spanish Ministerio de Economía y Competitividad under project PID2019104268GB-C21. References [1]. R. Chipman, W-S. Lam, G. Young, Polarized Light and Optical System, CRC Press, New York, 2018. [2]. V. Tuchin, L. Wang, D. Zymnayakov, Optical Polarization in Biomedical Applications, Springer, 2006. [3]. M. Wakaki, Optical Materials and Applications, CRC Press, Boca Raton, 2013. [4]. D. Goldstein, Polarized Light, Marcel Dekker Inc., 2003. 176 Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré Beams and a CCD Camera [5]. R. Azzam, Mueller-matrix measurement using the four-detector photopolarimeter, Optics Letters, Vol. 11, Issue 5, 1986, pp. 270-272. [6]. D. Goldstein, Mueller matrix dual-rotating retarder polarimeter, Optics Letters, Vol. 31, Issue 31, 1992, pp. 6676-6683. [7]. J. Bueno, Polarimetry using liquid-crystal variable retarders: theory and calibration, Journal of Optics A: Pure and Applied Optics, Vol. 2, Issue 3, 2000, 216. [8]. K. Twietmeyer, R. Chipman, Optimization of Mueller matrix polarimeters in the presence of error sources, Optics Express, Vol. 16, Issue 15, 2008, pp. 11589-11603. [9]. R. Azzam, Stokes-vector and Mueller-matrix polarimetry, Journal of the Optical Society of America A, Vol. A 33, Issue 7, 2016, pp. 1396-1408. [10]. G. Piquero, R. Martínez-Herrero, J.C.G. de Sande, M. Santarsiero, Synthesis and characterization of non-uniformly totally polarized light beams: tutorial, Journal of the Optical Society of America A, Vol. 37, Issue 4, April 2020, pp. 591-605. [11]. J. C. G. de Sande, G. Piquero, J. Suarez-Bermejo, M. Santarsiero, Beams with propagationinvariant transverse polarization pattern, arXiv.org, arXiv:2102.00024v, 29 January 2021. [12]. J. C. G. de Sande, M. Santarsiero, G. Piquero, Spirally polarized beams for polarimetry measurements of deterministics and homogeneous samples, Optics and Lasers in Engineering, Vol. 91, 2017, pp. 97-105. [13]. J. C. G. de Sande, G. Piquero, M. Santarsiero, Polarimetry with azimuthally polarized light, Optics Communications, Vol. 410, 2018, pp. 961-965. [14]. J. Suárez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Mueller matrix polarimetry using full Poincaré beams, Optics and Lasers in Engineering, Vol. 122, 2019, pp. 134-141. [15]. A. Beckley, T. Brown, M. Alonso, Full Poincaré beams, Optics Express, Vol. 18, Issue 10, 2010, pp. 10777-10785. [16]. E. J. Galvez, S. Khadka, W. H. Schubert, S. Nomoto, Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light, Applied Optics, Vol. 51, 2012, pp. 2925-2934. [17]. G. Piquero, L. Monroy, M. Santarsiero, M. Alonzo, J. C. G. de Sande, Synthesis of full Poincaré beams by means of uniaxial crystals, Journal of Optics, Vol. 20, Issue 6, 2018, 065602. [18]. G. Piquero, J. Vargas-Balbuena, Non-uniformly polarized beams across their transverse profiles: An introductory study for undergraduate optics courses, European Journal of Physics, Vol. 25, 2004, pp. 793-800. [19]. J. Suarez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Analysis of the errors in polarimetry with full Poincaré beams, in Proceedings of the Photonics & Electromagnetics Research Symposium – Spring (PIERS-SPRING’19), Rome, 2019, pp. 1-7. [20]. J. Suarez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Simulation of Mueller matrix polarimetry with full Poincaré beams and a CCD camera, in Proceedings of the 3rd International Conference on Optics, Photonics and Lasers (OPAL'20), Tenerife, 2020. [21]. R. Chipman, Polarimetry, in Handbook of Optics, McGraw-Hill, 2010. [22]. D. Layden, F. Wood, I. Vitkin, Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach, Optics Express, Vol. 20, Issue 18, 2012, pp. 20466-20481. [23]. K. Irie, A. E. McKinnon, K. Unsworth, I. M. Woodhead, A technique for evaluation of CCD video-camera noise, IEEE Transactions on Circuits and Systems for Video Technology, Vol. 18, Issue 2, 2008, pp. 280-284. [24]. L. Chen, X. Zhang, J. Lin, D. Sha, Signal-to-noise ratio evaluation of a CCD camera, Optics & Laser Technology, Vol. 41, Issue 5, 2009, pp. 574-579. [25]. J. Dai, F. Goudail, M. Boffety, J. Gao, Estimation precision of full polarimetric parameters in the presence of additive and Poisson noise, Optics Express, Vol. 26, Issue 26, 2018, pp. 34081-34093. 177 Advances in Optics: Reviews. Book Series, Vol. 5 [26]. J. Zallat, M. Stoll, Optimal configurations for imaging polarimeters: impacts of image noise and systematic errors, Journal of Optics A: Pure and Applied Optics, Vol. 8, Issue 9, 2006, pp. 807-814. [27]. G. López-Morales, M. M. Sánchez-López, A. Lizana, I. Moreno, J. Campos, Mueller matrix polarimetric imaging analysis of optical components for the generation of cylindrical vector beams, Crystals, Vol. 10, Issue 12, 2020, 1155. [28]. N. C. Bruce, J. M. López-Téllez, O. G. Rodríguez-Herrera, Permitted experimental errors for optimized variable-retarder Mueller-matrix polarimeters, Optics Express, Vol. 410, 2018, pp. 13693-13704. 178 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications Chapter 4 The Optical Bidirectional Lambertian Conductance Law with Applications Antonio Parretta1 4.1. Prologue The study of the optical properties of a generic element, or system, when subjected to Lambertian irradiation, that is when each point of its optical terminal is source of light with constant and unpolarized radiance, is not common in the literature. However, this has been one of our topic themes of research at the University of Ferrara for several years, under the direction of Prof. Giuliano Martinelli (†), aimed at the design, fabrication and optical characterization of photovoltaic type solar concentrators (SC) [1-5]. The use of Lambertian radiation in solar concentrators is unusual, as these devices are designed to concentrate the quasi-collimated direct component of solar radiation. However, it happened that, on the occasion of a work session dedicated to the optical characterization of a truncated and squared compound parabolic solar concentrator [6], I illuminated, by chance, one of them from the inside with a laser beam, while its exit opening, at that moment, was closed by a white Lambertian diffuser (see Fig. 4.1(a)). On that occasion, I observed that the laser light, reflected by a small portion of the diffuser, after crossing the solar concentrator and “exiting” from its “entrance” opening, was projected on the laboratory wall, forming an image with a particular shape with square contours, the same of the entrance opening itself. After this first observation, I made other experiments to explain the meaning of the luminous figure projected on the laboratory wall. In the meantime, I realized a new optical configuration to better study the phenomenon, placing a punched white screen between the laser and the concentrator, to avoid any interference with the path of light (see Fig. 4.1(b)). Furthermore, I expanded the laser beam with a beam expander in order to illuminate the entire surface of the diffuser facing inside the concentrator (see Fig. 4.2(b)). Fig. 4.1(b) shows the luminous figure formed by the reflected light on the holed screen. Antonio Parretta Physics and Earth Science Department, University of Ferrara, Academy of Sciences of Ferrara, Italy 179 Advances in Optics: Reviews. Book Series, Vol. 5 a) b) Fig. 4.1. (a) A truncated and squared compound parabolic concentrator (TS-CPC), tapped with a white diffuser (ld), is illuminated by a laser beam; (b) Image formed on the punched screen by the reflected laser light. a) b) Fig. 4.2. (a) After being expanded by the beam expander (be), the laser beam crosses the punched planar screen (ps) and strikes on the full exposed area of the diffuser (ld); (b) Control of the expanded laser beam size. After thinking to this phenomenon, it became clear that the luminous figure on the screen, image of the angular distribution of the radiance of the emitted light, corresponded to the angular distribution of the optical efficiency that the concentrator would show if it were illuminated, from the entrance opening, with a collimated light. Thanks to this discovery, and after understanding the importance it could have on the work of optical characterization of solar concentrators, the configuration of the optical apparatus was further improved until it assumed the final aspect shown in Fig. 4.3(a). Here, the Lambertian light to be used to feed the concentrator is produced by a lamp illuminating a first integrating sphere which in turn illuminates a second integrating sphere, having an opening directly coupled to the exit opening of the solar concentrator. The use of a double integrating sphere is necessary to supply the SC by a well-shaped Lambertian, unpolarized light with 90° divergence. The light exiting from the input opening of the SC is then projected on a far, white screen normally used to project slides. The luminous figure diffused by the white screen is then recorded by a CCD aligned with the solar concentrator. The elaboration of the CCD image (see Fig. 4.3(b)), by an appropriate software [6], allows finally to derive the angular distribution of radiance and therefore the angular distribution of the optical efficiency. To get the absolute optical efficiency, it is necessary to know the 180 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications optical efficiency at 0°, the on-axis efficiency. This is obtained by elaborating the image recorded after facing the CCD directly on the solar concentrator, as shown in Fig. 4.4 (a). Fig. 4.4 (b) shows an example of image produced by projecting the inverse light from the Rondine® concentrator [7-14]. a) b) Fig. 4.3. (a) Final version of the configuration of the optical apparatus for recording the radiance of the emitted light in the inverse way. The lamp (ls) is a source of white light; (b) Image of the luminous figure projected on the planar screen (ps). The on-axis optical efficiency is obtained by the ratio between the average intensity of the full image in the yellow contour and the average intensity of the Lambertian source in the red contour [6]. This technique has been successfully applied to all lab-made concentrators, which were characterized in few days, when, adopting the traditional method, it would have taken more than a month [6, 15-28]. This, in short, is the story I lived putting together the Lambertian light and a solar concentrator. But this story does not finish here, because it gave rise to the idea of investigating the optical behavior of a solar concentrator when it is irradiated with Lambertian light from one of its openings, regardless of the study of its optical efficiency. And this has given rise to the definition of new optical quantities related to the transmission, reflection and absorption of light [29-37]. From here it was easy to imagine a further development of the theory in which the two openings of the SC are simultaneously irradiated with Lambertian light [6, 29]. Hence the idea of formulating an optical law that relates the net flux crossing the SC with the differential radiance at its ends through a quantity defined as "optical conductance" [6, 29]. But this study does not consider only solar concentrators, or at least not only solar concentrators with a canonical form, because the law of optical conductance has a universal character and can be applied to any optical system for which the principle of optical reversibility is valid [38, 39]. In this work, I have chosen, therefore, some simple optical elements to apply the law of optical conductance. Some of them are non-imaging solar concentrators of the CPC type, well designed to concentrate light within a defined angle with maximum efficiency; others, simpler, are the so-called “light cones” (LC), often used in refractive solar concentrators as secondary concentrating elements or for reducing the dispersion of rays in the focal region when an accidental misalignment of the solar tracker could lead to a damage to the walls. Other elements, such as “solar tunnels” (ST), are considered here because they allow us to go beyond the field of solar concentration and to explore other fields such as that of the transport of sunlight from outdoor to indoor. Last but not least, I thought about the use of some of the mentioned 181 Advances in Optics: Reviews. Book Series, Vol. 5 optical elements as light sources when irradiated by Lambertian light in the reverse way. What kind of light do they emit? How their radiance is distributed in space? Can they be used in the field of lighting? I will try to answer these questions in the second part of this work. However, before presenting the results of optical simulations, it is useful to report in the following section the fundamental concepts that are the basis of the operation of canonical solar concentrators, because the actual work originates from them, and because their knowledge will allow us to better understand everything that will be exposed later. a) b) Fig. 4.4. (a) Experimental configuration used for the measurement of the on-axis optical efficiency of a solar concentrator. Unlike the configuration of Fig. 4.3 (a), now the CCD camera is aligned with the optical axis and facing the concentrator inlet opening, from which the inverse light exits; (b) Image of the inlet opening, with the two contours used to calculate the on-axis efficiency. List of Symbols (as they appear in the text):  𝐶𝑔𝑒𝑜 , geometrical concentration ratio;  𝐶𝑜𝑝𝑡 , optical concentration ratio;  𝐸𝑖𝑛 , input irradiance;  𝐸𝑜𝑢𝑡 , output irradiance;  𝜂𝑜𝑝𝑡 , 𝜂𝑑𝑖𝑟 , optical efficiency under collimated radiation;  𝐴𝑖𝑛 , 𝐴1 , inlet opening area;  𝐴𝑜𝑢𝑡 , 𝐴2 , outlet opening area;   𝑖𝑛 , flux at input;   𝑜𝑢𝑡 , flux at output;   𝑆 , angular divergence of Sun;   𝑎𝑐𝑐 , acceptance angle;  𝑛𝑖𝑛 , index of refraction at input;  𝑛𝑜𝑢𝑡 , index of refraction at output;   , polar angle; 182 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications  , azimuthal angle;  𝐿𝑑𝑖𝑟 , direct radiance;  𝐿𝑖𝑛𝑣 , inverse radiance; 𝑖𝑛  𝛷𝑑𝑖𝑟 , direct flux at input;  𝑜𝑢𝑡  𝛷𝑑𝑖𝑟 , 𝛷𝑑𝑖𝑟 , transmitted direct flux; 𝑖𝑛  𝛷𝑖𝑛𝑣 , inverse flux at input;  𝑜𝑢𝑡 , 𝛷𝑖𝑛𝑣 , transmitted inverse flux;  𝛷𝑖𝑛𝑣  𝐿𝑖𝑛𝑣 , transmitted inverse radiance;   𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , direct Lambertian transmittance;   𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 , inverse Lambertian transmittance;  ̅̅̅̅̅ 𝐿𝑜𝑢𝑡 , average direct radiance at output;  𝑑𝑖𝑟 ̅̅̅̅̅ 𝐿𝑜𝑢𝑡 𝑖𝑛𝑣 , average inverse radiance at output;   𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , direct Lambertian reflectance;   𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , direct Lambertian absorbance;   𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 , inverse Lambertian reflectance;   𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 , inverse Lambertian absorbance; 𝑛𝑒𝑡  𝛷𝑑𝑖𝑟 , direct net flux; 𝑛𝑒𝑡  𝛷𝑖𝑛𝑣 , inverse net flux;  𝛥𝛷, net flux;  𝛥𝐿, difference of radiance;  Q, transmitted heat;  H, heat flux;  𝐶𝑡ℎ , thermal conductance;  𝑅𝑡ℎ , thermal resistance;  𝛥𝑇, difference of temperature;  I, electrical current;  𝑅𝑒𝑙 , electrical resistance;  G, electrical conductance;  𝛥𝑉, difference of electrical potential; 𝑙𝑎𝑚𝑏  𝐺𝑑𝑖𝑟 , direct Lambertian conductance; 𝑙𝑎𝑚𝑏  𝐺𝑖𝑛𝑣 , inverse Lambertian conductance; 183 Advances in Optics: Reviews. Book Series, Vol. 5  𝐺 𝑙𝑎𝑚𝑏 , optical (bidirectional) Lambertian conductance;  𝐴1 , area of input (1) terminal;  𝐴2 , area of output (2) terminal; 𝑙𝑎𝑚𝑏 , direct Lambertian transmittance;   12 𝑙𝑎𝑚𝑏   21 , inverse Lambertian transmittance;  𝑙𝑎𝑚𝑏 , Lambertian transmittance; 𝑙𝑎𝑚𝑏  𝐺𝑠𝑝𝑒𝑐 , specific Lambertian conductance;  a, radius of inlet opening;  a’, radius of outlet opening;  f, focal length of CPC parabolic profile;  l, length (of CPCs, LCs or STs);  𝑅𝑆 , surface reflectance of the optical element;  𝑅𝑠𝑝ℎ , surface reflectance of the integrating sphere;   𝑀 , angular aperture of the spherical screen;  R, radius of the spherical screen;  𝛥𝛺, elemental solid angle;  𝛥, elemental flux;  𝛥𝑆, elemental area;  𝐸𝑝𝑟𝑜 , projected irradiance;  𝐿1 , radiance at optical terminal (1);  𝐿2 , radiance at optical terminal (2);  𝐸𝑑𝑖𝑟 , direct irradiance;  𝐸𝑑𝑖𝑓𝑓 , diffuse irradiance;  𝐸𝑑𝑖𝑟,⊥ , normal direct irradiance;  𝐸𝑑𝑖𝑓𝑓,⊥ , normal diffuse irradiance;  𝑑𝑖𝑟 , direct flux;  𝑑𝑖𝑓𝑓 , diffuse flux;  𝐿𝑑𝑖𝑟 , direct radiance;  𝐿𝑑𝑖𝑓𝑓 , diffuse radiance;  AM, Air Mass. 184 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 4.2. Introduction to Solar Concentrators and Their Characterization A photovoltaic solar concentrator is generally composed of a series of units assembled in such a way as to efficiently transmit the solar radiation to a certain surface. Each element of the system is composed of an optical part that collects the solar radiation, mainly that of the direct component of the Sun, through an input optical terminal, and conveys it with increased flux density on an output optical terminal, where a photovoltaic receiver, consisting of one or more solar cells transforming the concentrated solar radiation into electricity, is placed. A fundamental part of the photovoltaic concentrator (CPV) is therefore the optical unit, transmitting the solar radiation from the input terminal to the output terminal with the maximum possible efficiency. The areas of the two terminals determine the geometric concentration ratio, 𝐶𝑔𝑒𝑜 , that is the maximum achievable ratio between the average density of the outgoing flux and the density of the incoming flux. This ratio is called the optical concentration ratio, 𝐶𝑜𝑝𝑡 . We will have therefore: 𝐶𝑔𝑒𝑜 ≥ 𝐸𝑜𝑢𝑡 ⁄𝐸𝑖𝑛 = 𝐶𝑜𝑝𝑡 (4.1) 𝐶𝑜𝑝𝑡 = 𝐸𝑜𝑢𝑡 ⁄ 𝐸𝑖𝑛 = 𝜂𝑜𝑝𝑡  𝐶𝑔𝑒𝑜 = 𝜂𝑜𝑝𝑡  (𝐴𝑖𝑛 ⁄ 𝐴𝑜𝑢𝑡 ), (4.2) 𝜂𝑜𝑝𝑡 =  𝑜𝑢𝑡 ⁄  𝑖𝑛 , (4.3) 𝐶𝑜𝑝𝑡 = 𝐸𝑜𝑢𝑡 ⁄ 𝐸𝑖𝑛 = ( 𝑜𝑢𝑡 ⁄ 𝑖𝑛 )  (𝐴𝑖𝑛 ⁄ 𝐴𝑜𝑢𝑡 ) (4.4) The equal sign holds when all the radiation facing the input terminal is transferred to the output terminal, that is when the optical efficiency, 𝜂𝑜𝑝𝑡 = 1.0. We will have therefore: where 𝐴𝑖𝑛 and 𝐴𝑜𝑢𝑡 are the areas of the input and output terminals, respectively, and: with  𝑜𝑢𝑡 and  𝑖𝑛 outgoing and incoming fluxes (in W), respectively. Ultimately, we can write: The optical unit, however, must not only transfer the radiation with the maximum efficiency; it must also transfer it uniformly onto the PV receiver, because this is the condition that minimize the ohmic losses [3, 40-43], maximizing the conversion efficiency with the same incident flux. This, for example, is not a stringent condition in the case of thermodynamic solar concentrators. However, we will not dwell here on the functional conditions of the PV receiver, because they are beyond the subject of this work, all centered on the study of the optical properties. Solar concentrators essentially work with the direct component of solar radiation, that is with the radiation emitted by the solar disk, and then are operational only when the sky is clear. Direct solar radiation can be compared to a quasi-parallel light beam, with a small angular divergence,  𝑆 =  0.27°, emitted by an unpolarized Lambertian source. The solar concentrators, in turn, are characterized by a certain acceptance angle,  𝑎𝑐𝑐 , generally defined as the angle at which the solar radiation is collected with an efficiency of 50 % [44-52]. Acceptance angles  𝑎𝑐𝑐 are almost always much greater than the angular dispersion  𝑆 . It is important to consider a fundamental relationship between acceptance 185 Advances in Optics: Reviews. Book Series, Vol. 5 angle and optical concentration ratio, 𝐶 𝑜𝑝𝑡 , when these conditions apply: concentrators in air (𝑛 𝑖𝑛 = 1, 𝑛 𝑜𝑢𝑡 = 1), as will be all those tested in the present work, 𝜂𝑜𝑝𝑡 = 1 (ideal concentrator) and maximum divergence (90°) for the outgoing rays: 𝐶𝑜𝑝𝑡 = 1⁄(sin 𝑎𝑐𝑐 )2 (4.5) 𝐶𝑜𝑝𝑡 = 1⁄(sin 𝑆 )2 ≈ 46.000 (4.6) Eq. (4.5) establishes a maximum limit to the optical solar concentration, given by the angle of acceptance of the concentrator, even if the solar angular dispersion  𝑆 is very small: the optical concentration ratio 𝐶 𝑜𝑝𝑡 is independent of  𝑆 , but depends only on  𝑎𝑐𝑐 . Recurring  𝑎𝑐𝑐 values for a PV concentrator range from about 20° downwards to about 2.5°, corresponding to optical concentrations from about 10× up to about 500× [53]. People do not operate at higher concentration levels, even if, in theory, the direct component of the Sun could be concentrated up to: If one tried to overcome this limit by designing a concentrator with ratio (𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 ) > 𝐶𝑜𝑝𝑡 , it would be useless because it would automatically reduce the optical efficiency (see Eq. (4.4)). Although concentrating sunlight is advantageous for the purpose of reducing the size of the photovoltaic receiver, and therefore its cost, it is clear that the more sunlight is concentrated, the smaller will be its acceptance angle, and therefore the more precise and more expensive will have to be the pointing system, that is the solar tracker. Furthermore, the temperature of the receiver would rise with increasing concentration, as dictated by the Stefan-Boltzmann law, and then the tracking system would have to be designed increasingly sophisticated for the heat dissipation. This is what happens when the SC is irradiated by the direct solar radiation. The entry on the scene of an unpolarized Lambertian radiation, extended to an angle of 90°, occurs with the introduction of the Inverse Lambertian Irradiation Method (ILIM), already illustrated in the prologue, which is part of an optical characterization technique of the optical unit of a PV concentrator 1 . This characterization technique has been extensively illustrated in previous works [6, 15-19, 22-29, 32-37] and here I limit myself to summarizing its functioning. As we have previously seen, the optical figure of a SC is its optical efficiency curve defined with respect to a parallel light beam oriented at a certain polar angle  and azimuthal angle  with respect to the optical axis of the concentrator (for optical systems with cylindrical symmetry it is sufficient the use of only the polar angle  ). So, it comes down to finding the 𝜂𝑜𝑝𝑡 () function. Well, it has been demonstrated [6, 29], as discussed in the Prologue, that the profile of this efficiency is the same as that of the radiance emitted by the concentrator from its entrance opening when it is illuminated in the opposite direction, that is, on its exit opening, with an unpolarized and uniform Lambertian light with 90° divergence. If then the measurement of the profile 1 Some acronyms used here for the optical methods or for the optical quantities are slightly different from those used so far in that I have added “I” as “Irradiation”. Other optical quantities are defined here for the first time. 186 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications of the inverse radiance is added to that of the on-axis efficiency, the absolute curve of the optical efficiency will be completely determined. The concept underlying the ILIM method can be summarized as follows: if the SC receives light well from a certain spatial direction (, ) when operating in the direct mode, it will emit light equally well from the entrance opening towards the same direction (, ) when operating in the inverse mode. The inverse mode is applied irradiating the exit aperture with uniform and unpolarized Lambertian light, and the only required condition is the validity of the optical reversibility principle for the tested optical system [38, 39]. The ILIM is then applied simply by projecting the inverse light on a white, Lambertian screen (see Fig. 4.3), recording the intensity map and from this, by the use of a correction function [6], translating it into an optical efficiency map, centered on the optical axis, and from which it can be extracted the profile along any chosen azimuthal direction, if the concentrator is not of cylindrical symmetry. If the concentrator has cylindrical symmetry, then the intensity map will be first symmetrized around the optical axis itself before transforming it into an optical efficiency map. The ILIM method thus proves to be very simple to apply and very fast. It has been an important innovation in the field of optical characterization of SCs because the traditional method, called DLIM (Direct Lambertian Irradiation Method) [15-17, 28, 30, 31, 33, 34, 37] is much more elaborate, and its experimental apparatus more expensive: it involves the measurement of the incoming flux of a collimated beam and its outgoing flux through a series of discrete measurements at different polar  and, eventually, azimuthal angles , the more numerous the higher the angular resolution of the optical efficiency to be achieved. The latter is obtained, for each point, by the ratio between the outgoing and the incoming flux, respectively. Therefore, the DILM provides a series of discrete values, while the ILIM method produces, with a single measurement, a continuous map of values, whose resolution depends only on the resolution of the webcam, or CCD camera, used to record the intensity map of the reverse light projected on the Lambertian screen. It is useful to remember here that the ILIM method has undergone an evolution to eliminate the limitations related to the distance between the inverse source and the screen. This is referred to as the “Parretta-Herrero method” (PH-Method) [22, 27]. This is a summary of the state of the art of the optical characterization of a SC, and the use of a Lambertian source is made only for this purpose, which becomes propaedeutic to what will be explained below. At this point we open a new research scenario which consists in the use of a Lambertian radiation as a source for which to study different, new optical properties of the SC. Basically, the collimated light beam is replaced by an unpolarized Lambertian light of 90° angular divergence, and with respect to this we will study all the optical properties we have already studied for the collimated incoming beams [6, 29]. New optical quantities will be defined, the most important of which is the light transmission property summarized in the quantity: “Optical Bidirectional Lambertian Conductance” (OBLC), or simply “Optical Lambertian Conductance” (OLC), from which the corresponding law is derived. All this will be investigated not only on the innovative solar concentrators, like the nonimaging concentrators of the CPC type, and also on the simpler light cones, but will be extended to a new category of optical elements, the solar tunnels, generally used to transfer sunlight from outdoors to indoors. 187 Advances in Optics: Reviews. Book Series, Vol. 5 4.3. Theory of the Bidirectional Lambertian Irradiation Method (BLIM) When a solar concentrator (SC) is irradiated at its input opening from all the directions in space, we say that it is subjected to Lambertian irradiation with 90° angular divergence, then all the points of the opening are source of unpolarized light with constant radiance 𝐿𝑑𝑖𝑟 . Let us consider, for simplicity, only solar concentrators with cylindrical symmetry; all the optical quantities will be expressed only as a function of the polar angle . In the following, when necessary, the dependence of the optical quantities on the azimuthal 𝑖𝑛 , and angle  will be suitably specified. The flux incident on the entrance opening, 𝛷𝑑𝑖𝑟  that transmitted by the exit opening, 𝛷𝑑𝑖𝑟 , can be expressed, respectively, as follows: 𝜋 2𝜋 𝑖𝑛 𝛷𝑑𝑖𝑟 = 𝐿𝑑𝑖𝑟 ∙ 𝐴𝑖𝑛 ∙ ∫0 𝑑𝜑 ∙ ∫02 𝑑 ∙ sin ∙ cos =  ∙ 𝐴𝑖𝑛 ∙ 𝐿𝑑𝑖𝑟 , 𝜋⁄2  𝛷𝑑𝑖𝑟 = 2 𝜋 ∙ 𝐴𝑖𝑛 ∙ 𝐿𝑑𝑖𝑟 ∙ ∫0 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃), (4.7) (4.8) where 𝐴𝑖𝑛 is the SC input area and 𝑑𝑖𝑟 (𝜃) is the optical efficiency of the SC when it is irradiated by collimated light. This method of irradiating a SC is the Direct Lambertian Irradiation Method (DLIM). Similar equations give the incident and transmitted fluxes,  𝑖𝑛 , when a Lambertian source of radiance 𝐿𝑖𝑛𝑣 is applied to the outlet 𝛷𝑖𝑛𝑣 and 𝛷𝑖𝑛𝑣 opening. /2 2𝜋 𝑖𝑛 𝛷𝑖𝑛𝑣 = 𝐿𝑖𝑛𝑣 ∙ 𝐴𝑜𝑢𝑡 ∙ ∫0 𝑑𝜑 ∙ ∫0 𝜋⁄2  𝛷𝑖𝑛𝑣 = 2 𝜋 ∙ 𝐴𝑖𝑛 ∙ ∫0 𝑑 ∙ sin ∙ cos =  ∙ 𝐿𝑖𝑛𝑣 ∙ 𝐴𝑜𝑢𝑡 , 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃), (4.9) (4.10) where 𝐴𝑜𝑢𝑡 is the area of the SC exit opening and 𝐿𝑖𝑛𝑣 (𝜃) is the radiance of light inversely emitted towards direction (𝜃). This method of irradiating a SC is the Inverse Lambertian Irradiation Method (ILIM). For DLIM and ILIM, the ratio of output to input fluxes gives the “direct Lambertian transmittance”  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , and the “inverse Lambertian transmittance” 𝑙𝑎𝑚𝑏  𝑖𝑛𝑣 , respectively: 𝜋⁄2  ⁄ 𝑖𝑛  𝑙𝑎𝑚𝑏 𝛷𝑑𝑖𝑟 = 2 ∙ ∫0 = 𝛷𝑑𝑖𝑟 𝑑𝑖𝑟 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃), 𝜋⁄2  ⁄ 𝑖𝑛 = 𝛷𝑖𝑛𝑣  𝑙𝑎𝑚𝑏 𝛷𝑖𝑛𝑣 = (2 ∙ 𝐶𝑔𝑒𝑜 ⁄𝐿𝑖𝑛𝑣 ) ∙ ∫0 𝑖𝑛𝑣 (4.11) 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃), (4.12) where 𝐿𝑖𝑛𝑣 (𝜃) is the radiance of light emitted by the entrance opening of the SC. It must be considered, of course, that, although the SC is fed with Lambertian light, the transmitted one has not necessarily a constant radiance (this topic will be fully covered in Section 4.6), and then we will consider an average value of it:  𝑜𝑢𝑡 ̅̅̅̅̅ 𝐿 𝑑𝑖𝑟 = 𝛷𝑑𝑖𝑟 ⁄( ∙ 𝐴𝑜𝑢𝑡 ) = 188 𝜋⁄2 = 2 ∙ 𝐿𝑑𝑖𝑟 ∙ 𝐶𝑔𝑒𝑜 ∙ ∫0 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃), (4.13) Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 𝜋 ⁄2  𝑜𝑢𝑡 ̅̅̅̅̅ 𝐿 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃) 𝑖𝑛𝑣 = 𝛷𝑖𝑛𝑣 ⁄( ∙ 𝐴𝑖𝑛 ) = 2 ∙ ∫0 (4.14) It can be demonstrated [6, 29] that, as long as the reversibility principle applies [38, 39], the two transmittances are related as follows: 𝑙𝑎𝑚𝑏 = 𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 = 𝐶𝑔𝑒𝑜 ,  𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 ⁄ 𝑑𝑖𝑟 (4.15) where 𝐶𝑔𝑒𝑜 is the geometrical concentration ratio. Eq. (4.15) tells us that the “transparency” of the SC under Lambertian light is not symmetrical, being 𝐶𝑔𝑒𝑜 times higher when the SC is irradiated in the reverse way. There are other new optical quantities that can be defined when we irradiate the SC with Lambertian light. Two of them arise from the direct irradiation method (DLIM) and represent, one the fraction of the input flux back reflected from the inlet opening,  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , and the other the fraction of the input flux absorbed by the inner surface of the SC,  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , which are, respectively: 𝜋⁄2  𝑖𝑛 = 𝛷𝑑𝑖𝑟 ⁄𝛷𝑑𝑖𝑟  𝑙𝑎𝑚𝑏 = 2 ∙ ∫0 𝑑𝑖𝑟 𝜋⁄2  ⁄ 𝑖𝑛  𝑙𝑎𝑚𝑏 𝛷𝑑𝑖𝑟 = 2 ∙ ∫0 = 𝛷𝑑𝑖𝑟 𝑑𝑖𝑟 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃), 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃) (4.16) (4.17) We define these two quantities related to DLIM, respectively: the “direct Lambertian 𝑙𝑎𝑚𝑏 reflectance”  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 , and the “direct Lambertian absorptance”  𝑑𝑖𝑟 . In a similar way we define the two quantities that arise from the inverse Lambertian irradiation of the SC (ILIM): the “inverse Lambertian reflectance”  𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 , and the 𝑙𝑎𝑚𝑏 “inverse Lambertian absorptance”  𝑖𝑛𝑣 , given by, respectively: 𝜋⁄2  𝑖𝑛 = 𝛷𝑖𝑛𝑣 ⁄𝛷𝑖𝑛𝑣  𝑙𝑎𝑚𝑏 = 2 ∙ ∫0 𝑖𝑛𝑣 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑖𝑛𝑣 (𝜃), 𝜋⁄2  ⁄ 𝑖𝑛 𝛷𝑖𝑛𝑣 = 2 ∙ ∫0  𝑙𝑎𝑚𝑏 = 𝛷𝑖𝑛𝑣 𝑖𝑛𝑣 𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑖𝑛𝑣 (𝜃) (4.18) (4.19) The theory of the Lambertian irradiation of a SC, discussed so far, can be pushed further forward by imagining to feed the SC simultaneously on both openings with two different Lambertian sources, of radiance 𝐿𝑑𝑖𝑟 and 𝐿𝑖𝑛𝑣 , respectively (see Fig. 4.5). Putting 𝐿 = 𝐿𝑑𝑖𝑟 − 𝐿𝑖𝑛𝑣 , we can write for the net flux through the SC in the direct direction: 𝑜𝑢𝑡 𝑜𝑢𝑡 𝑖𝑛 𝑖𝑛 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏  =  𝑛𝑒𝑡 𝑑𝑖𝑟 =  𝑑𝑖𝑟 −  𝑖𝑛𝑣 =  𝑑𝑖𝑟   𝑑𝑖𝑟 −  𝑖𝑛𝑣   𝑖𝑛𝑣 = … 𝑖𝑛 𝑖𝑛 … =  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟  ( 𝑑𝑖𝑟 − 𝐶𝑔𝑒𝑜   𝑖𝑛𝑣 ) = … 𝑙𝑎𝑚𝑏 … =  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟  (  𝐿𝑑𝑖𝑟  𝐴𝑖𝑛 − 𝐶𝑔𝑒𝑜    𝐿𝑖𝑛𝑣  𝐴𝑜𝑢𝑡 ) = (  𝐴𝑖𝑛   𝑑𝑖𝑟 )  𝐿, (4.20) and, putting 𝐿 = 𝐿𝑖𝑛𝑣 − 𝐿𝑑𝑖𝑟 , for the net flux through the SC in the reverse direction: 189 Advances in Optics: Reviews. Book Series, Vol. 5 𝑜𝑢𝑡 𝑜𝑢𝑡 𝑙𝑎𝑚𝑏  =  𝑛𝑒𝑡 𝑖𝑛𝑣 =  𝑖𝑛𝑣 −  𝑑𝑖𝑟 = (  𝐴𝑜𝑢𝑡   𝑖𝑛𝑣 )  𝐿 (4.21) Fig. 4.5. It is represented schematically the bidirectional Lambertian irradiation (BLI), applied to a cylindrically symmetric solar concentrator (SC), sum of the direct Lambertian irradiation (DLI) with radiance 𝐿𝑑𝑖𝑟 and the inverse Lambertian irradiation (ILI) with radiance 𝐿𝑖𝑛𝑣 . Eqs. (4.20) and (4.21), taking account of Eq. (4.15), give the same result and are similar to the formula used for describing the heat transmission, 𝑄 ⁄𝑡 = 𝐻 = (𝐶𝑡ℎ  𝑆) 𝑇 = (𝑆⁄𝑅𝑡ℎ ) 𝑇, where Q is the heat transmitted over time t, H the heat flux (W), 𝐶𝑡ℎ the thermal conductance (𝑊 ⁄𝑚2  𝐾), S the heat-crossed area, 𝑅𝑡ℎ the thermal resistance ( 𝑚2  𝐾⁄ 𝑊 ), and 𝑇 the temperature difference, or the current transmission, 𝐼 = (1⁄𝑅𝑒𝑙 ) 𝑉 = 𝐺  𝑉 , where 𝑅𝑒𝑙 is the electrical resistance, G the electrical conductance and 𝑉 the potential difference. It is natural, therefore, to define a "Direct Lambertian Conductance" and an "Inverse Lambertian Conductance" of the SC, respectively, as follows: 𝑙𝑎𝑚𝑏 𝐺𝑑𝑖𝑟 = (  𝐴𝑖𝑛   𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 ), 𝑙𝑎𝑚𝑏 𝐺𝑖𝑛𝑣 = (  𝐴𝑜𝑢𝑡   𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 ) (4.22’) (4.22’’) Being the two conductances equal from Eq. (4.15), we can define an "Optical Bidirectional Lambertian Conductance" (OBLC), or simply an “Optical Lambertian Conductance”, OLC, which can be referred indifferently to the entrance or to the exit opening of the SC and can be put in the form: 𝐺 𝑙𝑎𝑚𝑏 = (  𝐴)  𝑙𝑎𝑚𝑏 (sr  m2), (4.23) that is: “Lambertian Conductance” = “étendue” x “Lambertian transmittance”. By comparing Eq. (4.15) with Eqs. (4.22), we observe that the asymmetry of the Lambertian transmittance is cancelled out by the symmetry of the Lambertian conductance. By virtue of the symmetry between the input and output terminals of the SC from the point of view of the Lambertian conductance, we can look at the SC as a generic optical element, a box 190 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications with an unknown internal structure, provided with a first optical terminal (1) and a second optical terminal (2) (see Fig. 4.6). Its Lambertian optical conductance, equal in both directions 1  2 and 2  1, can then be expressed as: 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝐺 𝑙𝑎𝑚𝑏 =   𝐴1   12 =   𝐴2   21 , (4.24)  𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) = 𝐺 𝑙𝑎𝑚𝑏  𝐿 (4.25) 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 and  21 are the Lambertian optical transmittances in directions 12 and where  12 21, respectively. The optical element of Fig. 4.6 is crossed by a net flux 𝛷𝑛𝑒𝑡 , in direction 1  2 if 𝐿 > 0 or in direction 2  1 if 𝐿 < 0, given by: Fig. 4.6. It is represented a generic optical element bidirectionally irradiated with radiances 𝐿1 and 𝐿2 at the optical terminals (1) and (2). The Lambertian conductance depends on the shape of the optical element, on its internal surface properties and on its dimensions. Now, however, it would be better to deal with a quantity characterizing the optical conductance properties of an element regardless of its absolute dimensions, which would allow us to make a comparison between elements of different shape and surface state more effectively. We then introduce a quantity that is independent of the size of one of the two optical terminals, for example the terminal (1), which in solar concentrators corresponds to the entrance opening of the light, and which is larger than the terminal (2), that of the exit of concentrated light. We then define the 𝑙𝑎𝑚𝑏 "Specific Optical Bidirectional Lambertian Conductance", 𝐺 𝑠𝑝𝑒𝑐 , (SOBLC), or simply the “Specific Optical Lambertian Conductance”, SOLC, which refers to the surface unit of the terminal (1) and which is equal to: 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 = (⁄𝐶𝑔𝑒𝑜 )   21 𝐺 𝑠𝑝𝑒𝑐 = 𝐺 𝑙𝑎𝑚𝑏 ⁄𝐴1 =    12 (4.26) 𝑙𝑎𝑚𝑏 𝐺 𝑠𝑝𝑒𝑐 will have the dimension of steradian (sr). If we now refer to the thermal equivalent 𝑙𝑎𝑚𝑏 of optical conduction, we can equate 𝐺 𝑠𝑝𝑒𝑐 with the thermal conductance, because this is defined specific with respect to the area crossed by the heat, while, if we refer to the electrical equivalent of optical conduction, we can equate 𝐺 𝑙𝑎𝑚𝑏 with the electrical conductance, which does not refer to unit of area. In the following, when we will have to deal with optical elements of different shapes and sizes, we will compare them starting 𝑙𝑎𝑚𝑏 from the quantity 𝐺 𝑠𝑝𝑒𝑐 . Then, to know the effective optical conductance of an element, 𝑙𝑎𝑚𝑏 we will just multiply 𝐺 𝑠𝑝𝑒𝑐 by the surface area of the terminal (1), 𝐴1 . 191 Advances in Optics: Reviews. Book Series, Vol. 5 4.4. Experimental 4.4.1. Optical Elements under Test The optical simulations were carried out on three types of optical elements: i) Nonimaging solar concentrators of the type CPC (Compound Parabolic Concentrators) [46, 50-52, 55]; ii) Light cones (LC); iii) Solar tunnels (ST). All them have a cylindrical symmetry. The Lambertian transmittance and the specific Lambertian conductance depend on their shape and surface state, whereas the Lambertian conductance depends also on their dimensions. The Three-dimensional CPCs (3D-CPCs) are well-known nonimaging reflective concentrators with parabolic profile, developed by Winston to efficiently collect Čerenkov radiation in high energy experiments [52]. Since then, the nonimaging CPCs have been widely used to concentrate sunlight [46, 50, 51]. The 3D-CPCs are characterized by a step-like optical transmission efficiency curve, allowing the efficient collection of collimated light from 0° to a maximum polar angle, called the acceptance angle 𝜃𝑎𝑐𝑐 . In this chapter we will consider only 3D-CPCs with the inlet and outlet openings in the air and with ideal shape, which means that the maximum divergence of the transmitted rays will be of 90° when the 3D-CPC is irradiated at entrance opening by a collimated beam incident at 𝜃 = 𝜃𝑎𝑐𝑐 . A 3D-CPC is characterized by the following parameters: a = radius of entrance opening; 𝑎′ = radius of exit opening; l = length; 𝜃𝑎𝑐𝑐 = acceptance (or tilt) angle; f = focal length of the parabolic profile. An ideal (canonical) 3D-CPC is completely determined by two of the above five parameters, related by the following basic relationships: 𝑓 = 𝑎′ (1 + sin 𝑎𝑐𝑐 ), 𝑎′ = 𝑎 sin 𝑎𝑐𝑐 , 𝑙 = (𝑎 + 𝑎′ ) ctg 𝑎𝑐𝑐 (4.27) (4.28) (4.29) In this work, we analyze in detail four different CPCs, with 𝑎𝑐𝑐 = 2.5°, 5°, 10° and 20°. Other CPCs are used only for comparison purpose. All the significant parameters of these CPCs are reported in Table 4.1. The reported acceptance angles refer to an ideal specular surface (𝑅𝑆 = 100 %), while for diffusive surfaces the acceptance angles are reported in Table 4.3. The optical elements reported in Table 4.1 are only models and their dimensions, chosen arbitrarily, should not necessarily be associated with real components. As it can be seen, the inlet and outlet openings are only a few millimeters in radius, but the entry window opening radius is of approximately 12 mm for all the elements, which is a measure I have used in the past for CPCs and which, for convenience, I have kept in this work. All the optical quantities do not depend on the size of the element, but on its shape, if we exclude the optical Lambertian conductance. On the other hand, the optical Lambertian conductance can be brought back to the desired real value by applying a scale factor on the area of the inlet opening. We have also added the areas of the entrance and exit openings in Table 4.1, which can be used to calculate the optical Lambertian conductance 𝐺 𝑙𝑎𝑚𝑏 when the Lambertian transmittance is known. Fig. 4.7(a) shows, as an example, as the open CPC10, with diffuse reflectance 𝑅𝑆 = 50 %, appears when 192 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications raytraced under an inverse irradiation with Lambertian light of 1k rays. The different colours of the rays are the effect of the different attenuation after impact on the absorbing surface. The red rays, the less attenuated ones, are transmitted at lower angles just because they undergo less internal reflections, while the blue ones, the more attenuated, are transmitted and reflected at high angles because they carry out more internal reflections. Table 4.1. Significant parameters of some CPCs used in the simulations. The acceptance angles are referred to specularly mirrored surface. 𝜃𝑎𝑐𝑐 (°) a (m) a’ (m) f (m) l (m) 𝐴1 (m2 ) 𝐴2 (m2 ) Cgeo CPC2.5 CPC5 CPC10 CPC20 CPC30 CPC40 2.5 12.03410−3 0.524910−3 0.547810−3 287.6510−3 454.9610−6 0.865610−6 525.58 5 12.03410−3 1.048810−3 1.140210−3 149.5410−3 454.9610−6 3.455910−6 131.65 10 12.03210−3 2.089310−3 2.452110−3 80.08610−3 454.810−6 13.71410−6 33.16 20 12.03410−3 4.115910−3 5.52310−3 44.37110−3 454.9610−6 5.32210−5 8.55 30 12.010−3 6.010−3 9.010−3 31.17710−3 452.3910−6 113.1010−6 4.0 40 12.03410−3 7.73510−3 12.70710−3 23.56010−3 454.9610−6 187.9810−6 2.42 Y X Z X (a) Z (b) Y X Z (c) Fig. 4.7. a) Inverse Raytracing of the CPC10 with Lambertian light; b) Direct raytracing of the LC3 with Lambertian light; c) Raytracing of the ST3 with Lambertian light. The surface reflectance is 𝑅𝑆 = 50 % diffuse for all the elements. The light cones (LCs) have the shape of truncated cones and can be considered as very simple solar concentrators. In this chapter, we will test four series of LCs with fixed inlet opening radius 𝑎 = 12 mm, outlet opening radius 𝑎′ = 6 mm (1st series), 𝑎′ = 5 mm (2nd series), 𝑎′ = 4 mm (3rd series) and 𝑎′ = 3 mm (4th series). The length of the LCs is variable and depends on the needs that arise during the simulations. The parameters of some LCs are reported in Table 4.2. 193 Advances in Optics: Reviews. Book Series, Vol. 5 Table 4.2. Significant parameters of some LCs used in the simulations. The acceptance angles are referred to specularly mirrored surface. Next to the name of the light cone the series in Roman numerals is indicated. 𝜃𝑎𝑐𝑐 (°) a(m) a’(m) l(m) 𝐴1 (m2 ) 𝐴2 (m2 ) Cgeo 𝜃𝑎𝑐𝑐 (°) a(m) a’(m) l(m) 𝐴1 (m2 ) 𝐴2 (m2 ) Cgeo 𝜃𝑎𝑐𝑐 (°) a(m) a’(m) l(m) 𝐴1 (m2 ) 𝐴2 (m2 ) Cgeo LC1(I) 27.2 LC2(I) 29.3 LC3(I) 29.7 LC4(I) 30.0 LC5(II) 27.8 LC6(II) 23.0 12.010−3 6.010−3 12.010−3 4.523910−4 1.13110−4 4.0 LC7(II) 24.2 12.010−3 6.010−3 24.010−3 4.523910−4 1.13110−4 4.0 LC8(II) 24.6 12.010−3 6.010−3 48.010−3 4.523910−4 1.13110−4 4.0 LC9(III) 33.8 12.010−3 6.010−3 96.010−3 4.523910−4 1.13110−4 4.0 LC10(III) 16.7 12.010−3 5.010−3 12.010−3 4.523910−4 7.85410−5 5.76 LC11(III) 18.8 12.010−3 5.010−3 24.010−3 4.523910−4 7.85410−5 5.76 LC12(III) 19.2 12.010−3 5.010−3 48.010−3 4.523910−4 7.85410−5 5.76 LC13(IV) 39.4 12.010−3 5.010−3 144.010−3 4.523910−4 7.85410−5 5.76 LC14(IV) 20.0 12.010−3 4.010−3 12.010−3 4.523910−4 5.02610−5 9.0 LC15(IV) 13.0 12.010−3 4.010−3 24.010−3 4.523910−4 5.02610−5 9.0 LC16(IV) 14.0 12.010−3 4.010−3 48.010−3 4.523910−4 5.02610−5 9.0 12.010−3 4.010−3 96.010−3 4.523910−4 5.02610−5 9.0 12.010−3 3.010−3 12.010−3 4.523910−4 2.82710−5 16.0 12.010−3 3.010−3 24.010−3 4.523910−4 2.82710−5 16.0 12.010−3 3.010−3 48.010−3 4.523910−4 2.82710−5 16.0 12.010−3 3.010−3 96.010−3 4.523910−4 2.82710−5 16.0 Fig. 4.7(b) shows as it appears the open LC3, with diffuse reflectance 𝑅𝑆 = 50 %, when raytraced under a direct irradiation with Lambertian light of 1k rays. The same considerations made previously apply; moreover, it is possible to note the scarcity of the transmitted rays and the abundance of reflected rays when the irradiation is of direct type, due to the narrowing of the section in the direct direction. The solar tunnels (STs) are simple cylinders with opening radius 𝑟1 = 12 mm and variable length l. We have simulated the following STs: ST1 (l = 12 mm); ST2 (l = 24 mm); ST3 (l = 48 mm); ST4 (l = 96 mm). All the STs have: 𝐴1 = 𝐴2 = 4.524  10−4 m2 and 𝐶𝑔𝑒𝑜 = 1.0 . Fig. 4.7(c) shows as it appears the solar tunnel ST3, with diffuse reflectance 𝑅𝑆 = 50 %, when raytraced under Lambertian light of 1 k rays. The appearance of rays is similar to that of the LC3. 4.4.2. Simulation Methods In each simulation, a certain number of rays was chosen to obtain optical maps with a sufficiently high S/N ratio. Each ray was assigned arbitrarily the power of 1 mW. The optical simulations, in fact, have the sole purpose of demonstrating the validity of the 194 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications hypotheses put forward in this paper and, as regards references to real situations from the point of view of dimensions and flux densities, we will expressly refer to Section 4.7. The optical simulations were made by using a commercial optical code [56]. For simulating the Lambertian transmittance, 𝑙𝑎𝑚𝑏 , one of the schemes of Fig. 4.8 can be 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 , and that in Fig. 4.8 (b) for measuring  21 . used; that in Fig. 4.8 (a) for measuring  12 The two quantities are related by Eq. (4.15), and then it is sufficient to measure just one of them, but measuring both allows to confirm the exposed theory. The chosen optical element, a light cone (LC1), is closed on both openings by two ideal absorbers, (a 1) and (a2). In Fig. 4.8(a), a Lambertian source, of flux 1,𝑖𝑛 , is imposed on the opening (w1), of area 𝐴1 . A part of the flux, 1 is back reflected and measured by (a1), a part, 𝑆 , is absorbed by the internal surface, and a part, 2, is transmitted to opening (w2), of area 𝐴2 . We have then for the incident radiance at opening (w1), 𝐿1 , for the direct lambertian 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 transmittance,  12 , and for the direct lambertian conductance, 𝐺 12 , respectively: 𝐿1 = 1,𝑖𝑛 /(  𝐴1 ), 𝑙𝑎𝑚𝑏 = 2 /1,𝑖𝑛 ,  12 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝐺 12 = (  𝐴1 )  12 (4.30) (4.31) (4.32) 𝑙𝑎𝑚𝑏 The same scheme is adopted to calculate  21 (see Fig. 4.8(b)), with the difference that now the Lambertian source, of flux 2,𝑖𝑛 and radiance 𝐿2 , is placed on the opening (w2). Fig. 4.8. Schematic representation of the DLIM for measuring  𝑙𝑎𝑚𝑏 (a) and of the ILIM 12 (b). for measuring  𝑙𝑎𝑚𝑏 21 We then have, in an equivalent manner to the above, the quantities: 𝐿2 = 2,𝑖𝑛 /(  𝐴2 ), 𝑙𝑎𝑚𝑏  21 = 1 /2,𝑖𝑛 , 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝐺 21 = (  𝐴2 )  21 (4.33) (4.34) (4.35) As an example, we report below the calculation of the direct and inverse Lambertian 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 transmittances,  12 and  21 of the light cone (LC1), of l = 12 mm length, which will be used in the next section to demonstrate the validity of the OLC law (Eq. (4.25)). In 195 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 4.9 are shown the raytracings of DLIM simulations. By applying Eq. (4.31), we obtain 𝑙𝑎𝑚𝑏 different values of  12 , which tend, at increasing the number of rays in the raytracing 𝑙𝑎𝑚𝑏 = 1.0 (no rays absorbed process, clearly to the value of 0.25 (see Fig. 4.10). So, as  21 due to 𝑅𝑆 = 100 %, no rays reflected due to the shape of LC and to the specular surface), 𝑙𝑎𝑚𝑏 and  12 = 0.25, we can confirm the validity of Eq. (4.15): 𝑙𝑎𝑚𝑏 = 1⁄0.25 = 4.0 = 𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 = 𝐶𝑔𝑒𝑜 𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 ⁄𝑑𝑖𝑟 (4.15) Fig. 4.9. Example of raytracing of the light cone (LC1) irradiated on the opening (w 1) by a Lambertian source, following the DLIM for measuring  𝑙𝑎𝑚𝑏 12 ; (a) 𝑅𝑆 = 100 % specular; (b) 𝑅𝑆 = 80 % specular. 0,254 0,254 0,252 0,252 0,250 0,250 0,248 0,248 12lamb 0,246 0,246 Light cone LC1 DLIM 0,244 0,244 0,242 0,242 0,240 0,240 0,238 0,238 0,236 0,236 0,234 0,234 0,232 0,232 0,0 500,0k 1,0M 1,5M 2,0M 2,5M 3,0M 3,5M 4,0M Number of rays, N on the number of rays used in the simulation process. It is Fig. 4.10. Dependence of  𝑙𝑎𝑚𝑏 12 interesting to note the peculiar trend of the result, with an overcoming of the equilibrium value and a subsequent exponential decay. A good accuracy is achieved already with 200 k rays. 4.4.3. Measurement of the Inverse Radiance For the measurement, through optical simulation, of the radiance coming out from one of the openings of the optical element, when the opposite opening is irradiated by Lambertian light, I have adopted the configuration shown in Fig. 4.11. In this figure, it is shown the example of the irradiation of a solar concentrator SC in the inverse way. The light 196 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications projected by SC is intercepted by an absorbing spherical screen (ss), of radius R, centered on the center O of the opening aperture of the SC. The angular aperture of the spherical screen,  𝑀 , is adjusted according to the light emission characteristics. For specular surfaces a limited portion of the screen is sufficient, while for diffusive surfaces it is necessary to set up a maximum aperture (90°). The radius R of the spherical screen (ss) has been chosen one hundred times greater than that of the diameter of the exit aperture of the rays, in order to have a high angular resolution on the measured radiance [6]. The irradiance produced on the screen in the  direction is 𝐸( ). We then consider the surface element 𝛥𝑆 on the screen in the  direction. It forms, with respect to the center O, the solid angle 𝛥𝛺. The flux emitted by SC within this angle 𝛥𝛺 is given by 𝛥. The radiance emitted by SC in direction  is then given by: 𝐿𝑜𝑢𝑡 () = 𝛥 ⁄ (𝐴1  𝑐𝑜𝑠  𝛥𝛺), (4.36) 𝐿𝑜𝑢𝑡 () = 𝐸( ) 𝛥𝑆 ⁄ (𝐴1  𝑐𝑜𝑠  (𝛥𝑆⁄𝑅 2 )) = 𝐸( ) 𝑅 2 ⁄𝐴1  𝑐𝑜𝑠 (4.37) where 𝐴1 is the area of the entrance opening of the SC. We have then: Fig. 4.11. Scheme describing the emission of light, of radiance 𝐿𝑜𝑢𝑡 (), from the entrance opening of the SC, irradiated in inverse way by a Lambertian source of radiance 𝐿𝑖𝑛𝑣 , collected by a spherical screen (ss). The radiance 𝐿𝑜𝑢𝑡 () can then be derived from the measurement of 𝐸(). The optical simulation code, however, does not show the function 𝐸(), but its projection on the x/y plane, equal to: 𝐸𝑝𝑟𝑜 () = 𝐸() / cos. Then, the program shows a circular map of radius 𝑅 sin 𝑀 of only the portion of 𝐸() bounded by the angle  𝑀 ,. From Eq. (4.37) we obtain for the emitted radiance: 𝐿𝑜𝑢𝑡 () = 𝐸𝑝𝑟𝑜 ( )  𝑐𝑜𝑠  𝑅 2 ⁄𝐴1  𝑐𝑜𝑠 = 𝐸𝑝𝑟𝑜 ( )  𝑅 2 ⁄𝐴1 (4.38) Being 𝐸𝑝𝑟𝑜 () proportional to 𝐿𝑜𝑢𝑡 (), its map gives the map of the emitted radiance up to the constant 𝑅 2⁄𝐴1 . Since all the elements considered have a cylindrical symmetry, the 𝐿𝑜𝑢𝑡 () can be symmetrized with respect to the optical axis to improve the signal-to-noise ratio. 197 Advances in Optics: Reviews. Book Series, Vol. 5 From the symmetrized map, any radial profile is then extracted which, normalized at  = 0°, provides the normalized 𝐿𝑛𝑜𝑟𝑚 (), which will be reported in detail for the various optical elements in Section 4.6. Fig. 4.12 shows an example of raytracing of the light cone LC2 for the measurement of the emitted radiance under reverse Lambertian irradiation. (a) Specular surface 𝑅𝑆 = 100 %; (b) diffusive surface 𝑅𝑆 = 100 %. The beneficial effect of rotational symmetry is shown in Fig. 4.13. Fig. 4.12. Example of raytracing of the LC2 light cone for measuring the exit radiance under inverse Lambertian irradiation. (a) Specular surface; (b) diffused surface. (a) (b) Fig. 4.13. Map of the irradiance on the spherical screen of LC2 after projection on the x/y plane: (a) without rotational symmetry; (b) with rotational symmetry. 4.4.4. Test of the Optical Bidirectional Lambertian Conductance (OBLC) Law To apply the theory of BLIM, the configurations illustrated in Fig. 4.14 were adopted. In Fig. 4.14(a) the optical element (LC), with specular surface reflectance 𝑅𝑆 = 100 %, is closed at its inlet opening (w1), of area 𝐴1 , with an ideal absorber (a1), while its outlet 198 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications opening (w2), of area 𝐴2 , is coupled to an integrating sphere (is), with 𝑅𝑠𝑝ℎ = 99 % diffuse surface reflectance, provided with another ideal absorber (a4) on the opening (w4), of area 𝐴4 . The ideal absorbers (a1) and (a4) measure, respectively, the inverse flux 1 on the inlet opening (w1) of (LC), and the flux 4 = 𝐸4  𝐴4 , where 𝐸4 is the uniform irradiance incident on the internal surface of the sphere. At the center of the integrating sphere, an ideal diffusive cone was placed in order to diffuse the light coming out of the (LC), which is not Lambertian but rather directional (see Section 4.6), and which would disturb the measurement of 𝐸4 . On window (w1), a Lambertian light source, of radiance 𝐿1 = 1,𝑖𝑛 /(  𝐴1 ), is applied, whose flux 1,𝑖𝑛 is partly reflected by (LC) and then absorbed by (a1) as 1 , and partly transmitted by (LC) to the integrating sphere as 𝑠𝑝ℎ ; this last flux being uniformly distributed on the sphere surface with irradiance 𝐸4 . Being 𝑅𝑆 = 100%, no flux is absorbed by the (LC). On the opening (w2) the radiance 𝐿2 = 𝐸2 / = 𝐸4 / is produced. Since both the Lambertian conductance, after the measurements described in Fig. 4.8(a), and the radiance difference 𝐿 = 𝐿1 − 𝐿2 , are known, the net flux 𝛷 𝑛𝑒𝑡 crossing the (LC) can be obtained applying Eq. (4.32). Fig. 4.14. Schematic representation of the bidirectional Lambertian irradiation method (BLIM) applied to a light cone (LC). (a) A Lambertian source is applied to the inlet opening of the optical element (LC) and its outlet opening is coupled to an integrating sphere; (b) A Lambertian source is applied to the outlet opening of the LC and its inlet opening is coupled to an integrating sphere. Fig. 4.14(b) shows the configuration used when applying an inverse light source on the (LC) outlet opening (w2), of flux 2,𝑖𝑛 and radiance 𝐿2 = 2,𝑖𝑛 /(  𝐴2 ). In absence of absorbed flux, and thanks to the particular shape of the truncated cone and to the specular surface, we have no reflected flux from the (LC) in the direction 1  2, and then the total flux 2,𝑖𝑛 is transmitted to (w1) and to the (is), where it is uniformly distributed on its surface with irradiance 𝐸3 . On the opening (w1) the radiance 𝐿1 = 𝐸1 / = 𝐸3 / is produced. Since both the Lambertian conductance, after the measurements described in Fig. 4.8 (b), and the radiance difference 𝐿 = 𝐿1 − 𝐿2 , are known, the net flux 𝛷𝑛𝑒𝑡 crossing the (LC) in the 2  1 direction can be obtained applying Eq. (4.35). An example of 𝛷𝑛𝑒𝑡 calculation is reported below for the light cone (LC1) of l = 12mm length, irradiated in direct mode (DLIM) (see Fig. 4.14(a)). The radii of window (w4) and of the sphere are 10 mm and 200 mm, respectively. The results of a raytracing are: 1,𝑖𝑛 = 50 W; 1 = 37.776 W; 4 = 0.6784 W; flux on the sphere surface, 𝑠𝑝ℎ = 10.893 W. We obtain: 𝐿1 = 3.5181104 W/m2sr; 𝐿2 = 687.32 W/m2sr; 𝐿 = 3.4494  104 W/m2sr. 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 Since all inverse rays are transmitted, we have:  21 = 1,  12 =1/𝐶𝑔𝑒𝑜 = 0.25, and 199 Advances in Optics: Reviews. Book Series, Vol. 5 therefore: 𝐺 𝑙𝑎𝑚𝑏 =   𝐴2 =   𝐴1  0.25 = 3.5530510-4 m2sr. From Eq. (4.25) we obtain for the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 12.256 W. To check the accuracy of this value, it is sufficient to make the sum of the direct flux (taken positive) and inverse flux (taken negative) crossing the opening (w1). In fact, as the (LC1) is not absorbing, the net flux must necessarily be: 𝛷𝑛𝑒𝑡 = 1,𝑖𝑛 − 1 = 12.224 W, which results in good agreement with the 12.256 W value obtained applying Eq. (4.25) (difference of only 0.3 %). The resulting difference is due to the limited number of rays (50 k) supported by the optical code and to numerical approximations. There is no way to assign an error to these measurements, other than to increase the number of rays in raytracing, but it happens that this number has an intrinsic limit in the optical code (see the result of Fig. 4.5). Another example of 𝛷𝑛𝑒𝑡 calculation is reported here for the same light cone (LC1) as above, irradiated in reverse mode (ILIM) (see Fig. 4.14(b)). The results of a raytracing are: 2,𝑖𝑛 = 40 W; 2 = 0.80689 W; 3 = 2.074 W; 𝑠𝑝ℎ = 35.024 W. To calculate 𝐿1 , we use the flux 𝑠𝑝ℎ on the sphere surface free from the windows (w1) and (w3), and of area 𝐴𝑠𝑝ℎ . We have therefore: 1 = (𝑠𝑝ℎ 100)(𝐴1 ⁄𝐴𝑠𝑝ℎ ). We obtain: 1 = 3.157 W, 𝐿1 = 1⁄( 𝐴1 ) = 2.2213103 W/m2sr; 𝐿2 = 1.1258105 W/m2sr; 𝐿 = ‒ 1.1036  104 W/m2sr. We have, as previously: 𝐺 𝑙𝑎𝑚𝑏 = 3.5530510-4 m2sr. From Eq. (4.25) we obtain for the net flux crossing the (LC1) in direction 2  1: 𝛷𝑛𝑒𝑡 = 39.211 W. To check the accuracy of this value, it is sufficient to make the relative sum of the direct and inverse fluxes crossing the opening (w2). As the (LC1) is not absorbing, the net flux must necessarily be: 𝛷𝑛𝑒𝑡 = 2,𝑖𝑛 − 2 = 39.193 W, in good agreement with the 39.211 W value (difference of only 0.05%). Like above, the small difference is due to the limited number of rays (40k) supported by the optical code and to numerical approximations. It is possible to adopt a different configuration to apply the BLIM method, that is to optically couple the two openings of the (LC) with two integrating spheres, as shown in Fig. 4.15. This method, however, does not allow to verify the accuracy of the 𝛷𝑛𝑒𝑡 value calculated by applying Eq. (4.25), as it is not possible to place an ideal absorber on one of the openings of the (LC). To apply a Lambertian source on one of the openings of the (LC), one integrating sphere must be illuminated by introducing a collimated light source (ls) which impacts on a diffuser placed at the center of the sphere. In Fig. 4.15 the (ls) is introduced in the left sphere, in this way replicating the configuration of Fig.4.14 (a). The results of a raytracing are: 𝑅𝑠𝑝ℎ1 = 𝑅𝑠𝑝ℎ2 = 99 % diffuse; 1,𝑖𝑛 = 30 W; 3 = 1.6757 W; 4 = 0.027329 W; flux on the sphere (is1) surface, 𝑠𝑝ℎ1 = 26.150 W; flux on the sphere (is2) surface, 𝑠𝑝ℎ2 = 0.50121 W. By calculating the radiance 𝐿1 from 𝐸3 = 3 /𝐴3 and the radiance 𝐿2 from 𝐸4 = 4 /𝐴4 , we obtain: 𝐿1 =1.697835103 W/m2sr and 𝐿2 = 27.69 W/m2sr; 𝐿 = 1.670145103 W/m2sr. 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 = 0.25, and then: 𝐺 12 =   𝐴1  0.25 = 3.5530510-4 m2sr. We have again:  12 𝑙𝑎𝑚𝑏 From Eq. (4.25) we obtain for the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 𝐺 12  𝐿 = 0.59341 W. If, as an alternative, we calculate 𝐿1 starting from the irradiance 𝐸𝑠𝑝ℎ1 = 𝑠𝑝ℎ1 /𝐴𝑠𝑝ℎ1 on the (is1) surface, where 𝐴𝑠𝑝ℎ1 is the area of the net (is1) surface, and 𝐿2 starting from the irradiance 𝐸𝑠𝑝ℎ2 = 𝑠𝑝ℎ2 /𝐴𝑠𝑝ℎ2 on the (is2) surface, where 𝐴𝑠𝑝ℎ2 is the area of the net (is2) surface, we would obtain: 𝐿1 = 1.65849103 W/m2sr; 𝐿2 = 𝑙𝑎𝑚𝑏  𝐿 = 31.7665 W/m2sr; 𝐿 = 1.670145  103 W/m2sr, and finally: 𝛷𝑛𝑒𝑡 = 𝐺 12 200 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 0.57798 W. The discrepancy between 𝛷𝑛𝑒𝑡 = 0.59341 W and 0.57798 W (~3%) is to be attributed to the raytracing process, whose result is closely related to the number of rays used, which cannot be pushed beyond a certain value, the processing limit of the optical code. Fig. 4.16 shows an example of raytracing with few rays of the BLIM method applied with two integrating spheres and illuminating the left sphere. Fig. 4.15. Schematic representation of the bidirectional Lambertian irradiation method (BLIM) applied to a light cone (LC). The left sphere is illuminated by the collimated light source (ls). Fig. 4.16. Example of raytracing of the BLIM method applied by using two integrating spheres. The higher density of rays can be seen on the left sphere, where the light source (s l) is located. It is interesting to see what happens when the same light source (ls), with the same intensity (30 W), is placed in the right sphere (is2), that is if the net flux through (LC) changes or not (see Fig. 4.17). The results of a raytracing are: 3 = 0.031466 W; 4 = 1.6704 W; 𝑠𝑝ℎ1 = 0.45363 W; 𝑠𝑝ℎ2 = 26.199 W. The flux distribution is now almost mirrored. By calculating the radiance 𝐿1 from 𝐸3 = 3 /𝐴3 and the radiance 𝐿2 from 𝐸4 = 4 /𝐴4 , we obtain: 𝐿1 = 31.88175 W/m2sr and 𝐿2 = 1.69247103 W/m2sr; 𝐿 = 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝐿1 − 𝐿2 = ‒1.66059103 W/m2sr. We have again: 𝐺 21 = 𝐺 12 = 3.55305 10-4 m2sr. 𝑙𝑎𝑚𝑏 From Eq. (4.25) we obtain for the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 𝐺 12  𝐿 = ‒0.59002 W. The minus sign indicates that the flux is flowing in the 2  1 direction. If we calculate 𝐿1 starting from the irradiance 𝐸𝑠𝑝ℎ1 = 𝑠𝑝ℎ1 /𝐴𝑠𝑝ℎ1 on the (is1) surface, and 𝐿2 starting from the irradiance 𝐸𝑠𝑝ℎ2 = 𝑠𝑝ℎ2 /𝐴𝑠𝑝ℎ2 on the (is2) surface, we would obtain: 𝐿1 =28.7702 W/m2sr; 𝐿2 = 1.66048103 W/m2sr; 𝐿 = ‒1.631712  103 W/m2sr, 𝑙𝑎𝑚𝑏 and finally: 𝛷𝑛𝑒𝑡 = 𝐺 21  𝐿 = ‒ 0.57975 W. The difference between 201 Advances in Optics: Reviews. Book Series, Vol. 5 𝛷𝑛𝑒𝑡 = ‒0.59002 W and ‒0.57975 W is of ~2%. We obtain finally that, moving from the configuration of Fig. 4.15 to that of Fig. 4.17, that is reversing the orientation of the (LC), and applying at input the same flux (30W), the net flux flowing through the (LC) in the two directions is practically the same. Fig. 4.17. Schematic representation of the bidirectional Lambertian irradiation method (BLIM) applied to a light cone (LC). The right sphere is illuminated by the collimated light source (ls). 4.5. Optical Conductance Simulation Results 4.5.1. Nonimaging Solar Concentrators (CPC) The optical simulations of this section were made for specular or Lambertian diffusive internal surface, with total reflectivity for monochromatic light ranging from 10 % to 𝑙𝑎𝑚𝑏 , of a CPC with 𝑎𝑐𝑐 = 10° (CPC10), 100 %. The direct Lambertian transmittance,  12 as function of reflectance of a specular surface, simulated following the DLIM described in Fig. 4.8 (a), is reported, in percent, in Fig. 4.18. It is relatively low, but increases monotonically with increasing surface reflectance 𝑅𝑆 , as expected, On the same graph, it 𝑙𝑎𝑚𝑏 /𝐶𝑔𝑒𝑜 , simulated following the is reported the inverse Lambertian transmittance  21 𝑙𝑎𝑚𝑏 , demonstrating the validity ILIM described in Fig. 4.8 (b), which perfectly overlaps  12 of Eq. (4.15). For a specular surface with 𝑅𝑆 = 100 %, all rays incident within 𝑎𝑐𝑐 = 10° 𝑙𝑎𝑚𝑏 cross the CPC, giving the maximum value of  12 : sin2𝑎𝑐𝑐 = 3.0154 10-2, as obtained from Eq. (4.11) integrating the angle  from 0° to 10°. Such a low optical transmittance value is not surprising, as the CPC10 only transmits light incident within 10° of the optical axis, very little compared to light at input having 90° divergence. The corresponding 𝑙𝑎𝑚𝑏 , being just (  𝐴1 ) = 1.4294 10-3 times OBLC, 𝐺 𝑙𝑎𝑚𝑏 , has the same behavior of  12 𝑙𝑎𝑚𝑏  12 (see Eq. (4.22’)). Although solar concentrators normally work with specular surface, it can be interesting to divert our attention from their application and consider the effect that also a diffusive surface can have on its transmittance properties. In Fig. 4.18 it is shown that a diffusive surface strongly hinders the transmission of Lambertian light up to the highest surface reflectance values and that, also with 𝑅𝑆 = 100 %, the diffusive surface exhibits a Lambertian transmittance lower than that of a specular surface. The simulated direct 202 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 Lambertian reflectances  12 of the CPC10 with specular and and absorbances  12 diffusive surface, following DLIM, are shown in Fig. 4.19. We note that the absorbances dominate the reflectances at low-medium values of 𝑅𝑆 . Furthermore, the Lambertian reflectance and absorbance curves show an opposite behavior, which is natural considering the low level of Lambertian transmittance. The Lambertian reflectance and absorbance referred to specular surface remain flat up to high values of 𝑅𝑆 and then rapidly converge to ~ 1 and to ~ 0, respectively. Those referring to diffusive surface, on the contrary, converge to ~ 1 and to ~ 0, respectively, in a softer way. Similar trends for 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 the quantities  12 ,  12 have been observed for the CPC2.5, CPC5 and and  12 CPC20. In the following we will report separately the single optical quantities to see the effect that the shape of the different CPCs has on them. CPC10, DLIM and ILIM tau_dir_spec tau_dir_diff tau_inv_spec(/Cgeo) tau_inv_diff(/Cgeo) 3,0 12lamb, 21lamb/Cgeo (%) 2,5 2,0 spec 1,5 1,0 diff 0,5 0,0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.18. Direct and inverse (/Cgeo) Lambertian transmittance, in percent, of the CPC10 vs. the reflectance, in percent, of specular and diffusive surface. 100 12lamb (%), 12lamb (%) 80 CPC10, DLIM rho_dir_spec alpha_dir_spec rho_dir_diff alpha_dir_diff 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.19. Direct Lambertian reflectance and absorbance, in percent, of the CPC10 vs. the reflectance, in percent, of specular and diffusive surface. Method: DLIM. 203 Advances in Optics: Reviews. Book Series, Vol. 5 𝑙𝑎𝑚𝑏 We start with the most relevant quantity, the specific Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐 , 𝑙𝑎𝑚𝑏 whose knowledge allows us to derive the Lambertian conductance 𝐺 of an element of the same shape and of any size, simply multiplying it by the area of the entrance 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 simply dividing it by . The 𝐺 𝑠𝑝𝑒𝑐 calculated applying the opening, or to derive  12 DLIM method are reported in Fig. 4.20 for four tested CPCs. We can note, apart the 𝑙𝑎𝑚𝑏 increasing of 𝐺 𝑠𝑝𝑒𝑐 with 𝑅𝑆 already anticipated by Fig. 4.18, also its evident increase moving from low-𝑎𝑐𝑐 to high-𝑎𝑐𝑐 CPCs, a natural trend considering that the increase of 𝑎𝑐𝑐 means greater dimensions of the exit opening with respect to the entrance opening, 𝑙𝑎𝑚𝑏 and then higher optical transparency. We note further the fact that the increasing of 𝐺 𝑠𝑝𝑒𝑐 with 𝑅𝑆 following a near-linear trend tends to diverge the curves. But there is another 𝑙𝑎𝑚𝑏 interesting data to be analyzed in the 𝐺 𝑠𝑝𝑒𝑐 curves, namely the value reached by the curves for 𝑅𝑆 = 100 %. At this condition, considering that the surface is specular, the 𝑙𝑎𝑚𝑏 must necessarily be equal to 1. In fact, due to the inverse Lambertian transmittance  21 shape of any CPC, any ray direct from the exit to the entrance opening crosses the CPC 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 = 𝜋⁄𝐶𝑔𝑒𝑜 . These undisturbed. So, as  21 = 1, we have 𝐺 𝑙𝑎𝑚𝑏 =   𝐴2 and 𝐺 𝑠𝑝𝑒𝑐 are, effectively, the values that we find in Fig. 4.20 taking into account the 𝐶𝑔𝑒𝑜 values reported in Table 4.1. 0,40 CPC20 CPC10 CPC5 CPC2.5 Specular surface 0,35 0,30 Gspeclamb (sr) 0,25 0,20 0,15 0,10 0,05 0,00 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.20. The specific bidirectional Lambertian conductance of four tested CPCs as function of reflectance of a specular surface. 𝑙𝑎𝑚𝑏 Fig. 4.21 reports the direct Lambertian reflectance  12 of the four CPCs. It is interesting to note that, at high 𝑅𝑆 values, the reflectance increases as 𝑎𝑐𝑐 increases, that is as the exit opening increases, which seems strange at a first glance. In reality, an increase in 𝑎𝑐𝑐 involves a reduction in length of the CPC and then a lower optical path of light, a lower number of internal reflections, a lower surface absorption and, ultimately, a higher reflectance, being reflectance and absorbance almost complementary (see Fig. 4.22) due to the relatively low values of transmittance. 204 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 100 CPC20 CPC10 CPC5 CPC2.5 Specular surface 12lamb (%) 80 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.21. The direct Lambertian reflectance of the four tested CPCs, as function of reflectance of a specular surface. 100 80 CPC2.5 CPC5 CPC10 CPC20 Specular surface 12lamb (%) 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.22. The direct Lambertian absorbance of the four tested CPCs, as function of reflectance of a specular surface. Although the use of diffusive surfaces is not contemplated in traditional solar concentrators, be they thermal, thermodynamic or photovoltaic, it may still be interesting to see if the use of diffusive surface can give rise to unexpected results that can eventually be exploited for other types of application. Indeed, an interesting result arises from the comparison of Lambertian reflectance and of Lambertian absorbance in the various tested CPCs. The two quantities are reported in Figs. 4.23 and 4.24, respectively. The unexpected and surprising result is the invariance, respect to 𝑎𝑐𝑐 , of the curves of Lambertian reflectance and of Lambertian absorbance. This property seems therefore intrinsic to the particular shape of these nonimaging solar concentrators. 205 Advances in Optics: Reviews. Book Series, Vol. 5 100 CPC20 CPC10 CPC5 CPC2.5 Diffusive surface 12lamb (%) 80 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.23. The direct Lambertian reflectance of the four tested CPCs, as function of reflectance of a diffusive surface. 100 80 12lamb (%) 60 CPC2.5 CPC5 CPC10 CPC20 Diffusive surface 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.24. The direct Lambertian absorbance of the five tested CPCs, as function of reflectance of a diffusive surface. 4.5.2. Light Cones (LC) In this section we report only the results obtained by simulating the first series of LCs (see Table 4.2). The simulated direct Lambertian transmittance, reflectance and absorbance of a light cone (LC2) of 24 mm length, for specular and diffuse surface, are shown in Fig. 4.25. Given the geometric similarity between a CPC and a truncated cone (both have an inlet opening greater than the outlet opening and the cross section of the element is reduced monotonously going from the inlet to the outlet opening), the optical quantities shown in Fig. 4.25 for the light cone LC2 resemble those shown in Figs. 4.18 and 4.19 for the CPC10. That is, we find a Lambertian transmittance that grows with 𝑅𝑆 for both a specular and a diffusive surface, where the second remains lower than the first; then we find a Lambertian reflectance that grows with 𝑅𝑆 for both a specular and a diffusive surface, where however the second remains higher than the first one, that is, there is a situation opposite to that observed for transmittance; finally we find a Lambertian 206 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications absorbance that slowly decreases with the increase of 𝑅𝑆 both with a specular and a diffusive surface, where the second remains lower than the first one, as it happens in the case of transmittance. We again find, as we found for the CPC10, that Lambertian reflectance shows qualitatively an opposite behavior respect to that of Lambertian absorbance. We find a similar behavior in the other light cones, LC1, LC3 and LC4, and then we do not report here graphs similar to that of Fig. 4.25. However, as made before describing the CPCs, we find it useful to compare the same optical quantity in the four 𝑙𝑎𝑚𝑏 different light cones. The specific bidirectional Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐 , calculated applying the DLIM method, of the light cones LC1, LC2, LC3 and LC4 vs. the 𝑙𝑎𝑚𝑏 𝑅𝑆 of a specular surface, is reported in Fig. 4.26. We note, apart the increasing of 𝐺 𝑠𝑝𝑒𝑐 with 𝑅𝑆 already seen for the CPCs in Fig. 4.20, also its evident increase at decreasing the LC length. 12lamb, 12lamb, 12lamb, (%) 100 80 LC2_DLIM tau_spec rho_spec alpha_spec tau_diff rho_diff alpha_diff 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.25. Lambertian transmittance, reflectance and absorbance of the LC2, of 24 mm length, are shown as function of specular and diffuse surface reflectance. 0,9 LC1 LC2 LC3 LC4 Specular surface 0,8 0,7 Gspeclamb (sr) 0,6 0,5 0,4 0,3 0,2 0,1 0,0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.26. The specific bidirectional Lambertian conductance of the four tested light cones is reported as function of reflectance of the specular surface. The low index "spec" in the symbol 𝑙𝑎𝑚𝑏 means "specific" and not "specular". 𝐺 𝑠𝑝𝑒𝑐 207 Advances in Optics: Reviews. Book Series, Vol. 5 𝑙𝑎𝑚𝑏 But the most interesting aspect to note in the 𝐺 𝑠𝑝𝑒𝑐 curves is their convergence to the same maximum value in correspondence to 𝑅𝑆 = 100 %, a value around 0.8 sr. The 𝑙𝑎𝑚𝑏 explanation of this result is the same used to comment the 𝐺 𝑠𝑝𝑒𝑐 's results of the CPCs, namely that, considering that the surface is specular and that the inverse Lambertian 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 must necessarily be equal to 1, we finally obtain 𝐺 𝑠𝑝𝑒𝑐 = 𝜋⁄𝐶𝑔𝑒𝑜 . transmittance  21 But now, differently from the CPCs, 𝐶𝑔𝑒𝑜 is the same for all the light cones, and then the 𝑙𝑎𝑚𝑏 𝐺 𝑠𝑝𝑒𝑐 at 𝑅𝑆 = 100 % must be equal to 𝜋⁄4 ≃ 0.7854. We can affirm, therefore, that the optical "transparency" to Lambertian light of a light cone with a surface equal to an ideal mirror (𝑅𝑆 = 100 %) is independent of its length when the dimensions of the entrance and exit opening are kept unchanged. This is a really interesting result because not so intuitive. Things are different when the inner surface of the light cones is diffusive and Lambertian, as it is shown in Fig. 4.27. In this regard, I remember that a Lambertian diffusive surface is such when the total reflectance is independent of the incidence angle of the light and 𝑙𝑎𝑚𝑏 when the radiance of the diffused light is constant [57, 58]. The quantity 𝐺 𝑠𝑝𝑒𝑐 now tends, as 𝑅𝑆 tends to 100 %, to different values for the different models of light cones. 𝑙𝑎𝑚𝑏 as function of 𝑅𝑆 of a specular Fig. 4.28 shows the direct Lambertian reflectance  12 𝑙𝑎𝑚𝑏 surface for the four light cones. The direct Lambertian absorbance  12 is shown in Fig. 4.29. It is interesting to observe their opposite behavior and the convergence of all 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 curves, for both  12 , at the two ends of the range of values of 𝑅𝑆 . Within and  12 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 these two extremes, the  12 and  12 curves are clearly distinguishable and well separated. If we now turn to consider diffusive surfaces, their effect is to compact the curves, even if not to overlap them as it has been observed with the CPCs concentrators. 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 for diffusive surface are shown in Figs. 4.30 and 4.31, The curves of  12 and  12 respectively. 0,7 LC1 LC2 LC3 LC4 Diffusive surface 0,6 Gspeclamb (sr) 0,5 0,4 0,3 0,2 0,1 0,0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.27. The specific bidirectional Lambertian conductance of the four tested light cones is reported as function of reflectance of the diffusive surface. 208 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 80 LC1 LC2 LC3 LC4 Specular surface 12lamb (%) 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.28. Direct Lambertian reflectance of the four tested light cones as function of reflectance of a specular surface. 100 80 12lamb (%) 60 LC4 LC3 LC2 LC1 Specular surface 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.29. Direct Lambertian absorbance of the four tested light cones as function of reflectance of a specular surface. 100 LC1 LC2 LC3 LC4 Diffusive surface 80 12lamb (%) 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.30. Direct Lambertian reflectance of the four tested light cones as function of reflectance of a diffusive surface. 209 Advances in Optics: Reviews. Book Series, Vol. 5 100 80 12lamb (%) 60 40 LC4 LC3 LC2 LC1 Diffusive surface 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.31. Direct Lambertian absorbance of the four tested light cones as function of reflectance of a diffusive surface. 4.5.3. Solar Tunnels (ST) The simulated direct Lambertian transmittance, reflectance and absorbance of a solar tunnel (ST3) of 48 mm length, for specular and diffuse surface, are shown in Fig. 4.32. 100 80 ST3, DLIM tau_spec rho_spec alpha_spec tau_diff rho_diff alpha_diff 12lamb (%) 60 40 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.32. Lambertian transmittance, reflectance and absorbance of the solar tunnel ST3, of 48 mm length, as function of specular and diffuse surface reflectance. From Fig. 4.32 we first note, as expected, that the specular Lambertian reflectance is identically equal to zero, since there is no possibility for a light beam to be reflected back. A notable result is that the Lambertian absorbance is practically the same for specular and diffusive surface; soon we will see if this also happens for the other STs. We also note that the Lambertian transmittance for specular surface is significantly greater than that for diffusive surface, a result that has been reported for all the tested optical components, because the diffusive surface facilitates the back reflection of the rays. In the case of CPCs and LCs, however, the back reflection of light rays was also favored by the particular 210 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications profile of the surface, while in this case, with a flat profile of the surface, the only means that allows the back reflection of light rays is just the surface when it is diffusive. Similar behaviors have been found with the other solar tunnels. So, let us see how the single optical quantities change passing from one solar tunnel to another. Let us start, as we have done so far, by considering the specific Lambertian conductance, which summarizes the transmissive properties of the optical element with respect to a Lambertian irradiation, and which is proportional to both the Lambertian transmittance and conductance, which 𝑙𝑎𝑚𝑏 we can therefore avoid plotting. The specific Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐 , calculated applying the DLIM method, of the solar tunnels ST1, ST2, ST3 and ST4 vs. the 𝑅𝑆 for a 𝑙𝑎𝑚𝑏 specular surface, is reported in Fig. 4.33. We note the increasing of 𝐺 𝑠𝑝𝑒𝑐 at increasing of 𝑅𝑆 and at decreasing of the ST length, as we have observed in both CPCs and LCs. It 𝑙𝑎𝑚𝑏 is interesting to note the convergence of 𝐺 𝑠𝑝𝑒𝑐 to the same maximum value of  (sr) for 𝑅𝑆 = 100 %. In this case, in fact, as 𝐶𝑔𝑒𝑜 = 1.0 and the surface is specular, the Lambertian transmittance is unitary in the two directions, and then we have: 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 =    21 𝐺 𝑠𝑝𝑒𝑐 =    12 = (⁄𝐶𝑔𝑒𝑜 )  =  21 3.5 ST1 ST2 ST3 ST4 Specular surface 3.0 2.5 Gspeclamb (sr) (4.39) 2.0 1.5 1.0 0.5 0.0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.33. Specific bidirectional Lambertian conductance of the four tested solar tunnels 𝑙𝑎𝑚𝑏 as function of reflectance of the specular surface. The low index "spec" of 𝐺 𝑠𝑝𝑒𝑐 means "specific". Things are different when the inner surface of the solar tunnels is diffusive and 𝑙𝑎𝑚𝑏 Lambertian, as it is shown in Fig. 4.34. We again note the increasing of 𝐺 𝑠𝑝𝑒𝑐 at 𝑙𝑎𝑚𝑏 increasing of 𝑅𝑆 and at decreasing of the ST length, but the quantity 𝐺 𝑠𝑝𝑒𝑐 now tends, as 𝑅𝑆 tends to 100 %, to different values for the different models of solar tunnels. 𝑙𝑎𝑚𝑏 of the four solar tunnels as Fig. 4.35 shows the direct Lambertian reflectance  12 function of 𝑅𝑆 for specular and diffusive surfaces. The data for the specular surface were 𝑙𝑎𝑚𝑏 reported equally although identically null. The direct Lambertian absorbance  12 is shown in Fig. 4.36 for both specular and diffusive surfaces. For the Lambertian reflectance at diffusive surface, we now find that it starts the same, near zero, for the four STs at low values of 𝑅𝑆 , but then diverges as 𝑅𝑆 increases, differently from what has been observed 211 Advances in Optics: Reviews. Book Series, Vol. 5 both in the CPCs and in the LCs. We have no more the convergence of curves at the two extremes of the range of 𝑅𝑆 values. A similar but opposite behavior is found in the Lambertian absorbance (Fig. 4.36), which starts the same, zero, for the four STs at 𝑅𝑆 = 100 % and then diverges as 𝑅𝑆 decreases. What we now observe is that the diffusive surface no longer compacts the Lambertian reflectance or absorbance curves. In particular, we note an interesting result, that is that the Lambertian absorbance changes very little when passing from the specular to the diffusive surface, while remaining well different in the different solar tunnels. If we now want to see more accurately what happens to the absorbance curves, we note that, starting from ST1 and going towards the longer solar tunnels, we find the following: the absorbance curve for diffusive surface exceeds that for specular surface in ST1; the same happens in the ST2; in ST3 the two curves overlap; finally, in the ST4 the two curves swap places. So, there is a precise evolution of the two 𝑙𝑎𝑚𝑏 as the length of the solar tunnel increases. curves of  12 2,5 ST1 ST2 ST3 ST4 Diffusive surface 2,0 Gspeclamb (sr) 1,5 1,0 0,5 0,0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.34. The specific Lambertian conductance of the four tested solar tunnels is reported as function of reflectance of the diffusive surface. 80 ST4_diff ST3_diff ST2_diff ST1_diff ST4_spec ST3_spec ST2_spec ST1_spec 70 60 12lamb (%) 50 diff 40 30 20 spec 10 0 -10 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.35. Direct Lambertian reflectance of the four tested solar tunnels as function of reflectance of specular and diffusive surface. 212 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 100 80 12lamb (%) 60 40 alpha_diff_ST4 alpha_spec_ST4 alpha_diff_ST3 alpha_spec_ST3 alpha_diff_ST2 alpha_spec_ST2 alpha_diff_ST1 alpha_spec_ST1 20 0 0 20 40 60 80 100 Surface reflectance, RS (%) Fig. 4.36. Direct Lambertian absorbance of the four tested solar tunnels as function of reflectance of specular and diffusive surfaces. 4.6. Exit Radiance Simulation Results In the Prologue, I explained that, applying to one of the two openings of the optical element a Lambertian radiation, the radiation coming out from the opposite opening is not necessarily Lambertian. The inverse Lambertian method, ILIM, in fact, is based precisely on the fact that the angular distribution of the radiance leaving the entrance opening when the exit one is irradiated with Lambertian light, is the same as that of the angular distribution of the light collecting efficiency of a collimated beam incident on the entrance opening. And this efficiency distribution is certainly restricted in angular terms to the extent that the acceptance angle is defined, which, as we know, is always less than about 20°. Hence, if we consider a SC, whose surface is certainly specular, its inverse radiance will be very different from the incident Lambertian one, which has an angular divergence of 90°. If the SC had a diffusive surface, then the inverse radiance exiting the entrance opening could have a higher angular divergence, up to 90°. In this Section, I am interested to exploit how is the radiance distributed out of the inlet opening of an optical element, when the exit opening is inversely irradiated with a Lambertian radiation. The aim is to consider the optical element as a source of light, whose emitted radiance will be distributed in space differently according to the properties imposed on the internal surface. The reason why we will only examine the radiance exiting the entrance opening is that 𝑙𝑎𝑚𝑏 is 𝐶𝑔𝑒𝑜 times larger than the direct one, the inverse Lambertian transmittance  21 𝑙𝑎𝑚𝑏  12 ; and this is an important aspect if we want to have a source that emits light with high efficiency. For all the optical elements, we will consider surfaces of specular and diffusive type with 𝑅𝑆 = 100 %. 4.6.1. Nonimaging Solar Concentrators (CPC) The angular distribution of the radiance emitted by four CPCs is shown in Fig. 4.37. As already discussed in Section 4.2, these concentrators have an optical efficiency, with 213 Advances in Optics: Reviews. Book Series, Vol. 5 respect to a collimated beam entering the aperture (1), of step-like shape, and this efficiency curve can be obtained, thanks to the ILIM method, by irradiating the aperture (2) (the smallest one) with Lambertian light and measuring the radiance exiting the aperture (1), which is what we did with the simulations shown in Fig. 4.37. The radiance is step-like shaped and the on / off transition occurs exactly at the angle of acceptance 𝑎𝑐𝑐 = 2.5°, 5°, 10° and 20° for the CPCs: CPC2.5, CPC5, CPC10 and CPC20, respectively. Similar step-like curves were obtained with the CPC30 and CPC40. The exit radiance curves were obtained with a specular 𝑅𝑆 = 100 %, which is the condition very close to that in which solar concentrators operate. Real reflective solar concentrators work with slightly lower reflectances (about 95 %), but this does not change the results much. Here we will always report the normalized output radiance, 𝐿𝑛𝑜𝑟𝑚 () (see Section 4.4.3), because we are mainly interested in studying its angular distribution, more than its absolute intensity. The actual value of radiance at 0° will depend on the flux entered into the aperture (2) that is on the imposed inverse radiance 𝐿𝑖𝑛𝑣 , but this is an arbitrary choice. 1,2 CPC20 CPC10 CPC5 CPC2.5 Normalized radiance 1,0 Inverse irradiation, RS = 100% specular 0,8 0,6 length 0,4 0,2 0,0 -30 -20 -10 0 Exit angle, 1,out (°) 10 20 30 Fig. 4.37. Angular distribution of the exit radiance from four CPCs with specular surface, when irradiated by a Lambertian radiation incident on the exit aperture. Here we are considering solar concentrators which are designed to minimize the reflection losses, and then the internal surface is necessarily specular. However, it is interesting to see how the angular distribution of radiance changes if the surface is diffusive. In order not to field too many parameters, I have chosen to consider only a diffusive Lambertian surface with 𝑅𝑆 = 100 %. Fig. 4.38 shows the radiance exiting all the CPCs of Table 4.1 with diffusive surface. The first thing to note is the loss of the step-like profile and the emergence of a Lorentzian-type shape that extends up to 90° The measured acceptance angles for six CPCs are reported in Table 4.3, and are compared to the ones for specular surface. From Table 4.3 we note the widening of the curve for CPC2.5, CPC5 and CPC10, but the shrinking of the curve for the CPC20, CPC30 and CPC40 when moving from the specular to the diffusive surface. 214 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications Normalized radiance 1,0 Inverse irradiation. RS=100%diffuse CPC40 CPC30 CPC20 CPC10 CPC5 CPC2.5 0,8 0,6 length 0,4 0,2 0,0 -80 -60 -40 -20 0 20 40 60 80 Exit angle, 1,out (°) Fig. 4.38. Angular distribution of the exit radiance from the CPCs with diffusive surface, when irradiated by a Lambertian radiation incident on the exit aperture. Table 4.3. Acceptance angles of the various CPCs with specular and diffusive surface with 𝑅𝑆 = 100 %. Element CPC2.5 CPC5 CPC10 CPC20 CPC30 CPC40 𝒂𝒄𝒄 (°) Specular Diffusive 2.5 4.1 5.0 8.1 10.0 11.6 20.0 19.0 30.0 27.0 40.0 35.2 𝜟𝒂𝒄𝒄 (%) Spec  Diff +64 +62 +16 ‒5 ‒10 ‒12 4.6.2. Light Cones (LC) The choice of studying the angular distribution of the inverse radiance emitted by light cones characterized by the same entrance and exit apertures, that is by the same 𝐶𝑔𝑒𝑜 , but of different length, has led to very interesting and unexpected results. Already the analysis of the radiance emitted by the four LCs studied in Section 4.5.2, LC1-LC4, of which we have learned about their optical transmission and conduction properties, has shown a first interesting result (see Fig. 4.39). The radiance profile for LC1 is very particular, as it is triangular in shape. Moving to the longer CPCs, the profile narrows at the sides, while it widens at the top, flattening and becoming of the step-like type, particularly for the LC4, whose profile closely resembles that of a CPC with specular surface and 𝑎𝑐𝑐 = 30°. Here it must be considered that, if an LC4 has a step-like profile similar to an ideal CPC, this means that, if it were used as a solar concentrator, it would show an optical efficiency curve equal to that of the emitted radiance, that is step-like, because this is what the ILIM method teaches, and therefore similar to that of an ideal CPC. 215 Advances in Optics: Reviews. Book Series, Vol. 5 Furthermore, it has been found that the on-axis direct optical efficiency of the LC4, 𝜂𝑑𝑖𝑟 (0°), is equal to unity, which causes the LC4 to behave almost exactly like a CPC30, that is a CPC with 𝑎𝑐𝑐 = 30°, even if with different dimensional values. The other interesting aspect that arises from Fig. 4.39 is the fact that the width at half height of the curves, that is the acceptance angle, remains almost constant and around 30°. Probably the almost constant 𝑎𝑐𝑐 is due to the fact that these light cones have the same 𝐶𝑔𝑒𝑜 = 4. It would be interesting, therefore, to deepen these results, to find out how things go with light cones still characterized by the same 𝐶𝑔𝑒𝑜 , but different from 4. So I thought to increase the range of optical elements that characterize the first series with 𝐶𝑔𝑒𝑜 = 4, of dimensions: a = 12, a' = 6, l = variable (which from now on we will indicate with the triad of values: (a, a', l) = (12, 6, l)), by introducing three other series of elements, the second one with dimensional parameters (a, a', l) = (12, 5, l) and 𝐶𝑔𝑒𝑜 = 5.76, the third one with dimensional parameters (a, a', l) = (12, 4, l) and 𝐶𝑔𝑒𝑜 = 9, and the forth one with dimensional parameters (a, a', l) = (12, 3, l) and 𝐶𝑔𝑒𝑜 = 16. Since Fig. 4.39 shows with LC1 (12, 6, 12) a triangular shape, which becomes a step-like shape as the length increases, I have expanded this first series of elements, adding light cones shorter than LC1 and longer than LC4. If we start from low values of length and gradually increase it, we find a very wide and trapezoidal curve which tends to shrink gradually, keeping the sides of the trapezium almost parallel, until it becomes a triangular curve in correspondence of LC1 (see Fig. 4.40a). There is therefore a first phase in which the curve narrows, while maintaining a trapezoidal shape, and LC1 is the final element of this shrinking process, with triangular shape. Fig. 4.39. Angular distribution of the exit radiance from the light cones with specular surface, when irradiated by a Lambertian radiation incident on the exit aperture. Starting from LC1, the increase in the length of the light cone does not change the width of the curve very much, rather it transforms into an increasingly squared curve (see Fig. 4.40b). This curve will tend to a perfect rectangular shape by increasing the length arbitrarily, because, having imposed a unitary and specular surface reflectance, no losses 216 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications are expected during the path of the light. Before moving on to the results obtained with the other three series of optical elements, it is worthwhile to focus on some aspects of these curves that deserve to be analyzed. The first one is the trend of the width at half height of the curves, that is of 𝑎𝑐𝑐 , as a function of the length of the light cone. There is therefore a first phase in which 𝑎𝑐𝑐 is reduced, which we can call the “trapezoidal phase”, ending with the triangular curve. Then there is the “squaring phase”, which begins with the triangular curve and ends with a quasi-squared curve. This phase is characterized by an almost constant 𝑎𝑐𝑐 , which must be related to the 𝐶𝑔𝑒𝑜 of the series. It is precisely the fact that these two phases of the curve exist, with the transition at the triangular curve, that I show the radiance profiles in two separate graphs, one for the trapezoidal phase (see Fig. 4.40 a) and the other for the squaring phase (see Fig. 4.40 b). What happens to the other series is qualitatively the same as that described so far, with the difference that the value of the length of the light cone changes at the trapezoidal / squaring transition (see Figs. 4.41, 4.42 and 4.43). We call this length as the “critical length”, 𝑙𝐶 (𝐶𝑔𝑒𝑜 ) a function of 𝐶𝑔𝑒𝑜 only. At the critical length, we then identify a “critical acceptance angle”, also a function of 𝐶𝑔𝑒𝑜 : 𝑎𝑐𝑐,𝐶 (𝐶𝑔𝑒𝑜 ). 1,0 1,0 length 0,8 Normalized radiance Normalized radiance 0,8 0,6 0,4 1st series LC(12/6/8) LC(12/6/9) LC(12/6/12) Rs=100% spec. 0,2 length 0,6 1st series LC(12/6/144) LC(12/6/96) LC(12/6/48) LC(12/6/24) LC(12/6/12) Rs=100% spec. 0,4 0,2 0,0 0,0 -80 -60 -40 -20 0 20 Exit angle, 1,out (°) (a) 40 60 80 -60 -40 -20 0 20 40 60 Exit angle, 1,out (°) (b) Fig. 4.40. Profiles of the normalized inverse radiance of the first series of LCs with dimensions (12/6/l). (a) “Trapezoidal” phase with length varying from 8 to 12 mm; (b) “Squaring” phase with length varying from 12 to 144 mm. In Fig. 4.44 are reported the values of 𝑎𝑐𝑐 as function of the LC length for the four series of simulations. As we can see, the trend of 𝑎𝑐𝑐 is the same for the four series. We observe at first a sharp drop of the angle (the trapezoidal phase), the reaching of a minimum followed by a small and quick recovery (the transition phase), and finally a stabilization, albeit associated with a very slight growth (the squaring phase). Significant parameters can be extracted in correspondence of the minimum of the curves, to which we associate the critical acceptance angle 𝑎𝑐𝑐,𝐶 and the critical length 𝑙𝐶 , and in correspondence of high LC lengths, to which we associate, for simplicity, the angle to infinity,  (𝐼𝑁𝐹 in the figure), which however can be only approximated. The three significant parameters for the four series of simulations on the light cones are reported in Fig. 4.45 217 Advances in Optics: Reviews. Book Series, Vol. 5 as function of 𝐶𝑔𝑒𝑜 . From Fig. 4.45 we see that the critical acceptance angle decreases as the geometric concentration ratio of the light cone increases; the same trend is observed for the angle to infinity, which is slightly higher than it. The critical length, on the other hand, definitely increases as the geometric concentration ratio increases. These results can be advantageously exploited if, for example, a light source with constant radiance within a certain exit angle is to be realized. If a tighter exit angle is desired, the geometric concentration ratio and also the length of the LC will have to be increased. A discussion about this application is reported in Section 4.7.2. 1,0 1,0 length 0,8 Normalized radiance Normalized radiance 0,8 0,6 2nd series LC(12/5/10) LC(12/5/12) LC(12/5/16) LC(12/5/18) Rs=100% spec. 0,4 0,2 length 0,6 2nd series LC(12/5/144) LC(12/5/72) LC(12/5/48) LC(12/5/36) LC(12/5/24) LC(12/5/20) LC(12/5/18) Rs=100% spec. 0,4 0,2 0,0 0,0 -60 -40 -20 0 20 40 -50 60 -40 -30 -20 -10 0 10 20 30 40 50 Exit angle, 1,out (°) Exit angle, 1,out (°) (a) (b) Fig. 4.41. Profiles of the normalized inverse radiance of the second series of LCs with dimensions (12/5/l). (a) “Trapezoidal” phase with length varying from 10 to 18 mm; (b) “Squaring” phase with length varying from 18 to 144 mm. 1,0 1,0 0,8 0,8 Normalized radiance Normalized radiance length 0,6 0,4 3rd series LC(12/4/12) LC(12/4/16) LC(12/4/20) LC(12/4/24) Rs=100% spec. 0,2 length 0,6 3rd series LC(12/4/144) LC(12/4/96) LC(12/4/48) LC(12/4/36) LC(12/4/24) Rs=100% spec. 0,4 0,2 0,0 0,0 -60 -40 -20 0 Exit angle, 1,out (°) (a) 20 40 60 -40 -30 -20 -10 0 10 20 30 40 Exit angle, 1,out (°) (b) Fig. 4.42. Profiles of the normalized inverse radiance of the third series of LCs with dimensions (12/4/l). (a) “Trapezoidal” phase with length varying from 12 to 24 mm; (b) “Squaring” phase with length varying from 24 to 144 mm. 218 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 1,0 1,0 length Normalized radiance Normalized radiance 0,8 0,6 0,4 4th series LC(12/3/16) LC(12/3/20) LC(12/3/24) LC(12/3/36) Rs=100% spec. 0,2 0,8 length 0,6 4th series LC(12/3/144) LC(12/3/96) LC(12/3/72) LC(12/3/48) LC(12/3/36) Rs=100% spec. 0,4 0,2 0,0 0,0 -50 -40 -30 -20 -10 0 10 20 30 40 50 -30 -20 -10 0 10 20 30 Exit angle, 1,out (°) Exit angle, 1,out (°) (a) (b) Fig. 4.43. Profiles of the normalized inverse radiance of the fourth series of LCs with dimensions (12/3/l). (a) “Trapezoidal” phase with length varying from 16 to 36 mm; (b) “Squaring” phase with length varying from 36 to 144 mm. 70 Light cones 1st series (Cgeo = 4) Acceptance angle, acc (°) 60 2nd series (Cgeo = 5,76) 50 3rd series (Cgeo = 9) 4th series (Cgeo = 16) 40 30 20 10 0 0 20 40 60 80 100 120 140 160 Length, l (mm) Fig. 4.44. Acceptance angles of the four series of simulations as function of LC length. But there is one last aspect to consider: the curves of normalized radiance, in the “squaring phase”, all intersect on a common point, which has as coordinates a certain angle of light output, 1,𝑜𝑢𝑡 , and a certain level of normalized radiance. I don't know the meaning to this point, and therefore I limit myself to reporting the coordinates for the four series of measures: 1st series: (30.2°; 0.46), 2nd series: (25.0°; 0.44); 3rd series: (20.0°; 0.40); 4th series: (14.8°; 0.42). As we can see, these points, where the curves converge, lie below the point where the acceptance angle is measured (at half height of the curve), and have an angular coordinate slightly higher than 𝑎𝑐𝑐,𝐶 and  , , and a coordinate of normalized radiance less than 0.5. 219 40 40 35 35 30 30 25 25 20 20 15 15 Angle to infinity Critical angle Critical length 10 5 lC (mm) acc,C (°), INF (°) Advances in Optics: Reviews. Book Series, Vol. 5 10 5 0 0 4 6 8 Cgeo 10 12 14 16 Fig. 4.45. The three significant parameters extracted by the curves of exit radiance from the four series of simulations on the light cones. If we want to see now what are the effects that a diffusive surface has on the distribution of the exit radiance, we limit ourselves to examine only the light cones LC1-LC4, whose normalized radiance curves in presence of a specular surface are shown in Fig. 4.39. We find a substantial change in the curves, as shown in Fig. 4.46. At first, we notice that the curves extend up to 90° angles, as always happens in the presence of diffusive surfaces, and that their flat top decreases in width as the length increases. Also, the width of the radiance curves at half height narrows as the length of the LC increases, similarly to what happened for the CPCs, although these two elements are of different shape. The acceptance angle values measured for the radiance curves of Fig. 4.46 are reported in Table 4.4, compared with those relating to the specular surface. We find that the diffusive surface has the effect of significantly widening the LC1 curve; the enlargement is reducing moving to LC2 and transforms in a narrowing for LC3 and LC4. A similar trend, even if less pronounced, is found in the CPCs (see Table 4.3). Summarizing, the diffusive surface, in the light cones, has shown a very strong effect on the acceptance angle, much more than in the CPCs. Table 4.4. Acceptance angles of four LCs with specular and diffusive surface. Element LC1 LC2 LC3 LC4 220 𝒂𝒄𝒄 (°) Specular 27.2 29.3 29.7 30.0 Diffusive 51.0 34.2 20.2 10.7 𝜟𝒂𝒄𝒄 (%) Spec  Diff +87.5 +16.7 -32.0 -64.3 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications Fig. 4.46. Angular distribution of the exit radiance from the light cones with diffusive surface, when irradiated by a Lambertian radiation incident on the exit aperture. 4.6.3. Solar Tunnels (ST) The inverse radiance curves for the STs are quite particular. For specular surface, the radiance curves overlap exactly. They are all of the Lambertian type with 90° angular divergence. The curves shown in Fig. 4.47 show a drop in intensity from 70°, while we expect to see a flat curve up to 90°. In reality this is an effect of the limited number of rays used for the raytracing, as already observed in a previous work [31]. This has been easily demonstrated by placing a Lambertian source at the input window of the SC of Fig. 4.11. The profile of the emitted radiance results exactly the same of that reported in Fig. 4.47. Normalized radiance 1,0 0,9 ST4 ST3 ST2 ST1 Inverse irradiation RS = 100% specular 0,8 0,7 0,6 0,5 -100 -80 -60 -40 -20 0 20 40 60 80 100 Exit angle, 1,out (°) Fig. 4.47. Angular distribution of the exit radiance from the solar tunnels with specular surface, when irradiated by a Lambertian radiation incident on the exit aperture. 221 Advances in Optics: Reviews. Book Series, Vol. 5 This is the only case where the inverse radiance is Lambertian, like the applied one. When the surface is diffusive, the radiance curves change drastically and the outgoing radiation is no longer Lambertian (see Fig. 4.48). The curves, no longer flat, take on a shape that, being almost semi-circular for ST1, gets closer and closer to a Lorentzian one, with a gradual narrowing of the curve. The acceptance angles measured for the radiance curves with specular and diffuse surface are reported in Table 4.5, where we find that all the curves of the STs with diffusive surface undergo a relative reduction in width that is increasingly accentuated as the length increases. Fig. 4.48. Angular distribution of the exit radiance from the solar tunnels with diffusive surface, when irradiated by a Lambertian radiation incident on the exit aperture. Table 4.5. Acceptance angles of the various STs with specular and diffusive surface. Element ST1 ST2 ST3 ST4 𝒂𝒄𝒄 (°) Specular 90 90 90 90 Diffusive 61.5 44.4 24.9 14.4 𝜟𝒂𝒄𝒄 (%) Spec  Diff -31.7 -50.7 -72.3 -84.0 4.7. Short Discussion of Practical Applications. Although the present work is substantially theoretical, we can identify two topics on which to carry out practical considerations. The first concerns the calculation of the net flux crossing a solar tunnel, an element widely used in practice, to be installed on the roof of a building to convey sunlight from the outside to the internal environment. The second concerns the use of the optical elements CPCs or LCs as light sources to create particular light beams. 222 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications 4.7.1. Net Flux Crossing a Solar Tunnel An example of a solar tunnel applied to the collection of sunlight to be sent to a domestic environment is shown in Fig. 4.49. A transparent and diffusive dome (dm) is placed horizontally on the roof of a house. A solar tunnel, connected to the dome, crosses vertically the roof and reaches the inside of the house, in a generic room. At the end of the solar tunnel there is a diffuser (df), which distributes the sunlight all around. The room is modelled as a homogeneous box whose internal surface has a reflectance 𝑅𝑤 . To apply the law of Lambertian conductance we must evaluate the solar radiance 𝐿1 incident on the dome and the radiance 𝐿2 present inside the box and incident on the diffuser of the solar tunnel. Even if it is the diffuser that emits most of the light, this one, once distributed inside the room, produces a diffused light that hits the diffuser with an 𝐿2 radiance. Fig. 4.49. Simplified scheme of a room illuminated by the light diffused from the solar tunnel (ST) and coming from the sunlight incident on the dome (dm). The dome is irradiated by the light diffused from the sky and by the direct light of the Sun. As a first approximation, the light transferred by the dome to the solar tunnel can be transformed, due to the multiple reflections undergone, into diffused light whose flux must be equal to the total incident flux from the sky and the solar disk. At this point, we must evaluate the radiance produced by the direct component of the solar irradiance, 𝐸𝑑𝑖𝑟,⊥ , and of the radiance produced by the diffuse component on the horizontal plane, 𝐸𝑑𝑖𝑓𝑓 , for some sky conditions. Table 4.6 shows the direct and diffuse irradiances, very approximate, evaluated by our previous measurements [59]. The flux from the direct component of solar radiation, expressed as diffuse light, becomes:  𝑑𝑖𝑟 = 𝐴1  𝐸𝑑𝑖𝑟,⊥  𝑐𝑜𝑠 =   𝐴1  𝐿𝑑𝑖𝑟 , (4.40) where 𝐴1 is the cross-section area of the solar tunnel. The flux from the diffuse component of solar radiation is: 223 Advances in Optics: Reviews. Book Series, Vol. 5  𝑑𝑖𝑓𝑓 =   𝐴1  𝐿diff (4.41) In Eq. (4.41) the diffuse component of solar radiation is approximated to be of constant radiance. The total radiance due to the direct and diffuse components of solar radiation becomes: 𝐿 1 = (𝐸𝑑𝑖𝑟,⊥  𝑐𝑜𝑠) ⁄ + 𝐸𝑑𝑖𝑓𝑓,⊥ ⁄ = (4.42) = 𝐸𝑑𝑖𝑟,⊥ ⁄ (  𝐴𝑀) + 𝐸𝑑𝑖𝑓𝑓,⊥ ⁄ where we have placed: 𝑐𝑜𝑠 = 1/𝐴𝑀, with AM = Air Mass [60]. Following now what is reported in Table 4.6, we can calculate the total external radiance 𝐿 1 incident on the dome for the four examined sky conditions. The results of 𝐿 1 , calculated for the standard condition AM = 1.5, are reported in Table 4.7. Table 4.6. Irradiance (in W/m2) from the direct and the diffuse components of solar radiation. Component Direct Diffuse Clear sky ~ 900 ~ 100 Atmospheric conditions Partially cloudy Cloudy ~ 800 ~ 400 ~ 200 ~ 300 Covered 0 ~ 200 Table 4.7. The two components of the external radiance, and the total one, are calculated (in W / (m2  sr) starting from the corresponding estimated irradiances (in W / m2), and reported as a function of atmospheric conditions. Sky Condition Clear sky Partially cloudy Cloudy Covered 𝑬𝒅𝒊𝒓,⊥ (W/m2) ~ 900 ~ 800 ~ 400 ~0 𝑬𝒅𝒊𝒇𝒇,⊥ (W/m2) ~ 100 ~ 200 ~ 300 ~ 200 𝑳𝒅𝒊𝒓 (W/m2 sr) ~ 190 ~ 170 ~ 85 ~0 𝑳𝒅𝒊𝒇𝒇 (W/m2 sr) ~ 30 ~ 65 ~ 95 ~ 65 𝑳𝟏 (W/m2 sr) ~ 220 ~ 235 ~ 180 ~ 65  𝒏𝒆𝒕 (W) ~ 20 ~ 21 ~ 16 ~6 The internal radiance, 𝐿 2 , can be evaluated by making the following consideration. We set the average illuminance level in the room at 500 lx, which is high enough to read a book [61]. Simplifying sunlight as monochromatic light in the green (555 nm), the illuminance of 500 lx corresponds to an irradiance of 500/683 ≈ 0.7 (W/m2), and then to a radiance 𝐿 2 ≈ 0.7 /  ≈ 0.2 W / (m2  sr). This is the average radiance incident on the walls of the room. Taking into account the low average reflectance of the walls of a habitable room, the radiance incident on the diffuser (df) would be even smaller than 0.2 W / (m2  sr). We thus find that the internal radiance 𝐿 2 is incomparably smaller than 𝐿 1 for any sky condition (see Tab. 4.7), and then we can neglect it. If we now apply the law of optical Lambertian conductance, we find for the net flux through the ST: 224  𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) ≈ 𝐺 𝑙𝑎𝑚𝑏  𝐿1 (4.43) Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications To calculate the 𝐺 𝑙𝑎𝑚𝑏 conductance, we choose a real solar tunnel of the same shape as the ST4, that is with a length / diameter ratio = 4, but with a diameter of 20 cm (instead of 24 mm), and then of a length of 20  4 = 80 cm. We choose a specular surface with 98 % reflectivity, obtained covering the internal surface of the solar tunnel with the 3 M 𝑙𝑎𝑚𝑏 Radiant Mirror polymeric film. From Fig. 4.33 we obtain for ST4: 𝐺 𝑠𝑝𝑒𝑐 ≈ 2.8 sr. The 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 Lambertian conductance becomes therefore: 𝐺 = 𝐴1  𝐺 𝑠𝑝𝑒𝑐 ≈ 0.09 (m2  sr). From Eq. (4.43) we then obtain the net flux  𝑛𝑒𝑡 (in W) crossing the solar tunnel in the outdoor  indoor direction, reported in Table 4.7 as a function of the sky conditions. 4.7.2. Light Sources from CPCs and LCs The optical elements CPCs and LCs present very interesting inverse radiance distributions, so they can be used to realize light sources. CPCs with a specular surface, for example, if inversely irradiated with Lambertian light, are able to emit a bundle of light of constant radiance within a well-defined angle, equal to 𝑎𝑐𝑐 (see Fig. 4.37). Fig. 4.50(a) shows the schematic of a light source realized making use of a CPC. Light cones with a specular surface, too, are suitable to emit light with a step-like shaped radiance within a defined angle (see Fig. 4.39), if an appropriate length of the LC is chosen. Fig. 4.50(b) shows the schematic of a light source realized making use of the LC4 or longer LCs. Fig. 4.50. Configurations used to make light sources with constant radiance within a defined angle. (a) CPC supplied with light from a single integrating sphere; (b) LC supplied with a double integrating sphere. (lp) = lamp; (bf) = baffle; (is) = integrating sphere. To realize a light source with CPC or LC elements, however, it is necessary to supply the opening (2), the smaller one of the CPC or the LC, with a Lambertian light. This can be obtained as we have already done in Fig. 4.3 to study the emission of inverse radiance. In Fig. 4.50(a) a possible configuration is shown, in which a lamp source (lp) illuminates an integrating sphere (is), inside which a baffle (bf) allows to distribute the light well inside the (is). The opening of (is) from which Lambertian light is emitted is then connected with the opening (2) of the CPC, which is thus able to emit light with constant radiance within 225 Advances in Optics: Reviews. Book Series, Vol. 5 the acceptance angle. Alternatively, the lamp (lp) can supply two integrating spheres, (is1) and (is2), to obtain a better Lambertian distribution at the output of (is2), as shown in Fig. 4.50 (b). The internal surface of the two spheres must be highly reflective, at least 95 %, in order not to lose too much light by absorption. It is sufficient to make a multilayer coating of BaSO4 [6, 19]. 4.7.3. Bent Solar Tunnels In Section 4.5.3 we have studied the optical transmission properties of some linear solar tunnels, in particular the ST4 of 96 mm long. In this section I want to expand the study to bent solar tunnels, which can be encountered in practice. In particular, I am interested to know the effects that the presence of one, two or three bends has on the optical transmission properties of a linear ST. Therefore, starting from the ST4 as a basic unit, I modeled a solar tunnel made up of two ST4 joined by a 90° bend, a solar tunnel made up of three ST4 joined by two 90° bends, and finally a solar tunnel made up of four ST4 joined by three 90° bends (see Fig. 4.51). These three bent STs were then compared with three linear STs of the same length, in terms of Lambertian absorbance and transmittance. In order to consider only STs that have potential practical applications, I have chosen to study only mirror surfaces with high reflectivity, for example from 80 % to 100 %. In Fig. 4.51 the three types of bent STs are shown after a short raytracing done with a surface of 80 % reflectivity. The Lambertian source is applied to the openings (1) and the transmitted flux is measured at the openings (2). The color along the tubes indicates the intensity of the light, which fades from red to blue. Fig. 4.51. Examples of raytracing of three bent solar tunnels (ST4×2 (a); ST4×3 (b); ST4×4 (c)) used for simulating the Lambertian absorbance, transmittance and conductance. (1) is the entrance opening of the Lambertian source; (2) is the exit opening where the transmitted rays are collected and measured. 226 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications The bent STs consist of two, three or four ST4s (l = 96 mm each) and one, two or three bends, respectively. Each bend at 90° engages a length 𝑙 =   𝑅, where 𝑅 = 12 mm is the tunnel radius. Ultimately, the three tunnels of Fig. 4.51 (a), (b) and (c) will have a length of 229.7 mm, 363.4 mm and 497.1 mm, respectively. The effect of length of the 𝑙𝑎𝑚𝑏 straight solar tunnels on the Lambertian absorbance  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 and transmittance  𝑑𝑖𝑟 can be seen in Fig. 4.52, where the three STs, of increasing length, are indicated as ST4×2, ST4×3 and ST4×4. They were simulated using the direct Lambertian irradiation method (DLIM) (see Section 4.4.2). In Fig. 4.52 the Lambertian reflectance is missing because, being the surface specular, we have no rays reflected to inputs (1). The trend of  𝑙𝑎𝑚𝑏 𝑑𝑖𝑟 shows, as expected, that the increase of the surface reflectance 𝑅 from 80 % and  𝑙𝑎𝑚𝑏 𝑆 𝑑𝑖𝑟 to 100 % involves a reduction of the Lambertian absorbance, down to 0 %, and an increase of the Lambertian transmittance, up to 100 %. Opposite is the effect of the increase in the length of the solar tunnel on the two quantities (see the arrows in Fig. 4.52). Fig. 4.52 shows that the Lambertian transmittance is very low, close to zero, even though the surface reflectivity is quite high as 80 %. However, it must be considered that the Lambertian irradiation involves many rays close to the perpendicular to the wall surface, and this in turn involves a large number of reflections inside the tunnel. 100 100 length 80 alpha(ST4x4) alpha(ST4x3) alpha(ST4x2) tau(ST4x2) tau(ST4x3) tau(ST4x4) 60 40 60 40 dirlamb (%) dirlamb (%) 80 20 20 length 0 0 80 85 90 95 100 Surface reflectance, RS (%) Fig. 4.52. Lambertian absorbance and transmittance, as a function of surface reflectance, for three straight solar tunnels of different length. This is why the absorption on the wall can be high even though the reflectivity is high. Similar trends occur with the bent solar tunnels of Fig. 4.51 (see Fig. 4.53). Now, the Lambertian absorbance curves are a little shifted upwards and those of Lambertian transmittance are a little shifted downwards, due to the effects of the bends. We find that, even in the presence of bends in the tunnels, we have no back reflected rays. The average percentage loss observed for the Lambertian transmittance due to the presence of one, two or three bends, is: ≈ 4 %, ≈ 5 % and ≈ 6 %, respectively. This loss is not so relevant for high transmittance values, rather it is for low ones, for which there may be an absolute loss of ≈ 50 %. Naturally, these losses constitute gains for the Lambertian absorbance, since the two quantities are complementary. From the Figs. 4.52 and 4.53 we can see that, to have a Lambertian transmittance well above 50 %, it is necessary to have solar tunnels 227 Advances in Optics: Reviews. Book Series, Vol. 5 with very high surface reflectance, at least of 98 %, as in fact reported for the optical characteristics of commercial solar tunnels (see also the 3M Radiant Mirror cited in Section 4.7.1). From Fig. 4.52 we can see again that the effect that the solar tunnel elongation has on the Lambertian transmittance and absorbance is less when moving from ST4×3 to ST4×4, compared to when moving from ST4×2 to ST4×3. The same trend is observed in Fig. 4.53 when moving from two to three bends, compared to when moving from one to two bends. Finally, we report in Fig. 4.54 the direct Lambertian conductance, 𝑙𝑎𝑚𝑏 𝐺 𝑑𝑖𝑟 , calculated for real bent solar tunnels with internal radius of 10 cm, the same described in Section 4.7.1, and of the same shape as the model solar tunnels illustrated in Fig. 4.51. These real tunnels, taking into account the due proportions, will have lengths of: 1.914 m, 3.028 m and 4.142 m. length 100 100 80 alpha(3 bends) alpha(2 bends) alpha(1 bend) tau(1 bend) tau(2 bends) tau(3 bends) 60 40 60 dirlamb (%) dirlamb (%) 80 40 20 20 length 0 0 80 85 90 95 100 Surface reflectance, RS (%) Fig. 4.53. Lambertian absorbance and transmittance, as a function of surface reflectance, for three bent solar tunnels of different length. 10 Gdirlamb (m2*sr) 8 ST4x2(1 bend) ST4x3(2 bends) ST4x4(3 bends) R(tunnel) = 10 cm 6 length 4 2 0 80 85 90 95 100 Surface reflectance, RS (%) Fig. 4.54. Lambertian conductance, as a function of surface reflectance, for three bent real solar tunnels of different length. 228 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications A more effective way of studying how 90° bends change the Lambertian transmittance properties of a solar tunnel is to start from a solar tunnel of a certain length and then insert one, two and three bends along its path, at equal distances, keeping its total length the same. Starting from the usual solar tunnel model with a radius of 12 mm, the total length chosen is 497.1 mm, that of a solar tunnel made up of four ST4 joined by three 90° bends (see Fig. 4.51(c)). Considering that this bent solar tunnel and the straight one, of the same length, were already been simulated, it remains to simulate two other solar tunnels with one and two bends, as function of the reflectance of a specular surface from 80 % to 100 %. We report here only the effect that the bends have on the Lambertian optical conductance of the straight solar tunnel. In this way the real dimensions of the ST do not count, but only its shape, that is the ratio between its total length and the internal radius. In Fig. 4.55 it is shown the relative Lambertian optical conductance of model solar tunnels with R = 12 mm and l = 497.1 mm, equivalent to real solar tunnels with R = 10 cm and l = 4.142 m. The graph of Fig. 4.55 confirms what found previously, namely that the main effect on the reduction of conductance is given by the first bend, while the following two have a gradually smaller effect. Furthermore, the reflectance of the surface plays a fundamental role, since even with relatively high reflectances of 90 % strong reductions in conductance are obtained, and it is necessary to rise to 98 % the reflectance value to contain bending losses below 10 %. 100 Gdirlamb (rel) (%) 80 60 ST(straight) ST(1 bend) ST(2 bends) ST(3 bends) R(tunnel) = 10 cm l(tunnel) = 4.142 m 40 20 0 80 85 90 95 100 Surface reflectance, RS (%) Fig. 4.55. Optical Lambertian conductance of solar tunnels with same length, provided with one, two or three bends, calculated respect to that of a straight solar tunnel as a function of surface reflectance. 4.8. Some Insights into the Law of Optical Lambertian Conductance Eq. (4.25):  𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) = 𝐺 𝑙𝑎𝑚𝑏  𝐿, (4.25) 229 Advances in Optics: Reviews. Book Series, Vol. 5 where 𝐺 𝑙𝑎𝑚𝑏 is given by: 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏 𝐺 𝑙𝑎𝑚𝑏 =   𝐴1   12 =   𝐴2   21 , (4.24) 𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏  21 = 𝐶𝑔𝑒𝑜   12 , (4.15) 𝑙𝑎𝑚𝑏 (𝜆) 𝑙𝑎𝑚𝑏 = 𝐶𝑔𝑒𝑜   12 (𝜆),  21 (4.44)  𝑛𝑒𝑡 (𝜆) = 𝐺 𝑙𝑎𝑚𝑏 (𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆)] (4.45) 𝑙𝑎𝑚𝑏 (  𝑛𝑒𝑡 (, 𝜆) =   𝑠𝑖𝑛2 ()  𝐴1   12 , 𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆)] (4.46) 𝑙𝑎𝑚𝑏 ( , 𝜆) 𝐺 𝑙𝑎𝑚𝑏 (, 𝜆) =   𝑠𝑖𝑛2 ()  𝐴1   12 (4.47) 𝑙𝑎𝑚𝑏 (𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆) ]}  𝑛𝑒𝑡 (𝑠) =   𝐴1  {∫𝑠 𝑑𝜆   12 (4.48) and: is a generic expression that does not take into account the wavelength of the Lambertian light. If then we still hold expression (4.15) valid for any wavelength, we can write: In this case, Eq. (4.25) becomes: However, we can introduce a more general form of Eq. (4.45), which takes into account the fact that the angular divergence of the incoming light into (1) and (2) optical terminals can be different from the linear angle ±/2, and in general equal to ± (°). The law of Lambertian optical conductance becomes: with: The net flux expression does not take the canonical form of Eq. (4.25) when the light entering the optical terminals is distributed on a spectrum s, even if a term equal to the difference between the two radiances appears in it. The net flux takes in fact the form: 4.9. Conclusions This work originates from previous research in which the effects of the Lambertian irradiation, mainly with 90° angular divergence, of a solar concentrator (SC), were experimented. In particular, the inverse Lambertian irradiation, that is the irradiation of the opening normally used as exit opening of the concentrated light, has given rise to a new technique for characterizing the solar concentrators, called ILIM (Inverse Lambertian Irradiation Method). In previous works, this method was identified with the acronym ILM (Inverse Lambertian Method). The theoretical study that followed this technique has led to the definition of several new optical quantities. The reason is simple: SCs are used to concentrate a quasi-collimated bundle of light, and therefore the use of a Lambertian 230 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications radiation, instead of a collimated one, has found a free field from which new optical properties have arisen. In this chapter, the previous theoretical work is further developed up to the formalization of an optical law, called the “Optical Bidirectional Lambertian Conductance” law, or simply the “Optical Lambertian Conductance” law, which allows to treat in a new way the issues of optical transmittance of a solar concentrator. Basically, the Lambertian radiation, when applied to optical systems for which the principle of optical reversibility applies, allows us to introduce the concept of optical conductance, that is common in the field of heat and electrical transmission. Just as, in the field of heat transmission, thermal conductivity relates the heat flux to a temperature difference, and in electrical transmission the electrical conductance relates the current flow to the electric potential difference, in the Lambertian optical transmission of light the bidirectional Lambertian conductance relates a flux of radiation to a difference of radiance. The law of bidirectional Lambertian conductance is demonstrated in this work using simple optical elements such as light cones, shaped as truncated cones. Other optical elements studied here have been nonimaging solar concentrators of the CPC type (Compound Parabolic Concentrators) and solar tunnels, of cylindrical shape. For each of these elements, chosen with different characteristics of shape and size, numerous optical simulations were performed in which their optical transmittance, reflectance and absorbance were calculated under direct and inverse Lambertian irradiation, and as a function of the reflectivity of the specular or diffusive internal surface. Among the relevant results, not obvious, we can mention the invariability of reflectance and absorbance between the various ideal CPCs of different acceptance angles, when their surface is diffusive, while this invariability in the light cones is less pronounced. The Lambertian optical conductance was then calculated on these optical elements, and is used in the law of conductance to calculate the net flux crossing the optical element when the difference in radiance at its optical terminals is known. A separate study consisted in the simulation of the angular distribution of the inverse radiance, that is the radiance emitted when the smallest aperture (of CPCs or LCs) or one of the two apertures of STs, is irradiated with Lambertian light with 90° angular divergence. In this regard, it should be noted that the other condition that a Lambertian light beam must satisfy, in order to reproduce the data reported in this work, is that of being unpolarized light. Results of this type for CPCs are widely known, and here I recall the fact that the emitted inverse radiance has the typical step-like profile. Regarding the LCs, it has been found that, in a series of them characterized by the same geometric concentration ratio, but of different lengths, they show, beyond a critical length, an angular distribution of the emitted inverse radiance with a step-like profile, as in the CPCs, while maintaining almost unchanged, and this is the novelty, the angular interval of the distribution. This behavior has been confirmed by other three series of simulations. No relevant results were found for the solar tunnels. The last part of this work was devoted to possible practical applications of the theoretical and simulated results. An application that refers to the just mentioned law consisted in the calculation of the net flux crossing a solar tunnel applied on the roof of a building and connected with an internal environment. The net flux was estimated for a conventional solat tunnel exposed to four different sky conditions, from clear to full covered. The second application concerned the use of the results obtained on the angular distribution of the inverse radiance. The simulation results allowed us to design light sources characterized by a constant radiance and with the bundle of light contained within a very 231 Advances in Optics: Reviews. Book Series, Vol. 5 precise angular limit. These light sources can be created with CPCs or LCs suitably equipped with a conventional light source and one or two integrating spheres. As a last application, we have studied the effects that one, two or three bends at 90° have on the Lambertian absorptance, transmittance and conductance of solar tunnels. Acknowledgements I dedicate this work to the memory of Prof. Giuliano Martinelli, former director of the Physics Department of the University of Ferrara, and then Director of the Sensors and Semiconductors Laboratory, who, with his great generosity, foresight and managerial skills, opened his laboratories to other research institutions, allowing me to work with him, as an ENEA researcher associated at his university. Special thanks, never enough, go to my wife Antonietta, who, in the difficult period of pandemic, supported me morally, allowing me to dedicate all my energies exclusively to this work. References [1]. D. Vincenzi, A. Busato, M. Stefancich, G. Martinelli, Concentrating PV system based on spectral separation of solar radiation, Physica Status Solidi, Vol. 206, Issue 2, 2009, pp. 375-378. [2]. D. Vincenzi, S. Baricordi, M. Occhiali, M. Stefancich, A. Parretta, G. Martinelli, Measurement of Sun-Tracking Accuracy and Solar Irradiance through Multispectral Imaging, in Proceedings of Imaging and Applied Optics Congress, 2010, pp. 1-3. [3]. D. Vincenzi, M. Stefancich, S. Baricordi, M. Pasquini, F. Gualdi, G. Martinelli, A. Parretta, A. Antonini, Effects of Irradiance Distribution Unevenness on the Ohmic Losses of CPV Receivers, in Proceedings of the 24th European Photovoltaic Solar Energy Conference and Exhibition (EU PVSEC’09), Hamburg, Germany, 21-25 September 2009, pp. 725-728. [4]. A. Parretta, G. Martinelli, M. Stefancich, D. Vincenzi, R. Winston, Modelling of CPC-based photovoltaic concentrator, Proceedings of SPIE, Vol. 4829, Issue 1, 2002, pp. 1045-1047. [5]. G. Martinelli, M.A. Butturi, C. Malagù, M. Stefancich, D. Vincenzi, A. Antonini, F. Bizzi, A. Parretta, R. Winston, Composite dish systems for high efficiency concentrator photovoltaics, in Proceedings of the 4th ISES Europe Solar Congress (EuroSun’02), Bologna, Italy, 23-26 June 2002. [6]. A. Parretta, Optical Methods for the characterization of PV solar concentrators, Chapter 13, in Advances in Optics: Reviews (S. Y. Yurish, Ed.), Vol. 3, IFSA Publishing S. L., 2018, pp. 316-416. [7]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, M. Armani, A. Parretta, N. Baggio, Rondine® PV concentrators: Field results and developments, Progress in Photovoltaics, Vol. 17, 2009, pp. 451-459. [8]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, A. Parretta, N. Baggio, Rondine PV concentrators: Field results and innovations, in Proceedings of the 24th European Photovoltaic Solar Energy Conference (EU PVSEC’09), Hamburg, Germany, 21-25 Sept. 2009. [9]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, A. Parretta, N. Baggio, Rondine PV concentrators: Field results and progresses, in Proceedings of the 34th IEEE Photovoltaic Solar Energy Conference (PVSC’09), Philadelphia, US, 7-12 June 2009, pp. 001043-001047. 232 Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications [10]. A. Antonini, M. A. Butturi, P. Di Benedetto, E. Milan, D. Uderzo, P. Zurru, M. Stefancich, A. Parretta, M. Armani, N. Baggio, Field testing per la tecnologia di concentratori a disco e per moduli “Rondine”, in Proceedings of the “ZeroEmission Rome 2008” Conference, Rome, Italy, 1-4 October 2008 (in Italian). [11]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, M. Armani, A. Parretta, N. 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A. Parretta, A. Antonini, M. Butturi, P. Zurru, Optical simulation of Rondine® solar concentrators by two inverse characterization methods, Journal of Optics, Vol. 14, 2012, 125704. [23]. A. Parretta, A. Antonini, E. Bonfiglioli, M. Campa, D. Vincenzi, G. Martinelli, Il metodo inverso svela le proprietà dei concentratori solari, PV Technology, Vol. 3, Agosto/Settembre 2009, pp. 58-64 (in Italian). [24]. A. Parretta, A. Antonini, E. Milan, M. Stefancich, G. Martinelli, M. Armani, Optical efficiency of solar concentrators by a reverse optical path method, Optics Letters, Vol. 33, 2008, pp. 2044-2046. 233 Advances in Optics: Reviews. Book Series, Vol. 5 [25]. A. Parretta, A. Antonini, M. Stefancich, G. Martinelli, M. Armani, Optical characterization of CPC concentrator by an inverse illumination method, in Proceedings of the 22nd European Photovoltaic Solar Energy Conference (EU PVSEC’07), Fiera Milano, Milan, Italy, 3-7 Sept. 2007. [26]. A. Parretta, A. Antonini, M. Stefancich, G. 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Part II: Models of light collection of 3D-CPCs under direct and collimated beams, International Journal of Optics and Applications, Vol. 3, Issue 5, 2013, pp. 72-102. [31]. A. Parretta, Optics of solar concentrators. Part III: Models of light collection of 3D-CPCs under direct and Lambertian irradiation, International Journal of Optics and Applications, Vol. 5, Issue 3, 2015, pp. 82-102. [32]. A. Parretta, A. Antonini, Optics of solar concentrators. Part IV: Maps of transmitted and reflected rays in 3D-CPCs, International Journal of Optics and Applications, Vol. 7, Issue 1, 2017, pp. 13-30. [33]. A. Parretta, L. Zampierolo, D. Roncati, Theoretical aspects of light collection in solar concentrators, in Proceedings of the Conference on Optics for Solar Energy (SOLAR’10), The Westin La Paloma, Tucson, AZ, USA, 7-10 June 2010, p. STuE1. [34]. A. Parretta, E. Bonfiglioli, L. Zampierolo, M. Butturi, A. 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Aperiodic Zone Plates for Optical Tweezers and Optical Imaging Chapter 5 Aperiodic Zone Plates for Optical Tweezers and Optical Imaging Tian Xia, Shubo Cheng, Shaohua Tao1 5.1. Introduction Since the aperiodic sequences were introduced to design new kinds of zone plates (ZPs) [1], the aperiodic zone plates have found many special applications, e.g. lithograph [2], X-ray microscope [3], optical encryption [4], THz technology [5], optical tweezers [6] and optical imaging [7, 8]. The aperiodic zone plates have a series of foci along the optic axis, which form multiple optical traps used to trap particles at the different planes simultaneously [9]. Although a single beam designed by a beam shaping algorithm can generate multiple traps in threedimensional space, distortion made by reflection, refraction, or absorption of a trapped particle would affect the subsequent propagation of the beam. To trap multiple particles separately distributed in three-dimensional space, one can utilize a beam with tailorable three-dimensional intensity distribution and the self-reconstruction property [10, 11]. The Bessel beams and the Airy beams were used to simultaneously trap multiple particles [1214], but positions of the trapped particles could not be customized owing to the nondiffracting intensity profiles of the beams in the propagation direction. Recently, fractal zone plates (FZPs) were proposed to form two-dimensional photonic structures [1]. The fractal pattern has the self-similarity when the FZP is illuminated with a plane wave [15]. Besides having one main focus and other weaker foci, i.e., the so-called major foci, an FZP beam also has many subsidiary foci around the main focus and major foci along the optical axis. Moreover, as focal lengths of multiple foci provided by an FZP beam are determined by the structural parameters of the FZP, the axial distances between adjacent foci of the beam can be finely controlled with a fractional number of the segments of the fractal structure [16]. Thus, with the FZP beams one can realize not only the simultaneous trapping of multiple particles in different planes but also the customized trapping locations of trapped particles. Although optical trapping with a beam generated by the Devil's lens Shaohua Tao School of Physics and Electronics, Central South University, Changsha 410083, China 237 Advances in Optics: Reviews. Book Series, Vol. 5 characterized with a fractal structure was realized [17], the simultaneous manipulation of particles at different axial positions has not been reported. The aperiodic zone plates have all kinds of foci used for optical imaging. There are some zone plates which generate the high-intensity foci, i.e., a conventional composite Fresnel zone plate consists of a central zone plate with first-order focal length and outer zones with third-order focal length, which generate the high-intensity focus with the improved resolution [18]. A composite photon sieve can also suppress sidelobes, and produces a focus with slightly better resolution [19]. Moreover, composite fractal zone plates have the single high-intensity main foci with many subsidiary foci [20]. Composite fractional fractal zone plates generate the tailorable main foci with high intensity and subsidiary foci [21]. In addition, composite Thue-Morse zone plates generate two high-intensity foci with many subsidiary foci, which are applicable to produce two images with the low chromatic aberration [22]. Furthermore, modified Thue-Morse zone plates based on the Thue-Morse sequence of S = 3 provide two high-intensity foci located at the positions satisfying the fixed ratio of 3/5 [23]. Another interesting way to produce the high-intensity foci is to implement devil's lenses, which are the fractal zone plates with the gradient phases and have the high-intensity main foci with many subsidiary foci [24]. Moreover, kinoform generalized mean lenses generate twin high-intensity main foci with the generalized mean and the suppressed high-order foci along the optic axis [25]. Nevertheless, for the above composite zone plates, the first-order diffraction areas of the inner and outer parts are not coincident. Some radially modulated zone plates are used to acquire annular beams at the focal planes. For example, Fresnel zone plates with radially shifted phases create annular beams with designed radii at the focal planes [26]; radial phase modulated spiral zone plates generate optical vortices with tailorable radii at the focal plane [27]; annular beams produced by modified spiral zone plates have tailorable radii and segmented phase gradients [28]. However, these zone plates cannot produce high-intensity foci at the focal planes used for generating the clear images. Some photon sieves generate the first-order foci with suppressed high-order foci [29]. For example, fractal photon sieves have the low-intensity high-order foci and single main foci with many subsidiary foci in the firstorder focal area, which are applicable to produce the colourful images with the low chromatic aberration [30]; lacunar fractal photon sieves whose lacunarity is used to adjust the intensities of axial foci can also provide the self-similar foci in the first-order diffraction area and suppressed high-order foci [31]; generalized Fibonacci photon sieves based on the different initial seeds and iteration rules generate two equal-intensity foci with many ratios in the first-order focal area and low-intensity high-order foci [32]. However, the high-order foci for photon sieves still have low intensities, and these photon sieves cannot generate the first and second order foci only. The single first-order foci can be created by many single-focus zone plates. There are stagger-arranged zones for novel single-focus zone plates (SFZPs) [33] and single-focus photon sieves [34], whose transmittances are cosinusoidal to produce the first-order foci only. Annulus-sectorelement coded Gabor zone plates consist of annulus-sector-shaped nanometer structure apertures, so that this kind of zone plates have a cosinusoidal transmittance to generate the first-order foci only [35]. A single-focus spiral zone plate utilises the modified transmittance function based on the two-parameter modified sinusoidal apodization window to realise the single first-order focus along the optic axis [36]. Nevertheless, these 238 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging zone plates cannot generate the single main foci with many subsidiary foci and two equalintensity main foci with many subsidiary foci used to create the single and two colourful images, respectively. In this chapter, the aperiodic zone plates are used for optical tweezers and optical imaging. Firstly, FZP beam can simultaneously trap multiple particles positioned in different focal planes of the FZP beam. The dynamic manipulation of microparticles in optical tweezers will be implemented by changing Ns of the fractional FZPs. The symmetrical major foci around the main focus of the Thue-Morse zone plate (TMZP) [37] have the same intensity so that the microparticles can be trapped stably. Fibonacci zone plate (FiZP) [38] can simultaneously trap particles positioned in two different planes and move particles freely in the focal plane. The experimental results demonstrate that the FZP beam, the fractional FZP beam, the TMZP beam and the FiZP beam can achieve the simultaneous manipulation of particles at different axial positions. Secondly, generalized composite aperiodic zone plates (GCAZPs) are proposed to generate clearer images at focal planes. A modified single-focus fractal zone plate (MSFFZP) is proposed to generate the single main focus with many subsidiary foci or two equal-intensity main foci with many subsidiary foci, which are used to achieve optical colourful imaging. It is proved in the experiments that the GCAZP has the clearer images at the focal planes, and the MSFFZP produces the single or two colourful images with the low chromatic aberrations at the focal planes. 5.2. Theory of Aperiodic Zone Plates 5.2.1. Aperiodic Sequence Aperiodic sequences have self-similar properties and no connection with disorder. Compared with the periodic sequences, the aperiodic sequences have more special regular properties. The aperiodic sequences are a kind of substitution sequences, and constructed by unified substitution rules, which make the aperiodic sequences have the self-similarity. The aperiodic sequence is first an alphabetic system, which consists of a certain number of letters. Different letters are corresponding to the different substitution rules substituting the letters into the strings of letters. Based on the substitution rule, the initial sequence is substituted to generate the aperiodic sequence. Thus, this unified substitution rule makes the aperiodic sequence have the self-similarity. For example, the initial seeds of the Cantor, Thue-Morse and Fibonacci sequences are all ‘A’. The corresponding substitution rules are {A→ABA, B→BBB}, {A→AB, B→BA} and {A→AB, B→A}, respectively. 5.2.2. Method of Constructing Aperiodic Zone Plates The aperiodic zone plate can be constructed by the transmittance function NS     j  1 2  dS  q     tS , j  rect  , dS j 1   q   [37] (5.1) 239 Advances in Optics: Reviews. Book Series, Vol. 5 where    r a  , r is the radial coordinate of the zone plate, a is its maximum value, 2  0,1 is the normalized square radial coordinate, S is the order, and NS is the total count of the elements for the modified Fibonacci sequence of the S-th order. In Eq. (5.1), tS , j is “1” or “0” when the j-th element of the corresponding modified Fibonacci sequence is “A” or “B”, respectively. Thus, the sequence of each order consists of NS equal parts, and each part has the same length of dS  1 NS . rect[] presents the rectangular function, which is defined as  0 rect t     1 t  0.5 t  0.5 . (5.2) The binary functions of the amplitude-only and phase-only zone plates can be calculated by q   and q     , respectively. The aperiodic square zone plate can be generated with a transmission function p(x, y) [39] written as N     x y p  x, y    t j [rect  rect     2a j / N   2a j / N  j 1        x y rect   rect    2a ( j  1) / N   2a ( j  1) / N     ],  (5.3) where N is the total number of zones, the length of the outer zone is 2a, (x, y) is the Cartesian coordinate, and tj is “1” or “0” when the j-th element of the corresponding sequence is “A” or “B”, respectively. The aperiodic lens is defined with a linear variation between 0 and 2 at each subinterval {AB} of the aperiodic sequence, being 0 otherwise, and can be calculated in Eq. (5.4). m 1 n m 1  )/(  ) 2  (  F ( )   k k k 0 m 1 n   k k, others (5.4) where k is the number of elements of the aperiodic sequence. m and n are the ordinal numbers of two elements A and B for each subinterval {AB} of the corresponding aperiodic sequence, respectively. The transmittance function t (r, ) of the aperiodic lens is defined by q( , S )  exp[iF ( )], 240 (5.5) Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging The binary transmittance function tFSZP(r2, θ) of the aperiodic spiral zone plate can be calculated by Eq. (5.6).   r2     (3  2 j ) 1 (2 2 )   j L  1 , t FSZP (r 2 , )   f 0 else  where     3 / 2, f  (5.6) a2 , the wavelength λ is set as 532 nm, (r, θ)  (2 L1  2 M  3) represents the polar coordinate, L1 presents the topological charge and j presents the sequence number of the high-transmittance spiral zone. In fact, j is equal to the sequence number of the element “1” for the aperiodic sequence with the maximum of M. It should be noted that j is equal to -L1, -(L1-1), …-1 or 0 to make the first spiral zone complete. The transmittance of the spiral aperiodic zone plate can be obtained by [40] (5.7) t ( , )  q( )exp[im ], where m represents the topological charge. 5.2.3. Theory of Diffraction The diffraction distributions of aperiodic zone plates can be calculated by two means. One method is the Fresnel integral formula to find the analytical solution, and the other method is the angular plane-wave spectrum theory or the fast Fourier transform-based FresnelKirchhoff integral to find the numerical solution. The Fresnel approximation expressed in Eq. (5.8) [24] is used to calculate the crosssection intensity distribution illuminated with the monochromatic planar wave.  2  I  z, r      z  2 2  a 0   2   2 r0 r  p  r0  exp  i r0 J 0   r0 dr0 ,  z   z  (5.8) where the rotation invariant transmittance function of the aperiodic zone plate is p(r0), r0 is the radial coordinate of zone plate, λ and z present the wavelength and axial distance, respectively, r is radius at the diffraction plane, and J0() the Bessel function of the first kind of order zero. After the pupil function p(r0) in Eq. (5.8) is Hankel transformed, and Eq. (5.9) can be obtained [24]. I (u, v)  4 2u 2  1 0   2 q( )exp( i 2 u ) J m 4  uv d  , (5.9) 241 Advances in Optics: Reviews. Book Series, Vol. 5 where u  a 2 （ / 2 z）is the reduced axial coordinate, v = r/a is the normalized transverse coordinate. When v is equal to zero, the axial intensity distribution can be given in Eq. (5.10), I (u )  4 2u 2  1 0 q( )exp( i 2 u ) d  2 (5.10) Combining Eqs. (5.1) and (5.10) one can obtain Eq. (5.11) NS I (u )  i  tS , j (exp(i 2 ujN s )  exp(i 2 u ( j  1) N s )) 2 (5.11) j 1 Eq. (5.11) can be used to calculate the intensity at the arbitrary axial position. The transmittance of the aperiodic zone plate can be expressed by a matrix of Tp comprising 0 and  . The angular plane-wave spectrum theory expressed in Eqs. (5.12) and (5.13) [41] is used to analyze the focusing properties of the aperiodic zone plate, In Eq. (5.12), E p  iFT  FT Tp  H  , (5.12) 2 2  1  x  y  H ( x, y )  exp i 2 z 2         L L     (5.13) E p is the complex amplitude of the diffracted beam, FT , iFT , and H represent the Fourier transform, the inverse Fourier transform, and the transfer function, respectively. In Eq. (5.13), z ,  , and d are the propagation distance, the wavelength of light, and the maximum size of the ZP, respectively. x and y are the dimensionless coordinates of the sampling grids. The diffraction complex amplitude U ( R, , z) of the phase-only aperiodic zone plate expressed by a matrix of t (r, ) comprising 0 and π can be calculated by the fast Fourier transform-based Fresnel-Kirchhoff integral expressed in Eq. (5.14) [42]. U ( R, , z)  FT -1  FT [exp(i  t (r, ))]FT [h(  )], (5.14) where ( R, , z), FT , FT -1 and h(  )  exp(ikz) / (i z)  exp(ik  2 / (2 z)) present the cylindrical coordinate at the diffraction plane z, the Fourier transform, the inverse Fourier transform and the free-space impulse response [43], respectively. 242 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging 5.3. Aperiodic Zone Plates for Optical Tweezers 5.3.1. Fractal Zone Plate As an example, the construction procedure of a regular Cantor fractal structure is shown in Fig. 5.1(a) [1]. The segment is divided into an odd number of segments 2N-1 and the segments in the even-number positions are removed. Fig. 5.1(b) shows the associated FZP generated from the fractal structure with S = 3. Fig. 5.1. (a) Schematic of the one-dimensional fractal structure based on the Cantor sets, and (b) the associated FZP generated from the one-dimensional fractal structure with S = 3. The focal length of the main focus of an FZP beam can be expressed by Eq. (5.15) f (a, λ, N, S) = a2·[λ·(2N – 1)S]-1, (5.15) where a is the radius of the FZP, λ is the wavelength of the illuminating light, N is the number of the segments forming the fractal structure, and S is the fractal stage. In Eq. (5.15), when the FZP is illuminated with a collimated laser beam with a wavelength of 532 nm, the focal length of the main focus of the FZP with N = 3 and S = 3 is f = 222 mm. The axial irradiance of the FZP is shown in Fig. 2(a), where multiple foci with internal fractal properties along the z direction can be observed, and the main focus has the highest intensity and a distance of ~222 mm from the FZP. Furthermore, the additional foci surrounding the main focus are at distances of ~213 mm and 230 mm, respectively. Fig. 5.2(b) shows the calculated intensity distributions of the FZP beam in the x-z plane. In Fig. 5.2 the peaks around the main focus are marked as the first and second subsidiary foci, respectively. In the trapping experiments polystyrene beads with a diameter of about 3 μm were immersed in deionized water of refractive index of 1.33 and used as the manipulating objects. The binary phase-only FZP with N = 3 and S = 3 was loaded onto a spatial light modulator (SLM) to generate the FZP beam. We had added phase distribution of a blazed grating to the phase-only FZP for an off-axis output. The computer-generated hologram is shown in Fig. 5.3(a). The CCD-captured intensity distributions of the first subsidiary 243 Advances in Optics: Reviews. Book Series, Vol. 5 focus, the main focus, and the second subsidiary focus are shown in Fig. 5.3(b-d), respectively. Fig. 5.2. (a) Axial irradiance of the FZP with N = 3 and S = 3, and (b) the calculated intensity distribution of the FZP in the x-z plane, where the main focus and the two additional subsidiary foci are illustrated. Fig. 5.3. (a) Phase distribution of the FZP of N = 3 and S = 3, and (b-d) CCD-captured intensity distributions of the first subsidiary focus, the main focus, and the second subsidiary focus of the FZP beam at propagating distances of 21 cm, 22 cm, and 23 cm, respectively. In order to conveniently observe the 3D trapping of particles in multiple focal planes in the experiment, we arrange the optical path shown in Fig. 5.4(a), where the beam enters the sample cell at a small oblique angle. In Fig. 5.4(a) the dashed green line represents the collinear arrangement of optical components, and the red solid line shows the configuration of our trapping experiment. Fig. 5.4(a) shows that the focus of the objective locates just in the sample cell and the first subsidiary focus is on the focal plane of the objective, so the first subsidiary focus in the cell can be clearly observed. The observed first subsidiary focus in the view field of the CCD camera is shown in the right bottom of the square exposure region in Fig. 5.4(b). It is worth mentioning that the relative distance between the objective and the sample cell was fixed in the optical trapping. When the sample cell and the objective is shifted upward synchronously about 180 μm, the main focus shown in Fig. 5.4(c) can be observed clearly in the view field. Although the beam 244 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging has been shifted relatively toward the bottom of the sample cell, a dim image of the first subsidiary focus on the right of the main focus can still be observed in Fig. 5.4(c). The sample cell and the objective are further shifted upward synchronously until the second subsidiary focus appears clearly. The CCD-captured image of the second subsidiary focus is shown in Fig. 5.4(d). Similarly, a dim image of the main focus exists on the right of the second subsidiary focus. From Figs. 5.4(b-d) we can also observe that the locations of the three foci shift toward a fixed direction when the sample cell and the objective shift upward synchronously. Obviously, the displacement of the foci in the view field is resultant from the oblique incidence of the beam ray. When the first subsidiary focus is observed clearly in the view field and then the sample cell and the objective is shifted upward synchronously about l μm, the second subsidiary focus can be observed clearly. If the second subsidiary focus shifts a distance of d μm from the first subsidiary focus in the view field, the incident angle of the FZP beam on the sample cell can be expressed approximately by arctan(d/l). Fig. 5.4. (a) Schematic of the experimental setup, and (b-d) captured CCD frames of intensity distributions of the first subsidiary focus, the main focus, and the second subsidiary focus, respectively. The dashed arrows in (b-d) represent the shifting directions of the foci with synchronous shifts of the sample cell and the objective. In the experiment the sample cell and the objective were shifted upward synchronously until the first subsidiary focus was observed clearly in the view field of the CCD camera. Then the sample cell moved horizontally until particles appeared in the view field. When 245 Advances in Optics: Reviews. Book Series, Vol. 5 the laser power was about 800 mW, the polystyrene bead marked as 1 was trapped by the first subsidiary focus, and the trapping images are shown in Fig. 5.5(a-c). The particles highlighted with the white dashed rectangles are used as the reference. When the particle 1 had been trapped by the first subsidiary focus, the sample cell and the objective were shifted slowly upward synchronously. In the mean time we observed that the particle 1 trapped by the first subsidiary focus was shifted toward the bottom of the sample cell with the first subsidiary focus, and then the CCD images of the trapped particle 1 became blurred. The trapping images are shown in Fig. 5.5(c, d). When the sample cell and the objective were shifted upward synchronously until the main focus appeared on the focal plane of the objective, a cluster of particles marked as 2 was trapped. The trapping images are shown in Fig. 5.5(d-f). Then the sample cell and the objective were further shifted upward synchronously, we observed that the cluster of trapped particles was shifted toward the bottom of the sample cell with the main focus, and the particle 1 trapped by the first subsidiary focus was also attracted to the main focus for the higher intensity of the main focus. CCD images of the particle 1 trapped by the main focus are shown in Fig. 5.5(f-h). The sample cell and the objective were further shifted upward synchronously until the second subsidiary focus appeared on the focal plane of the objective. At that time the particle marked as 3 was trapped by the second subsidiary focus. The trapping images are shown in Fig. 5.5(g-i). Fig. 5.5. Experimental demonstration of simultaneous trapping with three foci of the FZP beam. (a-c) Trapping of the particle marked as 1 with the first subsidiary focus, (d-f) trapping of a cluster of particles marked as 2 with the main focus, and (g-i) trapping of the particle marked as 3 with the second subsidiary focus and the simultaneous trapping of the particles marked as 1 and 2 with the main focus, respectively. 246 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging From Fig. 5.5(i) we can find that the cluster of particles is still trapped by the main focus in the bottom of the sample cell. Due to the limited depth of the sample cell and short work distance of the objective, when the second subsidiary focus is located on the focal plane of the objective, the first subsidiary focus has moved out of the sample cell. Nonetheless, we can observe simultaneous trapping of particles by the second subsidiary focus and the main focus in the view field from Fig. 5.5(i). Hence, particles can be manipulated with the multiple foci in different planes. In the trapping experiments, the distance between the trapping positions of the first subsidiary focus and the second focus in the view field of the CCD camera was about 25 μm. When the first subsidiary focus appeared clearly in the view field, the sample cell and the objective were shifted upward synchronously about 380 μm until the second subsidiary focus appeared. Thus, the incident angle on the sample cell was about 0.066 radian. Figs. 5.6(a) and (b) show an example FZP with N = 3 and S = 3, and an example fractional FZP with N = 2.9 and S = 3, respectively. The axial locations of multiple foci and the axial distance between two neighbouring foci of a fractional FZP beam can be precisely adjusted by modifying the fractional number N. Fig. 5.6. FZPs with (a) N = 3 and S = 3 and (b) N = 2.9 and S = 3. The bright and dark segments correspond to the transmittances of 1 and 0, respectively. A collimated laser beam with a wavelength of 532 nm was applied in the following simulations and trapping experiments. The radii of the FZPs are set as 256 × 15 μm. The main foci in Figs. 5.7(a) and (b) have the calculated distances of 221.7 mm and 250.7 mm from the FZPs, respectively. The peaks around the respective main focus are marked as the first subsidiary focus and the second one, respectively. The average axial distance between two neighbouring foci (i.e., the main focus and subsidiary foci) of the FZP beam with N = 3 and S = 3 is about 8.5 mm and the counterpart of the fractional FZP with N = 2.9 and S = 3 is about 10.3 mm. It can be seen in Figs. 5.7(c) and (d) that when N is changed with a fractional increment, both the axial locations and the axial spacings of the foci of the FZP beam can be more precisely controlled. Thus, microparticles can be dynamically manipulated by changing Ns of the fractional FZPs. In fact, N can be modified with a variation as small as 0.001. For example, the main focus of the fractional FZP with N = 3.001 and S = 3 has shifting displacements of 0.2 mm in 247 Advances in Optics: Reviews. Book Series, Vol. 5 free space and 8 μm in the sample cell from that of the FZP with N = 3 and S = 3, respectively. Similarly, the corresponding subsidiary foci also have smaller shifting displacements. For comparisons, N can be modified with an increment of 1, e.g., from N = 2 to N = 3. The main focus of the FZP with N = 2 and S = 3 has shifting displacements of 804.9 mm in free space and 32.2 mm in the sample cell from that of the FZP with N = 3 and S = 3, respectively. Thus, microparticles can be dynamically manipulated by modifying N with an especially small fractional variation. For convenience we used the second subsidiary foci of the fractional FZPs to trap microparticles dynamically. Firstly, the FZPs with N = 3.001, 3.002, and 3.003 were loaded on the SLM sequentially every 5 seconds. When N was increased, the second subsidiary focus of the corresponding fractional FZP was shifted toward the bottom of the sample cell. At this time, it can be seen in Fig. 5.8(c)-(e) that the trapped particle becomes blurred. Then, the FZPs with N = 3.002, 3.001, and 3.0 were loaded on the SLM sequentially every 5 seconds. The second subsidiary focus of the corresponding fractional FZP would be shifted toward the top of the sample cell. The dynamic trappings are shown in Fig. 5.8 (f)-(h), where the trapped microparticle moves toward the original location and is in focus. Thus, the positions of the trapped particles can be dynamically controlled. Fig. 5.7. Axial irradiance of the FZPs with (a) N = 3, S = 3 and (b) N = 2.9, S = 3. Plots of (c) the axial location of the main focus from the SLM and (d) axial distance between the main focus and the second subsidiary focus of an FZP beam with S = 3 versus N. 248 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging Fig. 5.8. Experimental demonstration of dynamic trapping with the second subsidiary foci of the fractional FZPs with N = 3, 3.001, 3.002, and 3.003, as well as S = 3, respectively. (a) The second subsidiary focus of the FZP with N = 3 and S = 3, and (b) the trapped microparticle in the view field of the CCD camera. (c)-(e) Trapped particle becomes blurred with dynamic changing of the fractional FZPs with N increasing from 3.001 to 3.003. (f)-(h) Trapped particle becomes clear again with N decreasing from 3.002 to 3. 5.3.2. Fibonacci Zone Plate In Fig. 5.9, the tilted dashed line arrow shows the movement direction of the focus. The main reason is that the beam striking the CCD is titled. Fig. 5.9. (a) Hologram of FiZP with optical grating phase; (b) the first prime focus captured by CCD; (c) the second prime focus captured by CCD. In Fig. 5.10, the circle presents the main focal position, and the titled dashed arrow shows the movement direction of the focus. During the trapping process with the total time of 1 minute, the particle marked by the circle is from blurry to clear. When the particle is trapped stably, the particle is the clearest and on the focal plane. When the objective and sample cell move up 100 μm, the second focus is at the focal plane of the objective. In Fig. 5.11, the circle and titled dashed arrows present the second main focal position, and the movement direction of the second focus. As the depth of the sample cell is 100 μm, the particle gets rid of the trapping force of the first main focus, 249 Advances in Optics: Reviews. Book Series, Vol. 5 and is trapped by the second main focus. In Fig. 5.12, the dashed box presents the static referenced particles, and the horizontal and titled dashed arrows show the movement directions of the particle and focus, respectively. During the process when the main focus moves left 20 μm, the particle is trapped stably all the time and moves with the movement of the focus. Fig. 5.10. Longitudinal capture of particle by the first prime focus of the FiZP beam. Fig. 5.11. Longitudinal capture of the particle by the second prime focus of FiZP beam. Fig. 5.12. Capture and move of the particle in horizontal plane. 5.3.3. Thue-Morse Zone Plate In Fig. 5.13(e) and (f) v (v = x/a) is the normalized transverse coordinate. The main focal distance of the equivalent periodic ZP can be written as f = a2/(λ2S), where λ is the wavelength of incident light and λ = 532 nm, a = 1.15 mm is the radius of the TMZP, S is the order of the adoptive TM sequence. The foci of a TMZP are located around the axial point z = a2/(λ2S) symmetrically and the symmetrical foci have the same intensity. As the TMZP beams are found to have the self-reconstruction property, the trapped particles in a TMZP beam will not affect the subsequent propagation of the beam. Thus, the marked foci in Figs. 5.14(b) and (c) with the Arabic numerals are used to trap microparticles. Due to the oblique incidence of the beam ray, there exists a displacement 250 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging between the adjacent foci in the view field and the trapped microparticles by the different foci can also be seen in the view field. But when a trapped microparticle can be seen clearly in the focal plane of the objective, the others microparticles trapped in the rest foci are out of focus and cannot been clearly. The microparticle marked as 1 was trapped at the focal plane of the objective and the CCD-captured image is clear. The other marked microparticles were trapped in different planes and out of focus. Furthermore, in Figs. 5.15(a)-(f) the trapped microparticles at the foci marked as 1, 2, and 3 lined up in the view field, which can be marked with a dashed red line. In the figures the microparticle circled with the white dashed box is used as a reference. The results demonstrated that the trapped microparticles are not only trapped stably, but also can be manipulated simultaneously along a defined route in different planes. In fact, all the foci of the TMZP can trap and manipulate microparticles simultaneously in multiple planes, which will realize the 3D optical tweezers. Compared with the FZP, the experimental results demonstrate that the TMZP can be applied in 3D optical tweezers and the microparticles will be trapped more stably in the TMZP beam for the symmetry foci with the same intensity. Fig. 5.13. (a) and (b) The phase-only TMZP of order 6 and the corresponding equivalent periodic ZP, respectively; (c) and (d) the normalized axial irradiance versus the axial coordinate u for the TMZP of order 6 and the corresponding equivalent periodic zone plate with a = 1.15 mm, respectively; (e)-(f) the transverse irradiance provided by the TMZP of order 6 and the corresponding equivalent periodic zone plate with a = 1.15 mm, respectively. 251 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 5.14. (a) The phase-only TMZP of order 6 with a blazed grating, (b) the axial intensity distribution of the phase-only TMZP of order 6, (c) the axial intensity contour of the TMZP shown in (a) and (d) the optical path of the optical tweezers system. Fig. 5.15. (a)-(f) Manipulations of the trapped microparticles (1, 2, and 3) highlighted with the white arrows along the direction marked with the black arrows in the three different planes. 5.4. Aperiodic Zone Plates for Optical Imaging 5.4.1. Clear Imaging For simplicity, the GCAZP based on the Fibonacci sequence is taken for example to illustrate the construction method of the GCAZP. The GCAZP based on the Fibonacci sequence is composed of the FiZP with (r0 = 0, R = αa, S) and the Fibonacci radial zone 252 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging plate (RZP) with (r0 = βa, R = a, S). r0 and R respectively present the original radius and the outermost radius of the FiZP or the Fibonacci RZP, α is a positive number between 0 and 1/ 2, β is a positive number between 1/ 2 and 1, and S presents the order of the Fibonacci sequence. It should be noted that α and β satisfy the equation α2 + β2 = 1, and these zone plates are designed on the devices with the same size of a × a. The transmittance function qi   of the FiZP with (r0 = 0, R = αa, S), i.e., the inner part of the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (5.16), M    j  1 2  dS  qi     tS , j  rect  , dS j 1   (5.16) where    r / ( a )  represent the normalized square radial coordinate, r presenting the radial coordinate of the ZP ranges from 0 to αa, when the j-th element of the Fibonacci 2 sequence of S is “A” or “B”, the corresponding tS , j is “1” or “0”, and dS  1 M . The transmittance function qo   of the Fibonacci RZP with (r0 = βa, R = a, S), i.e., the outer part of the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (5.17), M    j  1 2  dS  qo     tS , j  rect  , dS j 1     (5.17) where    r a 2 - 2 /  2 is the normalized square radial coordinate, which has the   value between 0 and   1- 2 /  2 , and r is equal or greater than r0. It should be noted that the transmittance function of the GCAZP based on the Fibonacci sequence can be expressed as qGCAZP    qi    qo   , where    r a  is the normalized square radial coordinate. The FiZP with (r0 = 0, R = αa, S) generates the first-order focus located at the axial position z = (αa)2/(λM). λ and M are the wavelength and the number of the elements of the Fibonacci sequence of order S, respectively. The Fibonacci RZP with (r0 = βa, R = a, S) generates the first-order focus located at the axial position z = (1-β2)‧a2/(λM). Thus, as the result of the equation α2+β2 = 1, the FiZP with (r0 = 0, R = αa, S) and the Fibonacci RZP with (r0 = βa, R = a, S) generate the first-order focus with the same axial position. That is to say, the inner and outer parts have the coincident first-order diffraction area. In addition, the first-order focal axial position of the GCAZP can be adjustable by designing α and β. For simplicity, a, λ, α and β are set as 3.84 mm, 532 nm, 1/ 2 and 1 / 2, respectively. Fig. 5.16 illustrates the construction method of the GCAZP based on the Fibonacci sequence. 2 The modulation transfer function (MTF) can be obtained by the Fourier transform of the corresponding point spread function, which is calculated by the Fresnel diffraction 253 Advances in Optics: Reviews. Book Series, Vol. 5 integral formula [42]. It can be found in Fig. 5.17 (a) that the thick light and black lines are above the thin light and black lines, respectively. Thus, the images located at z = 330.05 and 532.82 mm generated by the GCAZP based on the Fibonacci sequence of S = 8 are clearer than those of the FiZP of S = 8, respectively. In Fig. 5.17 (b), the thick solid line represents the MTF at the main focal plane located at z = 495 mm of the GCAZP based on the Cantor sequence of S = 3, respectively. The thin solid line represents the MTF at the main focal plane located at z = 495 mm of the FZP of S = 3, respectively. It can be found in Fig. 5.17 (b) that the thick solid line is above the thin solid line. Thus, the image located at z = 495 mm generated by the GCAZP based on the Cantor sequence of S = 3 is clearer than that located at z = 495 mm of the FZP of S = 3. In Fig. 5.17(c), the thick light and black lines represent the MTFs at two focal planes located at z = 329.34 and 632.24 mm of the GCAZP based on the Thue-Morse sequence of S = 5, respectively. The thin light and black lines present the MTFs at two focal planes located at z = 329.34 and 632.24 mm of the TMZP of S = 5, respectively. It can be found in Fig. 5.17(c) that the thick light and black lines are above the thin light and black lines, respectively. Thus, the images located at z = 329.34 and 632.24 mm generated by the GCAZP based on the Thue-Morse sequence of S = 5 are clearer than those located at z = 329.34 and 632.24 mm of the TMZP of S = 5, respectively. Thus, the images generated by the GCAZP based on the different aperiodic sequences are clearer than those of the corresponding common aperiodic zone plate, respectively. Fig. 5.16. Binary profiles of (a) the FiZP with (r0 = 0, R = a / 2, S = 6), (b) the Fibonacci RZP with (r0 = a / 2, R = a, S = 6) and (c) the corresponding GCAZP. An imaging experimental system is shown in Fig. 5.18. A laser beam (λ = 532 nm) illuminated a beam expender, a lens and a target object (a letter U) at the focal plane of the lens, and then passed a collimator (a focal length of 550 mm and a diameter of 55 mm), a polarizer, a SLM (CAS MICROSTAR, TSLM07U-A, 1920 × 1080 pixels, 8.5 µm × 8.5 µm /pixel, and transmissive type) with the holograms of the above zone plates, and an analyzer, and was finally captured by a Complementary Metal Oxide Semiconductor (CMOS, Mindvision, MV-UBS300C). After the target object is removed from a light path, the focal intensity distribution can be captured. It can be seen in Figs. 5.19 (a), (b), (f) and (g) that the GCAZP based on the Fibonacci sequence produce two main foci with higher intensity than the corresponding FiZP. It can be seen in Figs. 5.19 (c) and (h) that the GCAZP based on the Cantor sequence produce the single main focus with higher intensity than the corresponding FZP. It can be seen in Figs. 5.19 (d), (e), (i) and (j) that the GCAZP based on the Thue-Morse sequence produce two main foci with higher intensity than the corresponding TMZP. Thus, the 254 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging GCAZPs based on the different aperiodic sequences can generate the main foci with higher intensity than the corresponding common aperiodic zone plates. It should be noted that as the diffraction of the SLM, the intensity distribution in the shorter diffraction distance has higher intensities than the same intensity distribution in the longer diffraction distance. Thus, it can be found in Fig. 5.19 that for the bifocal zone plates, the measured focus with the short focal length has higher intensity than that with the long focal length. Fig. 5.17. (a) MTFs of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of S = 8 at two main focal planes; (b) MTFs of the GCAZP based on the Cantor sequence of S = 3 and the FZP of S = 3 at the main focal plane; (c) MTFs of the GCAZP based on the Thue-Morse sequence of S = 5 and the TMZP of S = 5 at two main focal planes. Fig. 5.18. Experimental setup of the imaging system. After the target object is added to the light path, the generated images of these zone plates can be captured. It can be seen that the letters in Figs. 5.20 (a) and (b) have higher contrast than those in Figs. 5.20 (f) and (g), respectively. The main reason is that the GCAZP based on the Fibonacci sequence has twin foci with higher intensity than the corresponding FiZP. Thus, intensity differences between the letters and background in Figs. 5.20 (a) and (b) are bigger than those in Figs. 5.20(f) and (g), respectively. Therefore, the letters in Figs. 5.20 (a) and (b) are clearer. It can be seen that the image in Fig. 5.20 (c) have higher contrast than that in Fig. 5.20(h). The main reason is that the GCAZP based on the Cantor sequence has the single main focus with higher intensity than the corresponding FZP. Thus, it can be seen that intensity difference between the letter and background in Fig. 5.20 (c) is bigger than that in Fig. 5.20 (h), so that the letter in Fig. 5.20 (c) is clearer than that in Fig. 5.20 (h). It can be seen that the images in Figs. 5.20 (d) and (e) are clearer than those in Figs. 5.20 (i) and (j), respectively. The main reason is that the GCAZP based 255 Advances in Optics: Reviews. Book Series, Vol. 5 on the Thue-Morse sequence has twin foci with higher intensity than the corresponding TMZP. Thus, intensity differences between the letters and background in Figs. 5.20 (d) and (e) are bigger than those in Figs. 5.20 (i) and (j), respectively. Therefore, the letters in Figs. 5.20 (d) and (e) are clearer. Thus, the GCAZPs have clearer images than the corresponding common aperiodic zone plates. Fig. 5.19. Intensity distributions at z ≈ 330 and 530 mm of (a) and (b) the GCAZP based on the Fibonacci sequence of S = 8. Intensity distributions at z ≈ 330 and 530 mm of (f) and (g) the FiZP of S = 8. Intensity distributions at z ≈ 495 mm of (c) the GCAZP based on the Cantor sequence of S = 3 and (h) the FZP of S = 3. Intensity distributions at z ≈ 330 and 630 mm of (d) and (e) the GCAZP based on the Thue-Morse sequence of S = 5. Intensity distributions at z ≈ 330 and 630 mm of (i) and (j) the TMZP of S = 5. Fig. 5.20. Captured images at z ≈ 330 and 530 mm of (a) and (b) the GCAZP based on the Fibonacci sequence of S = 8. Captured images at z ≈ 330 and 530 mm of (f) and (g) the FiZP of S = 8. Captured images at z ≈ 495 mm of (c) the GCAZP based on the Cantor sequence of S = 3 and (h) the FZP of S = 3. Captured images at z ≈ 330 and 630 mm of (d) and (e) the GCAZP based on the Thue-Morse sequence of S = 5. Captured images at z ≈ 330 and 630 mm of (i) and (j) the TMZP of S = 5. 256 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging 5.4.2. Colourful Imaging The mth high-transmission zone for the MSFFZP consists of the upper and bottom half high-transmission zones. The inner and outer radii of the mth upper half high-transmission zone for the MSFFZP are rm1 and rm2 calculated by Eqs. (5.18) and (5.19) [33], respectively. The inner and outer radii of the mth bottom half high-transmission zone for the MSFFZP are rm3 and rm4 calculated by Eqs. (5.20) and (5.21) [33], respectively. rm1  f  [arccos(1  2 )  2m ]   rm 2  f  [arccos(1  2 )  2m + ]   rm 3  f  [arccos(3  2 )  2m ]   rm 4  f  [arccos(3  2 )  2m + ]   0  , 0  ,     2 ,     2 , (5.18) (5.19) (5.20) (5.21) where θ is the azimuthal angle, λ presents a wavelength and α is a number between 0 and 2, which is used to adjust the widths of high-transmission zones. Thus, for the MSFFZP of the radius a, the corresponding focal length f can be calculated by the expression f  a2 / [(M +3) ]. The MSFFZP is fractal according to the Cantor sequence of order S, which have the total elements of 3S. M  2n  3S represents the total number of hightransmission and low-transmission zones in the upper or bottom half part of the MSFFZP, where n is a positive integer. Moreover, n high-transmission and low-transmission zones or 2n low-transmission zones of the MSFFZP from inside out are corresponding to the single element, i.e., “1” or “0” of the Cantor sequence of S from left to right in turn, respectively. For simplicity, λ is set as 550 nm and the MSFFZP of a = 3.456 mm and M = 54, which is based on the Cantor sequence of S = 2 with nine elements [1], is taken for example to illustrate the focusing properties of the MSFFZP. Thus, the corresponding f calculated by the expression is 381 mm. It should be noted in this chapter that three hightransmission and low-transmission zones or six low-transmission zones of the MSFFZP from inside out are corresponding to the single element, i.e., “1” or “0” of the Cantor sequence of S = 2 from left to right in turn, respectively. It can be seen in Figs. 5.21(a-c) that the MSFFZPs with different values of α have two foci with the focal lengths f = 381 mm and f2 = 457.2 mm, respectively. Thus, the MSFFZP can generate two foci with many subsidiary foci along the optic axis. Moreover, we can find that the foci at z = 381 mm in Figs. 5.21(a) and (b) have higher intensities than those at z = 457.2 mm, respectively, and the focus at z = 457.2 mm in Fig. 5.21(c) has higher intensity than that at z = 381 mm. Therefore, two foci with many subsidiary foci of the MSFFZP may have the unequal intensities. The suppression ratio of the MSFFZP is defined as the ratio between the diffracted light intensities of the high-order and first-order 257 Advances in Optics: Reviews. Book Series, Vol. 5 main foci. In Fig. 5.21(a), there are the high-intensity first, second and fourth order foci for the MSFFZP, whose low second-order and fourth-order suppression ratios are -8.489 and -0.492 dB, respectively. Thus, besides the first-order focus, the MSFFZP in Fig. 5.21(a) have the second and fourth order foci. In Fig. 5.21(b), there are significantly lower intensities for the second and fourth order foci, whose main foci have the high suppression ratios of -32.261 and -31.515 dB, respectively, so there is the first-order focus only. In Fig. 5.21(c), the fourth-order focus has significantly lower intensity, but the second-order focus has high intensity, and the corresponding main focal suppression ratios are -17.055 and -2.811 dB, respectively. Therefore, the fourth-order suppression ratio is high, but the second-order suppression ratio is low. There are the first order focus and the second order focus along the optic axis only in Fig. 5.21(c), and the other high order diffraction foci are suppressed. Thus, different values of α, i.e., the different widths of high-transmission zones, have influence on the number of the high-order diffraction foci. Fig. 5.21. (a-c) Axial irradiances of the MSFFZPs of α = 0.2, 1 and 1.614, respectively. 2 The intensity non-uniformity of twin foci is defined as  =  I i  I i 1 I , where I i and I are the intensity of the i-th focus and the average intensity of two foci, respectively. We can find in Fig. 5.22 (a) that with the increase of α, the intensity of the focus at z = 381 mm firstly increases and then decreases, and the intensity of the focus at z = 457.2 mm also firstly increases and then decreases. Moreover, it can be also seen in Fig. 5.22 (a) that when α is equal to 1.03, the intensity of the focus at z = 381 mm has the maximum. Therefore, the single main focus with highest intensity is produced by the MSFFZP of α = 1.03 shown in Fig. 5.23(a). We can find in Fig. 5.22 (b) that with the increase of α,  of twin foci firstly decreases and then increases. In addition, it can be also seen in Fig. 5.22 (b) that when α is equal to 1.614,  of twin foci has the minimum. Thus, the MSFFZP of α = 1.614 shown in Fig. 5.23 (b) has two equal-intensity foci located at z = 381 and 457.2 mm. The second main focus locates at z = 457.2 mm. Moreover, the MSFFZP of α = 1.03 produces the single main focus with many subsidiary foci. The dense data in Fig. 5.22 have been zoomed in, and the small sub-figures are plotted to show the results. 258 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging The tristimulus values, X, Y and Z can be calculated by Eq. (5.22). A horizontal chromaticity coordinate x and a vertical chromaticity coordinate y can be calculated by Eq. (5.23). 2 X ( z )   I ( z;  ) S ( ) xd  , 1 2 Y ( z )   I ( z;  ) S ( ) yd  , (5.22) 1 2 Z ( z )   I ( z;  ) S ( ) zd  , 1 where (λ1, λ2) shows the wavelengths from 380 to 780 nm with a 10 nm interval, S(λ) is a spectral distribution of the standard illuminant C, and  x, y , z  presents sensitivity functions (CIE 1931). Fig. 5.22. (a) Relationships between α and intensities of two foci located at z = 381 and 457.2 mm produced by the MSFFZPs of different values of α; (b) Intensity non-uniformity of corresponding two foci. Fig. 5.23. (a) and (b) Phase distributions of the MSFFZPs of α = 1.03 and 1.614, respectively. 259 Advances in Optics: Reviews. Book Series, Vol. 5 X , X Y  Z . Y y X Y  Z x (5.23) In the simulations, the wavelengths from 380 to 780 nm with a 10 nm interval are used to calculate the chromaticity coordinate of the chromaticity diagram in Fig. 5.24 (c). The horizontal coordinates show the normalized axial distance normalized by the focal length of 381 mm of the SFZP under the illumination with the wavelength of 550 nm. The triangles, circles and squares in Fig. 5.24 (c) are corresponding to those in Fig. 5.24 (b), respectively. It can be seen in Fig. 5.24 (c) that the distances between the circles and the point C (0.31006, 0.31616) for the SFZP and the MSFFZP of α = 1.03 are 0.318 and 0.202, respectively. Moreover, the distances between the point C and the circles, which are corresponding to two foci located at z = 381 and 457.2 mm for the MSFFZP of α = 1.614, respectively, are 0.255 and 0.087, respectively. Thus, the distances between the circles and the point C for the MSFFZPs of α = 1.03 and 1.614 are shorter than that for the SFZP. When the distance between the point in the chromaticity diagram and the point C representing the white illuminant is shorter, the corresponding colourful image has the lower chromatic aberration [3]. Therefore, compared with the SFZP, the MSFFZP of α = 1.03 has the single image with the lower chromatic aberration at the single main focal plane, and the MSFFZP of α = 1.614 generates two images with the lower chromatic aberration at two main focal planes simultaneously. Thus, a distance of 0.318, which is largest, imply that the corresponding colourful image of the SFZP has high chromatic aberration. The MSFFZP has one or two colourful images with low chromatic aberration at the focal planes. However, the SFZP has the colourful image with high chromatic aberration at the focal plane. The experimental schematic is shown in Fig. 5.25. Firstly, a beam (λ = 650, 532 or 450 nm) is expended by a beam expander, and then passes a lens, a target object at the focal plane of the lens, a collimator (a focal length of 550 mm and a diameter of 55 mm), a polarizer, a splitting prism and a SLM (CAS MICROSTAR, FSLM-2K55-P, 1920 × 1080 pixels, 6.4 μm × 6.4 μm /pixel, and reflective type). Then, the reflective beam passes the splitting prism, and a Complementary Metal Oxide Semiconductor (CMOS, Mindvision, MV-UBS300C) camera is used to record the transverse intensity distribution. The original target is one of the horizontal lines of the group and element (2, 1) on the USAF 1951 resolution test chart, whose other transparent parts are covered with the printing paper. It should be noted be that for the phase-only SLM, the product of the wavelength and maximum phase is constant approximately. When λ is equal to 532 nm, the maximum phase is equal to 2π. Thus, the maximum phases of λ = 650 and 450 nm are 1.637π and 2.364π, respectively. The grey values of Gray images are proportional to the phases of the SLM approximately. Thus, for the SLM at λ = 650, 532 and 450 nm, the phase of π is corresponding to the grey values of 156, 127 and 108, respectively. The phase profiles at the different wavelengths are shown in Fig. 5.26. The grey level images of each line have the same grey bar. 260 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging Fig. 5.24. (a) Normalized axial intensity distributions under the illumination with the wavelengths 450, 550 and 650 nm, (b) the normalized axial illuminance Y and (c) the chromaticity diagram of the SFZP, the MSFFZP of α = 1.03 and 1.614, respectively. Fig. 5.25. Experimental system of the imaging system. It can be seen in Fig. 5.27 that there are the clearest images at the main focal planes for the above zone plates, and there are not the clear images at the other focal planes for the SFZP of a = 3.456 mm, but there are the clear images at the subsidiary focal planes for the MSFFZPs. Moreover, the MSFFZP of α = 1.03 has the colourful image at z = 395 mm, and the MSFFZP of α = 1.614 has two colourful images at z = 395 and 475 mm, respectively. However, the SFZP has not the colourful images at z = 395 mm. The main reason is that the SFZP has the single focus with the short focal length, but the subsidiary foci extend the focal length of the single main focus and two main foci for the MSFFZPs of α = 1.03 and 1.614, respectively. Thus, the MSFFZP has one or two colourful images at the focal planes. 261 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 5.26. Gray level images of (a-c) the SFZP of a = 3.456 mm, the MSFFZPs of α = 1.03 and 1.614 at the wavelength λ = 650 nm, respectively, and (d-f) the above zone plates at the wavelength λ = 532 nm, respectively, and (g-i) the above zone plates at the wavelength λ = 450 nm, respectively. 5.5. Conclusion In this chapter, we have provided an overview about the aperiodic zone plates based on optical tweezers and optical imaging. The method of constructing aperiodic zone plates and the theory of the diffraction are illustrated. We have proved that the FZP beam can simultaneously trap multiple particles positioned in different focal planes, and the dynamic manipulation of microparticles in optical tweezers is implemented by changing Ns of the fractional FZPs. Moreover, the symmetrical major foci around the main focus of the TMZP are used to trap particles stably and the Fibonacci zone plate can simultaneously trap particles positioned in two different planes and move particles freely in the focal plane. It is also proved in the experiments that the GCAZP has the clearer images at the focal planes, and the MSFFZP produces the single or two colourful images with the low chromatic aberrations at the focal planes. Aperiodic zone plates have a wide range for practical applications and can offer great potential developments of optical tweezers and optical imaging in the future. Acknowledgments The research was financially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11674401), the Natural Science Foundation of Hunan Province, China (Grant No. 2019JJ40358), and the Fundamental Research Funds for the Central Universities of Central South University, China (Grant No. 2020zzts043). 262 Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging Fig. 5.27. (a-c) Captured images at z = 395 mm for the SFZP of a = 3.456 mm at λ = 650, 532 and 450 nm, respectively; (d) The corresponding synthetic colourful image. (e-g) Captured images at z = 395 mm for the MSFFZP of α = 1.03 at λ = 650, 532 and 450 nm, respectively; (h) The corresponding synthetic colourful image; (i-k) Captured images at z = 395 mm for the MSFFZP of α = 1.614 at λ = 650, 532 and 450 nm, respectively; (l) The corresponding synthetic colourful image. 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Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Chapter 6 Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Boian A. Hristov1 6.1. Introduction Still the syntheses of optical systems is both an art and a science [1, pp. 3-4], [2, p. 409]. There is no procedure that will lead from a set of specifications to the successful design of an optical system [2, pp. 409-410]. From long ago it is well known that the designers use modern software and optimization programs. However these programs require a suitable starting design which is vitally important for a good end results [2]. Now-a-days there is no suitable computer program to create a useful lens design in the absence of the guidance of a skilled optical designer. In [3] and [4] we developed a rigorous analytical aberrations theory of centered optical systems containing spherical surfaces. It permits to correct rigorously aberrations for a strictly set aperture angles or view angles. But we are not able to correct strictly aberrations on the whole aperture or on the whole field of view. That’s why this theory does not allow to design diffraction-limited systems with high and super-high numerical apertures. In Section 6.2 we present the paraxial matrix transformations in a centered optical systems and find the rigorous analytical dependences between constructive parameters, paraxial characteristics and axial chromatic aberrations. We show how to correct strictly the axial chromatic aberration together with preset paraxial characteristics using some constructive Boyan Hristov Institute of Optical Materials and Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria 267 Advances in Optics: Reviews. Book Series, Vol. 5 parameters as free parameters. Further, we analyze the analytical connections between the parameters of real tangential rays in the system, paraxial matrix coefficients, sagittal radius of the conic surfaces and the so-called sagittal radius-coefficient (that is the ratio of the vertex to the sagittal radius). On this base we find rigorous analytical equation for correction of on-axis spherical aberration. It is important to note that the correction carries out for each aperture ray from zero to preset maximum aperture, using only one aspherical surface. So we develop a reliable algorithm for simultaneous correction of on-axis spherical and axial chromatic aberrations together with preset paraxial characteristics in the centered optical systems which contain conic surfaces. As a result, we calculate rigorously the coordinates of aspherical (or so-called multi-conical) surface and slope angle for each surface point. Thus we have all data for production of the aspherical optical component using CNC machines. There is no need to use optimization programs for the presentation of aspherical surfaces with the help of conic constant and polynomials (see [5]). In Section 6.3 we show variety of diffraction-limited lenses with high and super-high numerical apertures that have better image quality and higher numerical apertures than the best available up- to-now (see [6-11]). 6.2. Theoretical Basis 6.2.1. Paraxial Matrix Transformations and Axial Chromatic Aberration 6.2.1.1. Paraxial Matrix Transformations from Object to Image Space in a Single Optical Surface Let us consider a single refractive (or reflective) conical surface having serial number j in a rotationally symmetric optical system composed of 𝑘 conical surfaces. It has a vertex radius rj and separates two media of refraction (or reflection) index nj on the left and index nj+1 on the right. It is well known [2, p. 38] that the dependence between the object and paraxial image positions on a single spherical or conical surface is. 𝑛𝑗+1 ′ 𝑧𝑜𝑗 − 𝑛𝑗 𝑧𝑜𝑗 = 𝑚𝑜𝑗 𝑟𝑗 , (6.1) where zoj is the axial distance from the surface vertex to the object, zoj’ is the axial distance from the surface vertex to the paraxial image and m0j is the following refraction (or reflection) index difference 𝑚0𝑗 = 𝑛𝑗+1 − 𝑛𝑗 (6.2) The sign conventions for the radii, thicknesses and distances, which we shell observe here, are the same as it is in [2, pp. 35-36]. 268 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Transforming Eq. (6.1) for the axial image distance zoj’ we get 𝑟 ∗𝑛 ∗𝑧 ′ 𝑧𝑜𝑗 = 𝑚 𝑗 ∗𝑧𝑗+1+𝑟 0𝑗 ∗𝑛 𝑎𝑗 0𝑗 𝑗 (6.3) 𝑗 The axial image distance can be represented as a bilinear function 𝐴 ∗𝑧 +𝐵 ′ 𝑧𝑜𝑗 = 𝐶 𝑎𝑗∗𝑧 0𝑗+𝐷𝑎𝑗, (6.4) 𝐴𝑎𝑗 = 𝑟𝑗 𝑛𝑗+1 ; 𝐵𝑎𝑗 = 0; 𝐶𝑎𝑗 = 𝑚0𝑗 ; 𝐷𝑎𝑗 = 𝑟𝑗 𝑛𝑗 (6.5) 𝑎𝑗 where 0𝑗 𝑎𝑗 This bilinear function can be described by the following quadratic matrix Gaj 𝐺𝑎𝑗 = 𝐴𝑎𝑗 𝐶𝑎𝑗 𝐵𝑎𝑗 𝑟𝑗 ∗ 𝑛𝑗+1 = 𝑚 𝐷𝑎𝑗 0𝑗 The determinant Detaj of the matrix Gaj is 0 𝑟𝑗 ∗ 𝑛𝑗 𝐷𝑒𝑡𝑎𝑗 = 𝐴𝑎𝑗 ∗ 𝐷𝑎𝑗 − 𝐵𝑎𝑗 ∗ 𝐶𝑎𝑗 = 𝑛𝑗 ∗ 𝑛𝑗+1 ∗ 𝑟𝑗2 (6.6) (6.7) The matrix Gaj we shell call an axial surface transformation matrix (ASTM) of a single surface or a single axial transformation matrix. The coefficients Aaj, Baj, Caj and Daj we shell call the ASTM-coefficients respectively. Note that equations from (6.1) to (6.7) are correct in the case when the origins of the object and image coordinate system coincide with surface vertex. If we want to relocate the origin of the image coordinate system at a distance dj (for example to the next surface in the centered optical system), we use a quadratic matrix Dj having the following matrix coefficients: 𝐷𝑗 = 1 −𝑑𝑗 0 1 (6.8) The matrix Dj we will call an axial relocation matrix (ARM) of the dj thickness. The distance z’dj in the relocated at a distance dj coordinate system is 𝑧′𝑑𝑗 = 𝑧′𝑜𝑗 − 𝑑𝑗 (6.9) Substituting z’oj from Eq. (6.3) in Eq. (6.9) we get ′ =((𝑛𝑗+1 ∗ 𝑟𝑗 − 𝑑𝑗 ∗ 𝑚0𝑗 ) ∗ 𝑧𝑗 − 𝑛𝑗 ∗ 𝑟𝑗 ∗ 𝑑𝑗 )⁄(𝑚0𝑗 ∗ 𝑧𝑗 + 𝑟𝑗 ∗ 𝑛𝑗 ) (6.10) 𝑧𝑑𝑗 269 Advances in Optics: Reviews. Book Series, Vol. 5 It can be proved that the axial image distance z’dj in the relocated coordinate system is rigorously described with the help of the matrix Gdj that is a matrix product of the jth surface ATM and ARM of the jth thickness, i.e. 𝐺𝑑𝑗 = 𝐷𝑗 ∗ 𝐺𝑎𝑗 = 1 0 𝐴𝑑𝑗 0 = 𝑛𝑗 ∗ 𝑟𝑗 𝐶𝑑𝑗 −𝑑𝑗 𝑛𝑗+1 ∗ 𝑟𝑗 x 𝑚 0𝑗 1 𝐵𝑑𝑗 𝐷𝑑𝑗 (6.11) The matrix Gdj we call transformation-relocation matrix (TRM). The corresponding to the TRM bilinear function is 𝑧′𝑑𝑗 = 𝐴𝑑𝑗 ∗𝑧0𝑗 +𝐵𝑑𝑗 (6.12) 𝐶𝑑𝑗 ∗𝑧0𝑗 +𝐷𝑑𝑗 If we calculate TRM Gdj from Eq. (6.11) we obtain the following coefficients 𝐴𝑑𝑗 = 𝑛𝑗+1 ∗ 𝑟𝑗 -𝑑𝑗 ∗ 𝑚0𝑗 ; 𝐵𝑑𝑗 = -𝑛𝑗 ∗ 𝑟𝑗 ∗ 𝑑𝑗 ; 𝐶𝑑𝑗 = 𝑚0𝑗 ; 𝐷𝑑𝑗 = 𝑛𝑗 ∗ 𝑟𝑗 , (6.13) Comparing the coefficients Aaj, Baj, Caj, Daj from Eq. (6.13) with coefficients from bilinear function in Eq. (6.10) we can see that they are exactly the same. The matrix Gdj describes exactly the axial image distance z’dj in the relocated to the next surface coordinate system. 6.2.1.2. Exact Paraxial Matrix Transformations in a Centered Optical System of Conical Surfaces. Invention of Exact Analytical Dependences between Constructive Parameters and Paraxial Characteristics Let us discuss the first surface in the centered optical system and the first relocation to the second surface at a distance d1. Using Eqs. (6.11, 6.12 and 6.13) for the first surface we write where 𝐺𝑑1 = 𝐷1 𝐺𝑎1 = 1 −𝑑1 𝑛2 ∗ 𝑟1 x 𝑚01 0 1 𝐴 0 = 𝑑1 𝑛1 ∗ 𝑟1 𝐶𝑑1 𝐴𝑑1 = 𝑛2 ∗ 𝑟1 -𝑑1 ∗ 𝑚01 , 𝐵𝑑1 = - 𝑛𝑗 ∗ 𝑟1 ∗ 𝑑1 , 𝐶𝑑1 = 𝑚01 , 𝐷𝑑1 = 𝑛1 ∗ 𝑟1 𝐵𝑑1 , 𝐷𝑑1 (6.14) (6.15) The matrix Gd1 describes simultaneously the axial transformation of the first surface and the relocation at a distance d1. The corresponding to the first TRM bilinear function is 270 ′ 𝑧𝑑1 = 𝐴𝑑1 ∗𝑧01 +𝐵𝑑1 𝐶𝑑1 ∗𝑧01 +𝐷𝑑1 (6.16) Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Here zo1 is the object distance measured along the optical axis. It is clear that image distance z’d1 for the first surface is equal to the object distance zo2 for the second surface, i.e. ′ 𝑧𝑑1 = 𝑧02 (6.17) Applying Eq. (6.3) for the second surface and taking account of Eq. (6.17) we obtain ′ 𝑧02 = 𝐴 ∗𝑧 +𝐵 𝑛3 ∗𝑟2 𝑑1 01 𝑑1 𝐶𝑑1 ∗𝑧01 +𝐷𝑑1 𝐴𝑑1 ∗𝑧01 +𝐵𝑑1 𝑚02 +𝑛2 ∗𝑟2 𝐶𝑑1∗𝑧01 +𝐷𝑑1 𝐴 ∗𝑧 +𝐵 = 𝐶𝑎2∗𝑧01+𝐷𝑎2 𝑎2 01 𝑎2 (6.18) Transforming the right equality in Eq. (6.18) the coefficients Aa2, Ba2, Ca2, Da2 became as follows 𝐴𝑎2 = 𝑛3 ∗ 𝑟2 ∗ (𝑛2 ∗ 𝑟1 − 𝑑1 ∗ 𝑚01 ), (6.19) 𝐷𝑎2 = 𝑛1 ∗ 𝑟1 (𝑛2 ∗ 𝑟2 − 𝑚02 ∗ 𝑑1 ), (6.22) 𝐵𝑎2 = − 𝑛1 ∗ 𝑛3 ∗ 𝑟1 ∗ 𝑟2 ∗ 𝑑1 , 𝐶𝑎2 = 𝑚01 (𝑛2 ∗ 𝑟2 − 𝑑1 ∗ 𝑚02 ) + 𝑛2 ∗ 𝑟1 ∗ 𝑚02 , (6.20) (6.21) One can see from Eqs. (6.19)-(6.22) that every axial matrix coefficient Aa2, Ba2, Ca2, Da2 is linear function separately to each constructive parameter r1, r2, d1 in the system of two surfaces. It can be proved that the quadratic matrix ga2 in Eq. (6.23) describes exactly the bilinear function (6.18) without any approximations 𝑔𝑎2 = 𝐺𝑎2 ∗ 𝐷1 ∗ 𝐺𝑎1 (6.23) Multiplying the three matrices in Eq. (6.23) we receive exactly the same matrix coefficients as they are in Eqs. (6.19)-(6.22). The matrix ga2 describes exactly the axial image distance z’02 in the relocated to the second surface vertex coordinate system. In other words Eq. (6.23) is a matrix formula to calculate axial matrix coefficients Aa2, Ba2, Ca2, Da2 for an optical system of two conical surfaces when the origin of the object coordinate system coincides with the vertex of the first surface and the origin of the image coordinate system coincides with the vertex of the last surface. By analogy we can derive an exact matrix equation for an optical system of k conical surfaces and namely 𝑔𝑎𝑘 = 𝐺𝑎𝑘 ∗ ∏1𝑗=𝑘−1 𝐷𝑗 ∗ 𝐺𝑎𝑗 = 𝐴𝑎𝑘 𝐶𝑎𝑘 𝐵𝑎𝑘 𝐷𝑎𝑘 (6.24) 271 Advances in Optics: Reviews. Book Series, Vol. 5 The matrix gak we shell call axial transformation matrix for the whole system i.e. ATMS. It is easy to prove that the axial matrix coefficients Aak, Bak, Cak, Dak of the system are linear functions to each radius or axial thickness. But they are quadratic functions to each index of refraction except the first and the last one. The matrix gak corresponds to the next bilinear function of the axial image distance with the same axial matrix coefficients ′ 𝑧0𝑘 = 𝐴𝑎𝑘 ∗𝑧01 +𝐵𝑎𝑘 𝐶𝑎𝑘 ∗𝑧01 +𝐷𝑎𝑘 (6.25) The back focal length (bfl) of an optical system can be found as a limit of z’0k (see Eq. (6.25)) when z01 is tending to infinity 𝑏𝑓𝑙 = 𝐴𝑎𝑘 ⁄𝐶 𝑎𝑘 (6.26) If the axial image distance z’ok is tending to infinity (in other words the denominator of Eq. (6.25) tends to zero i.e. (Cak* z01+ Dak = 0) we can find the front focal length (ffl) as 𝑓𝑓𝑙 = − 𝐷𝑎𝑘 ⁄𝐶 𝑎𝑘 (6.27) Further on we shell derive formulas for the effective focal length f’ and for the lateral (or transverse) magnification M in an optical system of k surfaces. Using Eqs. (6.24)-(6.27) we can calculate exactly the four axial matrix coefficients the axial image distance for any object distance, the back focal length and the front focal length in each centered optical system of conical surfaces. Moreover Eqs. (6.24) and (6.25) have a remarkable capacity to give us an exact analytical relations between the paraxial characteristics (effective focal length, back focal length, front focal length and magnification) of optical system and its constructive parameters, such as vertex radiuses, thicknesses and refraction (or reflection) indexes– for example see Eqs. (6.19)-(6.22). Note that the axial object distance in Eq. (6.25) is measured from the first surface’s vertex to the object and the axial image distance is measured from the last surface’s vertex to the axial image. Let’s transpose the object coordinate system from the first surface’s vertex to the front focus, i.e. 𝑧01 = 𝑧 − 𝐷𝑎𝑘 ⁄𝐶 𝑎𝑘 (6.28) In addition the image coordinate system is transposed from the last surface vertex to the back focus 272 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures ′ 𝑧𝑜𝑘 = 𝑧′ + 𝐴𝑎𝑘 ⁄𝐶 𝑎𝑘 (6.29) In Eq. (6.28) z is the distance from the front focus to the object and in (6.29) z’ is the distance from the back focus to the axial image. Substituting z01 from (6.28) and z’0k from (6.29) in Eq. (6.25) after rearranging we have −𝑧𝑧 ′ = (𝐴𝑎𝑘 ∗ 𝐷𝑎𝑘 − 𝐵𝑎𝑘 ∗ 𝐶𝑎𝑘 ) ⁄𝐶 2 𝑎𝑘 (6.30) It can be proved that the axial determinant Detak of the matrix gak from Eq. (6.24), is 2 ∏𝑘𝑗=2 𝑛𝑗2 𝐷𝑒𝑡𝑎𝑘 = 𝐴𝑎𝑘 ∗ 𝐷𝑎𝑘 − 𝐵𝑎𝑘 ∗ 𝐶𝑎𝑘 = 𝑛1 ∗ 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑟𝑎𝑗 (6.31) Eq. (6.31) shows that the axial determinant in each optical system of conical surfaces is independent of axial distances between the optical surfaces. It depends on vertex radii of the system and refraction indices only. Using Newtonian [2, p. 30] equation and Eq. (6.31) we can rewrite Eq. (6.30) as −𝑧 ∗ 𝑧 ′ = (𝑛1 ∗ 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑟𝑗2 ∏𝑘𝑗=2 𝑛𝑗2 ) 𝑛1 ∗ (𝑓 ′ )2⁄ ⁄ 2 𝑛𝑘+1 = 𝐶 Here f’ is the effective focal length of the optical system. 𝑎𝑘 (6.32) From Eq. (6.32) we can find the effective focal length as follows 𝑓′ = ∏𝑘𝑗=1 𝑟𝑎𝑗 ∏𝑘+1 𝑗=2 𝑛𝑗⁄ 𝐶𝑎𝑘 (6.33) By means of Eq. (6.33) we transform (6.32) in the form −𝑧 ′ = 𝑓 ′ ∗ ∏𝑘𝑗=1 𝑟𝑗 ∗ 𝑛𝑗 ⁄𝑧 ∗ 𝐶𝑎𝑘 (6.34) −𝑧 ′ = 𝑀 ∗ ∏𝑘𝑗=1 𝑟𝑗 ∗ 𝑛𝑗 ⁄𝐶𝑎𝑘 (6.35) 𝑀 = 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑛𝑗∗ 𝑟𝑗 ⁄𝑛1∗ (𝐶𝑎𝑘 ∗ 𝑧01 + 𝐷𝑎𝑘 ) (6.36) It is well known that the fraction f’/z is nothing but the lateral (or transverse) magnification (M) of an optical system, i.e. Now we replace z’0k from (6.25) in (6.29). After that we substitute 𝑧 ′ from Eq. (6.29) in (6.35). In addition after rearranging of Eq. (6.35) we obtain rigorous formula for a lateral magnification M in a centered optical systems of conical surfaces If we want to design a centered optical system, composed of conical surfaces, with preset paraxial parameters (effective focal length, lateral magnification, back focal length or front focal length) we use Eqs. (6.26), (6.27), (6.33) and (6.36). 273 Advances in Optics: Reviews. Book Series, Vol. 5 Below we show a simple example how to find an exact analytical connection between the constructive parameters and paraxial characteristics of a single lens. For example we want to design a lens with preset effective focal length f’2 and back focal length bfl2. As a free parameters we use the two axial radii r1 and r2 respectively. Using Eqs. (6.33) and (6.26) for the case 𝑘 = 2 it can be written 𝑓2′ = 𝑟1 ∗ 𝑟2 ∗ 𝑛2 ∗ 𝑛3 ⁄𝐶𝑎2 , (6.37) 𝑏𝑓𝑙2 = 𝐴𝑎2 ⁄𝐶𝑎2 (6.38) 𝑟1 = 𝑑1 ∗ 𝑚01 ∗ 𝑓2′ ⁄𝑛2 ∗ (𝑓2′ − 𝑏𝑓𝑙2 ) (6.39) 𝑟2 = 𝑏𝑓𝑙2 ∗ 𝑑1 ∗ 𝑚01⁄(𝑛2 ∗ (𝑓2′ − 𝑏𝑓𝑙2 ) − 𝑑1 ) (6.40) Excluding Ca2 from Eqs. (6.37) and (6.38) with the help of (6.19) we get Now if we substitute r1 from Eqs. (6.39) in (6.37) or (6.38) we find Eqs. (6.39) and (6.40) allows us to find the numerical values of the two radii r1 and r2 so that the two preset paraxial characteristics f’2 and bfl2 are satisfied. Let’s the constructive parameters and preset paraxial characteristics of the lens are: n1 = n3 = 1; n2 = 1.8; d1 = 20 mm; f’2 = 100 mm; bfl2 = 60 mm; According to Eqs. (6.39) and (6.40) we calculate: r1 = 22.222222 mm; r2 = 18.461538 mm. If we wont to have bfl2 = 140 mm then the radii are: r1 = -22.222222 mm; r2 = -24.347826 mm. In conclusion of Section 6.2.1.2 we claim that: • Independently from the number of surfaces in the centered optical system, Eq. (6.24) gives a possibility through analytical computations to generate rigorous analytical dependences between the paraxial matrix coefficients and constructive parameters. If the latter are known, we can calculate the exact values of the matrix coefficients for the whole system; • Using Eqs. (6.24)-(6.27), (6.33) and (6.36) we can generate exact analytical dependences between the paraxial characteristics and constructive parameters. If the latter are known, we can calculate the exact values of the paraxial characteristics for the whole system. 6.2.1.3. Axial Image Chromatic Aberration It is well known that the index of refraction varies as a function of wavelength of light. For a given object distance z01 the axial chromatic aberration is a longitudinal variation of image position with wavelength. We define the axial chromatic aberration as a difference between the axial image distances of any two wavelength. 274 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Let us consider a spectral wavelength range with left and right ends. We shell denote the indices of refraction nj for the right wavelength with additional lower index r (i.e. njr) and for the left end with l (i.e. njl) respectively. In accordance with Eq. (6.25) we write for the right wavelength 𝑧′𝑟 = 𝐴𝑟 ∗𝑧01 +𝐵𝑟 𝐶𝑟 ∗𝑧01 +𝐷𝑟 , (6.41) where Ar, Br, Cr and Dr are axial matrix coefficients for the right wavelength in the optical system with k conical surfaces and z’r is the axial image distance for the right wavelength. By analogy for the left spectral end we have ∗𝑧01+𝐵𝑙 𝑧′𝑙 = 𝐴𝐶𝑙∗𝑧 +𝐷 𝑙 01 (6.42) 𝑙 Here Al, Bl, Cl and Dl are axial matrix coefficients for the left wavelength and z’l is the axial image distance for the same wavelength. Taking into account Eqs. (6.41) and (6.42) the axial chromatic aberration chr is 𝑐ℎ𝑟 = 𝑧𝑟′ − 𝑧𝑙′ = 𝐴𝑟 ∗𝑧01 +𝐵𝑟 𝐶𝑟 ∗𝑧01 +𝐷𝑟 − 𝐴𝑙 ∗𝑧01 +𝐵𝑙 𝐶𝑙 ∗𝑧01 +𝐷𝑙 = 2 +𝑢 ∗𝑧 +𝑢 𝑢2 ∗𝑧01 1 01 0 2 +𝑣 ∗𝑧 +𝑣 𝑣2 ∗𝑧01 1 01 0 (6.43) Eq. (6.43) shows that the axial chromatic aberration is a quadratic fractional function to the object distance z01. Using the Eq. (6.43) after symbolic computation of matrix coefficients we can find the coefficients u2, u1, uo,υ2, υ1, υ0 𝑢2 = 𝐴𝑟 ∗ 𝐶𝑙 − 𝐴𝑙 ∗ 𝐶𝑟 , (6.44) 𝑢0 = 𝐵𝑟 ∗ 𝐷𝑙 − 𝐵𝑙 ∗ 𝐷𝑟 , (6.46) 𝑣1 = 𝐶𝑟 ∗ 𝐷𝑙 + 𝐶𝑙 ∗ 𝐷𝑟 , (6.48) 𝑢1 = 𝐴𝑟 ∗ 𝐷𝑙 − 𝐴𝑙 ∗ 𝐷𝑟 + 𝐵𝑟 ∗ 𝐶𝑙 − 𝐵𝑙 ∗ 𝐶𝑟 , (6.45) 𝑣2 = 𝐶𝑟 ∗ 𝐶𝑙 , (6.47) 𝑣0 = 𝐷𝑟 ∗ 𝐷𝑙 (6.49) It is evident from Eqs. (6.44)-(6.49) that each coefficient of u2, u1, uo,υ2, υ1 and υ0 is a quadratic function to each constructive parameter (rj and dj) except the last vertex radius rk. Eq. (6.43) describes the axial chromatic aberration for any object distances z01 rigorously without any approximations. For the object distance tending to infinity we denote the axial chromatic aberration as chr ∞. 275 Advances in Optics: Reviews. Book Series, Vol. 5 It is easy to find it as a limit of the function (6.43) 𝑐ℎ𝑟∞ = 𝑢2 𝑣2 = 𝐴𝑟 ∗𝐶𝑙 −𝐴𝑙 ∗𝐶𝑟 𝐶𝑟 ∗𝐶𝑙 (6.50) The right equality of Eq. (6.50) describes analytically the axial chromatic aberration for an infinitely distant object. To correct the axial chromatic aberration in this case it is necessary to satisfy the next condition 𝑢2 = 𝐴𝑟 ∗ 𝐶𝑙 − 𝐴𝑙 ∗ 𝐶𝑟 = 0 (6.51) For the satisfaction of Eq. (6.51) in general we have to solve one quadratic equation in respect to one vertex radius rj or to one axial thickness dj of the optical system. There is one special case when we use as a free parameter the last vertex radius of the system. Then the correction of the chromatic aberration reduces to a solution of one linear equation. In these case Eq. (6.51), after analytical symbolic computations of matrix coefficients Ar, Cl, Al, Cr has the shape 𝑢2 = 𝑒2 ∗ 𝑟𝑘2 + 𝑒1 ∗ 𝑟𝑘 = 0 (6.52) 2 𝑒2 = 𝑒22 ∗ 𝑟𝑘−1 + 𝑒21 ∗ 𝑟𝑘−1 + 𝑒20 , (6.53) Grouping the coefficient u2 in respect to the different degrees of rk-1 we can find the analytical form of e2 and e1 2 𝑒1 = 𝑒12 ∗ 𝑟𝑘−1 + 𝑒11 ∗ 𝑟𝑘−1 + 𝑒10, (6.54) and the analytical form (or numerical value) of the sub-coefficients e22, e21, e20, e12, e11, e10. 6.2.1.4. Simultaneous Correction of Axial Image Chromatic Aberration and Guarantee of Preset Focal Length in a Centered Optical System Below we describe a special case for a correction of axial chromatic aberration at infinitely distant object and providing of preset focal length, when we use the two last vertex radii as a free parameters. For the chromatic aberration correction we use equations (6.52), (6.53) and (6.54). To provide the preset value of the focal length we formulate with the help of Eq. (6.33) a special function Ff 𝐹𝑓 = 𝑓 ′ ∗ 𝐶𝑎𝑘 − ∏𝑘𝑗=1 𝑟𝑎𝑗 ∏𝑘+1 𝑗=2 𝑛𝑗 = 0 (6.55) 𝐹𝑓 = ℎ1 ∗ 𝑟𝑘 + ℎ0 = 0, (6.56) After symbolic computation of Cak with the help of Eq. (6.24) and its substitution in Eq. (6.55) we obtain 276 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Grouping the coefficients h1 and h0 in respect to the different degrees of rk-1 we can find the analytical form of h1 and h0and the analytical form (or numerical values) of the sub-coefficients h11, h10, h01, hoo. ℎ1 = ℎ11 ∗ 𝑟𝑘−1 + ℎ10 , (6.57) ℎ0 = ℎ01 ∗ 𝑟𝑘−1 + ℎ00 (6.58) 3 2 + 𝑉2 ∗ 𝑟𝑘−1 + 𝑉1 ∗ 𝑟𝑘−1 + 𝑉0 𝑑𝑠𝑚 = 𝑉3 ∗ 𝑟𝑘−1 (6.59) For a simultaneous satisfaction of Eqs. (6.52) and (6.56) we use Sylvester’s matrices to exclude the unknown parameter rk. After excluding of rk we find the determinant of the Sylvester’s matrix dsm as follows Here 𝑉3 = 𝑒22 ∗ ℎ01 − 𝑒12 ∗ ℎ11 , (6.60) 𝑉1 = 𝑒20 ∗ ℎ01 − 𝑒11 ∗ ℎ10 − 𝑒10 ∗ ℎ11 + 𝑒21 ∗ ℎ00 , (6.62) 𝑉2 = 𝑒21 ∗ ℎ01 − 𝑒12 ∗ ℎ10 − 𝑒11 ∗ ℎ11 + 𝑒22 ∗ ℎ00 , (6.61) 𝑉0 = 𝑒20 ∗ ℎ00 − 𝑒10 ∗ ℎ10 (6.63) Knowing the sub-coefficients e22, e21, e20,e12, e11, e10, hoo, h11, h10, h01, hoo we can find the numerical values of V3, V2, V1, V0. Now we solve Eq. (6.59) in respect to rk-1. It is well known that the set of roots of Eq. (6.59), as roots of Sylvester’s matrix determinant, contains in itself all common roots of the equation system (6.56) and (6.52). Finally through a consecutive checking of the whole set of roots of Eq. (6.59) we find all mathematically existing common roots of the equation system (6.56) and (6.52). 6.2.2. Rigorous Analytical Function of the On-axes Spherical Aberration in a Centered Optical System Containing Conical Surfaces The discovery of an exact analytical connection between the sagittal radius in every point of the conical surface, the vertex radius, the incidence and slope angles of the rays is extremely important for developing of an exact analytical aberrations (such as on-axes spherical aberration, optical path difference, sagittal and tangential curvatures, astigmatism, distortion and field curvature) theory. 6.2.2.1. Sagittal Radius and Sagittal Radius-coefficient in a Single Conical Surface Sagittal radius in each conical surface point by definition is the distance (along the normal) from the point itself to the intersection of the normal with the optical axes. Let us consider a conical surface together with a Descartes coordinate system zy located in the surface vertex. The axes z coincides with surface axes. In this case the general equation for a conic section of the surface having a conic constant K and vertex radius r is 277 Advances in Optics: Reviews. Book Series, Vol. 5 𝑦 2 − 2 ∗ 𝑟 ∗ 𝑧 + (𝐾 + 1) ∗ 𝑧 2 = 0 (6.64) 𝑟𝑠 = 𝑦/ 𝑠𝑖𝑛 𝜑 (6.65) 𝑠𝑖𝑛 𝜑 = 1/√1 + 𝑐𝑜𝑡 2 𝜑 (6.66) 𝑟𝑠 = 𝑦 ∗ √1 + 𝑐𝑜𝑡 2 𝜑 (6.67) The sagittal radius rs in each surface section point can be presented as ratio between the y-coordinate of the point and sine of the normal slope φ. Now we present the sine of the angle thru the cotangent and substitute it in Eq. (6.65) Then On the other hand the derivative 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑧 from Eq. (6.64) is = 𝑐𝑜𝑡 𝜑 = 𝑟−𝑧∗(𝐾+1) 𝑦 (6.68) The combination between Eqs. (6.67), (6.68) and (6.64) after rearranging leads to 𝑟𝑠 = √𝑟 2 − 𝐾 ∗ 𝑦 2 (6.69) 𝑟 = 𝑟𝑠 ∗ √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑 (6.70) Finally for the vertex radius r we find 𝑟 The ratio 𝑟 between the vertex and sagittal radii we shell call a sagittal radius-coefficient ρs. Then 𝑠 𝜌𝑠 = 𝑟 𝑟𝑠 = √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑 (6.71) So we found a formula for the sagittal radius-coefficient ρs that will play a fundamental role in the rigorous analytical aberrations theory. As it is known [2, p. 515] the tangential radius rt in a conic surface is equal to 𝑟𝑡 = By placing r from the Eq. (6.70) in the previous formula we obtain 𝑟3 𝑟3 𝑟 𝑟𝑠3 . 𝑟2 𝑟 𝑠 𝑠 𝑟𝑡 = 𝑟𝑠2 = 𝑟 2 ∗𝜌 2 = 𝜌2 = 𝜌3 𝑠 𝑠 𝑠 The ratio between the vertex and tangential radii coefficient ρt i.e. 𝑟 278 𝑟 𝑟𝑡 𝑠 we shell call a tangential radius- 𝜌𝑡 = 𝑟 = 𝜌𝑠3 = (1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑) ∗ √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑 𝑡 (6.72) Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures So we found exact and simple formulas for the sagittal and tangential radii (rs and rt) and also for the sagittal and tangential radius-coefficients (ρs and ρt). The next task is to find exact dependence between the z-coordinate of a surface point and the sagittal radius-coefficient in the same point. For this purpose we use the known formula [2, p. 514]. 𝑧= 𝑟−√𝑟 2 −(𝐾+1)∗𝑦 2 (6.73) 𝐾+1 Substituting r from the Eq. (6.70) and y from (6.65) in (6.73) we get 𝑧= 𝑟𝑠 ∗𝜌𝑠 −√𝑟𝑠2 ∗𝜌𝑠2 −(𝐾+1)∗𝑟𝑠2 ∗𝑠𝑖𝑛2 𝜑 𝐾+1 = 𝑟𝑠 ∗ 𝜌𝑠 −√𝜌𝑠2 −(𝐾+1)∗𝑠𝑖𝑛2 𝜑 𝐾+1 (6.74) It can be proved that the last radicand in Eq. (6.74) is equal to As a result we obtain √𝜌𝑠2 − (𝐾 + 1) ∗ 𝑠𝑖𝑛2 𝜑 = cos 𝜑 𝑧 = 𝑟𝑠 ∗ 𝜌𝑠 −𝑐𝑜𝑠 𝜑 𝐾+1 = 𝑟𝑠 ∗ 𝑠𝑖𝑛2 𝜑 𝜌𝑠 +𝑐𝑜𝑠 𝜑 The ratio between the z-coordinate and sagittal radius z-coordinate zs. 𝑧𝑠 = 𝑧 𝑟𝑠 = 𝜌𝑠 −𝑐𝑜𝑠 𝜑 𝐾+1 = (6.75) 𝑠𝑖𝑛2 𝜑 𝜌𝑠 +𝑐𝑜𝑠 𝜑 (6.76) 𝑧 𝑟𝑠 we will call a relative (6.77) So with the help Eqs. (6.65), (6.71) and (6.76) we can correctly describe the conic surface using the sagittal radius-coefficient 𝜌𝑠 , the conic constant K and the slope of the normal 𝜑 as intermediate parameters. If we transform Eq. (6.70) the sagittal radius becomes 𝑟𝑠 = 𝑟 √1+𝐾∗𝑠𝑖𝑛2 𝜑 (6.78) We estimate the sagittal radius as a function of the parameters φ and K at constant vertex radius r = 100 mm. The result is presented in Figs. 6.1, 6.1a, 6.1b. On Fig. 6.1 we show five graphics of the sagittal radius for different values of K that correspond as follows: number 1 to K = 100, number 2 to K = 10, number 3 to K = 3, number 4 to K = 1, and number 5 to K = 0.2 respectively. The sagittal radius is always equal to the vertex radius at φ = 0. The bigger is the conic constant, the smaller the sagittal radiuses are in the whole range of the slope angles. When the conic constant tends to zero 279 Advances in Optics: Reviews. Book Series, Vol. 5 (i.e. the conic surface becomes sphere) then the function of rs is turning in a straight horizontal line. This means that the sagittal radius is constant and equal to the vertex one in the whole range of the slope angles from zero to 90 deg. Fig. 6.1. Sagittal radius rs as a function of the slope φ in an oblate spheroid with vertex radius 𝑟 = 100 mm and conic constant 0 < K < infinity. On Fig. 6.1a we present an example for an elliptical surface. Here the sagittal radius is bigger than the vertex radius and it increases significantly for 40 < φ < 85 deg. On Fig. 6.1b we show an analogical graphic of the sagittal radius for a paraboloid. The special thing here is that the sagittal radius (as well as the two coordinates 𝑦 and z) tends rapidly to infinity for φ tending to 90 deg. If we go back to Eq. (6.78) we will see that radius-coefficient ρs in this case is zero. Now we nullify the radius-coefficient and find 1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑 =0, −1 𝑠𝑖𝑛 𝜑∞ = √ 𝐾 (6.79) (6.80) Analyzing the Eq. (6.80) we can say: 280 • If the conic constant is equal to -1 the sagittal radius tends to infinity in a paraboloid at φ = 90 deg only; • When the conic constant is in the range of -1>K> -∞ the sagittal radius approaches infinity for a strictly defined angle φ∞ at each value of the conic constant (see Fig. 6.2). The angle φ∞ we will call boundary slope. In these cases of hyperboloids the slope angle φ varies from zero to the boundary slope. Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Fig. 6.1a. Sagittal radius rs as a function of the slope 𝜑 in an ellipsoid with vertex radius 𝑟 = 100 mm and conic constant K = -0.8. Fig. 6.1b. Sagittal radius rs as a function of the slope φ in a paraboloid with vertex radius 𝑟 = 100mm and conic constant K = -1. Fig. 6.2. The boundary slope φ∞ as function of the conic constant K. 281 Advances in Optics: Reviews. Book Series, Vol. 5 On the Fig. 6.2a. a graphic for a sagittal radius as a function of the slope angle in a hyperboloid with a conic constant K = -2 is presented. In this case the boundary slope is 45 deg and the sagittal radius is approaching infinity at φ = 45 deg. Fig. 6.2a. The sagittal radius as function of the slope φ in a hyperboloid with vertex radius 𝑟 = 100 mm and conic constant K = -2. 6.2.2.2. Connections between the Parameters of the Tangential Rays and the Paraxial Matrix Transformations on a Single Conic Surface We demonstrate an analytical connection between the parameters of the incident and refracted tangential rays and paraxial matrix transformations on a single conic surface. On the Fig. 6.3 is shown a conic section of the conical surface S with the index j and vertex in the point V. The indices of refraction before and after the surface are nj and nj+1. The surface point N has coordinates yj and zj. The tangential ray R (it can be a chief ray or aperture one) falls on the surface at point N with an incidence angle εj and refraction angle ε’j. The incident ray R and the refracted one R’ form with the optical axis angle wj, wj+1 and cross the axis at points P and P’ respectively. The normal to the conic surface at N crosses the optical axis at the point C under the angle φj. The rectangular projection of N on the axis z is in the point No. It can be proved that the axial distances VP and VP’ are equal to 𝑉𝑃 = 𝛼𝑗 ∗ 𝑟𝑠𝑗 , (6.81) 𝑉𝑃′ = 𝛽𝑗 ∗ 𝑟𝑠𝑗 , 𝛼𝑗 = 𝑧𝑠𝑗 + ∆𝑗 ∗ 𝑐𝑜𝑠 𝑤𝑗 , 𝑠𝑖𝑛 𝜑 ∆𝑗 = 𝑠𝑖𝑛 𝑤𝑗 = 282 (6.82) (6.83) 𝑠𝑖𝑛(𝑤𝑗 −𝜀𝑗 ) , (6.84) 𝛽𝑗 = 𝑧𝑠𝑗 + 𝛿𝑗 ∗ 𝑐𝑜𝑠 𝑤𝑗+1 , (6.85) 𝑗 𝑠𝑖𝑛 𝑤𝑗 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures 𝛿𝑗 = 𝑠𝑖𝑛 𝜑𝑗 𝑠𝑖𝑛 𝑤𝑗+1 = 𝑠𝑖𝑛(𝑤𝑗+1 −𝜀𝑗′ ) 𝑠𝑖𝑛 𝑤𝑗+1 (6.86) Fig. 6.3. Conic surface transformation of an incident tangential ray into a refracted ray and their connections with the paraxial transformations. It is obvious from Eqs. (6.83) and (6.85) that the variables αj and βj depend on the indices of refraction (nj, nj+1), ray’s parameters (εj, wj, φj) and the sagittal radius coefficient ρs. Note that zsj in Eqs. (6.83) and (6.85) is the familiar to us already relative z-coordinate (see Eq. (6.77)). Let the points A0 and A’0 be the object and paraxial image points respectively. The object and image distances are zoj and z’oj. The two distances are measured from the point V along the optical axis. Further on we will use Eq. (6.4) and rewrite the paraxial matrix coefficients from Eq. (6.5) according to Eq. (6.71) i.e. 𝐴𝑗 = 𝑟𝑠𝑗 ∗ 𝜌𝑗 ∗ 𝑛𝑗+1 , 𝐵𝑗 = 0, 𝐶𝑗 = 𝑚𝑗0 , 𝐷𝑗 = 𝑟𝑠𝑗 ∗ 𝜌𝑗 ∗ 𝑛𝑗 (6.87) 𝑧𝑝𝑗 = 𝑧0𝑗 − 𝑟𝑠𝑗 ∗ 𝛼𝑗 , (6.88) ′ ′ − 𝑟𝑠𝑗 ∗ 𝛽𝑗 = 𝑧0𝑗 𝑧𝑝𝑗 (6.89) Now we relocate the object coordinate system from the vertex V to point P and the image coordinate system to point P’, respectively. Then the object distance zpj (measured from the point P) in the new object coordinate system is and the paraxial image distance z’pj (measured from the point P’) in a new image coordinate system becomes 283 Advances in Optics: Reviews. Book Series, Vol. 5 According to Eqs. (6.88), (6.89) and (6.3) the paraxial image distance turns into 𝐴 ∗𝑧 Here +𝐵 ∗𝑟 ′ 𝑧𝑝𝑗 = 𝑟𝑠𝑗 ∗ 𝐶 𝑝𝑗∗𝑧 𝑝𝑗+𝐷 𝑝𝑗∗𝑟 𝑠𝑗 , (6.90) 𝐴𝑝𝑗 = 𝑛𝑗+1 ∗ 𝜌𝑗 − 𝑚𝑗0 ∗ 𝛽𝑗 , (6.91) 𝐵𝑝𝑗 = 𝑛𝑗+1 ∗ 𝜌𝑗 ∗ 𝛼𝑗 − 𝛽𝑗 ∗ 𝐷𝑝𝑗 , (6.92) 𝐶𝑝𝑗 = 𝑚𝑗0 , (6.93) 𝐷𝑝𝑗 = 𝑛𝑗 ∗ 𝜌𝑗 + 𝑚𝑗0 ∗ 𝛼𝑗 (6.94) 𝑝𝑗 𝑝𝑗 𝑝𝑗 𝑧𝑗 Substituting the parameters αj, βj and zsj from Eqs. (6.83), (6.85) and (6.77) in Eq. (6.91). (6.92), (6.93) and (6.94) after boring transformations and rearrangements (but without any approximations) we obtain 𝐴𝑝𝑗 = (𝑛𝑗+1 ∗ 𝜌𝑗2 + 𝑝1𝑗 ∗ 𝜌𝑗 + 𝑝0𝑗 ) ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 ), 𝐵𝑝𝑗 = 𝑞3𝑗 ∗ 𝜌𝑗3 + 𝑞2𝑗 ∗ 𝜌𝑗2 + 𝑞1𝑗 ∗ 𝜌𝑗 + 𝑞0𝑗 , 2 𝐶𝑝𝑗 = 𝑚𝑗0 ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 ) , 𝐷𝑝𝑗 = (𝑛𝑗 ∗ 𝜌𝑗2 + 𝑙1𝑗 ∗ 𝜌𝑗 + 𝑙0𝑗 ) ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 ) (6.95) (6.96) (6.97) (6.98) The sub-coefficients p1j, p0j, q3j, q2j, q1j, qoj, l1j and loj in the Eqs. (6.95), (6.96), (6.97) and (6.98) depend on the indices of refractions and the ray parameters only. They are shown in Table 6.1. Table 6.1. Formulas for the sub-coefficients in the paraxial matrix coefficients of a single conic surface. 284 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures The coefficients Apj, Bpj, Dpj are polynomial of third degree in respect to the sagittal radiuscoefficient, while Cpj is of second degree. We proved that the paraxial matrix Gpj in the Eq. (6.99) describes correctly the bilinear function in Eq. (6.90) 𝐺𝑝𝑗 = 𝐴𝑝𝑗 𝐶𝑝𝑗 𝐵𝑝𝑗 𝐷𝑝𝑗 (6.99) Because Eq. (6.90) describes correctly the paraxial image distance trough the parameters of the real tangential ray in one conic surface we will call the matrix Gpj by ‘surface paraxial real ray transformation matrix’ (SPRRTM). If the surface is spherical, the matrix coefficients in Eq. (6.99) became much simpler as they are in Eqs. (6.95) -(6.98) and namely Here 𝐴𝑝𝑗 = 𝑛𝑗+1 − 𝑚0𝑗 ∗ 𝛿0𝑗 , (6.100) 𝐶𝑝𝑗 = 𝑚0𝑗 , (6.102) 𝐵𝑝𝑗 = 𝑛𝑗+1 ∗ ∆0𝑗 − 𝐷𝑗 ∗ 𝛿0𝑗 , (6.101) 𝐷𝑝𝑗 = 𝑛𝑗 + 𝑚0𝑗 ∆0𝑗 (6.103) 𝑠𝑖𝑛 𝜀 ∆0𝑗 = 1 − 𝑠𝑖𝑛 𝑤𝑗 ; 𝛿𝑜𝑗 = 1 − 𝑗 𝑠𝑖𝑛 𝜀𝑗′ 𝑤𝑗+1 (6.104) (6.105) 6.2.2.3. Paraxial Matrix Coefficients Linked to the Real Tangential Rays in a System of Centered Conic Surfaces If we add to one conic surface having an index (j) another surface with an index (j+1) then the paraxial image from the first surface appears as an object to the second one. In this case we can write ′ 𝑧𝑝(𝑗+1) = 𝑧𝑝𝑗 (6.106) Further we analyze the optical system of k surfaces. Applying Eq. (6.90) for the first surface we have 𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1 ∗𝑟𝑠1 ′ 𝑧𝑝1 = 𝑟𝑠1 ∗ 𝐶 𝑝1 ∗𝑧𝑝1 +𝐷𝑝1 ∗𝑟𝑧1 (6.107) Taking into account Eqs. (6.106) and (6.90) for the second surface we write 285 Advances in Optics: Reviews. Book Series, Vol. 5 𝑧′𝑝2 = 𝑟𝑠2 ∗ 𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1∗𝑟𝑠1 𝐴𝑝2 ∗𝑟𝑠1 ∗𝐶 ∗𝑧 +𝐷 ∗𝑟 +𝐵𝑝2 ∗𝑟𝑠2 𝑝1 𝑝1 𝑝1 𝑧1 𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1∗𝑟𝑠1 𝐶𝑝2 ∗𝑟𝑠1 ∗𝐶 ∗𝑧 +𝐷 ∗𝑟 +𝐷𝑝2 ∗𝑟𝑧2 𝑝1 𝑝1 𝑝1 𝑧1 𝐴 ∗𝑧 +𝑟 ∗𝐵 = 𝑟𝑠2 ∗ 𝐶𝑝∗𝑧𝑝1+𝑟𝑠1∗𝐷𝑝 𝑝 𝑝1 𝑠1 𝑝 (6.108) After performing the right equality in Eq. (6.108) and adequate arrangement we find where 𝐴𝑝 = 𝐴𝑝1 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 + 𝐵𝑝2 ∗ 𝐶𝑝1 , (6.109) 𝐶𝑝 = 𝐴𝑝1 ∗ 𝐶𝑝2 ∗ 𝑥𝑠1 + 𝐷𝑝2 ∗ 𝐶𝑝1, (6.111) 𝐵𝑝 = 𝐵𝑝1 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 + 𝐵𝑝2 ∗ 𝐷𝑝1 , (6.110) 𝐷𝑝 = 𝐵𝑝1 ∗ 𝐶𝑝2 ∗ 𝑥𝑠1 + 𝐷𝑝1 ∗ 𝐷𝑝2 , (6.112) 𝑟 (6.113) 𝑥𝑠1 = 𝑟𝑠1 𝑠2 Now let us assign to the paraxial transformation for the first surface (see Eq. (6.107)) the next quadratic matrix Gp1 and for the second one the matrix Gp2 then 𝐺𝑝1 = 𝐺𝑝2 = 𝐴𝑝1 𝐶𝑝1 𝐵𝑝1 , 𝐷𝑝1 𝐴𝑝2 ∗ 𝑥𝑠1 𝐶𝑝2 ∗ 𝑥𝑠1 (6.114) 𝐵𝑝2 𝐷𝑝2 (6.115) The paraxial coefficients Ap, Bp, Cp, Dp for the two surfaces in the right equality of Eq. (6.108) can be fined through the matrix gp as a matrix product of the two matrices Gp1 and Gp2, i.e. 𝑔𝑝 = 𝐺𝑝2 ∗ 𝐺𝑝1 (6.116) 𝑔𝑝𝑘 = ∏2𝑗=𝑘 𝐺𝑝𝑗 ∗ 𝐺𝑝1 (6.117) By analogy, for a system of k surfaces we prove that The matrix gpk is so called paraxial real ray transformation matrix of the system of k centered conic surfaces, i.e. (PRRTMS). In these case the matrix Gp1 is the same as it is in the Eq. (6.114) and Gpj for k ≥ j = 2 is as follows 𝐺𝑝𝑗 = 𝐴𝑝𝑗 ∗ 𝑥𝑠(𝑗−1) 𝐶𝑝𝑗 ∗ 𝑥𝑠(𝑗−1) 𝐵𝑝𝑗 𝐷𝑝𝑗 (6.118) The ratio between the two adjacent sagittal radii in an optical system we will call sagittal relative parameter xsj 286 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures 𝑥𝑠𝑗 = 𝑟 𝑟𝑠𝑗 (6.119) 𝑠(𝑗+1) The matrix gpk from (6.117) belongs to the next bilinear function 𝐴 ∗𝑧𝑝1 +𝐵𝑝𝑘 ∗𝑟𝑠1 ′ 𝑧𝑝𝑘 = 𝑟𝑠𝑘 ∗ 𝐶 𝑝𝑘∗𝑧 𝑝𝑘 𝑝1 +𝐷𝑝𝑘 ∗𝑟𝑧1 (6.120) Here z’pk is the paraxial image distance. It is measured from the axial intersection point of the tangential ray after the last surface to the paraxial image. The object distance is zp1 and rsk, rs1 are the last and the first sagittal radii respectively in the optical system. Using Eq. (6.117) after symbolic computations we find in analytical form (see for example Eqs. (6.109)-(6.112)) the four paraxial matrix coefficients for a system of k surfaces. Each matrix coefficient is a linear function of each relative parameter. Furthermore you can calculate, if it is necessary, the numerical values of the coefficients or their detached parts. This gives a possibility to satisfy preset conditions to the system using the sagittal relative parameters or sagittal radius-coefficients as a free parameters. For an infinity distant object the image distance 𝑧′𝑝𝑘∞ becomes 𝐴 ′ 𝑧𝑝𝑘∞ = 𝑟𝑠𝑘 ∗ 𝐶𝑝𝑘 (6.121) 𝑑𝑗 = 𝑟𝑠𝑗 ∗ 𝛽𝑗 − 𝑟𝑠(𝑗+1) ∗ 𝛼𝑗+1 , (6.122) 𝑑𝑗 = 𝑟𝑗 ∗ 𝛿0𝑗 − 𝑟𝑗+1 ∗ ∆0(𝑗+1) (6.123) 𝑝𝑘 It can be proved that the axial distance dj between the surface with number j and the one with number (j+1) in a system of conic surfaces is and for a system of spherical surfaces it becomes Knowing the matrix coefficients one can estimate the paraxial image distance in a system of k conic surfaces using the paraxial image function (see Eqs. (6.120) and (6.121)). The Eqs. (6.117) - (6.123) are the basic formulas both for the symbolic paraxial computations and for the numeric paraxial calculations in a centered optical system. 6.2.2.4. Geometrical Interpretation of the Paraxial Image Function and Its Connection with On-axis Spherical Aberration So far, we found rigorous analytical connections between the paraxial matrix coefficients and parameters of the real tangential rays which pass throughout the optical system. One can consider these real rays as chief rays or aperture ones. Let us analyze the latter case. By definition the longitudinal spherical aberration is a variation of focus with aperture. In this sense the distance zp1 (see Eq. (6.120)) is in fact the longitudinal object spherical 287 Advances in Optics: Reviews. Book Series, Vol. 5 aberration dz with the opposite sign, while the distance z’pk is the longitudinal image spherical aberration dz’ with the opposite sign too ′ 𝑑𝑧 = −𝑧𝑝1 , 𝑑𝑧 ′ = −𝑧𝑝𝑘 (6.124) Using the Eq. (6.124) we can rewrite (6.120) and (6.121) as follows 𝑑𝑧𝑟′ = ′ 𝑑𝑧 ′ 𝑟𝑠𝑘 𝐴 ∗𝑑𝑧𝑟 −𝐵𝑝𝑘 = −𝐶𝑝𝑘 ∗𝑑𝑧 𝐴 ′ 𝑑𝑧𝑟∞ = − 𝐶𝑝𝑘 𝑝𝑘 𝑟 +𝐷𝑝𝑘 , 𝑝𝑘 Here dz’r and 𝑑𝑧𝑟∞ are the so called image relative aberrations. The ratio ‘relative object aberration’ dzr. (6.125) (6.126) 𝑑𝑧 𝑟𝑠1 we will call It is evident from Eq. (6.125) that the relative image aberration dz’r is bilinear transformation of the relative object aberration dzr. If the object space of the optical system is aberration free, then the relative image aberration for an object at finite distances is equal to 𝐵 𝑑𝑧𝑟′ = − 𝐷𝑝𝑘 (6.127) 𝐵𝑝𝑘 = 0 (6.128) 𝐴𝑝𝑘 = 0 (6.129) 𝑝𝑘 To remove or correct the on-axis image aberration at finite object distance we have to nullify the numerator of Eq. (6.127) i.e. For an infinitely distant object the correction of spherical aberration according Eq. (6.126) is It is clear from Eqs. (6.128), (6.129), (6.109), (6.110), that the on-axis spherical aberration correction both for a finite object and for an infinitely distant object leads to solution of one linear equation in respect to one arbitrarily chosen relative parameter of the system or to solution of one cubic equation (see Eqs. (6.95) and (6.96)) in respect to one arbitrarily chosen sagittal radius-coefficient. 6.3. Simple Examples of Analytical Aberrations Correction for an Infinitely Distant Object In this paragraph we intentionally show ones of the simplest examples for the sole purpose of clear distinguishing the analytical computations from the numerical calculations in the practical applications of our rigorous analytical aberration theory. 288 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures 6.3.1. Analytical Correction of Axial Chromatic Aberration Together with Preset Focal Length in a System of Three Surfaces (Case of the so Called Doublets) Let the constructive parameters of the doublet be: three axial radii, two thicknesses and two glasses with indices of refraction for the design wavelength n2, n3. The doublet is immersed in air i.e. n1 = n4 = 1. To correct the axial chromatic aberration we have to satisfy Eqs. (6.51) or (6.52) using Eqs. (6.53) and (6.54). In our case for k = 3 they are 𝑢2 = 𝑒2 ∗ 𝑟32 + 𝑒1 ∗ 𝑟3 = 0, (6.130) 𝑒1 = 𝑒12 ∗ 𝑟22 + 𝑒11 ∗ 𝑟2 + 𝑒10 (6.132) 𝑒2 = 𝑒22 ∗ 𝑟22 + 𝑒21 ∗ 𝑟2 + 𝑒20 , (6.131) The indices of refraction for the left spectral end are n2l, n3l and for the right end are n2r,n3r respectively. At this stage we do not know the numerical values of the sub-coefficients (e22, e21, e20, e12, e11, e10). Below we shell show how to do this procedures. First we formulate the two ARM D1 and D2, the three ASTM G1r, G2r, G3r and ATMS gr for the right end of the wavelength spectrum and similarly for the left end in the system as follows 𝐷1 = [1 − 𝑑1 ; 01]; 𝐷2 = [1 − 𝑑2 ; 01], 𝐺1𝑟 = [𝑛2𝑟 ∗ 𝑟1 0; 𝑛2𝑟 − 1𝑟1 ]; 𝐺2𝑟 = [𝑛3𝑟 ∗ 𝑟2 0; 𝑛3𝑟 − 𝑛2𝑟 𝑟2 ∗ 𝑛2𝑟 ], 𝐺3𝑟 = [𝑟3 0; 1 − 𝑛3𝑟 𝑟3 ], 𝑔3𝑟 = 𝐺3𝑟 ∗ 𝐷2 ∗ 𝐺2𝑟 ∗ 𝐷1 ∗ 𝐺1𝑟 ; 𝐴3𝑟 = 𝑔3𝑟 (1,1); 𝐶3𝑟 = 𝑔3𝑟 (2,1), 𝐺1𝑙 = [𝑛2𝑙 ∗ 𝑟1 0; 𝑛2𝑙 − 1𝑟1 ]; 𝐺2𝑙 = [𝑛3𝑙 ∗ 𝑟2 0; 𝑛3𝑙 − 𝑛2𝑙 𝑟2 ∗ 𝑛2𝑙 ], 𝐺3𝑙 = [𝑟3 0; 1 − 𝑛3𝑙 𝑟3 ], 𝑔3𝑙 = 𝐺3𝑙 ∗ 𝐷2 ∗ 𝐺2𝑙 ∗ 𝐷1 ∗ 𝐺1𝑙 ; 𝐴3𝑙 = 𝑔3𝑙 (1,1); 𝐶3𝑙 = 𝑔3𝑙 (2,1) After symbolic computation of above shown matrices gr and gl we receive the four matrix coefficients A3r, C3r, A3l, C3l. Placing the latter coefficients in Eq. (6.130) after grouping on r3 we find the coefficients e2 and e1. Now we can group each coefficient on r2 and calculate the numerical values of the sub-coefficients e22, e21, e20, e12, e11, e10 for the system of three surfaces. It is clear that the just listed coefficients depend on r1, d1, d2, n2r, n3r, n2l and n3l. By analogy, we form the three ASTM and ATMS for the design wavelength in the system of three surfaces. To guarantee the preset focal length f’ we have to satisfy Eq. (6.56). After grouping on r3 and r2 we can calculate the numerical values of the sub-coefficients h11, h10, h01, h00. The just listed coefficients depend on r1, d1, d2, n2, n3. Further we proceed to search the common roots of the Eqs. (6.52) and (6.56). For this purpose we shell use Eq. (6.59) which in the case of three surfaces is 289 Advances in Optics: Reviews. Book Series, Vol. 5 𝑑𝑠𝑚 = 𝑉3 ∗ 𝑟23 + 𝑉2 ∗ 𝑟22 + 𝑉1 ∗ 𝑟2 + 𝑉0 (6.133) Knowing the numerical values of all sub-coefficients we calculate coefficients 𝑉3 , 𝑉2 , 𝑉1 , 𝑉0 in Eq. (6.133). Now we solve Eq. (6.133) in respect to r2 and check which of the roots for r2 satisfy simultaneously the Eqs. (6.52) and (6.56). Varying the lens parameters r1, d1, d2 and indices of refraction of the two glasses we can find all mathematically existing doublets with corrected axial chromatic aberration and preset focal length. In Table 6.2 we present a very small sample of calculations for the doublets of Schott-glass N-PK51 and Ohara-glass S-NPH2. The doublets have effective focal length of 20 mm and are corrected for two wavelengths 486 and 656 nm. In this table are shown the preset parameters r1, d1, d2 and calculated results for the two radii r2 and r3. In the last column is shown the residual of the axial chromatic aberration (𝑐ℎ𝑟∞) according Eq. (6.50). Table 6.2. Doublets with effective focal length of 20 mm and corrected axial chromatic aberration for wavelengths 486 and 656 nm at object in infinity. 𝒓1 ,mm 𝒅1 , mm 8 𝒅2 , mm 2 𝒓2 , mm -34.06914 𝒓3 , mm 829.4354 𝒄𝒉𝒓∞, mm 9 8 3 -38.24711 355.8315 -1.77e-15 9 8 4 -43.55748 219.0632 5.33e-15 10 9 4 -26.3199 -74.22728 3.55e-15 10 9 5 -28.81828 -80.90122 -1.78e-15 10 9 6 -31.85959 - 89.60543 -1.78e-15 12 10 2 -15.5144 -26.28416 -3.55e-15 12 10 3 -16.13771 -26.49421 -3.55e-15 12 10 4 -16.83818 -26.71472 3.55e-15 16 11 3 -11.62321 -16.1855 -5.33e-15 16 11 4 -11.72118 -16.17704 0 16 11 5 -11.82786 -16.16852 0 18 13 5 -10.10692 -13.86927 -1.78e-15 18 13 6 -10.05058 -13.85794 -1.78e-15 18 13 7 -10.07277 -13.8467 -1.78e-15 9 290 7.1e-15 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures The research shows that the residual chromatic abberations for the two wavelengths are less than 10-14 mm. We saw above that all calculations are analytical and in this sense “exact”. That is why we claim that the correction is rigorous analytical correction. 6.3.2. Analytical Correction of On-axis Spherical Aberration in a Systems of Two and Three Surfaces 6.3.2.1. Systems of Two Surfaces Let us discuss a simple system of two surfaces immersed in air. It has three constructive parameters (two vertex radii r1, r2 and axial distance d1). The index of refraction is n2. Let us imagine that the optical designer wants to design a lens with preset focal length f’2, back focal length bfl2, image numerical aperture NA and corrected on-axis spherical aberration for a given wavelength so that the residual aberration on the whole aperture is diffraction limited. To achieve the above requirements first we use Eqs. (6.39) and (6.40) in order to satisfy the requirements for f’2 and bfl2. In such a way we find the precise values of the vertex radii r1 and r2. Note that the two radii are calculated at concrete values of d1. This means that the vertex of the first surface is situated at given axial distance d1 from the vertex of the second surface. In our case we will call the first vertex as a common vertex for all points of the first aspherical surface. To correct the on-axis spherical aberration we use the sagittal radius-coefficient ρ1 of the first surface as a free parameter to satisfy Eq. (6.129). For this purpose we write the surface paraxial real ray transformation matrices (SPRRTM) for the first aspherical surface Gp1 and for the second spherical surface Gp2: 𝐺𝑝1 = [𝐴𝑝1 𝐵𝑝1 ; 𝐶𝑝1 𝐷𝑝1 ] (6.134) 𝐺𝑝2 = [𝐴𝑝2 ∗ 𝑥𝑠1 𝐵𝑝2 ; 𝐶𝑝2 ∗ 𝑥𝑠1 𝐷𝑝2 ] (6.135) The matrix coefficients for the first aspherical surface Ap1, Bp1, Cp1, Dp1 in Eq. (6.134) and its sub-coefficients (see Table 6.1) are described by Eqs. (6.95), (6.96), (6.97), (6.98) at index j = 1. The matrix coefficients for the second spherical surface Ap2, Bp2, Cp2, Dp2 and its sub-coefficients (see Eqs. (6.104), (6.105)) are described by Eqs. (6.100), (6.101), (6.102), (6.103) at index j = 2. After that the paraxial real ray transformation matrix of the system gp2 (according Eq. (6.117)) became 𝑔𝑝2 = 𝐺𝑝2 ∗ 𝐺𝑝1 ; 𝐴𝑝𝑘 = 𝑔𝑝2 (1,1) (6.136) 3 2 𝐴𝑝𝑘 = 𝑈3 ∗ 𝜌𝑠1 + 𝑈2 ∗ 𝜌𝑠1 + 𝑈1 ∗ 𝜌𝑠1 + 𝑈0 = 0 (6.137) Performing the symbolic computations according Eqs. (6.134)-(6.136) we receive for Apk a polynomial of the third degree in respect to the sagittal radius-coefficient ρ1 and namely 291 Advances in Optics: Reviews. Book Series, Vol. 5 The analytical formulas for the coefficients of Eq. (6.137) are 𝑈3 = 𝑛2 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 , (6.138) 𝑈2 = 𝐵𝑝2 ∗ 𝑚02 + 𝐴𝑝2 ∗ (𝑝11 + 𝑛2 ∗ 𝑐𝑜𝑠 𝜑1 ) ∗ 𝑥𝑠1 , (6.139) 𝑈0 = 𝐵𝑝2 ∗ 𝑚01 ∗ 𝑐𝑜𝑠 2 𝜑1 + 𝐴𝑝2 ∗ 𝑐𝑜𝑠 𝜑1 ∗ 𝑥𝑠1 (6.141) 𝑈1 = 𝐵𝑝2 ∗ 𝑚01 ∗ 𝑐𝑜𝑠 𝜑1 + 𝐴𝑝2 ∗ (𝑝01 + 𝑝11 ∗ 𝑐𝑜𝑠 𝜑1 ) ∗ 𝑥𝑠1 , (6.140) The latter coefficients depend on the tangential ray parameters ε1, ε2, w1, w2, w3, ε‘1, ε’2, n2 and on the relative parameter xs1. By definition the numerical aperture NA for our case is 𝑁𝐴 = 𝑠𝑖𝑛 𝑤3𝑚𝑎𝑥 (6.142) We find the aperture angle w3max from Eq. (6.142) at given NA. The back focal length, w3max and r2 are already known, so we can find the refracted and incidence angles on the second surface ε’2, ε2 and w2. The angle w1 between the aperture ray and optical axis for an infinitely distant object is zero. Knowing w2 and w1 we find the exact value of the incidence and refraction angles ε1 and ε’1. In this way we calculate all sub-coefficients and matrix coefficients for the first and second surface. As a result we can calculate the numerical values of U3, U2, U1 and U0. Now we solve the Eq. (6.137) and find its roots for ρs1. After that it is easy to calculate the sagittal radius rs1 from Eq. (6.78), the relative z-coordinate zs1 from the third equality of Eq. (6.77), the coordinates y1,z1 from Eq. (6.65) and Eq. (6.76), and finally the conic constant K1. Thus we calculated the parameters (K1, y1, z1,rs1) of one point which is infinitely small part of one conic surface according Eq. (6.64). The parameters calculated above refer to the intersection point of the aperture ray and the first aspherical surface. Varying the aperture angle w3 from zero to the w3max, it is easy to find all points of the first aspherical surface so that the on-axis spherical aberration is corrected on the whale aperture. But the studies have shown that the conic constant in each point is different as well as the vertex of the conic surface of each point is different from the common vertex. The latter means that the 𝑧-coordinate of each point needs to be corrected so that the vertex of each point to coincides with the common vertex. To do this we need one additional free parameter. How to avoid this problem you can see in the next subheading. 6.3.2.2. Multi-conical Surfaces in the Centered Optical Systems Eqs. (6.137)-(6.141) show that the relative parameter xs1 can be used to solve the problem mentioned above. The combination between Eqs. (6.122) and (6.123) give us the possibility to write in this case a formula for d1 𝑑1 = 𝑟𝑠1 ∗ 𝛽1 − 𝑟2 ∗ ∆02 (6.143) Using Eqs. (6.85) and (6.86) for the case of j = 1 and Eq. (6.104) for the case of j = 2 after substitution of β1 and ∆02 in (6.143) we can convert Eq. (6.143) into 292 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures 𝑥𝑠1 = 𝐿1 ∗(𝜌1 +𝑐𝑜𝑠 𝜑1 ) 𝑎1 ∗𝜌1 +𝑏1 (6.144) The parameters a1, b1 can be found in Table 6.1 and parameter L1 is 𝐿1 = ∆02 + 𝑑1 𝑟2 (6.145) The Eqs. (6.144) and (6.145) provide the condition each point of the first aspherical surface to be an infinitely small part of a conic surface whose vertex coincides with the common vertex. Now we put xs1 from Eq. (6.144) in the coefficients U3, U2, U1 and U0 of Eq. (6.137) and get 4 3 2 𝐴𝑝𝑘 = 𝑇4 ∗ 𝜌𝑠1 + 𝑇3 ∗ 𝜌𝑠1 + 𝑇2 ∗ 𝜌𝑠1 + 𝑇1 ∗ 𝜌𝑠1 + 𝑇0 = 0 (6.146) In fact the Eq. (6.146) is a sufficient condition to find the common roots of equation system (6.137) and (6.144). Now varying the aperture angle w3 from zero to the w3max, we find the sagittal radius-coefficient ρs1 from Eq. (6.146) and rs1, zs1, y1, z1, K1 as it was mentioned above. So we form a surface in the plane of y1 and z1 that we will call ‘multiconical surface’. Generally speaking, the multi-conical surface is a locus of points each of which is a point from a conic surface that has a common vertex but a different conic constant. 6.3.2.3. Examples of On-axis Spherical Aberration Correction in the Systems of Two Surfaces We used the described in Subsections 6.3.2.1, 6.3.2.2 algorithm for the spherical aberration correction and designed a series of lenses maid of fused silica with high and supper-high apertures (see Table 6.3) for high power Nd:YAG laser applications. The first surface of the lenses is multi-conical and the second is spherical or plane. The lenses have the next characteristics and parameters: D - diameter; EFL – effective focal length; NA – numerical aperture; r1 – vertex radius of the first surface; r2 – radius of the spherical surface; d1 – axial thickness; CW – correction wavelength; te – edge thickness. All the lenses of the Table 6.3 are diffraction limited for the correction wavelength and have high or supper-high apertures. The supper-high precision multi-conical designs decrease laser spot size and maintain high power per area. The comparative analysis shows that our lenses maintain a power per area about two times better than the best existing ones (see [6-9]). On the Fig. 6.4 we show the transverse on-axis spherical aberration for the correction wavelength of 350 nm. The lens characteristics and parameters are: r1 = 9.63 mm; r2 – plane; d1 = 12.5 mm; D = 25 mm; EFL = 21 mm; NA = 0.65. The residual aberration is estimated to be less than 1. 10-14 mm. 293 Advances in Optics: Reviews. Book Series, Vol. 5 In Table 6.4 We show a series of super-high numerical aperture diffraction limited plan-convex lenses of N-SF5 SCHOTT- glass. The designations here are the same as in the Table 6.3. The numerical apertures of the lenses are in the range of NA = 0.9 to NA = 0.97. The half of them are corrected for wavelength 587.6 nm and the other – for 780 nm. The diameters vary from 25 mm to 75 mm and the effective focal lengths are between 11.5 mm and 37.5 mm. The residuals of on-axis spherical aberration for the corrected wavelengths in all lenses are less than 5.10-14 mm i.e. they have a diffractionlimited image quality for the corrected wavelengths. On the Fig. 6.5. We show the transverse on-axis spherical aberration as function of the aperture angle in a lens with the next characteristics and parameters: D = 50 mm; EFL = 22.5 mm; NA = 0.95; r1 = 15.136 mm; d = 27.4 mm; te = 1.93 mm. We see that the aberration is very small and it is less than 1.5.10-14 mm. Table 6.3. High and super-high aperture diffraction limited lenses of FUSED SILICA for high power Nd:YAG laser applications. D, mm EFL, mm NA CW, nm r1,mm 16 13.5 0.65 355 6.19 16 13.5 0.65 532 6.19 16 13.5 0.65 1064 6.19 25 18 0.7 355 8.9 25 18 0.7 632 25 18 0.7 25 16 25 d1, mm te, mm 9 2.36 9 2.35 9 2.35 -59.35 13.5 2.5 8.9 -59.35 13.5 2.39 1064 8.9 -59.35 13.5 2.31 0.75 355 7.89 -44.5 14.4 2.32 16 0.75 532 7.89 -44.5 14.4 2.13 25 16 0.75 1064 7.89 -44.5 14.4 1.99 23 15 0.8 355 7.33 -40 15 3.73 23 15 0.8 532 7.33 -40 15 3.08 23 15 0.8 1064 7.33 -40 15 2.98 25 16 0.85 355 7.65 -57.5 16.5 2.66 25 16 0.85 532 7.65 -57.5 16.5 2.46 25 16 0.85 1064 7.65 -57.5 16.5 2.34 25 16.5 0.95 532 7.565 20 2.19 294 r2, mm ∞ ∞ ∞ ∞ Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Fig. 6.4. Transverse on-axis spherical aberration as function of the aperture angle in a plan-convex lens of FUSED SILICA corrected for 350 nm with NA = 0.65. Table 6.4. Super-high numerical aperture diffraction limited lenses of N-SF5 SCHOTT- glass. NA CW, nm r1, mm r2, mm d, mm te, mm 25 EFL, mm 12.5 0.90 587.6 8.409 plane 13.3 1.89 25 12.5 0.90 780 8.409 plane 13.3 1.77 25 11.5 0.95 587.6 7.737 plane 14 1.49 25 11.5 0.95 780 7.737 plane 14 1.66 25 11.5 0.97 587.6 7.737 plane 15 2.36 25 11.5 0.97 780 7.737 plane 15 2.61 30 15 0.90 587.6 15.6 plane 15.6 1.96 30 15 0.90 780 15.6 plane 15.6 1.81 30 14 0.95 587.6 9.418 plane 17 2.03 30 14 0.95 780 9.418 plane 17 1.96 30 13 0.97 587.6 8.75 plane 17.7 1.76 30 13 0.97 780 8.75 plane 18 1.77 40 20 0.90 587.6 13.454 plane 20.2 2.08 40 20 0.90 780 13.454 plane 20.2 1.88 40 18 0.95 587.6 12.11 plane 22.4 1.97 40 18 0.95 780 12.11 plane 22.8 2.02 40 18 0.97 587.6 12.11 plane 23 2.49 40 18 0.97 780 12.11 plane 23.6 2.63 50 24 0.90 587.6 16.145 plane 25.6 1.91 D, mm 295 Advances in Optics: Reviews. Book Series, Vol. 5 Table 6.4. Continued. NA CW, nm r1, mm r2, mm d, mm te, mm 50 EFL, mm 24 0.90 780 16.145 plane 25.6 1.62 50 22.5 0.95 587.6 15.136 plane 27.4 1.93 50 22.5 0.95 780 15.136 plane 27.8 1.91 50 22 0.97 587.6 14.8 plane 28 1.94 50 22 0.97 780 14.8 plane 28.4 1.89 60 30 0.90 587.6 20.181 plane 29 1.98 60 30 0.90 780 20.181 plane 29 1.68 60 27 0.95 587.6 18.163 plane 32.4 1.90 60 27 0.95 780 18.163 plane 33 1.98 60 26.5 0.97 587.6 17.827 plane 33 1.90 60 26.5 0.97 780 17.827 plane 33.4 1.78 75 37.5 0.90 587.6 25.227 plane 38 4.02 75 37.5 0.90 780 25.227 plane 38 3.65 D, mm On the Figs. 6.6 and 6.7 we show similar diffraction-limited lenses with numerical apertures NA = 0.97 and different diameters and correction wavelengths. The residual spherical aberrations in these lenses are very small too. Fig. 6.5. Transverse on-axis spherical aberration as function of the aperture angle in a lens of glass N-SF5 with D = 50 mm, NA = 0.95 and corrected wavelength 587.6 nm. 296 Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures Fig. 6.6. Transverse on-axis spherical aberration as function of the aperture angle in a lens of glass N-SF5 with D = 25 mm, NA = 0.97 and corrected wavelength 780 nm. Fig. 6.7. Transverse on-axis spherical aberration as function of the aperture angle in a lens of glass N-SF5 with D = 25 mm, NA = 0.97 and corrected wavelength 780 nm. 6.3.2.4. Examples of On-axis Spherical Aberration Correction in the Systems of Three Surfaces By analogy with the Subsections 6.3.1, 6.3.2.1 and 6.3.2.2 we designed series of doublets with corrected on-axis spherical aberration for 587.6 nm and chromatic aberration for 486 nm and 656 nm. The firs surface of the doublets is multi-conical. On Fig. 6.8 we show the transverse on-axis spherical aberration for a doublet with the next characteristics and constructive parameters: EFL = 15 mm; D = 20 mm; NA = 0.75; r1 = 7.58 mm; r2 = 203.435 mm; r3 = 48.243 mm; d1 = 10.7 mm; d2 = 8.1 mm. The on-axis spherical aberration on the whole aperture is less than 4.10-14 mm. The doublet has on-axis a diffraction-limited image quality for the correction wavelength of 587.6 nm. The maximum on-axis spherical aberration for 486 nm is less than 0.0088 mm and for 656 nm it is less than 0.006 mm. The comparative analysis shows that this doublet has a numerical aperture which is one and a half times larger than the best existing ones (see [10] and [11]). 297 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 6.8. Transverse on-axis spherical aberration as function of the aperture angle in a doublet of Schott-glass N-PK51 and Ohara-glass S-NPH2 with EFL = 15 mm, D = 20 mm, NA = 0.75. 6.4. Conclusions Based on the above presented text, tables and figures we can make the following conclusions: 298 • We found rigorous analytical dependences between the paraxial characteristics and constructive parameters in centered optical systems by the help of matrix transformations. This gives us the possibility to design optical systems with preset paraxial characteristics strictly calculating the necessary values of the constructive parameters such as vertex radii, axial thicknesses and refraction indices. The differences between the preset values of the paraxial characteristics and those, evaluated through our rigorous algorithms, constructive parameters, are less than 1.10-14 mm; • By analogy we found rigorous dependences between the axial chromatic aberrations and constructive parameters. Thus one can design optical systems with preset paraxial characteristics together with corrected axial chromatic aberrations without any optimization programs. The residual axial chromatic aberration between the preset left and right wavelengths in most cases is less than 1. 10-14 mm (see the last column in Table 6.2); • Formulas have been developed, which connect analytically the parameters of the real tangential rays with the sagittal radius-coefficient of the conical surface in the centered optical systems. They gave us the possibility to create a reliable rigorous algorithm for on-axis spherical aberration correction on the whole aperture, using one multi-conical surface only. The residual on-axis transverse spherical aberration in most cases is less than 5.10-14 mm; • The developed above theoretical base allows the creation of simple computer program that guarantees: rigorous correction of on-axis spherical aberration, correction of axial chromatic aberration and guarantee preset paraxial Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrations in the Centered Conical Optical Systems with High and Super-high Numerical Apertures characteristics in optical systems independently from the number of surfaces. The program calculates strictly all technological data (such as y-coordinates and zcoordinates, slope of the normal in each point of the surface, etc.) for the production of optical components with aspherical profiles using CNC machines. Neither an optimization software nor a skilled or experienced designer are needed for the design of diffraction-limited (for one wavelength) optical systems with high or super-high numerical apertures. References [1]. J. Sasian, Introduction to Lens Design, Cambridge University Press, New York, Melbourne, New Delhi, Singapore, 2019. [2]. W. J. Smith, Modern Optical Engineering, McGraw Hill, New York, Chicago, San Francisco, Lisbon, London, Madrid, Mexico City, Milan, New Delhi, San Juan, Seoul, Singapore, Sydney & Toronto, 2008. [3]. B. A. Hristov, Development of optical design algorithms on the base of the exact (all orders) geometrical aberration theory, Proceedings of SPIE, Vol. 8167, 2011, 81670E. [4]. B. A. Hristov, Exact analytical theory of aberrations of centered optical systems, Optical Review, Vol. 20, Issue 5, 2013, pp. 395-419. [5]. C. Chen, Methods of solving aspheric singlets and cemented doublets with given primary aberrations, Applied Optics, Vol. 53, Issue 29, 2014, pp. 202-212. [6]. Precision UV Fused Silica Aspheric Lenses, http://www.asphericon.com [7]. Fused Silica Aspheres, http://www.edmundoptics.com [8]. NIR Ultrafast Optimized Fused Silica Plano-Convex Lenses, http://www.newport.com [9]. Aspheric Lenses, http://www.optoaxis.com [10]. Aberration Corrected Lenses, http://www.Ulooptics.com [11]. Precision Aspherized Achromatic Lens, http://www.edmundoptics.com 299 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry Chapter 7 Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry Ksenia Vlasova, Alexandre Makarov and Nikolai Andreev1 7.1. Introduction The problem of detecting low concentrations of chemical elements in various transparent materials by thermo-optical absorption measurement has a long history [1]. The possibility to measure concentration of inclusions by measuring the absorption of the material under consideration from connected with the resonant behavior of the spectral dependence of the inclusion absorption and its linear dependence on the inclusion concentration. An example of such measurements in ultrapure quartz glasses (UQGs) is described elsewhere [2]. In this paper the results of optical radiation absorption measurements were used to calculate the OH group concentration. The sample transmission was measured at the maximum of the absorption line of the OH group (2720 nm) associated with the vibrations of this group in quartz glass. The measurement was performed by means of Fourier transform infrared (FTIR) spectroscopy with an absorption detection limit of about 10-2 cm-1. To measure the quartz matrix contribution, a thermal radiation source with a continuous spectrum in the resonance absorption region was exploited. Measuring the spectral absorption dependence outside the resonant absorption line of the inclusion made it possible to calculate accurately the quartz glass matrix contribution by interpolation of results to the resonant absorption region. In this example, the possibility Ksenia Vlasova Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia 301 Advances in Optics: Reviews. Book Series, Vol. 5 of measuring a low concentration of an inclusion is related to the strong absorption at the center resonance frequency of its absorption contour, which is available for measurement by the traditional direct method. However, e. g., for metallic inclusions, the absorption spectrum of which lies in the visible or near-infrared range, the absorption value is no greater than 10-5 cm-1 – 10-7 cm-1 at the concentrations of interest of ~10 ppb – 1 ppt. In this case, indirect thermooptical methods are exploited to measure absorption. The idea of thermooptical methods is to heat the test sample with a focused laser beam at the resonant frequency of an absorbing inclusion and observing the refractive index response of the medium resulting from changes in its temperature. This response is detected by a probe laser beam with a wavelength different from the heating laser wavelength. Fig. 7.1 shows an example of one of the most widely used sensitive absorption measurement schemes based on a thermo-optical method called Photothermal Common-path Interferometry (PCI) [3]. This scheme has the following features: 1. The probe laser beam crosses the region of the heated by the pump laser test sample at a certain angle. The angle is selected from consideration of the suppression degree of the heating radiation penetration into the probe beam recording channel. 2. The pulse-periodic structure of the heating radiation is created by a mechanical modulator in the form of an opaque rotating disc with holes cut therein for the CW radiation transmission. 3. Changes in the probe beam spatial structure are detected by means of a diaphragm set up in front of the photodiode in a certain cross-section after the sample, and a lock-in amplifier. 4. A common feature of lock-in amplifying is the absence of information on the detected signal temporal form. Fig. 7.1. Schematic block diagram of the PCI method for absorption measurement. Obviously, the sensitivity of such a method depends, among other things, on the dn/dT value of the test sample. Therefore, at present, the method has found the greatest application in the measurement of inclusion concentrations in liquids having high dn/dT values. Detectable levels from ppb (10-9) to ppt (10-12) weight fractions have been achieved when measuring the absorption values at the level of 10-9 cm-1 at wavelengths close to the resonant frequencies of absorbing chemical elements [1]. In this method, the inclusion concentration calculation from the result of measuring the absorption at the resonant 302 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry frequency is organically related to the separation of the UQG matrix contribution from the measured absorption. In this chapter, we will discuss the determination of inclusion concentrations in quartz glasses and optical crystals. The specificity of the problem of determining low inclusion concentrations by thermooptical measurements in these dielectrics is related to the low value of their dn/dT, which differs by about one or two orders of magnitude from the corresponding values in liquids. E. g., for quartz glass dn/dT = 0.96×10-5 K-1 [4] is about 80 times less than for CS2 (dn/dT = -8×10-4 K-1) and about 60 times less than for CCl4 (dn/dT = -6.1×10-4 K-1). 7.2. Relevance To date, the problem of detecting inclusions in different materials remains relevant from the point of view of studying the influence of impurities on the properties of manufactured materials and, ultimately, their quality control. We will discuss the possibility of applying a thermo-optical method for measuring low absorption to determine the inclusion concentrations in UQG and synthetic crystalline quartz (SCQ) which are widely used in different fields of science and technology. UQG underlies all information technologies in the modern world. UQGs are used in the production of optical fibers [5], UV radiation projection optics [6, 7], the design of UV lasers which play an important role in the production of printed circuit boards [8-10] and integrated electronic microcircuits [11, 12]. The quality of UQGs is also important in the production technology of large integrated circuits. In this case, UQGs are employed for the manufacture of crucibles in which silicon melts [13]. SCQs grown by hydrothermal method can be utilized in the manufacture of quartz glasses as a raw material. In addition, SCQs are also exploited in laser technology for the manufacture of large-aperture polarizing optical elements and windows, as well as lenses for the output stages of high-power CW lasers [14]. In both cases, low absorption at the laser radiation wavelength is required, which is directly related to the concentration of impurities of different chemical elements. In addition to quality control of SCQs, the concentration measurement of polluting chemical elements is also relevant in the process of development of the technology for growing crystals intended frequency conversion of high-power CW lasers using nonlinear optics methods, because the efficiency of nonlinear optical processes is limited by parasitic radiation heating of crystals [15] due to the presence of such inclusions. Obviously, one of the necessary conditions for improving the ultrapure materials production technology is the possibility of measuring impurity concentrations at a level better than the minimal level achieved at this stage of production technology development. The state of this issue for UQG can be understood from the information provided, for 303 Advances in Optics: Reviews. Book Series, Vol. 5 example, on the website of Heraeus company, the world leader in UQG production [16]. Fig. 7.2, taken from this website, shows a table of UQG characteristics (F series glasses). It can be seen that the concentration of polluting chemical elements is below the detection limit of one of the most sensitive Inductively Coupled Plasma Mass Spectrometry (ICP-MS) method [17] (see “Trace impurities”). Fig. 7.2. Table of typical material parameters for quartz glasses of various Heraeus brands. Source: [16]. On the other hand, the production of glass with trace impurities below ICP-MS detection limit does not mean at all that there is no need to continue improving the UQG purification technology. As an example in which a further decreasing in the concentration of impurities relative to the achieved level is relevant, we can mention the technology of growing ultrapure silicon crystals for microelectronics. UQG is used for the manufacture of crucibles in which silicon melts to grow crystalline silicon for the production of large integrated circuits and solar cells. The quality of the crystals grown critically depends on the impurities concentrations in the crucible material [13, 18, 19]. The concentration value level (e.g., for iron), affects the quality of the grown crystals, is below the sensitivity of the ICP-MS and Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LAICP-MS) methods, which does not exceed 10 ppb [17, 20]. Thus, the insufficient sensitivity of methods for determining the impurities concentration in UQGs produced today makes it necessary to increase the sensitivity of methods for determining the concentration of different chemical elements in UQGs и SCQs. 304 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry 7.3. State of the Art In this chapter, we will discuss the possibility of impurity concentration measurements in quartz glass and crystalline quartz close to the detection limits, and for some chemical elements significantly exceeding them, of one of the most sensitive and widely used LA-ICP-MS method. This possibility is associated with ultra-low absorption measurements at discrete wavelengths ranging from near UV to IR utilizing a thermooptical method. From the above example of OH group concentration measurement, it is clear that the absorption coefficients of both the UQG matrix and a chemical element at the resonant frequency need to be known in order to calculate the concentration exploiting results of absorption measurement. These values are calculated allowing for the results of measurements at several frequencies. In the above example, a source of thermal radiation with a continuous spectrum in the resonant absorption region of the desired inclusion was employed to measure the contribution of the UQG matrix. In the case of utilizing laser sources generating at a certain wavelength, such measurements are performed using radiation with an additional test wavelength, shifted from the line center to the region of weaker ion absorption. Further, the actual ion absorption is calculated under the assumption of a weak wavelength dependence of the UQG matrix absorption. In a more complex case of a notable variation in the UQG matrix absorption at the width of the line contour, the absorption is measured at one more wavelength, shifted from the line center in the opposite direction from the first one to construct an absorption value linear interpolation to the line center. Thus, it is easy to understand that in order to correctly calculate the chemical element concentration, the absorption measurement must be accurate enough to reflect the variations made by the chemical element in the matrix absorption. If the absorption coefficient is changed in such a way that the contribution made by the inclusion in the UQG absorption is less than the random deviations of the measured values, then it becomes impossible to calculate the desired concentration. Therefore, the parameter determining the measurement accuracy is the ratio of the signal level to its noise component level. The successful solution of the task is related to the possibility of the thermooptical method to provide measurements at the absorption level of the UQG matrix with a signal-to-noise ratio large enough to reliably detect the variations in the absorption of desired chemical element. Exploiting the data presented in the literature, we will estimate the UQG matrix absorption level and the acceptable noise level for these measurements. For the evaluation of the UQG matrix absorption in the spectral region of the absorption lines of metal ions, the theoretical estimates presented in Fig. 7.3 can be employed (Fig. 3.3 from [21]). We obtain that the absorption coefficient of the UQG matrix varies in the range from 4×10-6 cm-1 at the wavelength of 400 nm (resonant for Fe3+ ions) to 0.9×10-7 cm-1 at the 305 Advances in Optics: Reviews. Book Series, Vol. 5 wavelength of 1100 nm (resonant for Fe2+ ions). I. e., the absorption measurement method should allow the absorption measurement of 10-7 cm-1 with a sufficiently high signal-tonoise ratio. To estimate the ratio tolerable value, one should consider the characteristic values of the minimum concentrations of some chemical elements detected by LA-ICP-MS in UQG (Fig. 7.4). Fig. 7.3. The calculated attenuation spectra for some of the loss mechanisms contributing to the overall fiber attenuation (dashed and dotted lines). Source: [21]. Fig. 7.4. The LA-ICP-MS detection limit for determining the concentration of inclusions in quartz glass. Source: [17]. It follows from the Fig. 7.4 that the detection limit of different chemical elements in UQG varies widely from 18 ppb for Al, Ti, and Fe to 1.5 ppt for Cu. The LA-ICP-MS detection limit for polluting inclusions in SCQ, as shown in Table 7.1, is much worse, ranging from 14 ppm for Ca to 10 ppb for Pb. This detection limit is even worse for ICP-MS. 306 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry Using the presented data, it is possible to calculate the changes that are made by some chemical elements into the UQG absorption, if the absorption of these elements at a given concentration is known from literature. The absorption values of some of the most common metal ion inclusions are presented in the first three columns of Table 7.2. The data are given for resonant frequency and inclusion concentration of 1 ppb. The fourth column shows the LA-ICP-MS detection limits of concentrations of the elements (data taken from Fig. 7.4). The fifth column shows the calculation results of the element absorption values at concentrations corresponding to the LA-ICP-MS detection limit from the third column. The sixth one shows a theoretical estimate of the UQG matrix absorption coefficients at the corresponding wavelengths. Table 7.1. ICP-MS and LA-ICP-MS detection limits (ppm) for different chemical elements in crystalline quartz [22-24]. ICP-MS [22] LA-ICP-MS [23] LA-ICP-MS [24] Li 0.5 0.55 0.22 B 1.28 Na 100 5.5 6.4 Al 100 6 6 K 100 3.2 2.3 Ca 100 14 29 Ti 5 1.2 0.66 Mn 0.3 0.07 Fe 1 1.6 0.52 Pb 0.01 - Table 7.2. Absorption of some of the most common metal ions (first column) at a concentration of 1 ppb (third column) at resonance frequency (second column) and corresponding absorption coefficient (fifth column) at the detection limit concentration (fourth column). The data of the first three columns are from [21], the data of the fourth column are from Fig. 7.2, the data of the fifth column are the result of the calculation based on the data of the third and fourth columns. The sixth column presents a theoretical estimate of the UQG matrix absorption coefficients [21]. Ion Peak wavelength (nm) Absorption coefficient, one part in 109 (dB/km) LA-ICPMS detection limit (ppb) Cr3+ C2+ Cu2+ Fe2+ Fe3+ Ni2+ Mn3+ V4+ 625 685 850 1100 400 650 460 725 1.6 0.1 1.1 0.68 0.15 0.1 0.2 2.7 0.1 1.5·10-3 17.8 3.16 0.75 0.013 Absorption coefficients at concentration equal to detection limit (×106 cm-1) 0.37 0.0039 27.8 1.1 0.34 0.08 Theoretical estimate of the UQG matrix absorption (×106 cm-1) ≈1.4 ≈1 0.26 0.09 ≥3 ≈1.2 ≥2.8 0.69 We now collate the sensitivity of the LA-ICP-MS method and widely used PCI method, utilizing the fifth column data. To estimate PCI method detection limit of absorption measurements in UQG, we employ the measurement data of Suprasil 300 glass provided 307 Advances in Optics: Reviews. Book Series, Vol. 5 by Heraeus, the world's leading manufacturer of UQGs. According to the company's data (see Fig. 7.5, taken from [25]), the absorption coefficient of Suprasil 300 is 𝛼1064 ≈ (2-8)×10-7 cm-1 or 5×10-7 cm-1 ± 60 % at a wavelength of 1064 nm. The scatter of values most likely indicates that the measurements were performed at the extreme level sensitivity. I. e., according to the accepted definition, the detection limit (the absorption value obtained under experimental conditions when the values measured are equal to their noise component amplitude) will be 𝛼1064 ≈ 5×10-7 cm-1. Fig. 7.5. Absorption in quartz glasses of different Heraeus brands at a wavelength of 1064 nm. Source: [25]. It is now possible to compare the absorption values (fifth column) of the elements having concentration corresponding to the detection limit with the value of the PCI method detection limit when measuring the absorption of 0.5×10-6 cm-1. In this case, without any generality loss, we assume that the sensitivity of the PCI absorption measurements at the wavelengths of the centers of the absorption lines of the ions under consideration does not depend on the wavelength. The comparison shows that among the considered ions, only two can change the UQG absorption by the values available for detection by the PCI method, which has the signal-to-noise ratio equal to 1 when measuring the absorption ~0.5×10-6 cm-1. These are Fe2+ (detection limit is 18 ppb) and Fe3 (detection limit is 3.2 ppb) ions. In the presence of Fe2+ and Fe3+ ions with the minimum detectable concentration by mass spectroscopy methods, the variation in the UQG matrix absorption should be 50 times and 12 times, respectively, higher than the UQG matrix absorption. It follows that PCI concentration measurements are available for these elements, since when measuring total absorption, the signal-to-noise ratio will be 50/1 in the first case and 12/1 in the second. All other considered ions will vary the UQG absorption by values that are beyond the PCI measurement capabilities. 308 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry In addition to metal ions presence, the UQG absorption can increase due to the presence of the OH groups. In the spectral range of absorption of metal ions, the third vibration overtone of OH group (absorption maximum at a wavelength of 940 nm in SiO2 matrix) will distort the UQG matrix absorption value. The absorption value contributed by OH groups to the UQG matrix absorption can be obtained employing the dependence of the OH-related absorption on the wavelength shown in Fig. 7.6 [26]. From the data in the figure, we obtain that the absorption is 1050 Db/km or 1050 2.3×10-6 cm-1 = 2.4×10-3 cm-1 at a wavelength of 940 nm (the third overton of the vibrational line of 2720 nm) at the OH group concentration of 700 ppm. E. g., if the OH concentration in the Suprasil 300 glass is 0.2 ppm, then the absorption will be 2.4×10-3 cm-1×0.2/700 = 6.9×10-7 cm-1. This absorption is slightly higher than the PCI method detectable level, and under some conditions, the result of such a measurement can be utilized to determine the concentration of about 0.2 ppm. Fig. 7.6. Absorption spectral dependence for silica fibers with high (700 ppm, F100) and low (0.2 ppm, F300) concentrations of OH groups. Source: [26]. The above analysis shows that in some cases the absorption values detected by the PCI method can be determined by inclusions of chemical elements, the concentration of which lies beyond the detection limits of mass spectroscopy methods. However, the main problem with thermooptical measurements is that the concentration measurements of even one impurity requires an absorption measurement at least two wavelengths (as, e.g., in the case of the OH group concentration measuring [2] described above). But the case where the inclusion absorption spectrum is determined by an isolated absorption line is unique. Real glasses contain many chemical elements with overlapping contours of the absorption spectrum. So, when discussing the issue of the quality of UQGs, an estimate of the acceptable concentrations of more than ten chemical elements is given (see Fig. 7.7) [27, 28]. Thus, in general, the determination of impurity concentrations involves the problem of calculating the concentration of N chemical elements by measuring the absorption at a 309 Advances in Optics: Reviews. Book Series, Vol. 5 number of selected wavelengths. This raises the question of the optimal choice of a set of wavelengths, improving the concentration calculation accuracy, based on the wavelength dependences of the absorption of all considered elements and the results of measuring the UQG absorption at the selected wavelengths. Generally, the problem of measuring the concentrations of the chemical elements is integrated and it consists of three, in fact, independent components, the successful solution of which influences the solution of general problem. Fig. 7.7. ICP-MS for quartz glasses of different Heraeus brands. Source: [27]. First, the dependences of the absorption coefficients of the considered chemical elements on the wavelength and on their concentration in UQG should be known. The possibility of obtaining such dependences for UQG was demonstrated elsewhere [29], where the availability of the controlled doping procedure with different chemical elements up to absorption levels ≥0.01 cm-1 in the process of UQG fabricating was shown. This makes it possible to measure the absorption of this level utilizing traditional Fourier spectroscopy method. Spectral dependences of absorption contours were obtained for a number of chemical elements. The resonance values of the absorption coefficients of some chemical elements in UQG, which we employed in the Table 7.2, are given in [21]. It should be noted that we are not aware of the dependences of the absorption coefficients of the considered chemical elements on the wavelength and on their concentration in SCQ, as well as in nonlinear crystals. Moreover, the procedure for measuring them in SCQ is apparently very problematic due to the impossibility of incorporating foreign chemical elements into the crystal structure during the crystal growth. Second, a detailed solution to the mathematical problem of calculating the concentration of N chemical elements by absorption measurement at a number of selected wavelengths is needed. This problem is not trivial and it requires serious consideration. We have not found the solution to this problem in the literature. Although there are studies with similar 310 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry content and formulations of the problem. E. g., the spectral dependence of the transmission associated with the concentration of the Fe2+ ion was calculated from the measured spectral dependence of the near-IR region transmission of silicate glass with unknown concentrations of inclusions [30]. Both the transmission wavelength of the OH group and the silicate glass matrix, which has a complex chemical composition, were exploited in the calculations. Also, for silicate glass the spectral dependencies related to Fe3+, Fe2+, and OH groups and glass matrix were calculated by measured spectral transmission in the visible and UV spectral regions [31, 32]. Third, a physical tool is needed for sufficiently accurate absorption measurements at a certain number of selected wavelengths, determined by solving a mathematical problem. It is quite obvious that from the point of view of the formulated problem, solving the first two problems is not of great importance without a sufficiently reliable prospect of a successful solution to the problem of creating a physical tool for measuring ultra-low absorption with sufficient accuracy at an independent set of wavelengths. The measurement accuracy requirement can only be met with a sufficiently high sensitivity, i.e., at a sufficiently low detection limit value relative to the measured absorption value. The ratio of the detection limits of the mass spectroscopy method and the PCI method indicates the prospects of further development of the PCI method in the direction of increasing the sensitivity in the interests of the task at hand. 7.4. Time-resolved Photothermal Common-path Interferometry Scheme It can be seen from the above analysis that the PCI method for measuring absorption is one of the most sensitive of the known thermooptical methods and it can be useful in measuring the concentrations of different inclusions in UQGs. However, its sensitivity is clearly not enough for a comprehensive solution to the problem of determining the concentration at the ICP-MS detection level. Nevertheless, as we have shown, the given sensitivity of absorption measurements using the PCI scheme is not the limit for thermooptical methods. In our recent papers [33-35], a modification of this scheme, called Time-resolved Photothermal Common-path Interferometry (TPCI), has been introduced. The sensitivity of this scheme in the current implementation is two orders of magnitude higher than its PCI prototype. Fig. 7.8 shows the electron-optical block diagram of the experimental setup. The operation principle of this scheme has been discussed in detail elsewhere [33-35]. Here we will only point out the main features of the TPCI scheme that distinguish it from its predecessor PCI scheme, which made it more sensitive and functional. First of all, it is the heating and probe beams coaxiality in the test samples. In the PCI scheme, the beams are crossed at a low angle (see Fig. 7.1) to get rid of the influence of powerful heating radiation on the measuring photodiode [36]. Although the angle 311 Advances in Optics: Reviews. Book Series, Vol. 5 intersection of the beams improves the spatial resolution of the absorption measurement scheme, it significantly reduces sensitivity. The possibility of implementing a coaxial scheme is associated with the using of a space-frequency filter. It includes an interference M3 mirror for the heating laser radiation initial filtering, a DPr dispersion prism and a M5 mirror (schlieren), which cuts off radiation at a wavelength of 1070 nm. This filter effectively filters out heating radiation and, in coaxial geometry, makes it possible to achieve its suppression coefficient of 10-9 on the photodiode. Fig. 7.8. Electron-optical block diagram of the TPCI method. The coaxial configuration enables the use of video cameras (not shown in the figure) to control and adjust the alignment of the beams at the initial scheme setup and in the case of its misalignment. It ensures the reproducibility of the measurement results. In addition, the coaxial propagation of beams allowed us to develop a theory linking the magnitude of the signal observed during TPCI measurements with all existing parameters, both the test sample and the measuring setup. The obtained analytical formulae make it possible to calculate the required absorption in the test sample from the results of similar measurements in a reference sample with known absorption. These formulae were obtained by asymptotic expansions with respect to the dimensionless small parameters of solutions of the following problems: 1) Theory of deformation in a given nonuniform temperature field arising when the sample is heated by focused laser radiation, and the calculation of a thermo-optical response, i.e., spatially inhomogeneous change in the material refractive index when exposed to the heat of focused radiation; 2) Theory of probe beam diffraction on the spatial-temporal refractive index inhomogeneity, associated 312 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry with the arising stresses, and 3) Interference of the constant component of the probe beam with its time-varying component; this interference forms the measured photodiode signal. The accuracy of the calculations is sufficient for practical purposes, which is due to the extreme smallness of the dimensionless parameters, in order of magnitude equal to ~10-4. The details can be found elsewhere [34, 35]. The obtained analytical formulae, given below, make it possible to use any optical sample with known absorption and thermo-optical coefficients as a reference sample for calibration. An important conclusion from the developed theory [34, 35, 37] is that the allowance for the influence of spatially inhomogeneous stresses arising in a local heated region leads to a replacement of the dn/dT value by the so-called thermo-optical parameter P [38] in the commonplace expressions for the thermo-optical response, which are valid for liquids and gases [1]. This parameter depends on the sample physical parameters and, if the sample is anisotropic, on its orientation relative to the propagation direction and radiation polarization. Thus, the expression for the measured signal equal to the diaphragmaveraged time-varying component of the probe radiation power P~ can be written as (7.1) 𝑃~ (𝑡) = 𝐾 ∗ 𝑃 ∗ 𝛥𝑇(𝑡) Here, ΔT is the temperature change in the heating beam center, equal to [34] 𝑇(𝑡) = 𝑊ℎ (𝑡) 2 ∗𝑐∗𝜌 αℎ , 𝜋𝑎ℎ (7.2) in the absence of heat diffusion from the heated area. Here, αℎ is the absorption (cm-1) of the test sample at the heating laser wavelength; 𝑊ℎ (𝑡) is the current heating radiation energy density; c is the heat capacity; ρ is the test sample density; ah is the heating beam radius in the sample at the 1/e intensity level. The expression for the thermo-optical parameter obtained in case of isotropic sample [35] 𝜕𝑛 𝜕𝑛 𝑃 = (𝜕𝑇) − 𝜌 𝛼𝑇 ∗𝑛30 1+𝜈 4 1−𝜈 (7.3) ∗ (𝑝11 +𝑝12 ) Here, (𝜕𝑇 )𝜌 is the temperature change in the refractive index at a constant density, αT is the thermal expansion coefficient, 𝑝11 and 𝑝12 are the components of the photoelasticity tensor, and ν is the Poisson's ratio. The expression for the thermo-optical parameter obtained for the case where the radiation propagates along the crystallographic axis C for a crystal sample of class 32 (symmetry of crystalline alpha-quartz) 𝜕𝑛 𝑃 = ( 𝜕𝑇𝑜 )𝜌 − 𝑛03 ∗ 𝑝11 +𝑝12 4 ∗ 𝑇 (𝐶11 +𝐶12 )∗𝛼⏊ +𝐶13 ∗𝛼∥𝑇 𝐶11 , (7.4) 313 Advances in Optics: Reviews. Book Series, Vol. 5 𝑇 where С11 and С12 are the stiffness tensor coefficients, 𝛼⏊ is the thermal expansion coefficient across the C axis, 𝛼∥𝑇 is the thermal expansion coefficient along the C axis. Formulae for calculating the thermo-optical parameter for other configurations of the mutual arrangement of the crystal axes and the direction of radiation propagation can be found elsewhere [35]. Function K, which allows for the beam configuration shown in Fig. 7.9, has the form 𝐾 = 𝑃𝑝𝑟 ∗ 𝑧ℎ𝑝𝑟 2 2 2∗𝑎𝑝𝑟 𝑧 exp[𝛼𝑒𝑓𝑓 ∗(𝑧−𝑧0 )]∗ [exp[−𝑅 2 ∗𝑓1(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 )]−1] 𝑑𝑧)], 𝑓2(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 )∗𝑓3(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) 0 ∗ [−𝐼𝑚 (∫𝑧 1 (7.5) where Ppr is the time-independent probe beam power, R is the radius (normalized to ah) of the projection of the diaphragm installed in front of the photodiode. The functions 𝑓𝑖(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) are the complex functions of real variables, the meaning and designations of which are shown in Fig. 7.9. 𝑓1(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) = 1 −2i∗[(z−z1 )− 𝑓2(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) = = 𝑎2ℎ 2∗𝑎2𝑝𝑟 𝐿∗𝑛 𝑝𝑟 ]+B(z) + 𝑎2 − 𝛥𝑝𝑟 ∗ 𝑎2ℎ + 𝑝𝑟 𝐵(𝑧) = 2 𝑎2 𝜆ℎ ∗𝑀2 ) ) + (1 + 𝑖 ∗ 𝑧 ∗ 2ℎ ), 𝑎𝑝𝑟 𝜆𝑝𝑟 2 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 2∗𝑧𝑝𝑟 𝑎2 𝐿∗𝑛 ]) + 2∗𝑎ℎ2 (−2 ∗ 𝑖 ∗ [𝑧 − 𝑧1 − 𝑧 𝑝𝑟 𝜆 ∗𝑀2 2 2 𝑎2 ) )∗(1+𝑖∗𝑧∗ 2ℎ ) (1+[𝑧+𝛥𝑝𝑟 ] ∗( ℎ𝜆 𝑎𝑝𝑟 𝑝𝑟 ℎ_𝑝𝑟 , 2 2 2 𝑎2 ℎ ∗(1+[𝑧+𝛥 ]2 ∗(𝜆ℎ ∗𝑀 ) )+((1+𝑖∗𝑧∗ 𝑎ℎ ) 𝑝𝑟 2 2 𝜆𝑝𝑟 𝑎𝑝𝑟 2∗𝑎𝑝𝑟 𝛥𝑝𝑟 = 𝛥0_𝑝𝑟 ∗𝑛 𝛼𝑒𝑓𝑓 = (𝛼ℎ + 𝐿 𝑧ℎ_𝑝𝑟 𝛼𝑝𝑟 2 , ) ∗ 𝑧ℎ_𝑝𝑟 , 𝑠𝑎𝑚𝑝𝑙𝑒 𝑧0 = − 2∗𝑧 − 𝛥𝑝𝑟 , 314 , 𝑧1 = ℎ_𝑝𝑟 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 2∗𝑧ℎ_𝑝𝑟 − 𝛥𝑝𝑟 (7.6) (7.7) ∗ (1 + [𝑧 + 𝛥𝑝𝑟 ] ∗ ( 𝑓3(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) = = (1 − 𝑖 ∗ [ 𝑧 L∗n zh_pr 2 𝑎ℎ 2∗𝑎2 𝑝𝑟 𝑎2 Lsample L∗n 1−i( −Δpr ∗ 2ℎ + ) zpr 𝑎𝑝𝑟 2∗zpr ] + 𝐵(𝑧)) (7.8) (7.9) (7.10) (7.11) (7.12) (7.13) Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry Here, 𝛼ℎ and 𝛼𝑝𝑟 are the absorption coefficients (cm-1) of the heating and probe radiation, taking into account of which is essential for the calibration sample. 𝑀2 is the beam quality factor describing the difference between the waist length of a real beam and the diffractive Rayleigh length of a Gaussian beam. In our experiments, the heating beam 𝑀2 value was 1.08. Fig. 7.9. The layout of caustics of the heat and probe coaxial beams in the sample: 1 is the test sample, 2 is the caustic of the probe beam with power Ppr (r, z), and 3 is the caustic of the heat beam with power Ph (r, z, t). D is the projection of the diaphragm installed in front of the photodiode. Source: [35]. The above analytical expression for the K function can be calculated in an elementary way utilizing any interactive computing system of the Mathcad type for a personal computer. Fig. 7.10 shows an example of a theoretical calculation for the dependence of the power of the probe beam time-varying component on the diaphragm size before the photodiode and the distance L (see Fig. 7.9). K8 glass and heavy water were exploited as samples. Matching of theoretical curves with experimental points for K8 glass is ensured by a suitable choice of a calibration coefficient of proportionality between the theoretically determined time-varying component of the probe beam power, which falls on the photodiode cathode through the diaphragm, and the measured signal. The compliance of experimental points obtained for heavy water with theoretical curves (Fig. 7.10) was achieved by using the same calibration factor in the recalculation procedure. Thus, the theoretical calculation of the coefficient, which depends both on the sample material properties and dimensions and on the experimental setup parameters, makes it possible to significantly simplify the calibration process and use the same reference sample with known material parameters and absorption for all measurements. This eliminates the need to perform additional measurements due to IR Fourier spectrometer [3] or conduct the sample doping procedure, which in general is not possible for all materials [39]. Another important advantage of the TPCI scheme is the temporal resolution of the signal obtained from the photodiode, which makes it possible to record in the observed signal 315 Advances in Optics: Reviews. Book Series, Vol. 5 the component associated with the refractive index nonlinearity in the field of the heating beam. It allows to separate it from the thermooptical contribution, which significantly increases the validity of the results of calculating the absorption value when processing the shape of the observed oscillograms recorded at the limiting sensitivities. Fig. 7.10. Pulse amplitude of the time-varying component of the probe beam observed on the oscilloscope and the corresponding theoretical calculation depending on the distance L (cm) of the aperture projection to the output ends of the following samples: 4-cm-long K8 glass (solid curves) and heavy water with a layer thickness of 1.8 cm (dashed curves). Experimental points: the diamond ◊ is the diaphragm diameter of 0.1 cm, the square □ is the diameter of 0.13 cm, the plus sign + is the diameter of 0.18 cm, and the circle ○ is the diameter of 0.052 cm. The aperture magnification factor in the projection section is 6.7. The calibration coefficient of the y axis is the same for all curves. Source: [34]. Fig. 7.11 shows for comparison two normalized oscillograms obtained with the KU-1 quartz glass with the absorption of ≥ 10-5 cm-1 and UQG Suprasil 311 with the absorption of 2.6×10-6 cm-1. The difference between the shapes of the presented oscillograms is associated with a notable contribution of the refractive index surge in Suprasil 311 due to the refractive index nonlinearity in the heating laser field in comparison with the signal thermooptical component. It is obvious that the separation of the contributions of different mechanisms of the refractive index change is especially important in measurements at the limiting sensitivities. Fig. 7.12 shows an example of such measurements conducted for SCQ (LLC “Quartz Technology Company” [40]) with an absorption of ≈10-7 cm-1. The oscillograms show a tendency of temporal saturation of the thermooptical signal due to heat diffusion from the heated area. For this particular example, taking this effect into account gives a two-fold correction that increases the calculated absorption. In addition, a signal surge is observed due to the refractive index change in the field of a heating laser rectangular pulse. Separation of the effects is performed by measuring the thermooptical contribution at the moment immediately after the end of the rectangular heating pulse. Due to the decrease in signal because of saturation, the signal-to-noise ratio at the measurement point is about 25/1. 316 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry Fig. 7.11. Oscillograms of typical signals obtained by measuring the absorption coefficient in quartz glasses: Suprasil 311, KU-1; cooling – decrease of signal due to heat diffusion after the end of heating. Source: [35]. Fig. 7.12. Oscillograms of pulse signals obtained by observing the absorption coefficient. The absorption coefficient is estimated in the range of 10 -7 – 5×10-7 cm-1 in synthetic crystalline quartz. Source: [35]. The described measurement procedure in the TPCI scheme became possible as a result of the implementation of electronic temporal modulation of the heating radiation power, which is performed by modulating the pump diode currents in a heating CW Yb-doped fiber laser. Electronic modulation ensures the practical absence of a jitter when synchronizing heating pulses and digital oscilloscope sweeps. This allows averaging over a large number of pulses (≥104) in a repetitively sequence of heating radiation pulses. Averaging leads to the suppression of the signal noise component and increases the measurement sensitivity by two orders of magnitude. Furthermore, the absence of the influence of jitter on the oscillogram shape at large averaging numbers gives an undistorted signal from the photodiode, which makes it possible to correctly separate the contributions of absorption and the refractive index nonlinearity. Electronic modulation 317 Advances in Optics: Reviews. Book Series, Vol. 5 also makes it possible to promptly adjust the duration and repetition rate of heating pulses in conditions when the effect of heat diffusion from the heated area is significant. This function is very useful for measurements in quartz glasses and quartz crystals at extreme sensitivities. In addition, the use of a digital oscilloscope allows a detailed study of changes in the oscillogram with an accuracy much greater than 1/50 of the signal amplitude. This property is based on the ADC increasing resolution effect when averaging noisy signals [41, 42]. The effect is associated with the ratio of the ADC bit width (8 bits for a standard ADC) and the CPU bit capacity (usually 64 bits). This property is especially important when measuring small deviations of the absorption of UQG from the absorption of the matrix caused by the presence of a foreign chemical element with a concentration at the detection limit level of the mass spectroscopy method. The possibilities of measurements using this scheme are demonstrated in Fig. 7.13, which shows the variations in the absorption coefficient in the UQG samples. Various experimental points were obtained by moving the samples relative to the heating beam axis. The size of the spatial area of the absorption coefficient measurement was determined by the size of the heating beam waist, which was 3 cm × 10-2 cm. The direction of the heating beam propagation was along the layers formed in the Suprasil 300 (Heraeus) and SK-1310 (Ohara) glasses production process utilizing the vapor-phase hydrolysis technology: outside vapor deposition and vapor axial deposition, respectively. In particular, it follows from the figure, that the absorption coefficient can differ greatly in different regions of the sample made of such glasses. The relative root-mean-square (RMS) deviation of the absorption values was ±70 % for Suprasil 300 glass, and ±50 % for SK-1310 glass. Here, in contrast to the indicated glasses, the 3D optically isotropic Suprasil 311 glass demonstrates a significantly smaller spread of the absorption values with an relative RMS deviation of the absorption values of ±3.4 %. Fig. 7.13. Variations in the measuring absorption when the samples are displaced in the transverse direction with respect to the laser heating beam, with a step of 64.5 μm. Source: [35]. 318 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry During the measurement process, the most transparent SK-1310 was found to contain areas with a minimum absorption close to 10-7 cm-1. This leads to the important conclusion that the absorption of the UQG matrix does not exceed 10-7 cm-1 at a wavelength of 1070 nm. The conclusion confirms the theoretical estimate of UQG absorption given in Fig. 7.3. This scheme has been used to demonstrate the absorption measurement in the most transparent UQGs at a level of 10-7 cm-1 at a wavelength of 1070 nm. In these measurements the heating pulse duration was τp = 1.24 ms, which was 2.4 times less than the diffusion time from the heated area of the UQG 𝜏𝑑𝑖𝑓𝑓 = 𝑐∗𝜌∗𝑎ℎ 2 4∗𝛬 = 3 𝑚𝑠 (7.14) in our experiments (here 𝑐 is the heat capacity, 𝜌 is the density, and 𝛬 is the thermal conductivity) [43]. The temporal shape of the observed thermooptical signal almost does not differ qualitatively from the shape for Suprasil 311 glass (Fig. 7.11), obtained with a heating pulse duration of 100 μs. This is due to the relatively weak influence of the heat diffusion effect at such a temporal relation τp and τdiff. In contrast to the PCI method, for which the measurement of the absorption coefficient of 5×10-7 cm-1 is given with a spread of ±60 %, the absorption coefficient of 10-7 cm-1 was measured with a signal-to-noise ratio of 50/1 in these experiments. I.e., the above measurement was performed with a high excess above the detection limit. As noted above, the limiting sensitivity of the TPCI scheme for a given heating pulse power is realized at its durations close to the time τdiff, when the distortion of the signal shape associated with thermal diffusion begins to appear, but the corrections associated with its influence are not too large, and taking them into account does not introduce a notable error in absorption coefficient determination [44]. An example of the result of such measurements is shown in Fig. 7.12. The measurements were conducted with the SCQ, the heat diffusion time was τdiff = 670 μs, and the heating pulse duration was τp = 450 μs. The signal-to-noise ratio at the measurement point was about 25/1. The correction due to heat diffusion is easily found experimentally, and the error associated with such procedures is at an acceptable level. For UQG [44], a situation similar to that shown in Fig. 7.12 can be realized with a heating pulse duration τp = 2 ms and an absorption 𝛼h = 5×10-8 cm-1. But it should be noted that we do not know UQGs with such an absorption level, and, therefore, this situation has not been realized experimentally. The above estimate was obtained by recalculating the results of the experiment carried out for the SCQ. In this case, without any generality loss, we assumed that the geometric parameters of the beams and samples of the UQG and SCQ, as well as the K function of the samples, were the same. The pulses shown in the Fig. 7.12 were obtained with the mutual orientation of the crystal and the polarization direction of the beams, for which the thermo-optical parameters P of the SCQ and UQG were approximately the same (≈0.96·10-5 K-1) [35]. As follows from (7.1), in order for 319 Advances in Optics: Reviews. Book Series, Vol. 5 such pictures to be observed on the oscilloscope screen, it is necessary that the time dependences of temperature ΔT (t) be similar in measurements in SCQ and UQG. For this purpose, it is necessary to increase the duration of the rectangular heating pulse in UQG to 0.67 of the heat diffusion time equal to 3 ms, i.e. up to 2 ms. In this case, the pulse energy will increase by 1.6 times, and taking into account the ratio of the values of the product with c×ρ, the height of the pulses will be preserved with a decrease in the absorption coefficient to 𝛼h = 5×10-8 cm-1. Then, taking into account that the noise component of the measured signal is associated with the noise characteristics of the probe laser, as well as with the choice of the scheme parameters for detecting the probe radiation time-varying component (i.e. the noise component does not depend on the physical parameters of the samples), we find out that the signal-to-noise ratio when measuring the thermo-optical component will remain 25/1. Measurement with such a heating pulse duration in real UQGs with an absorption of 10-7 cm-1 will lead to an increase in the signal on the oscilloscope screen by a factor of 2 while maintaining the level of the noise component. That is, formally, the signal-to-noise ratio will be 50/1. This is still not enough to detect an increase in the UQG matrix absorption at a wavelength of 850 nm due to the presence of Cu2+ ions with a concentration at the detection limit level of the mass spectrometric method, which is about 4×10-9 cm-1, i.e. 1/50 of the matrix absorption. Further appropriate to make the explanations regarding measuring accuracy using the TPCI scheme. This issue has two aspects. As mentioned above, Fig. 7.13 shows the results of the change in the absorption coefficient when the sample was displaced across the laser beams. In this series of measurements, the accuracy of relative measurements of absorption coefficient at different sample positions is determined by the accuracy of pulse height measurement on a digital oscilloscope screen. This accuracy can be estimated by considering the shape of the averaged pulse, shown, for example, in Fig. 7.12. It can be seen that after averaging over the number of pulses N ~ 104 in a pulse-periodic sequence, the pulse has a noise component ~50/1 in this case. Hence, the accuracy of the pulse amplitude measurement can be estimated as ±1 %. That is, the accuracy of relative measurements of absorption coefficient for different positions of the sample is of the order of 1-2 %. The question of the accuracy of measuring the absolute absorption coefficients values is more complicated. This measurement error has several components, the main of which are associated with the measurement calibration method we use [34]. In order to calibrate we use calculations based on the theory of the probe beam diffraction on time-varying and spatial-heterogeneous deformations due to heating the test sample by pulsed laser radiation. Comparison of calculations with experimental results is shown in Fig. 7.10. This allows us to compare the oscilloscope readings obtained for the test sample with the readings obtained for the reference sample, which has different geometric and physical parameters from the test one (linear thermal expansion coefficient, piezo-optical coefficients, refractive index, dn/dT). These physical parameters, measured with a certain accuracy, are given in various papers. For a reference sample made of K8 glass (BK-7 analog), manufactured in the USSR, we use the data of the reference book [45], in which these values are given with an accuracy of ±5 % guaranteed by the USSR standards of 320 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry measurements. In addition, inaccuracies in some of our measurements contribute to the error in the absolute values of absorption. One of them, for example, measuring the magnification factor of the projection lens L3, shown in Fig. 7.8, as well as measuring the diameter of the diaphragm installed in front of the photodiode. In addition, inaccuracies of our settings for the location of the samples, as well as the photodiode diaphragm relative to the optical elements of the scheme, contribute to the error. There are also errors associated with inaccuracy of determining the parameters of laser beams. All of these errors are not of a fundamental nature and minimized by a more careful approach when conducting experiments. The calculation of the resulting error is beyond the scope of published results (although it is important when comparing the results of independent absorption measurements in different samples). The aim of our work is to demonstrate the high sensitivity of measurements using the proposed scheme. Nevertheless, now we estimate the absolute systematic error (which is the same for all points in Fig. 7.13) of the measured absorption values as ±15 % in our experiments. In addition, there is still the problem of measuring ultra-low absorption at a freely chosen set of wavelengths, dictated by the problem of obtaining the highest accuracy of concentration calculation. The problem is that the reported sensitivities of the TPCI scheme were obtained using a CW single mode fiber laser (1070 nm) as a heating one. It operated in a pulse-periodic mode by modulating the current of the pump diodes. The pulsed radiation peak power was ≤ 100 W, the average power was ≤ 2 W. Such lasers, generating repetitively pulsed radiation with a peak power sufficient for measurement using this method (from 10 W to 100 W), operate at their frequencies unrelated to the values from Table 7.2. The choice of such lasers is limited and, in addition, these lasers are rather bulky and expensive to exploit in one device. It would be possible to provide a sufficiently wide choice of discrete heating laser frequencies using single-mode CW diode lasers currently available on the laser market [46]. For these lasers, in addition to a wide range of generated wavelengths, there is the possibility of electronic control to form radiation in the form of sequences of pulses with a duration from 10 to 1000 μs and a repetition rate from 10 to 100 Hz, which are optimal for measurements in UQGs and SCQs. The problem of the practical implementation of the proposed solution to utilize diode lasers is the low peak power of pulses generated by them, ranging from 0.1 to 1 W, more than two orders of magnitude lower than the peak power of heating lasers used in the TPCI method. This will lead to a proportional decrease in the absorption detection level in the implemented TPCI scheme. Based on the studies [14, 35, 44], where absorption measurement 10-7 cm-1 with a signal-to-noise ratio of 50/1 and 100 W of peak heating power was realized, it can be concluded that the use of a 1 W diode laser will reduce the sensitivity of the scheme by two orders of magnitude, and the possibility of correct concentration determination will be possible only for isolated iron atoms. 321 Advances in Optics: Reviews. Book Series, Vol. 5 7.5. The Prospect of Further Increasing of the TPCI Scheme Sensitivity We will further show that the scheme has great potential for increasing sensitivity, as the level achieved is not a limit and can be increased by more than two orders of magnitude by optimization. This possibility opens the way to measuring concentrations and detecting traces of polluting chemical elements with a better sensitivity than the most sensitive mass spectroscopy method. First of all, we note that in this optical scheme, the measurement sensitivity is limited not by the technical possibilities of amplifying the electric signal from the photodiode, but by the noise level of different nature, in which the signal observed on the oscilloscope screen drowns. On the other hand, the signal level observed at a given absorption is related to the set of optical scheme parameters, which influences the K function. This function is a factor in the formula for determining the power of the time-varying component: P~ = K×P×ΔT. The above ratio clearly shows that an increase in the signal level is possible by optimizing the function K value with respect to the parameters of the probe beam at the given parameters of the heating beam. These include the waist radius apr, the mutual distance Δpr between the waist of the probe and heating beams, and also the probe laser power Ppr. Calculations have shown that the function K value optimization depending on the probe beam waist radius and the distance of the waist center leads to a 4.5-fold increase in its value relative to that realized in the presented scheme. In this case, the optimal probe beam geometry will be close to that described elsewhere [47]. In the cited paper, the ratio of the sample length and the size of the heating beam waist differs from ours and corresponds to the situation of a thin thermal lens. Further, the most obvious way to increase the signal level, as follows from the above formula for P~, is to increase the probe laser power Ppr. Theoretically, from the condition of smallness of the distortion it introduces into the sample heated region parameters, this increase is limited by the value of the heating pulse peak power, which is about 1 W in a scheme with semiconductor lasers. I. e., a 10-fold increase in the probe laser power to 20 mW is theoretically acceptable. However, as our recent studies have shown, such an increase in the probe laser power requires a special study of the noise component of its power. As an example, below is a study of a commercial He-Ne laser, which we exploited as a probe laser in our scheme. This laser with a power Plaser = 2 mW operated at a wavelength of 630 nm. According to data specifications, the power of the variable noise component should have been Pnoise = 10-3×Plaser in the frequency band up to 50 MHz. The gain was limited to the 50 kHz band in our scheme, i.e., Pnoise was supposed to be three orders of magnitude less than the specifications value. To study the effect of noise on the sensitivity in our measurements and the prospects for reducing this influence, we have studied the power of the laser radiation noise component. For this purpose, laser radiation was sent directly to the photodiode unit shown in Fig. 7.8, 322 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry bypassing all the optical scheme elements. The constant current component through the diode was changed by means of light filters. Fig. 7.14 shows the oscillogram of the time-varying component of a signal for all implemented values of the constant component. The signal had a clear periodic structure with a 50 Hz network voltage, indicating the purely technical nature of the observed noise. The RMS deviation value of voltage fluctuations was 1.5×10-4 of the constant component, which means that the specification level corresponding to the ratio Pnoise = 10-6 ×Plaser was exceeded by 150 times. In this regard, we have made some modernization of the laser tube power supply. As a result, the RMS noise deviation decreased by approximately 7 times. The corresponding oscillogram is shown in Fig. 7.15. Further research was undertaken into the nature of these noises. Fig. 7.16 shows the photodiode electrical circuit. The measurement showed that the quantum efficiency of our photodiode was 70 % at the probe radiation frequency, i.e., 1.4 photons of probe radiation were required for each electron transferring in the circuit. Fig. 7.14. Typical oscillogram of the time-varying component of the signal from the photodiode illuminated by a probe laser. In the study of the noise source of this circuit, the output signal was amplified with an amplification coefficient Kamp = 104 (80 dB) in a band from 16 Hz to 50 kHz. The noise power was measured employing a digital oscilloscope, due to which, by mathematical processing, the RMS deviation of the signal noise component σ was calculated at a time interval of 4 ms. There were several sources of noise in the scheme. 1. Amplifier noise. 2. Output resistance thermal noise. The sum of the magnitudes of these noises does not depend on the power of the probe laser radiation incident on the photodiode. According 323 Advances in Optics: Reviews. Book Series, Vol. 5 to our measurements, the noise corresponds to the RMS deviation σ1+2 = 34 mV. 3. Shot noise of a laser-illuminated photodiode associated with the discrete nature of the flowing current. The resulting fluctuations of the flowing current can be estimated exploiting a simple formula obtained by solving the Bernoulli mathematical problem [48]. Fig. 7.15. Oscillogram of the time-varying component of the signal from the photodiode illuminated by the probe laser radiation after the power supply upgrade. Fig. 7.16. Photodiode electrical circuit. ̅̅̅̅̅ 𝛥𝐼2 𝐼2 = 𝑒∗𝛥𝜈 , 𝐼 (7.15) or for the magnitude of the noise voltage at the load resistance expressed through the voltage constant component V √𝛥𝑉 2 𝑠ℎ𝑜𝑡 ~𝛽 ∗ √𝑉 ∗ 𝑅𝑙𝑜𝑎𝑑 ∗ 𝑒 ∗ 𝛥𝜈 324 (7.16) Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry (Here, 𝑒 = 1.6 × 10−19 C is the electron charge, I is the current through the diode caused by laser radiation, and 𝛥𝜈 = 5 × 104 Hz is the frequency band of the fluctuating current, the β coefficient was introduced to obtain a correspondence between the RMS deviation experimental values and its theoretical values obtained using the formula (7.3) below, which describes the dependence of the measured noise RMS deviation on the probe laser power incident on the photodiode). The RMS deviation value of the shot noise depends on the constant component of the voltage on the load, which occurs when the diode is illuminated by laser radiation. This voltage was 140 mV when measured in a real scheme. The formula implies the relation 𝜎3 = β · 19 𝑚𝑉. 4. The noise component of the laser radiation power, proportional to the constant component of the voltage on the load. The measurement result was 𝜎4 = 8 𝑚𝑉 at a voltage of 140 mV. I. e., the relation σ4(V) = V*8/140 is valid. It should be note that even after upgrading the power supply, the ratio of the laser noise power corresponding to the RMS deviation σ4 = 11 mV to the laser power corresponding to 140 mV constant component will be equal to 6×10-6, i.e., 6 times worse than the value according to the specification data of this commercial laser. Allowing for the statistical independence of the noise sources, the following formula is valid for calculating the RMS deviation of the total noise σsum 𝜎𝑠𝑢𝑚 2 = 𝛴𝜎𝑖 2 , (7.17) where σi is the RMS deviation of different sources of noise components. Taking the above data into account, a formula describing the change in the total RMS deviation with an increase in the probe laser power was obtained 𝜎𝑡𝑒𝑜𝑟 = √ 𝑉 2 ∗0.0082 0.1402 + 0.0342 + 𝑉 ∗ 4 ∗ 𝑒 ∗ 𝛥𝜈 ∗ 3300 ∗ 𝑅𝑙𝑜𝑎𝑑 2 (7.18) The validity of this ratio has been verified experimentally. The voltage of the constant component V was varied due to the light filters shown in Fig. 7.16. Fig. 7.17 shows the ratio of the experimental values of the total RMS deviation and the values obtained from formula (7.18). The best agreement between the data of the given formula and the experimental data is obtained at β = 2. In this case, the standard deviation of the experimental values from the theoretical ones is ±1.8 %, which indicates that the values of all noise components were correctly taken into account in formula (7.18). As we noted above, the most obvious way to increase the signal level, as follows from the formula for P~, is to increase the probe laser power Ppr. However, as the laser power increases, the RMS deviation of the signal noise component increases due to the laser noise and laser radiation-induced shot noise of the diode. Based on the above results of the study of the probe radiation noise component, it is possible to determine the signal-tonoise ratio increasing, which will occur with an increase in the probe laser power passing through the photodiode diaphragm. Fig. 7.16 shows the dependence of the ratio of the valid signal to the RMS deviation of the noise component with an increase in the constant component of the signal from the photodiode associated with the power of the laser radiation. Sto noise is: 325 Advances in Optics: Reviews. Book Series, Vol. 5 𝑆𝑡𝑜 𝑛𝑜𝑖𝑠𝑒 = 𝐴∗𝑉 , 𝜎𝑡𝑒𝑜𝑟 (7.19) where A = 0.72 is the experimentally determined coefficient of coupling of the signal value in measurements with maximum sensitivity with the value of the constant component of the voltage at the photodiode. Fig. 7.18 shows that an increase in the constant component of the output voltage V to 1.5 V (it corresponds to an increase in the probe laser power from 2 mW to 20 mW) will increase the signal-to-noise ratio up to about 7.3, i.e., about 3 times. In addition, the value of 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 after averaging over N = 104 pulses, as shown by the experiment, will increase by 56 times, i.e., it will grow to 408. In turn, the implementation of the optimal probe beam geometry, which increases the signal at a given absorption by 4.5 times, will increase the value of 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 to ~ 1840. This signal-to-noise ratio corresponds to measuring the UQG absorption coefficient equal to 5×10-8 cm-1. When measuring the deviations of the UQG absorption coefficient from the absorption level of the glass matrix ~2.6×10-7 cm-1, the 𝑆𝑡𝑜 𝑛𝑜𝑖𝑠𝑒 value will increase five times. Fig. 7.17. Ratio of the experimental values of the total RMS deviation σexp and the values σteor obtained from (7.18). Fig. 7.18. Dependence of the ratio of the valid signal to the RMS deviation of the noise component with an increase in the constant component of the signal from the photodiode V, associated with the laser radiation power. 326 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry If single-mode diode lasers are exploited as heated lasers with a peak power 100 times less, 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 will be ~90. The deviation of absorption due to the presence of copper ions Cu2+ with a concentration at the detection limit level (1.2 ppt) will be 4×10-9 cm-1, i.e., ~10-2 of the UQG matrix absorption. In this case, the RMS deviation of the noise component voltage will be approximately equal to the deviation caused by the presence of copper ions, which, in fact, means that the measurement will correspond to the detection limit level. Thus, it is possible to summarize the analysis of the possibilities for increasing the sensitivity of the TPCI scheme in the problem of measuring the concentration of chemical inclusions in UQGs. Implementing the above-described changes in the case of the utilization of diode lasers as sources of heating radiation will make it possible to measure concentrations of polluting chemical elements in UQGs having absorption lines over the entire generation range of diode lasers, with a detection sensitivity equal (e. g., for the Cu2+ ion) or significantly exceeding (e. g., for the Fe3+ ion) LA-ICP-MS detection limit. On the other hand, in the TPCI method, it is possible to exploit heating radiation sources with a peak power of pulsed radiation of ~ 100 W and an average power of ~1 W, operating at a given set of wavelengths, dictated by the problem of obtaining the highest accuracy in calculating concentrations from the data on absorption measurements. This will lead to an exhaustive solution of the problem of measuring the concentration of chemical elements in UQG at a level of ~10 – 1 ppt. The creation of such radiation sources is possible by efficient nonlinear optical frequency conversions of laser radiation of CW fiber lasers operating in a repetitively pulsed mode with a peak pulse power of ~1 kW. The idea of obtaining a wide range of generated wavelengths utilizing nonlinear optical conversions is to combine the effects of stimulated Raman and Brillouin scattering with methods of frequency multiplication and summation in nonlinear crystals [49, 50]. A wide range of frequency shifts implemented employing SRS in crystals and SBS in liquids enables to fill the frequency interval in the visible and IR range quite tightly. The possibility of achieving the efficiency of such processes at pulsed powers of ~600 W in nonlinear crystals was demonstrated by the example of frequency doubling of fiber laser radiation [15]. The efficiency of frequency conversion exploiting SRS and SBS processes in resonators and optical fibers has been demonstrated in many papers [51-57]. It is worth making a remark about the accuracy of measuring the calibration coefficients when utilizing different diode lasers generating at different frequencies. Obviously, when implementing high-sensitive measurements of inclusion concentrations, it is necessary to ensure a sufficiently high accuracy of measuring the ratios of the calibration coefficients for various lasers used. When changing the diode laser, the calibration factor may vary due to the variation in the parameter of the heating laser beam (waist length, beam waist width, and waist position), as well as due to the heating laser power. Although these changes are taken into account in calculations using the above formulae for calculating the K function, however, errors associated with inaccuracy in determining these parameters will lead to an increase in the minimum detectable concentration of inclusions. 327 Advances in Optics: Reviews. Book Series, Vol. 5 Minimum detectable concentrations for different chemical elements vary greatly due to different absorption coefficients. The known values of the coefficients for some chemical elements at a concentration of 1 ppb are shown in Table 7.2. Calculation of the limiting concentrations available for measurement requires special experiments to study the specific values of the indicated errors and the development of special calibration procedures to minimize them. This work is currently at the stage of calculations. 7.6. Conclusions We have presented a brief overview of the modern problem of measuring ultra-low concentrations of inclusions in UQG and SCQ. Further improvement of the technology for the production of these materials requires the development of measurement methods, since the currently existing measurement limits for the concentrations of various impurity elements do not make it possible to determine their content in the purest samples. As an alternative to the mass spectroscopy methods exploited today, we have proposed to take advantage of the possibilities of measuring ultra-low absorption using the new TPCI method [34, 35] to determine the inclusion concentrations polluting ultrapure glasses and crystals. The solution to the range of problems required to implement this proposal is closest to completion for ultrapure quartz glasses. In crystals, this proposal should be further analyzed in terms of obtaining calibration spectral absorption profiles of various chemical elements in crystals of interest. Allowing for the achieved sensitivity in measuring the absorption in UQGs, we analyzed the nature of noise in the measuring scheme and calculated the optical scheme optimal geometry. As a result, we have shown that the scheme sensitivity implemented so far opens up the possibility of measuring absorption at almost any given set of wavelengths in the optical and near-IR ranges by utilizing diode lasers as heating lasers. The given analysis of the concentration values of inclusions of chemical elements having absorption lines in the entire generation range of diode lasers showed that exploiting the TPCI scheme for measuring absorption at different wavelengths makes it possible to measure inclusion concentrations with a recording sensitivity equal to (e. g., Cu2+ ions) or significantly exceeding (e. g., Fe3+ ions) LA-ICP-MS detection limit. We hope that the presented study will serve as an impetus for the further development of this direction of measuring the concentrations of various extraneous chemical elements. This will expand the possibilities for improving the technology for the production of ultrapure quartz glasses and crystals. Acknowledgements The research was supported by the Ministry of Science and Higher Education of the Russian Federation, state assignment for the Institute of Applied Physics RAS, project № 0035-2019-0012. 328 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry References [1]. R. D. Snook, R. D. Lowe, Thermal lens spectrometry – A review, Analyst, Vol. 120, Issue 8, 1995, pp. 2051-2068. [2]. B. Sloots, Measuring the low oh content in quartz glass, Vibrational Spectroscopy, Vol. 48, Issue 1, 2008, pp. 158-161. [3]. M. 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Takke, IR grade fused silica for high-power laser applications, in Proceedings of the International Conference on Optics and Optoelectronics (ICOL’14), Dehradun, India, 5-8 March, 2014. [26]. O. Humbach, H. Fabian, U. Grzesik, U. Haken, W. Heitmann, Analysis of oh absorption bands in synthetic silica, Journal of Non-Crystalline Solids, Vol. 203, 2007, pp. 19-26. [27]. Heraeus Web Portal, https://www.heraeus.com/media/media/hca/doc_hca/ products_and_solutions_8/tubes/Quartz_Tubes_EN.pdf [28]. Ohara Web Portal, https://oharacorp.com/pdf/SK1310.pdf [29]. P. C. Schultz, Optical absorption of the transition elements in vitreous silica, Journal of the American Ceramic Society, Vol. 57, Issue 7, 1974, pp. 309-313. [30]. E. N. Boulos, L. B. Glebov, T. V. Smirnova, Absorption of iron and water in the Na2O-CaOMgO-SiO2 glasses. I. Separation of ferrous and hydroxyl spectra in the near IR region, Journal of Non-Crystalline Solids, Vol. 221, Issues 2-3, 1997, pp. 213-221. [31]. L. Glebov, E. 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Andreev, High-sensitive absorption measurement in ultrapure quartz glasses and crystals using time-resolved photothermal common-pass interferometry and its possible prospects, Journal of Applied Physics, Vol. 129, Issue 4, 2021, 043101. [36]. A. Alexandrovski, M. Fejer, R. Route, Optical Absorption Measurements in Sapphire, Stanford Photo-Thermal Solutions (SPTS), https://www.stan-pts.com/G000242-00.pdf 330 Chapter 7. Ultra-low Light Absorption Measurement in the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry [37]. K. V. Vlasova, A. I. Makarov, N. F. Andreev, A. Yu. Konstantinov, Theory of short-pulse photothermal common-path interferometry in isotropic dielectrics, Uspekhi Prikladnoi Fiziki, Vol. 5, Issue 4, 2017, pp. 313-328 (in Russian). [38]. Yu. A. Ananiev, N. I. Grishmanova, Deformation of active elements and thermo-optical constants of Nd-glass, Journal of Applied Spectroscopy, Vol. 12, 1970, pp. 668-691. [39]. A. Alexandrovski, A. S. Markosyan, H. Cai, M. M. Fejer, Photothermal measurements of absorption in LBO with a "proxy pump" calibration technique, Proceedings of SPIE, Vol. 10447, 2017, 104471D. [40]. LLC Quartz Technology Web Portal, http://quartztech.ru/eng.html [41]. Dr. Lindsay Wilson’s personal website, http://www.imajeenyus.com/electronics/ 20120908_improving_adc_resolution/index.shtml [42]. Microchip Web Portal, https://www.microchip.com/wwwAppNotes/AppNotes.aspx?appnote = en591540 [43]. J. Shen, R. D. Lowe, R. D. Snook, A model for CW laser induced mode-mismatched dual-beam thermal lens spectrometry, Chemical Physics, Vol. 165, Issue 2, 1992, pp. 385-396. [44]. K. Vlasova, A. Makarov, N. Andreev, A. Konovalov, Minimum absorption coefficient available for measurements using time-resolved photothermal common-path interferometry on the example of synthetic crystalline quartz, Sensors & Transducers, Vol. 233, Issue 5, May 2019, pp. 6-14. [45]. UdSSR-DDR, Optisches Glas, VEB Jenaer Glaswerk Schott&Gen, 1978 (in German). [46]. Laser Diode Source Web Portal, https://www.laserdiodesource.com/laser-diode-wavelength/ 635nm-780nm-mw/600nm-795nm [47]. T. Berthoud, N. Delorme, P. Mauchien, Beam geometry optimization in dual-beam thermal lensing spectrometry, Analytical Chemistry, Vol. 57, Issue 7, 1985, pp. 1216-1219. [48]. S. M. Rytov, Introduction to Statistical Radiophysics. Part 1, Nauka, 1976 (in Russian). [49]. G. I. Freidman, G. A. Pasmanik, A. A. Shilov, D. G. Peterson, Some new possibilities to design frequency-tunable lidars in the IR (2-14 μm), Proceedings of SPIE, Vol. 2366, 1995, pp. 403-407. [50]. G. A. Pasmanik, E. J. Shklovsky, G. I. Freidman, V. V. Lozhkarev, A. Z. Matveyev, A. A. Shilov, I. V. Yakovlev, D. G. Peterson, J. K. Partin, Development of eye-safe IR lidar emitter and detector technologies, Proceedings of SPIE, Vol. 3065, 1997, pp. 286-293. [51]. R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, R. P. Mildren, Efficient Raman frequency conversion of high-power fiber lasers in diamond, Laser & Photonics Reviews, Vol. 9, Issue 4, 2015, pp. 405-411. [52]. T. T. Basiev, V. V. Osiko, A. M. Prokhorov, E. M. Dianov, Crystalline and fiber Raman lasers, in Solid-State Mid-Infrared Laser Sources. Topics in Applied Physics (I. T. Sorokina, K. L. Vodopyanov, Eds.), Vol. 89, Springer, Berlin, Heidelberg, 2003. [53]. E. M. Dianov, M. V. Grekov, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, A. M. Prokhorov, CW high power 1.24 µm and 1.48 µm Raman lasers based on low loss phosphosilicate fiber, Electronics Letters, Vol. 33, Issue 18, 1997, pp. 1542-1544. [54]. A. A. Kaminskii, C. L. McCray, H. R. Lee, S. W. Lee, D. A. Temple, T. H. Chyba, W. D. Marsh, J. C. Barnes, A. N. Annanenkov, V. D. Legun, H. J. Eichler, G. M. A. Gad, K. Ueda, High efficiency nanosecond Raman lasers based on tetragonal PbWO4 crystals, Optics Communications, Vol. 183, Issues 1-4, 2000, pp. 277-287. [55]. H. Huang, H. Wang, S. Wang, D. Shen, Designable cascaded nonlinear optical frequency conversion integrating multiple nonlinear interactions in two KTiOAsO4 crystals, Optics Express, Vol. 26, Issue 2, 2018, pp. 642-650. 331 Advances in Optics: Reviews. Book Series, Vol. 5 [56]. M. Heinzig, G. Palma-Vega, T. Walbaum, T. Schreiber, R. Eberhardt, A. Tünnermann, Diamond Raman oscillator operating at 1178 nm, Optics Letters, Vol. 45, Issue 10, 2020, pp. 2898-2901. [57]. M. Heinzig, G. Palma-Vega, B. Yildiz, T. Walbaum, T. Schreiber, A. Tünnermann, Continuous-wave cascaded second Stokes diamond Raman laser at 1477 nm, Optics Letters, Vol. 46, Issue 5, 2021, pp. 1133-1136. 332 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Chapter 8 Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Renxian Li, Bing Wei, Shuhong Gong, Qun Wei, Ningning Song, Shu Zhang, Bojian Wei, Han Sun, Huan Tang and Liu Yang1 8.1. Introduction The light-sheet has been used in many fields because of its excellent characteristics such as only imaging the sample on the focal plane of the objective lens and slight phototoxicity [1], especially super-resolution imaging and optical manipulation. A light-sheet can be applied to imaging microscope [2], flow visualization [3], particle sizing [4] and optical clearing [5], etc. Since the light-sheet has many applications in the field of high-resolution imaging and optical manipulation, it is necessary for us to study the scattering and the mechanical effect theory of the light-sheet by the particles. Based on actual needs, this chapter will derive the basic theory of scattering and mechanical effects of spherical particles illuminated by the light-sheet. Airy light-sheet [6-8], Bessel pincer light-sheet [9, 10] and Gaussian light-sheet [11, 12] are three typical laser light-sheets. A light-sheet is equivalent to a two-dimensional state of the corresponding beam in space, at the same time retains part of the beam’s properties, for example, the non-diffraction [13, 14] and self-healing [15, 16] characteristics of the Airy beam and Bessel beam. We first derive the angular spectrum of Airy light-sheet, Bessel pincer light-sheet, and Gaussian light-sheet, and then generalize them to arbitrary light-sheet. After that, based on the advantages of the light-sheet mentioned above, this chapter will use Generalized Lorenz-Mie theory (GLMT) to study the interaction between three kinds of light-sheets and a spherical particle of arbitrary size, including scattering characteristics (scattering intensity, scattering, absorption, and extinction efficiencies) and mechanical effect(optical force and torque). We will focus on negative force and torque. Whether it is super-resolution imaging or optical manipulation of the light-sheet, the first problem to be solved is to study the scattering of the light-sheet by the particles. Therefore, it is necessary to develop the theory of the scattering of light by particles. In 2017, Mitri Renxian Li School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China 333 Advances in Optics: Reviews. Book Series, Vol. 5 [17] introduced the nonparaxial Bessel pincer light-sheet and calculated the radiation component of the electric field of this light-sheet. In the same year, Mitri [18] demonstrated an auto-focusing Bessel pincer light-sheet in his article, and analyzed the radiation components and propagation characteristics of the light-sheet. Cao et al. [19] used Mie theory and plane-wave spectrum to study the effect of the sphere radius, beam width, and other parameters on the angular distribution of the scattering Airy light-sheet. Pan et al. [20] used the above method to numerically study the scattering of an Airy lightsheet illuminating a chiral sphere. Zhai et al. [11] used numerical methods to study the light scattering characteristics of a sphere in a Gaussian light-sheet. The results show that the scattered light contains the structure and optical information of the sphere. GLMT is a rigorous theory that can calculate the scattering of arbitrary shape beams by homogeneous spheres of any size [9]. Later, GLMT has been also used to calculate the scattering problem of various types of scatters [21]. In the framework of GLMT, vector spherical wave functions (VSWFs), beam shape coefficients (BSCs), and angular spectrum decomposition method (ASDM) are crucial. Because arbitrary TE and TM polarized light-sheet can be equivalent to the superposition of countless different plane waves in the framework of the ASDM method, and then derived from the combination of VSWFs and BSCs. Finally, we also derive the expressions of the scattering intensity, scattering characteristics (such as extinction Qext , absorption Qabs and scattering Qsca efficiencies, etc.), force and torque of any light-sheet illuminating a spherical particle of arbitrary size. The energy efficiency reflects the scatter’s ability to transform the incident energy to scattering, extinction, or absorption energy. The manipulation of tiny particles by light can be used to detect, manipulate and locate biological systems. Because of its minimal damage to control particles, it can be used in the fields of cell biology [22], single molecule [23] and physics [24]. In 2016, Mitri [25] studied the mechanical effects of a non-paraxial Airy “acoustical-sheet” on a 2D circular structure. Numerical results show that the “acoustical-sheets" can move a particle in all directions. In 2017, Mitri [26] used the dipole approximation method to study the force and torque generated by the Bessel pincer light-sheet illuminating the subwavelength absorption sphere coated by a plasma layer under vacuum conditions. Numerical results show that during plasmon resonance, coating the optimal thickness of the plasma layer on the surface of the small sphere can enhance some components of the force and torque of Bessel pincer. The theoretical model of electromagnetic radiation force and torque on an ideal conducting cylinder is developed by Mitri [27], and the numerical calculation of the force and torque is carried out. Numerical calculation results show that the longitudinal and transverse components of the radiation force vector are positive. Song et al. [28] calculated the time-averaged optical spin torque (OST) applied by an Airy light-sheet on a sphere of arbitrary size. The mechanical effect of light is closely related to the characteristics of the incident light-sheet. The direction of force and torque can be changed by changing the parameters and the polarization of the beam [29]. In 2021, Mitri [30] introduced the circularly-polarized Airy light-sheet spinner tweezers, and considered their interaction with a sub-wavelength sphere. This kind of light-sheet generates longitudinal and transverse spin torques at the same time, which makes the particle rotate counterclockwise or clockwise due to different positions. The optical torque is caused by the transfer of the angular momentum of the incident light-sheet to the particles [31]. In 334 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) the following sections, we mainly study the optical spin torque (OST) exerted by a light-sheet on a spherical particle, which can cause the particle to rotate around its center of mass [32]. In this chapter, we have calculated the effects of parameters of TE and TM polarized light-sheet (the transverse scale ys , the modulation parameter  , the scaling parameter 0 , beam order l , and waist radius k0 ) and particle parameters (the size parameter ka of the dielectric sphere) on the scattering field, efficiencies, force, and torque, respectively. The research content of other sections are as follows. In Section 8.2, we will introduce the light-sheet theoretically. The first is the expansion of the angular spectrum of the TE and TM polarized light-sheets (Airy, Bessel, and Gaussian light-sheet). Then, the internal and scattered fields are also expressed in terms of BSCs and VSWFs using the multipole expansion. The general BSCs of laser light-sheet is derived from the expanded term. At the end of this section, the BSCs of Airy, Bessel, and Gaussian light-sheets are derived. In Section 8.3, we derive the expressions of the scattering (far-field scattering intensity, Qext , Qabs , and Qsca ) and mechanical properties (optical force and torque) of the laser light-sheet illuminating a dielectric sphere of arbitrary size. In Section 8.4, we numerically simulate the scattering and mechanical properties of Airy, Bessel, and Gaussian light-sheets. In Section 8.5, the theoretical derivation and numerical simulation are summarized. 8.2. Description of Light-sheet 8.2.1. Angular Spectrum The focal point of a polarized laser-sheet is ( y0 , z0 ), , and the direction of its wave vector is in the same direction as the z  axis in the half free space z > z0 of yz plane. The TE and TM-polarized laser-sheets are discussed in this paper, and the incident electric field TE = H xTE = 0), of TE-polarized laser sheet vibrates along the x  direction ( E TE y = Ez however the incident magnetic field of TM-polarized laser sheet vibrates along the = H zTM = ExTM = 0). By ignoring the contribution of the evanescent x  direction ( H TM y waves [33, 34], the unique non-zero component ExTE of the TE-polarized electric field can be expressed by the angular spectrum of the plane wave,  ExTE ( y, z ) = k  2 AxTE (q)e  ik [ qy  p ( z  z0 )] dq, (8.1) 2 1 where k = (  ) 2  / c is the wavenumber of surrounding medium, q = sin  sin  , p = cos (Fig. 8.1), AxTE (q) is the angular spectrum, and can be given by the Fourier transform of the initial electric field in the z = z0 plane 335 Advances in Optics: Reviews. Book Series, Vol. 5 AxTE (q) =  1 2   E ( y , z )e 2   ikqy 0 x (8.2) dy 2 Fig. 8.1. Definition of k , r , and the corresponding angles. The electromagnetic field’s rectangular component can be expressed as [35-38, 33]  ExTE ( y, z ) = k  2 AxTE (q)e  ik [ qy  p ( z  z0 )] (8.3) dq, 2 E TE y ( y , z ) = 0, (8.4) EzTE ( y, z) = 0, (8.5) H xTE ( y, z ) = 0, (8.6) k Z0 H TE y ( y, z ) = H zTE ( y, z ) =     mA k Z0 TE x 2  ( q )e ik [ qy  p ( z  z0 )] dq, (8.7) 2    qA 2  TE x ( q )e ik [ qy  p ( z  z0 )] dq, (8.8) 2 1 with the eit time dependence suppressed, Z 0 = (  /  ) 2 = E0 / H 0 . In general, the TE-polarized electromagnetic fields can be expressed in vector form as TE ETE ( y, z ) = ExTE e x  E TE y e y  Ez e z  = k  2 ATE (q)e  336 2 i[ qy  p ( z  z0 )] dq, (8.9) Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) with ATE (q) = AxTE (q)e x  AyTE (q)e y  AzTE e z = 1 = 2    [E 2  TE x 1 2   E 2  TE ( y, z0 )e  ikqy dy (8.10) 2 ( y, z0 )e x  E ( y, z0 )e y  E ( y, z0 )e z ]e TE y TE z  ikqy dy 2 Similarly, the magnetic fields can also be expressed as TE HTE ( y, z ) = H xTE ( y, z )e x  H TE y ( y , z )e y  H z ( y , z ) e z (8.11) When it is TM-polarized light, H xTM of the magnetic field vibrating in the x direction can be expressed as [33, 34], H xTM ( y, z ) =  1 ik [ qy  m ( z  z0 )] k  2 AxTM (q)e dq,  Z0 2 (8.12) where AxTM (q) is the angular spectrum of H xTM , and can be obtained from AxTM (q) = 1 2   H 2  TM x (8.13) ( y, z0 )e  ikqy dy 2 Similarly, its rectangular component can be expressed as [35-38, 33] H xTM ( y, z ) =  1 ik [ qy  m ( z  z0 )] k  2 AxTM (q)e dq, Z0  2 (8.14) H TM y ( y , z ) = 0, (8.15) H zTM ( y, z) = 0, (8.16) ExTM ( y, z ) = 0, (8.17)  E TM y ( y, z ) = k  2 mAxTM (q)e   dq, (8.18) 2 EzTM ( y, z ) = k  2 qAxTM (q)e  ik [ qy  m ( z  z0 )] ik [ qy  m ( z  z0 )] dq (8.19) 2 In general, the TM-polarized electromagnetic fields can be expressed in vector form as 337 Advances in Optics: Reviews. Book Series, Vol. 5  TM TM 2 ETM ( y, z ) = ExTM e x  E TM ( q )e y e y  Ez e z = k   A   = k  2 (me y  qe z ) A e  TM x ik [ qy  m ( z  z0 )] dq (8.20) 2 ik [ qy  m ( z  z0 )] dq, 2 with ATM (q) = (me y  qe z ) AxTM = (me y  qe z ) 1 2   H 2  TM x ( y, z0 )e  ikqy dy (8.21) 2 Similarly, the magnetic fields can also be expressed as H( y, z) = H xTM ( y, z)e x (8.22) 8.2.1.1. Airy Light-sheet It is assumed that the polarized type is TE, the initial transverse electric field of the Airy laser-sheet in the source plane z = z0 , shown in Fig. 8.2, is described as [39-41] Fig. 8.2. The graphical representation of the interaction of an Airy light-sheet propagating along the z -axis with a sphere.  y  y0    ( y  y0 )  ETE ( y, z0 ) = e x ExTE ( y, z0 ) = e x E0 Ai   exp  , ys  ys    (8.23) where E0 is the electric field amplitude, Airy function is denoted by Ai(), modulation parameter is  , and transverse scale is the ys . Substituting Eq. (8.23) into Eq. (8.10), 338 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) ATE (q) = e x AxTE (q) = e x   y  y0    ( y  y0 )  ikqy 1 E0  2 Ai  exp    e dy  ys 2  ys    2 (8.24) Considering the relation [42]   it 3  F (t ) =  2 Ai( x) exp( ixt )dx = exp     3  2 (8.25) Eq. (8.24) can be analytical solved, then the vector angular spectrum of Airy light-sheet is ATE (q) = e x AxTE (q) = e x   y  y0    ( y  y0 )   ikqy 1 E0  2 Ai  exp    e dy  ys 2  ys    2  (a  ikys q )3   ikqy0 y = e x s E0 exp  e 2 3   (8.26) When the polarization is TM, similarly, the vector angular spectrum of Airy light-sheet is ATM (q) = e x  (  ikys q)3  ik y y0 ys H 0 exp  e 2 3   (8.27) 8.2.1.2. Bessel Pincer-sheet Considering a Bessel pincer light-sheet with the parameters scaling parameter 0 , beam integer order n. . The initial transverse electric field (TE polarization) in the source plane z = z0 , shown in Fig. 8.3, is ETE  y, z0  = e x ExTE ( y, z0 ) = e x E0 jn 0 k ( y  y0 )  , (8.28) where spherical Bessel function of first kind is denoted by jn   . The angular spectrum calculated by Eq. (8.28) is ATE (q) = e x AxTE (q)  e x = ex   Re  q  i  02  q 2 2 2  0  q E0 k 0n  1 E0  2 jn  0 k ( y  y0 )  e ikqy dy  2 2  n   ikqy0 , e  (8.29) where Re denotes the real part of the complex. Similarly, the vector angular spectrum of TM-polarized Bessel pincer light-sheet is 339 Advances in Optics: Reviews. Book Series, Vol. 5 ATM (q) = e x k n 0     q  i  02  q 2 2 2 0  q  H0   e n  ikqy0 (8.30) Fig. 8.3. The graphical representation of the interaction of a Bessel pincer-sheet propagating along the z -axis with a sphere. 8.2.1.3. Gaussian Light-sheet We first discuss the case of traverse electric (TE) waves of Gaussian light-sheet. In the initial plane z = z0 , the electric field shown in Fig. 8.4, can be expressed as   y  y0 2   E ( y, z0 ) = e x E ( y, z0 ) = e x E0 exp    02   TE TE x (8.31) Fig. 8.4. The graphical representation of the interaction of a Gaussian light-sheet propagating along the z-axis with a sphere. After the Fourier transform of Eq. (8.31), the angular spectrum of is 340 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)    y  y0 2   ikqy 1 2 E0  exp   A  q  = e x A (q )  e x  e dy 2  2 02   TE TE x  2   ikqy  = e x E0 0 exp   0 k 2 q 2  e 0 2   4  (8.32) In the case of traverse magnetic (TM) waves of Gaussian light-sheet, the angular spectrum is ATM  q  = e x H 0  2  ikqy 0 exp   0 k 2 q 2  e 0 2   4  (8.33) 8.2.2. The BSCs (Beam Shape Coefficients) of Arbitrary Laser Light-sheet 8.2.2.1. Expansion of the Incident and Scattering Fields According to the generalized Lorenz-Mie theory (GLMT), incident field can be expanded by VSWFs as [43]  n pol (1)  Ei (r , , ) =   iEmn  pmpol, n N (1) mn ( kr )  qm , n M mn ( kr )  , (8.34) n =1 m =  n H i (r, , ) =   k n E  n =1 m =  n mn pol (1)  qmpol,n N (1)  mn ( kr )  pm , n M mn ( kr )  , (8.35) where pmpol, n and qmpol, n are the Beam shape Coefficients (BSCs), pol denotes different polarized case, i.e., TE or TM . 1/2 Emn  2n  1 (n  m)! = i E0   ,  n(n  1) (n  m)! n (8.36) the VSWFs are defined as L(mnj ) (k , r ) = 1  (mnj ) (k , r ) k =  mn (cos )e  i mn (cos  )e  er Pnm (cos ) zn( j ) (kr ) exp(im )  kr (8.37) 1 d ( j)  zn (kr )  exp(im ), k dr  341 Advances in Optics: Reviews. Book Series, Vol. 5 M (mnj ) (kr ) =   r (mnj ) (k , r )  = i mn (cos )e   mn (cos )e  zn( j ) ( kr )exp(im ), N (mnj ) (kr ) = (8.38) 1   M (mnj ) (kr ) k =  mn (cos )e  i mn (cos  )e  er n(n  1) Pnm (cos  ) 1 d  rzn( j ) ( kr )  exp(im )  kr dr  (8.39) zn( j ) (kr ) exp(im ), kr where (mnj ) (k , r ), scalar wave function, is the solution of Helmholtz equation, the zn( j ) (kr ) is the spherical Bessel function, Pnm (cos ) denotes the associated Legendre function, defined as Pnm ( x) = m nm n 1 2 m/ 2 d 2 2 m / 2 d Pn ( x )      x x x 1 1 = 1   dxnm      dxm , 2n n! (8.40) and dPnm (cos ) m m Pn (cos ),  mn  cos  =  mn  cos  = , d sin  (8.41) similar as Eqs. (8.34) and (8.35), the internal and scattered field can be expanded as  n (1) pol  E1 (r , , ) =   iEmn  d mpol, n N (1) mn ( kr )  cm , n M mn ( kr )  , (8.42) n =1 m =  n H1 (r , , ) =   k1 1  n E n =1 m =  n mn pol (1) cmpol,n N (1)  mn ( kr )  d m , n M mn ( kr )  , n pol (3)  Es (r , , ) =   iEmn  ampol,n N (3) mn ( kr )  bm , n M mn ( kr )  , (8.43) (8.44) n =1 m =  n H s ( r , ,  ) = k  n E  n =1 m =  n mn pol (3) bmpol,n N (3)  mn ( kr )  am , n M mn ( kr )  , (8.45) where ampol, n and bmpol,n are scattering field coefficients, cmpol,n and d mpol, n are internal field coefficients. 342 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 8.2.2.2. The Derivation of General BSCs of Laser Light-sheet According to the theory of electromagnetic field, the electric and magnetic fields satisfy the boundary condition of continuous tangential component on the surface of spherical particles Ei + Es   er = E1  er , Hi + Hs   er = H1  er (8.46) The scattering field (ampol,n , bmpol,n ) and internal jjfield ( cmpol,n , dmpol,n ) coefficients can be obtained by substituting the incident field, internal field and scattered field into Eq. (8.46) ampol,n = an pmpol,n , bmpol,n = bn qmpol,n , cmpol,n = cn qmpol,n , d mpol,n = d n pmpol,n (8.47) where an , bn , cn , dn is the traditional Mie scattering coefficients [44]  ms n (ms x) n' ( x)   n ( x) n' (ms x) an = ms n (ms x) n' ( x)   n ( x) n' (ms x)    (m x) n' ( x)  ms n ( x) n' (ms x) bn = n s  n (ms x) n' ( x)  ms n ( x) n' (ms x)  ,  ' ' ( ) ( ) ( ) ( ) m x x m x x      c = s n n s n n  n  (m x) ' ( x)  m  ( x) ' ( m x) n s n s n n s  ms n ( x) n' ( x)  ms n ( x) n' ( x)  d n = m  (m x) ' ( x)   ( x) ' (m x) s n s n n n s  (8.48) where x = ka, ms = ks / k ,  n ( x) = xjn ( x), n ( x) = xhn(1) ( x), k s is the wavenumber of the spherical particle, jn ( x) is the spherical Bessel function of the first kind. According to [45], BSCs can be obtained from: pmpol,n = qmpol,n =   2 i1 n 1/ 2 kr  mn er  E(r , , ) Pnm (cos )eim sin  d d ,     =0 =0 4 E0 jn (kr ) (8.49)  2 i  n Z 1/ 2 kr  mn er  H(r , , ) Pnm (cos )eim sin  d d ,     =0 =0 4 E0 jn (kr ) (8.50) where  2n  1 (n  m)!    n(n  1) (n  m)!  mn =  (8.51) 343 Advances in Optics: Reviews. Book Series, Vol. 5 An arbitrary beam propagating along z -axis with a focus point can be expressed using the vector angular spectrum of the plane waves Ε( x, y, z ) =       i  k x x  k y y  k z z  z0   A(k x , k y , z0 )e    =k 2 2    /2 =0 0 A( ,  )e ikr sin  sin  cos     ikr cos  cos e dk x dk y e  ik cos  z0 sin  cos  d d  (8.52) Corresponding magnetic field are H ( x, y , z ) = 1 i0    E( x, y, z ) (8.53) Substituting Eq. (8.52) into Eq. (8.49), considering the relations er = sin  cose x  sin  sin e y  cos e z , (8.54) A( ,  ) = Ax ( ,  )e x  Ay ( ,  )ey  Az ( ,  )e z , (8.55) then pmpol, n is pmpol, n e r  A( ,  ) k 2 Pnm (cos )e  im   2 2  / 2  ikr sin  sin  cos    cos  ik z    ikr cos  cos 0 = C0     e e e  d d  d  d   =0  =0  =0 0 sin  cos  sin     = C0   2 2     /2  =0  =0  =0 0 d d  d  d    sin  cos  Ax ( ,  )  sin  sin  Ay ( ,  )  cos  Az ( ,  )     ikr sin  sin  cos     ikr cos  cos   ik cos  z0 2   m im , e e  k Pn (cos )e e    sin  cos  sin     where C0 = i1 n 1/ 2 kr  mn 4 E0 jn (kr ) (8.56) Considering the following mathematical relations, cos   2 ixcos (   )  im  e sin   d =  i m eim 0 e 1  344 iJ m1 ( x)ei  iJ m1 ( x)e i   i  i   J m 1 ( x)e  J m 1 ( x)e   2 J m ( x)    (8.57) Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Then pmpol, n =  i m C0 k 2   2    /2 d d  d  e  ik cos  z0  =0  =0  =0 sin  cos  e  im Pnm (cos )eikr cos cos sin   sin  Ax ( ,  ) iJ m 1 (kr sin  sin  )ei  iJ m 1 (kr sin  sin  )e  i       i i       sin  Ay ( ,  )  J m 1 (kr sin  sin  )e  J m 1 ( kr sin  sin  )e       cos Az ( ,  )  2 J m (kr sin  sin  )     Note that the angular spectrum Az can be expressed by Ax and Ay ky k  Az ( ,  ) =   x Ax ( ,  )  Ay ( ,  )  kz  kz  sin cos sin sin       Ax ( ,  )  Ay ( ,  )    cos cos     Then pmpol,n =  i m C0 k 2   2  =0  /2   =0 d d  d  e =0  ik cos  z0 sin  e  im  iJ m 1 (kr sin  sin  )ei cos  sin  Ax ( ,  )    i  iJ m 1 (kr sin  sin  )e cos  sin  Ax ( ,  )    J (kr sin  sin  )ei cos  sin  Ay ( ,  )  Pnm (cos )eikr cos cos sin   m 1   i   J m 1 (kr sin  sin  )e cos  sin  Ay ( ,  )   2sin  cos  cos J ( kr sin  sin  ) A ( ,  )  m x   2sin sin cos ( sin sin )  J kr A       m y ( ,  )   (8.58) Considering the relation [46], J m 1 (kr sin  sin  ) = m J m (kr sin  sin  )  J m' (kr sin  sin  ), kr sin  sin  J m 1 (kr sin  sin  ) = m J m (kr sin  sin  )  J m' (kr sin  sin  ), kr sin  sin  (8.59) dJ m (kr sin  sin  ) dJ m ( kr sin  sin  ) 1 = d (kr sin  sin  ) kr cos  sin  d (8.60) and J m' (kr sin  sin  ) = The Eq. (8.59) becomes 345 Advances in Optics: Reviews. Book Series, Vol. 5 m J m ( kr sin  sin  )  kr sin  sin  dJ m ( kr sin  sin  ) 1  , kr cos  sin  d J m 1 ( kr sin  sin  ) = J m 1 ( kr sin  sin  ) = m J m ( kr sin  sin  )  kr sin  sin  dJ m ( kr sin  sin  ) 1  kr cos  sin  d (8.61) Substituting Eq. (8.61) into Eq. (8.58), after a series of derivation, pmpol, n =  i m C0 k 2    =0 2  /2   =0 d d  d  e =0  ik cos  z0 e  im sin  1  kr iei cos  Ax ( ,  )mJ m ( kr sin  sin  ) Pnm (cos  )eikr cos cos     iei sin  A ( ,  ) dJ m (kr sin  sin  ) P m (cos  )eikr cos  cos  x n   d  ie  i cos  A ( ,  ) mJ ( kr sin  sin  ) P m (cos  )eikr cos  cos  n x m     i  dJ m (kr sin  sin  ) m ikr cos  cos  Pn (cos  )e  ie sin  Ax ( ,  )  d  i  m ikr cos  cos   e cos  Ay ( ,  )mJ m ( kr sin  sin  ) Pn (cos  )e    dJ (kr sin  sin  ) m  ei sin  Ay ( ,  ) m  Pn (cos  )eikr cos  cos d    e  i cos  A ( ,  ) mJ (kr sin  sin  ) P m (cos  )eikr cos  cos  y m n   dJ m (kr sin  sin  ) m   i  ikr cos  cos  Pn (cos  )e  e sin  Ay ( ,  )  d  m ikr cos  cos    2sin  sin  cos  cos  krAx ( ,  ) J m ( kr sin  sin  ) Pn (cos  )e   2sin  sin  sin  cos  krA ( ,  ) J ( kr sin  sin  ) P m (cos  )eikr cos  cos  y m n   (8.62) Defining C = J m (kr sin  sin  ) Pnm (cos )eikr cos cos (8.63) So d  J m (kr sin  sin  )  d  dC  ikr cos cos / Pnm (cos )  ikr sin  cos  J m (kr sin  sin  ) e d (8.64) Substituting Eq. (8.64) into Eq. (8.62), after a series of derivation, 346 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) pmpol,n =  i m C0 k 2  2  =0  /2  d d  e  ik cos  z0 =0 e im 2 1  kr dC      =0 sin  d d i sin   Ax ( ,  )cos   Ay ( ,  )sin           sin cos ( , )cos ( , )sin C d m A A            x  y    =0 (8.65) Since   =0 sin  dC d d = d d   =0   =0 sin  C d = 2i n  m jn (kr ) dPnm (cos  ) , d sin  C d = 2i n  m jn (kr ) Pnm (cos  ) (8.66) The BSCs pmpol, n becomes pmpol,n = 4 i n C0 k 2  2   /2 d d  e  =0  =0  ik cos  z0 e im jn (kr )  kr i mn  cos    Ax ( ,  )cos   Ay ( ,  )sin        sin     mn  cos   cos   Ay ( ,  )cos   Ax ( ,  )sin       (8.67) After substituting Eq. (8.56) into Eq. (8.67), the final expression of pmn is pmpol,n = k 2 1/2 2  /2  ik cos  z0  im  mn   d d  e sin   e   =0 =0 E0   mn  cos    Ax cos   Ay sin    i mn  cos   cos   Ay cos   Ax sin    (8.68) The qmpol, n can be directly obtained from Eq. (8.68) by interchanging the functions  mn  cos  and  mn  cos   qmpol,n = k 2 1/2 2  /2  ik cos  z0  im  mn   d d  e sin   e   =0 =0 E0   mn  cos    Ax cos   Ay sin    i mn  cos   cos   Ay cos   Ax sin    (8.69) Eqs. (8.68) and (8.69) are general expressions for any vector beam, and can be applied to any beam by changing the angular spectrum Ax and Ay . 347 Advances in Optics: Reviews. Book Series, Vol. 5 Assuming the laser light-sheet is TE-polarization, there are Ax  0 and Ay = 0. And considering the difference between structured beam and light-sheet, removing k and sin  pmTE, n =   cos   cos   k 1/ 2 2  / 2  ik cos  z0  im e Ax    mn  mn   d d  e   =0  =0 E0  i mn  cos   cos  sin     cos   2 e im cos  d    =0 k 1/ 2  ik cos  z0  mn  Ax     mn  d e =  2  =0  im E0 sin  d    i mn  cos   cos   =0e   (8.70)  /2 Considering  =  pmTE, n  2 , the expressions of the BSCs pmTE, n is   ip 2 2   ikpz  mn (cos  ) AxE ( )e 0 cos  d  e   0 k 1/ 2   = i  mn  ip  0  E0 e 2  mn (cos  ) AxE ( )e  ikpz0 cos  d   2     (8.71) Similarly, the BSCs qmTE, n and pmTM, n , qmTM, n are given, respectively, as qmTE,n   ip 2 2   ikpz e  mn (cos  ) AxE ( )e 0 cos  d    0 k 1/ 2   = i  mn  ip  0 , E0 e 2  mn (cos  ) AxE ( )e  ikpz0 cos  d   2     (8.72) pmTM,n   ip 2 2   ikpz e  mn (cos  ) AxH ( )e 0 cos  d   0 k 1/ 2    =  mn   , ip 0  ikpz H0 e 2  mn (cos  ) AxH ( )e 0 cos  d   2     (8.73) qmTM,n   ip 2 2   ikpz e  mn (cos  ) AxH ( )e 0 cos  d    0 k 1/ 2   =  mn    ip 0 H0 e 2  mn (cos  ) AxH ( )e  ikpz0 cos  d   2   (8.74) Eqs. (8.71)-(8.74) are general expressions of BSCs for any vector laser light-sheet, and the BSCs of arbitrary light-sheet can be given by changing the vector angular spectrum AxE ( ) and AxH ( ), respectively. 348 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 8.2.2.3. Particular Structural Beam Laser Light-sheet Substituting Eq. (8.26) into Eqs. (8.71) and (8.72), the BSCs of TE-polarized Airy light-sheet are 1/ 2 pmTE, n = ik  mn 1/ 2 qmTE, n = ik  mn   ip 2 2   (  ikys q )3    e (cos ) exp exp  ik ( qy0  pz0 )  cos  d  mn     0 3 ys        , (8.75) 3 2  ip 0     iky q ( ) s e 2   mn (cos  ) exp  exp  ik ( qy0  pz0 )  cos  d      3 2       ip 2 2   (  ikys q )3  exp  ik ( qy0  pz0 )  cos  d  e 0  mn (cos  ) exp   3 ys        (8.76) 3 2  ip 0  (  ikys q )   2 e   mn (cos  ) exp   exp  ik ( qy0  pz0 )  cos  d    3 2     Similarly, the BSCs pmTM, n and qmTM, n are of TM-polarized Airy light-sheet are given, respectively, as 1/ 2 pmTM, n =  k  mn   ip 2 2   (  ikys q )3  exp  ik ( qy0  pz0 )  cos  d  e 0  mn (cos  ) exp   3 ys        , (8.77) 3 2  ip 0  (  ikys q )   2 e   mn (cos  ) exp   exp  ik ( qy0  pz0 )  cos  d    3 2     1/ 2 qmTM, n =  k  mn   ip 2 2   (  ikys q )3  e (cos ) exp exp  ik ( qy0  pz0 )  cos  d    mn     0 3 ys        , (8.78) 3 2  ip 0  (  ikys q )   2 e    mn (cos  ) exp   exp  ik ( qy0  pz0 )  d    3 2     Similarly, the BSCs of Bessel pincer-sheet are pmTE, n     n   ip 2 2  exp  ik ( qy0  pz0 )  cos  d  e   mn (cos  ) q  i  02  q 2  1/ 2 0 ik  mn   =   , 2 2 n n k 0  0  q  ip 2 0 2 2 e   mn (cos  ) q  i  0  q exp  ik ( qy0  pz0 )  cos  d     2 (8.79) q TE m,n     n   ip 2 2  2 2 e  (cos  )  q  i   q exp  ik ( qy0  pz0 )  cos  d  0  1/ 2  0 mn ik  mn   =   , n 2 2 n k 0  0  q  ip 2 0 2 2 e   mn (cos  ) q  i  0  q exp  ik ( qy0  pz0 )  cos  d     2 (8.80) 349 Advances in Optics: Reviews. Book Series, Vol. 5 pmTM, n     n   ip 2 2  e   mn (cos  ) q  i  02  q 2 exp  ik ( qy0  pz0 )  cos  d   1/ 2 0 k  mn     ,  n 2 2 n ip 0 k 0  0  q  2 2 e 2   mn (cos  ) q  i  0  q exp  ik ( qy0  pz0 )  cos  d     2 (8.81) q TM m,n     n   ip 2 2  2 2       (cos ) exp  ik ( qy0  pz0 )  cos  d   e q i q 0 mn  1/ 2  0 k  mn      , 2 2 n n k 0  0  q  ip 2 0 2 2 e   mn (cos  ) q  i  0  q exp  ik ( qy0  pz0 )  cos  d      2 (8.82) and the BSCs of Gaussian light-sheet are 1/ 2 pmTE,n = ik mn   ip     02 2 2  k q  exp  ik (qy0  pz0 )  cos  d  e 2 02 mn (cos  )exp   w0   4    ,  2 2   ip 2 0  0 2 2   e   mn (cos  )exp   4 k q  exp  ik (qy0  pz0 )  cos  d  2     (8.83) 1/ 2 qmTE,n = ik mn   ip 2 2   02 2 2  e k q  exp  ik ( qy0  pz0 )  cos  d   (cos )exp     mn  0 w0   4      2 2   ip 2 0  0 2 2   e   mn (cos  )exp   4 k q  exp  ik ( qy0  pz0 )  cos  d  2     (8.84) 1/ 2 pmTM,n = k mn   ip 2 2   02 2 2  k q  exp  ik (qy0  pz0 )  cos  d  e 0  mn (cos  )exp   w0   4    ,  2 2   ip 2 0  0 2 2   e   mn (cos  )exp   4 k q  exp  ik (qy0  pz0 )  cos  d  2     (8.85) 350 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 1/ 2 qmTM,n = k mn   ip 2 2   02 2 2  e k q  exp  ik ( qy0  pz0 )  cos  d   (cos )exp     mn  0 w0   4      2 2   ip 2 0  0 2 2   e   mn (cos  )exp   4 k q  exp  ik ( qy0  pz0 )  d  2     (8.86) 8.3 Light Scattering of Laser Light-sheet by Dielectric Sphere of Arbitrary Size 8.3.1. Far-field Scattering Intensity According to Eq. (8.44), the scattering field is expanded as  n pol (3)  Es (r , , ) =   iEmn  ampol,n N (3) mn ( kr )  bm , n M mn ( kr )  (8.87) n =1 m =  n (3) (3) The VSWFs M mn and N mn are expressed as (1) im M (3) mn ( kr ) =  i mn (cos  )e   mn (cos  )e  hn ( kr )e , (8.88) 1 d  rhn(1) (kr )  eim  =  mn (cos  )e  i mn (cos  )e  N (3) mn ( kr ) kr dr  h(1) (kr ) im e , e r n(n  1) Pnm (cos  ) n kr (8.89) where hn(1) is the spherical Hankel function of the first kind. Considering the far-field limit of the spherical Hankel function: (8.90) hn(1)  (i)n1 eikr / (kr ), Eqs. (8.88) and (8.89) become M (3) mn ( kr ) =   mn (cos )e  i mn (cos  )e  N (3) mn ( kr ) =   mn (cos  )e  i mn (cos  )e  1 ( i) n eikr eim , kr 1 ( i) n eikr eim kr (8.91) (8.92) Substituting Eqs. (8.91) and (8.92) into Eq. (8.87): Es = E e  E e , (8.93) 351 Advances in Optics: Reviews. Book Series, Vol. 5 E = i ikr  n i e   Emn (i ) n  ampol,n mn (cos  )  bmpol,n mn (cos )  eim = eikr S 2 , (8.94) kr kr n =1 m =  n E = 1 ikr  n 1 e   Emn (i ) n  ampol,n mn (cos )  bmpol,n mn (cos )  eim = eikr S1 , (8.95) kr kr n =1 m =  n where  n S1 =   Emn (i) n  ampol, n mn (cos )  bmpol,n mn (cos )  eim , (8.96) n =1 m =  n  n S2 =   Emn (i ) n  ampol,n mn (cos )  bmpol,n mn (cos )  eim (8.97) n =1 m =  n The dimensionless far-field scattering intensity can be evaluated as: I ( r , ,  ) = 1  2 2 S1 ( , )  S2 ( ,  )   k 2r 2  (8.98) For convenience, the dimensionless normalized far-field scattering intensity can be defined such that: I  ( , ) =  k 2 r 2  limI (r , , ) = S1 ( , )  S 2 ( , ) 2 r  2 (8.99) 8.3.2. Scattering, Extinction and Absorption Efficiencies The extinction and scattering cross sections are Cext = Wext / I 0 , Csca = Wsca / I 0 , where Wext and Wsca are extinguished and scattered energies by the dielectric sphere, respectively, and I 0 = E02 / (2* Z0 ) is a characteristic intensity factor, where E0 is the amplitude of the incident electric field, and Z 0 is the wave impedance of the surrounding medium. The dimensionless extinction and scattering efficiencies Qext = Cext / ( a2 ) and Qsca = Csca / ( a2 ), where a is the radius of the dielectric sphere, are expressed as [43]: Qext =  n 4  pmpol,n ampol,n*  qmpol,n bmpol,n *  , e R   2 2 k a n =1 m =  n (8.100) Qsca = 2 4  n  pol 2 a  bmpol,n  , 2 2    m,n  k a n =1 m =  n  (8.101) where the superscript symbol * denotes a complex conjugate. The absorption efficiency is given by the Qext and Qsca that 352 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) (8.102) Qext = Qabs  Qsca 8.3.3. Radiation Force and Torque In the framework of GLMT, the integral of a Maxwell’s stress tensor Γ is a typical calculation method of optical radiation force [47] F=  nˆ  (8.103) Γ dS , S where differential surface element is represented by dS , the unit vector is denoted by n̂ , and the n̂ is on the surface that is enfolding the dielectric sphere. Γ is defined as [47] 1  1  Γ = Re  E  E*   H  Η*   E  E*   H  H* I  , 2  2  (8.104) where unit tensor is I , a tensor product is marked by  . Substituting the total of incident and scattering EM fields ( E and H ) into Eq. (8.104) [48], the optical force components can be obtained as follows, Fx = Re  F1  , Fy = Im  F1  , Fz = Re  F2  , (8.105) where F1 2 = 2 | E0 |2 k 1  pol * pol pol * pol   (n  m)(n  m  1)  2  am ,n bm 1,n  bm ,n am 1,n         p pol q* pol  q pol p* pol  n(n  1) n =1 m =  n  m , n m 1, n m , n m 1, n     n 1 pol * pol   n(n  2)(n  m  1)(n  m  2)  2  ampol,n am* pol 1, n 1  bm , n bm 1, n 1      pol * pol 2 pol * pol  (n  1)  2n  1 2n  3    pm ,n pm 1,n 1  qm ,n qm 1,n 1  1  * pol pol   n(n  2)(n  m)(n  m  1)  2  ampol,n 1am* pol 1, n  bm , n 1bm 1, n  ,    pol * pol 2 pol * pol    (n  1)  2n  1 2n  3    pm ,n 1 pm 1, n  qm , n 1qm 1, n    F2 = 4 | E0 |2 k2      n(n  1)   a n n =1 m =  n  m pol * pol m,n m,n b   pmpol,n qm* pol  ,n 1  pol * pol   n(n  2)(n  m  1)(n  m  1)  2  ampol,n am* pol , n 1  bm , n bm , n 1   ,    pol * pol 2 pol * pol   (n  1)  2n  1 2n  3    pm,n pm,n 1  qm ,n qm ,n 1    353 Advances in Optics: Reviews. Book Series, Vol. 5 and 1 pol pol 1 pol pm , n , pm, n = pm , n , 2 2 1 pol pol 1 pol  qm , n , qm , n = qm , n 2 2 ampol, n = ampol, n  pol m,n b pol m,n =b (8.106) The integral of a time-averaged moment of the angular momentum flow tensor K is a feasible calculation method to calculate OST vector, that is [28] T =   nˆ  K dS , (8.107) 1  1  K = Re  E  E*   H  H*   E  E*   H  H*  I   r , 2  2  (8.108) S where [49-51] where position vector is denoted by r. Considering the background medium is lossless, the OST components are [52, 53] Tx = Re  N1  , Ty = Im  N1  , Tz = Re  N 2  , 2 N1 = 3 0 E0 k N2 =  2 2 0 E0 k3 (8.109) 1  [(n  m)(n  m  1)]2   n =1 m =  n  a pol a* pol  b pol b * pol  p pol p* pol  q pol q* pol m , n m 1, n m , n m 1, n  m,n m 1,n m,n m 1,n  n  2    m | a n n =1 m =  n |  | bmpol, n |2  | pmpol, n |2  | qmpol,n |2 pol 2 m,n     ,   (8.110) (8.111) 8.4. Numerical Results and Discussions Next, the simulation of dimensionless far-field scattering intensity, energy efficiencies, as well as radiation force, torque on a sphere are shown. Both the three types of incident light-sheet, such as Airy, Gaussian, Bessel pincer light-sheets, have been discussed. Besides, the TE-polarized and TM-polarized incident light-sheets are studied. The wavelength of the incident light is  =1.064  m, and the focused point of light sheets is ( y0 , z0 ) . The center point of the scatters (relative permeability is  = 1) is ( y, z)  (0,0). The amplitude of incident light is E0 = 1.0  10 6 V/m. 8.4.1. Airy light-sheet The influence of the parameters of dimensionless kys ,  of the Airy incident field and the ka (a is the radius of the sphere) of the sphere are considered. 354 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 8.4.1.1. Far-field Scattering Intensity Based on Eq. (8.99), the far-field scattering intensity of Airy light-sheet on a lossless dielectric sphere is studied as follows. Assuming the scatterer is a polystyrene sphere in vacuum, and the corresponding refractive index are n1 = 1.59 and n0 = 1.0, respectively. The scatterer is located at the main lobe of incident field. Fig. 8.5 shows the effect of kys on I  with TE-polarized incident field, where kys takes values as 1, 10, 50, respectively,  = 0.1, and ka =10. When kys changes from 1 to 10, the amplitude of I  becomes bigger. But the amplitude of I  in kys = 50 is same as it in kys = 10. It means when kys < ka, I  increases as kys increases. But when kys > ka, I  does not increase any more as kys increases. Fig. 8.6 is same as Fig. 8.5, but TM-polarized Airy light-sheet is paid attentions. It shows that the state of polarization has a minor effect on I . Fig. 8.5. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere. ka = 10,  = 0.1, kys varies as following list [1,10,50]. Fig. 8.6. The same as in Fig. 8.5, but the incident electromagnetic field is a TM-polarized. In Figs. 8.7 and 8.8, the effect of  is analyzed, where  is set as 0.05, 0.1, 0.3, respectively, and kys = 10, ka =10. Fig. 8.7 shows that all the maximum amplitude of I  is in the direction of wave propagation (  = 0 ). The change of  can not affect the I  . And the polarization of Airy light-sheet has little effect on I . 355 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.7. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere. ka = 10, kys = 10,  varies as following list [0.05,0.1,0.3]. Fig. 8.8. The same as in Fig. 8.7, but the incident electromagnetic field is a TM-polarized. Figs. 8.9 and 8.10 discuss the influence of size parameter ka, which are 2, 10, 20, respectively, and kys = 10,  = 0.1. Compared with Figs. 8.9 and 8.10, the type of polarization have an impact on the distribution of I  only when ka = 2. As ka increases, the maximum amplitude of I  increases. However, the distribution of I  has no change with different  . Fig. 8.9. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere.  = 0.1, kys = 10, ka varies as following list [2, 10, 20]. 356 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Fig. 8.10. The same as in Fig. 8.9, but the incident electromagnetic field is a TM-polarized. 8.4.1.2. Scattering, Extinction and Absorption Efficiencies Next, the energy efficiencies have been computed base on the Eqs. (8.100)-(8.102). The complex refractive index of the object (gold spherical particle) is n1 = 0.26  7.0i suspended in water (n0 = 1.33). The scatterer is also located at the main lobe of Airy lightsheet. And ka transforms from 0.01 to 66. Fig. 8.11 gives the calculated results for the scattering efficiency Qsca , investigating the effect of the type of polarization and different kys taking values as 5, 10, 15, respectively, where  = 0.1, ka varies in range (0,60). It is observed that Qsca of the three lines increases rapidity in the range of 0 < ka <1 in Fig. 8.11, but when ka >1, increasing ka makes Qsca decreases. And they are consistent in the range 0 < ka < 5. However, the plot of kys = 5 decrease quickly than the other two plots. And the two lines of kys being 10 and 15 are separated after ka >10. It means that Qsca is sensitive to the relative size of ka and kys . when ka > kys , Qsca increases as kys increases. The polarization of incident light also shows a minor effect on the kys in this case. Fig. 8.11. The Qsca of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1. 357 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.12 shows the extinction efficiency Qext , the parameters are same as Fig. 8.11. Since Qext is the sum of Qsca and Qabs and Qabs is much smaller than the Qsca , the plots of Qext and Qsca have the approximate trend and values. Fig. 8.12. The Qext of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1. Fig. 8.13 are the same Figs. 8.11 and 8.12, but it calculate Qabs . Compared with Qsca and Qext , the amplitude of Qabs is much smaller. Although the amplitude is different, the trend of Qabs and Qext , Qsca are similar. It increases when 0 < ka <1, and decrease in the range ka >1. When the ka >10, increasing kys leads Qabs become bigger. In the panel (b), where the incident EM field is TM polarization, Qabs decreases more quickly than it in panel (a), especially in ka >10. Fig. 8.13. The Qabs of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1. Figs. 8.14, 8.15, and 8.16 discussed the influence of  , which varies in the list [0.05, 0.1, 0.3]. And kys = 10, ka = 10. Similarly, the polarization of Airy light-sheet affects Qsca and Qext lightly, but for Qabs TM-polarized incident filed makes Qabs decreases more quickly. From Fig. 8.14, Qsca also increases as ka increases from 0 to 1. 358 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Then, Qsca deduces as ka increase. And the increasing of  cause Qsca decrease. Moreover, the same case happens in Fig. 8.15, and 8.16. Fig. 8.14. The Qsca of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10. Fig. 8.15. The Qext of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10. Fig. 8.16. The Qabs of an Airy light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10. 359 Advances in Optics: Reviews. Book Series, Vol. 5 8.4.1.3. Radiation Force and Torque Optical force on a spherical particle with refractive index n1 = 1.59 in vacuum (refractive index n0 = 1.0) is studies as follows. An lossless Rayleigh spherical particle ( ka = 0.1) is presented in Figs. 8.17-8.20. The optical force of different directions Fy and Fz is shown, with particularly emphasize the negative pulling force Fz < 0. The effect of kys ,  and the type of polarization is considered, where kys takes values as 5, 10, and  are 0.05, 0.1. Fig. 8.17. The optical force on a lossless Rayleigh spherical particle with TE-polarized Airy light, ka = 0.1,  = 0.1. kys varies as following list [5,10]. In Fig. 8.17, it is showing that different values of kys have an obvious influence on the amplitude and distribution. When kys increases from 5 to 10, the maximum amplitude of all the panels becomes smaller, but the distribution scope of force in the yz -plane is bigger. It means that the distribution scope of negative pulling force Fz increases too, which has important practical significance. Besides, in the case of the same parameters, the amplitude of Fz is smaller than Fy . Fig. 8.18 is the same case as Fig. 8.17, but incident light is TM-polarized. It is distinct that the maximum of radiation force when kys = 5 in panels (a), (b), (c) of Fig. 8.18, are smaller than that in Fig. 8.17. However, the distribution of optical force is as broad as it’s long. Fig. 8.19 displays the effect of  . When  changes from 0.05 to 0.1, the maximum amplitude of optical force decrease in all the Fy and Fz . And the distribution scope of Fz < 0 are smaller too. Similarly, when 360 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)  = 0.05 the maximum amplitude in Fig. 8.20 are smaller than the corresponding panels in Fig. 8.19. But the same situation do not happen in the case of  = 0.1. Fig. 8.18. Being same as Fig. 8.17, but is the Airy light-sheet is TM-polarized. Fig. 8.19. The optical force on a lossless Rayleigh spherical particle with TE-polarized Airy light, kys = 10.  varies as following list [0.05,0.1]. 361 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.20. Being same as Fig. 8.19, but is the Airy light-sheet is TM-polarized. The optical force on an lossless Mie spherical particle (ka = 10) is studied in Figs. 8.21-8.24. The lossless Mie spherical particle is located in z = 1  m of the yz -plane, and y varies in the range -25 to 25 ( m). Fig. 8.21 gives the simulated plots of Fy for different values of kys , where  = 0.1. It is shown that the maximum amplitude of positive Fy occurs when kys = 10. And in the position of y = 0, the direction of Fy is all towards the negative y -axis. Besides, the amplitude of these negative Fy increases as kys decreases. The amplitude of Fy on Mie particles illuminated by the TM-polarized incident field is slightly larger than it illuminated by the TE-polarized incident field. Fig. 8.22 is the same as Fig. 8.21, but is Fz . It is observed that all the directions of Fz is positive. As kys increases, Fz increases, which means the maximum amplitude of Fz is in the plot of kys = 15, different from Fy . Fig. 8.23 shows the effect of  on Fy , where  changes as following list 0.05, 0.1, 0.3, and kys = 10. As  increase, the maximum amplitude of Fy in the positive y -axis decreases. And Fy in panel (b) which is TM-polarized incident Airy light-sheet are bigger than it in panel (a) of TE polarization. Fig. 8.24 is same as Fig. 8.23, but is Fz . Similar to Fig. 8.23, increasing  makes Fz decreases. And the type of polarization of incident field has a minor effect on Fz . Then the optical torque on an absorptive spherical particle (n1 = 0.26  7i) in water (n0 =1.33) is calculated. A Rayleigh spherical particle (ka = 0.1) is first presented in Figs. 8.25 and 8.26, particularly analyzing negative Tx . Since the optical torque only occurs in the case of TM-polarization, and only Tx exists but Ty , Tz = 0 [28], we only 362 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) analysis the influence of  and kys on Tx . In Figs. 8.25, as kys increases, the trajectory of Tx become straight, and the amplitude of Tx in this calculated scope become smaller, but the distribution scope of negative Tx in panels (d), (e), (f) become bigger. The main lobe of Tx become wider too. Fig. 8.26 shows how  influence Tx . As  changes from 0.05 to 0.1, the amplitude of Tx has just a minor difference. The distribution scope of Tx becomes slightly smaller too. However, when  increases from 0.1 to 0.3, Tx decreases, the distribution scope of Tx are much smaller including negative Tx . Fig. 8.27 shows the optical torque on a Mie spherical particle (ka = 10) as 1D plots, where z = 1  m. Considering the different values of kys (5, 10, 15) and  (0.01, 0.1, 0.3). In panel (a), where  = 0.1, the maximum amplitude of positive Tx occurs when kys = 15, however the negative Tx occurs in kys = 10. In panel (b), where kys = 10, the maximum amplitude positive Tx does not change in different  . However, the amplitude of negative Tx located in about y = 0, increases as  decrease. Fig. 8.21. Fy on a lossless Mie spherical particle of Airy light-sheet with TE and TM polarizations. The values of kys are 5, 10, 15, respectively, and  = 0.1. Fig. 8.22. Being same as Fig. 8.21, but the optical force is Fz . 363 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.23. Fy on a lossless Mie spherical particle of Airy light-sheet with TE and TM polarizations. The values of  are 0.05,0.1, 0.3, respectively, and kys = 10. Fig. 8.24. Being same as Fig. 8.23, but the optical force is Fz . Fig. 8.25. Tx on an absorptive Rayleigh spherical particle with TM-polarized Airy light, where kys are 5, 10, 15, respectively, and  = 0.1. 364 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Fig. 8.26. Tx on an absorptive Rayleigh spherical particle with TM-polarized Airy light, where  are 0.05, 0.1, 0.3, respectively, and kys = 10. Fig. 8.27. Tx on an absorptive Mie spherical particle with TM-polarized Airy light. The values of kys are 5, 10, 15, respectively, and  = 0.1 in panel (a). But in panel (b)  takes values as 0.05, 0.1, 0.3, and kys = 10. 8.4.2. Bessel Pincer Light Sheet 8.4.2.1. Far-field Scattering Intensity Here, the I  on a dielectric sphere with a Bessel pincer light-sheet incident is analyzed. Assuming the object is a polystyrene sphere (refractive index is n1 = 1.59) in vacuum (refractive index is n0 = 1.0). Note that the amplitude of incident electric field is 365 Advances in Optics: Reviews. Book Series, Vol. 5 E0 = 1.0  10 [10, 45, 100]. 6 V/m. ka varies [2, 10, 20],  0 varies [0.2, 0.5, 1], and l varies Fig. 8.28 gives the far field scattering distribution ( I  ) under the TE-polarized Bessel pincer light-sheet. Panels (a)-(c) gives the I  of  0 = 0.5, l = 10 with different ka. In panel (a) (ka = 2), there are only two beam lobes (The left one is minor, and the right one is big). The I  is symmetrical along 0-  -axis. The gain value magnitude of I  is equal to 80. In panel (b) (ka = 10), there are a few beam lobes (like a small cat). Except for the the symmetrical main lobes on the right side, there are many side lobes symmetrically along 0-  -axis. The gain value magnitude of I  is equal to 90. In panel (c) (ka = 20), there are a few beam lobes (like a fried hairy cat). The gain value magnitude of I  is equal to 100. Fig. 8.28. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere.  0 = 0.5, l = 10, ka varies as following list [2,10, 20]. Fig. 8.29 gives the far field scattering distribution ( I  ) under the TM-polarized Bessel pincer light-sheet. Panels (a)-(c) gives the I  of  0 = 0.5, l =10 with different ka. In panel (a) (ka = 2 ), different from panel (a) of Fig. 8.28 there is no main beam lobes. The I  is symmetrical along 0-  -axis. The gain value magnitude of I  is equal to 80. In panel (b) (ka = 10) and panel (c) (ka = 20), the regularity of I  and the beam lobes are same the TE-polarized mode case of panels (b) and (c) in Fig. 8.28. Panels (a), (b) and (c) of Fig. 8.30 gives the far field scattering distribution ( I  ) of ka = 20, l = 10 with different  0 (0.2, 0.5, 100) under the TE-polarized Bessel pincer light-sheet. In panel (a) (0 = 0.2), there are a few beam lobes (like a small cat). The main lobes on the right side have the gain value magnitude of I  (equal to 80), there are many side lobes symmetrically along 0-  -axis. In panel (b) (0 = 0.5), there is only one pair of the main lobe distributing symmetrically, and less side lobes (comparing to panel (a)). The gain value magnitude of I  is equal to 100. In panel (c) (0 = 1.0), there are plenty 366 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) of main lobes with the same gain value magnitude (equal to 120). But there are less side lobes comparing to panel (b). Fig. 8.29. The same as Fig. 8.28, but the incident electromagnetic field is a TM-polarized. Fig. 8.30. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere. ka = 20, l = 10,  0 varies as following list [0.2,0.5,1.0]. Panels (a), (b) and (c) of Fig. 8.31 gives the far field scattering distribution ( I  ) of ka = 20, l = 10 with different  0 (0.2, 0.5, 100) under the TM-polarized Bessel pincer light-sheet. In panel (a) (0 = 0.2), there are a few beam lobes (like a small cat). The main lobes on the right side have the gain value magnitude of I  (equal to 80), there are many side lobes symmetrically along 0-  -axis. In panel (b), there is only one pair of the main lobe distributing symmetrically, and less side lobes (comparing to panel (a)). The gain value magnitude of I  is equal to 100. In panel (c) (0 = 0.5), there are plenty of main lobes with the same gain value magnitude (equal to 120). But there are less side lobes comparing to panel (b). The gain values and symmetrical characteristics is similar in two kinds of polarizations. Panels (a), (b) and (c) of Fig. 8.32 gives the far field scattering distribution ( I  ) of ka = 20,  0 = 0.5 with different l (10, 45, 100) under the TE-polarized Bessel pincer light-sheet. In panel (a) (l = 10), the one pair of the main lobe distribution. There are 367 Advances in Optics: Reviews. Book Series, Vol. 5 many side lobes distributing symmetrically. The maximal gain value magnitude is equal to 120. And there are a few lobe sides along 0-  -axis. In panel (b) (l = 45), the region of main lobes shrinks. There are many side lobes distributing symmetrically. The maximal gain value magnitude is equal to 110. Different in panel (a), there is little side lobes along 0-  -axis. In panel (g) (l = 100), the region of main lobes shrinks. There are many side lobes distributing symmetrically. The maximal gain value magnitude is equal to 120. Different in panel (b), there are shorter side lobes than that in panel (c). Fig. 8.31. The same as Fig. 8.30, but the incident electromagnetic field is a TM-polarized. Fig. 8.32. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere.  0 = 0.5, ka = 20, l varies as following list [10, 45,100]. Panels (a), (b) and (c) of Fig. 8.33 gives the far field scattering distribution ( I  ) of ka = 20,  0 = 0.5 with different l (10, 45, 100) under the TM-polarized Bessel pincer light-sheet. The cases of beam lobes, gain values and symmetrical characteristics is nearly to the situations in Fig. 8.32. In summary, 0 , l , and ka are sensitive to the I . Firstly, the gain value of lobe width remains same, but the lobe distribution shape becomes complex with symmetrical characteristic, as the increasing of dimensionless parameter ka. Secondly, the larger the scaling parameter  0 is, the greater the gain value of lobe width is. Meanwhile, the shape of main lobe distributes with more second lobes with the increasing of scaling parameter 368 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 0 . Lastly, the gain value of lobe width turns minor, and the main lobe appears little by little, while increasing the beam order l. Fig. 8.33. The same as Fig. 8.32, but the incident electromagnetic field is a TM-polarized. 8.4.2.2. Scattering, Extinction and Absorption Cross Section In this section, the scattering Qsca , extinction Qext and absorption efficiencies Qabs are analyzed. The complex refractive index of the object (gold spherical particle) is n1 = 0.26  7.0i suspended in water (n0 = 1.33). The ka of the sphere varies from 0 to 100. The panels (a) and (b) in Fig. 8.34 show that Qsca with Bessel pincer light-sheet incident, versus ka, with different l (10, 45, 100) for the scaling parameter  0 = 0.8. The Qsca in panel (a) with different l is under TE-polarized while panel (b) with TM-polarized. In panel (a), the maximum magnitude of Qsca is about 3.2 103 for l = 10, 0.5 103 for l = 45 and 3.2 108 for l = 100. When the value of order l = 10, there is a little peak with magnitude 0.25 103 at ka = 5. And then there is an abrupt increasing till maximum magnitude of Qsca near at ka = 12. subsequently, after a lower peak, the second maximum of Qsca appears about at ka = 19. And the third maximum magnitude of Qsca emerges at ka = 20. After ka larger than 20, the value of Qsca goes down till that converges. But for l = 45, the value of Qabs shows an rising trend until ka reaches 40, with a lower magnitude of Qsca . The trend towards convergence is relatively slow. However, the value of Qsca for l = 100 is almost 0, comparing to the value of that for l = 10 and l = 45. Panel (b) is the distribution of TM-polarized, which is similar with the distribution in panel (a). The difference of TM-polarization and TE-polarization is the location of the main peaks. The maximum magnitude of Qsca in panel (b) is further than that in panel (a), the same to the second maximum magnitude for l =10. The trends of Qsca in panel (b) is near that in panel (a) for l = 45 and l = 100. The panels (c) and (d) in Fig. 8.34 give the scattering efficiency Qsca versus ka, with different  0 (0.2, 0.5, 1.0) for the beam order l = 10. The Qsca in panel (c) with different 369 Advances in Optics: Reviews. Book Series, Vol. 5  0 is under TE-polarized while panel (d) with TM-polarized. In panel (c), the maximum magnitude of Qsca is about 3.7 103 for  0 = 0.2, 2.7 103 for  0 = 0.5 and 2.0 103 for  0 = 1.0. For  0 = 0.2, the maximum peak of Qsca is at ka = 10, with the second maximum peak at ka =16. And the trend downward to convergence is relatively quick. Comparing to  0 = 0.2, the location of maximal peak for  0 = 0.5 is at ka = 23, further than that of particle center. The downward trend for  0 = 0.5 is relatively steep than that of  0 = 0.2. For  0 = 1.0, the location of maximal peak with ka = 60, is the furthest among the value of 0 . But, the downward trend of Qsca is same to the case of  0 = 0.5. As for TM-polarized situation, the maximum magnitude of Qsca is lower than that of TE-polarization. For  0 = 0.2, there is one more peaks with maximum magnitude of Qsca = 3.1103. For  0 = 0.5, the maximum magnitude of Qsca is about 2.6 103. The trend of Qsca in panel (d) is similar with the trend in panel (c). From the standpoint of convergence, whatever situations in TE or TM-polarized, the rates of convergence is fast for  0 = 1.0, medium for  0 = 0.5, low for  0 = 0.2. And the amplitude of the peaks is close for the same scaling parameter  0 or beam order l. Fig. 8.34. The Qsca of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100]. The panels (a) and (b) in Fig. 8.35 show the scattering efficiency Qext versus ka, with different l (10, 45, 100) for the scaling parameter  0 = 0.8. The Qext in panel (a) with different l is under TE-polarized while panel (b) with TM-polarized. In panel (a), the 370 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) maximum magnitude of Qext is about 3.2 103 for l = 10, 0.5 103 for l = 45 and 3.2 108 for l = 100. And when the value of l is 10, there is a little peak with magnitude 0.25 103 at ka = 5. And then there is an abrupt increasing till maximum magnitude of Qext near at ka = 12. Subsequently, after a lower peak, the second maximum of Qext appears about at ka = 19. And the third maximum magnitude of Qext emerges at ka = 20. After ka larger than 20, the value of Qext goes down till that converges. But for l = 45, the value of Qext shows an rising trend until ka reaches 40, with a lower magnitude of Qext . The trend towards convergence is relatively slow. However, the value of Qext for l =100 is about 0, comparing to the value of that for l = 10 and l = 45. Panel (b) is the distribution of TM-polarized, which is similar with the distribution in panel (a). The difference of TM-polarization and TE-polarization is the location of the main peaks. Fig. 8.35. The Qext of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100]. The maximum magnitude of Qext in panel (b) is further than that in panel (a), the same to the second maximum magnitude for l = 10. The trends of Qext in panel (b) is near that in panel (a) for l = 45 and l =100. The panels (c) and (d) in Fig. 8.35 give the scattering efficiency Qext versus ka, with different  0 (0.2, 0.5, 1.0) for l = 10. The Qext in panel (c) (different 0 ) is under TE-polarized while panel (d) with TM-polarized. In panel (c), the maximum magnitude of Qext is about 3.7 103 for  0 = 0.2, 2.6 103 for  0 = 0.5 and 2.0 103 for  0 = 1.0. For  0 = 0.2, the maximum peak of Qext is at ka = 10, with the second maximum peak at ka = 16. And the trend downward to convergence is 371 Advances in Optics: Reviews. Book Series, Vol. 5 relatively quick. Comparing to  0 = 0.2, the location of maximal peak for  0 = 0.5 is at ka = 23, further than that of particle center. The downward trend for  0 = 0.5 is relatively steeper than that of  0 = 0.2. For  0 = 1.0, the location of maximal peak with ka = 60, is the furthest among the value of 0 . But, the downward trend of Qext is same to the case of  0 = 0.5. As for TM-polarized situation, the maximum magnitude of Qext is lower than that of TE-polarized. For  0 = 0.2, there is one more peaks with maximum magnitude of Qext = 3.1103. For  0 = 0.5, the maximum magnitude of Qext is about 2.6 103. The trend of Qext in panel (d) is similar with the trend in panel (c). From the standpoint of convergence, whatever situations in TE or TM-polarized, the rates of convergence is fast for  0 = 1.0, medium for  0 = 0.5, low for  0 = 0.2. And the amplitude of the peaks is close for the same  0 or l. Panels (a)-(d) in Fig. 8.36 show Qabs (given by Qsca  Qext ) under different l (l = 10, 45, 100 in panels (a) and (b)) and  0 ( 0 = 0.2, 0.5, 1.0 in panels (c) and (d)). As shown in panels (a) and (b) of Fig. 8.36, the value of Qabs with  0 = 0.8 but different l (10, 45, 100) is definitely different. For order l = 10 in panel (a), there is a obviously steep change from ka = 0 to ka = 20 under TE-polarization, but relatively slower than that in panel (b) with which ka varies from 0 to 10. For the situation of Qabs for ka > 20 but ka < 58 in panel (a), the downward rate of Qabs is relative low step by step till ka > 58. But for TM-polarized case, after ka > 10, the trend of Qabs is downward lower and lower. For l = 45 in TE-polarized case (panel (a)), there is a peak at ka = 58. And then, the downwards for ka > 58, go to converge slowly. For l = 45 in TM-polarized case (panel (b)), there is a period of steady trends for ka between 45 and 58. And then, the downwards is still continuous to converge. For l = 100, the value of Qabs is nearly to zero. Panels (c) and (d) of Fig. 8.36 gives the value of Qabs with l = 10 but different  0 (0.2, 0.5, 1.0). The changes of Qabs with different location of ka, still have the great effects. On the one hand, the maximal peak of Qabs is relevant different scaling parameter 0 . In panel (c), for scaling parameter  0 = 0.2, the location of peak is ka = 18,  0 = 0.5 for ka = 30, and  0 = 1.0 for ka = 58. In panel (d), for scaling parameter  0 = 0.2, the location of peak is ka = 15,  0 = 0.5 for ka = 19, and  0 = 1.0 for ka = 42. On the other hand, the polarization characteristic is influential. For  0 = 0.2, after ka = 20, Qabs in TE-polarized goes to a downward trend, relatively slower than that in TM-polarized. The same to  0 (0.5 and 1.0), the rate to obtain the maximal peak or the convergence in TM-polarized models quicker than that in TE-polarized mode. Based on the above simulation and analysis, it is obtained that the location of main peak and polarized mode is affected by  0 and l of Bessel pincer light-sheet. First, the bigger the l is, the closer the location of main peak to the source of particle is, but the smaller the amplitude of the main peak is. Secondly, the larger the scaling parameter  0 is, the 372 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) closer the location of main peak to the source of particle is, but the smaller the amplitude of the main peak is. Lastly, the convergence rate under TE-polarized mode is slower than that of under TM order. Fig. 8.36. The Qabs of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100]. 8.4.2.3. Radiation Force and Torque Here, the optical force (mainly focusing on Fy , Fz and Fz < 0) with different  0 and l of Bessel pincer light-sheet are discussed. The complex refractive index of the object is n1 = 1.59 in vacuum ( n0 = 1.0). The dimensionless parameter ka is chosen as ka = 0.1 and ka = 10. As can be gathered from the panels (a)-(l) of Fig. 8.37, Fy displaying positive values, interferes on the object, incoming of a TE-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l = 10 and 45. In panels (a) (0 = 0.2, l = 10) and (d) (0 = 0.2, l = 45), the different order l affects the distribution of Fy . It can be seen that in panel (a), there are some beam clusters along z -axis, and the beam focusing spot is about at z = 70 m. The magnitude of maximal force is 1020 N. It’s symmetric, and the width region of two bending arcs is  10,10  m along the y -axis. In panel (d), there is little clusters along z -axis( some margin about y = 30 m) , and the beam focusing spot doesn’t appear in our calculation region (the focusing spot is about z = 400 m in minor figure). The characteristic of symmetric didn’t appear in our calculation region, 373 Advances in Optics: Reviews. Book Series, Vol. 5 either.(width region is about  50,50  m along y -axis in minor figure). Panels (g) and (j) give the distribution of Fy with  0 = 0.5, the beam order l = 10 and 45, respectively. Identically, there are some clusters along z -axis, which the focusing spot is about at z =18 m . The magnitude of maximal force is 1019 N. It’s symmetric, and the width region of two bending arcs is  5,5 m along the y -axis. In panel (j), there is further focusing spot at z = 60 m, and the width region of two bending arcs is  15,15 m along the y -axis, symmetrically. Comparing to the margin cluster distributing region after the focusing spot in panels (g) and (j), we can see that the region is  30,30  m in panel (g), and the region is  12,12  m in panel (j). The distributing region after focusing in panel (g) is wider than that in panel (j). The panels (b), (e), (h) and (k) of Fig. 8.37 represents the plots for the longitudinal radiation force component Fz (the positive values), which interferes on the object, incoming of a TE-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l 10 and 45. In panel (b) (0 = 0.2, l = 10) and (e) (0 = 0.2, l = 45), the different order l effects the distribution of Fz . It can be seen that in panel (b), there are come beam clusters along z -axis, and the beam focusing spot is about at z = 70 m (the same with focusing spot in panel (a)). The magnitude of maximal force is 1021 N (smaller than that in panel (a)). It’s symmetric, and the width region of two bending arcs is  10,10  m along the y -axis (the same with distributing region in panel (a)). In panel (e), there is little clusters along z -axis (some margin about y = 30 m) , and the beam focusing spot doesn’t appear in our calculation region (the focusing spot is about z = 400 m in minor figure) (the same to focusing spot in panel (d)). The characteristic of symmetric didn’t appear in our calculation region, either.(width region is about  50,50  m along y -axis in minor figure) (the same with distributing region in panel (d)). Panels (h) and (k) give the distribution of Fz with  0 = 0.5, the beam order l = 10, 45, respectively. Identically, there are some clusters along z -axis, which the focusing spot is about at z =18 m. The magnitude of maximal force is 1019 N. It’s symmetric, and the width region of two bending arcs is  5,5 m along the y -axis. In panel (k), there is further focusing spot at z = 60 m, and the width region of two bending arcs is  15,15 m along the y -axis, symmetrically. Comparing to the margin cluster distributing region after the focusing spot in panels (h) and (k), we can see that the region is  30,30  m in panel (h), and the region is  12,12  m in panel (k). The distributing region after focusing in panel (h) is wider than that in panel (k). It is interesting that there are additional beam cluster along the margin beam in panels (b), (h) and (k), comparing to the cases of Fy . 374 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Panels (c), (f), (i) and (l) of Fig. 8.37 indicates the plots for the longitudinal radiation force component Fz (with negative values), which interferes on the object, incoming of a TE-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l 10 and 45. The object experiences a negative pulling longitudinal force; thus, the components affects in the opposite direction comparing to the direction of wave propagation. The influence of the scaling parameter  0 and the beam order l is similar to that in cases of Fz . Fig. 8.37. The optical acting on a dielectric sphere illuminated by a TE-polarized Bessel pincer light-sheet. ka = 0.1, alpha0 = [0.2,0.5], ł = [10, 45]. 375 Advances in Optics: Reviews. Book Series, Vol. 5 Panels (a)-(l) of Fig. 8.38, gives the transverse radiation force component Fy , but the object influenced by the incoming beam of a TM polarized Bessel pincer light sheet. In panel (a) ( 0 = 0.2, l =10) and (d) ( 0 = 0.2, l = 45), the different order l effects the distribution of Fy . Similar with the distribution of Fy under TE-polarized mode, the focusing spot is nearly at z = 70 m for panel (a) ( 0 = 0.2, l = 10), and z = 400 m for panel (d) ( 0 = 0.2, l = 45). The symmetrical characteristic still exists along y -axis. Fig. 8.38. The same to Fig. 8.37, but TM-polarized. The width region of two bending arcs is about  10,10 m for panel (a) ( 0 = 0.2, l = 10),  50,50  m for panel (b) ( 0 = 0.2, l = 45) along the y -axis. In the 376 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) same way, the distribution of Fy with  0 = 0.5 and the beam order l =10 and 45, is shown as panels (g) and (j). The location of focusing spot and the width region of bending arcs is the same as the cases in panels (g) and (j) of Fig. 8.37. The panels (b), (e), (h) and (k) of Fig. 8.38 illustrates the plots for the longitudinal radiation force component Fz (positive values), which interferes on the object, incoming of a TM-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l 10 and 45. The regularity for the location of focusing spot and the width region of bending arcs, is familar with the situations in panels (b), (e), (h)and (k) of Fig. 8.37. Panels (c), (f), (i) and (l) of Fig. 8.38 depicts the plots for the longitudinal radiation force component Fz (with negative values), which interferes on the object, incoming of a TM-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l = 10 and l = 45. In the same way, the components of optical force act in opposite direction of wave propagation, with the influence of  0 and l , similar to that in cases of Fz . Here, the transverse radiation force component Fy interfering on the object illuminated by a Bessel pincer light-sheet are analysis, chiefly focusing on the influences of 1D (one dimension) with different  0 and l in panels (a)-(d) of Fig. 8.39. The dimensionless size parameter is ka =10. The value of y is 4 m. Panel (a) gives the trend of the force component Fy acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet, as z changes under different  0 for l = 10. For  0 = 0.5 (the blue line), the amplitude of Fy is equal to zero along z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 35 m with maximal magnitude of Fy = 6  1017 N. There is a trough about at z = 50 m with minor magnitude of Fy = 11018 N. There is a period of second peak at the region 60,80 m along z -axis. After z > 85 m, Fy shows an upward trend. For  0 = 0.9 (the green line), Fy shows a downward trend till z = 40 m (the first trough with magnitude Fy = 0). There is a second crest (same magnitude with the first crest) at region  40,70 m along z -axis. The second trough is about at z = 70 m. After z > 80 m, there is negative and positive value of Fy , with some unstable changes. The maximal magnitude of Fy is 5.9 1017 N, while the minimal with 5.9 1017 N. Panel (b) gives the trend of the force component Fy acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different  0 for l =10. For  0 = 0.5 (the blue line), the amplitude of Fy is equal to zero along z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 35m with maximal magnitude of Fy = 6  1017. There is a trough about at z = 50 m with minor magnitude of 11018 N. There is a period of second peak at the region 60,80 m along z -axis. After z > 85 m, Fy shows an upward trend. For 0 = 9 (the green line), Fy shows a 377 Advances in Optics: Reviews. Book Series, Vol. 5 downward trend till z = 40 m (the first trough with magnitude Fy = 0 N). There is a second crest (same magnitude with the first crest) at region  40,70 m along z -axis. The second trough is about at z = 70 m. After z > 80 m, there is negative and positive value of Fy , with some unstable changes. The maximal magnitude of Fy is 5.9 1017 N, while the minimal with 5.9 1017 N. The differences between two polarized modes (panels (a) and (b)) is that the trends of Fy is stable for TM after z > 90 m. Panel (c) gives the trend of Fy acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. For different l (0, 5, 10), the value is positive, and the trend goes toward along z -axis. The difference is that the l = 0 (the blue line) changes rapidly, l = 5 changing medium, and l =10 changing slow. Panel (d) gives the trend of the force component Fy acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. The trends of Fy is similar to the cases in panel (c). Fig. 8.39. Optical force ( Fy ) with different  0 = [0.5,0.7,0.9] and l = [0,5,10] under TE and TM mode. Fig. 8.40 demonstrates the longitudinal radiation force component Fz acting on a dielectric sphere illuminated by Bessel pincer light-sheets are analysis, primarily focusing on the influences of 1D (one dimension) with different  0 and l. And the value of y = 4 m. Panel (a) gives the trend of the force component Fz acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet, as z changes under different  0 for l = 10. For  0 = 0.5 (the blue line), the amplitude of Fz is equal to zero along 378 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 42 m with maximal magnitude of Fz = 7 1017 N. There is a trough about at z = 60 m with minor magnitude of 11018 N. There is a period of second peak at the region 60,80 m along z -axis. After z > 85 m, Fz shows an upward trend. For  0 = 0.9 (the green line), Fz shows a downward trend till z = 60 m (the first trough with magnitude Fz = 0 N ). There is a second crest (same magnitude with the first crest) at region 40,75 m along z -axis. The second trough is about at z = 60 m. The second trough is about at z = 70 m. After z > 75 m, there is some unstable value changes of Fz . The maximal magnitude of Fz is 7.5 1017 N, while the minimal with 1.8 1017 N. Fig. 8.40. Optical force ( Fz ) with different  0 = [0.5,0.7,0.9] and l = [0,5,10] under TE and TM mode. Panel (b) gives the trend of the force component Fz acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different  0 for l =10. For 0 = 0.5 (the blue line), the amplitude of Fz is equal to zero along z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 42 m with maximal magnitude of Fz = 7 1017 N. There is a trough about at z = 60 m with minor magnitude of 11018 N. There is a period of second peak at the region 60,80 m along z -axis. After z > 85 m, Fy shows an upward trend. For  0 = 0.9 (the green line), Fz shows a downward trend till z = 60 m (the first trough with magnitude Fz = 0 N). There is a second crest (same magnitude with the first crest) at region  40,75 m along z -axis. 379 Advances in Optics: Reviews. Book Series, Vol. 5 The second trough is about at z = 60 m . The second trough is about at z = 70 m. After z > 75 m, there is some unstable value changes of Fz . The maximal magnitude of Fz is 7.5 1017 N, while the minimal with 1.8 1017 N. Panel (c) gives the trend of Fz acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. For different l (0, 5, 10), the value is positive, and the trend goes toward along z -axis. The difference is that the l = 0 (the blue line) changes rapidly, l = 5 changing medium, and 1 = 10 changing slow. Panel (d) gives the trend of the force component Fz acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. The trends of Fz is similar to the cases in panel (c). The spin torque can cause the object rotate around its center of mass in Figs. 8.41, 8.42. But only the non-vanishing axial spin torque component is Tx  0, The optical spin torque (mainly focusing on Tx , and Tx < 0) illuminated by a TM-polarized Bessel pincer light-sheet with different scaling parameter  0 and beam order l are discussed. The refractive index of the dielectric sphere is n1 = 0.26  7.0i, and that of surrounding medium is n0 = 1.33. The dimensionless parameter ka is chosen as ka = 0.1 and ka = 10. Fig. 8.41. Tx acting on a dielectric sphere illuminated by a TM-polarized Bessel pincer light-sheet. ka = 0.1,  0 = 0.2, l = [10, 45,100]. Fig. 8.41 demonstrates the optical spin torque component Tx (and Tx < 0, displaying positive values), interfering on the object, incoming of a TM-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and l = 10, 45, 100. Panels (a)-(c) is the Tx 380 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) distribution of different l. It can be seen that in panel (a) (l = 10), there are come beam clusters along z -axis, and the beam focusing spot is about at z = 70 m. The magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is  10,10  m along the y -axis. In panel (b) (l = 45), there is little clusters along z -axis (some margin about y = 30 m), and the beam focusing spot doesn’t appear in our calculation region (the focusing spot is about z = 400 m in minor figure). The characteristic of symmetric didn’t appear in our calculation region, either. (The width region of that is about  50,50  m along y -axis in minor figure). In panel (c) (l = 100), there are a few horizontal lines, which means the zero value. From the minor figure, there are some beam clusters along z -axis, and the beam focusing spot is about at z = 800 m. The magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is  100,100 m along the y -axis. After the focusing spot, the margin beam region is  30,30  m in panel (a) (l = 10),  100,100 m in panel (b) (l = 45) and  30,30  m in panel (c) (l = 100). The negative component (panels (d)-(f)) nearly has the same regularities in cases (panels (a)-(c)) of Tx . Fig. 8.42. Tx acting on a dielectric sphere illuminated by a TM-polarized Bessel pincer light-sheet. ka = 0.1,  0 = [0.2,0.5,1.0], l = 10. Fig. 8.42 depicts the optical spin torque component Tx (and Tx < 0, displaying positive values), interfering on the object, incoming of a TM-polarized mode Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2, 0.5, 1.0, and l = 10. Panels (a)-(c) is the Tx distribution of different 0 . It can be seen that in panel (a) (0 = 0.2), there are come beam clusters along z -axis, and the beam focusing spot is about at z = 70 m. The 381 Advances in Optics: Reviews. Book Series, Vol. 5 magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is  10,10  m along the y -axis. In panel (b) (l = 0.5), there are some clusters along z -axis, which the focusing spot is about at z =18 m. The magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is  5,5 m along the y -axis. In panel (c) (0 = 1.0), there are some clusters along z -axis, which the focusing spot is about at z = 2 m. The magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is  1,1 m along the y -axis. The negative component Tx (panels(d)-(f)) nearly has the same cases (panels(a)-(c)) of Tx . Here, Tx acting on an absorptive object illuminated by Bessel pincer light-sheets are analysis, basically focusing on the influences of 1D (one dimension) with different  0 and l in panels (a(d) of Fig. 8.43. The dimensionless parameter is chosen as ka = 10. And the value of y is 4 m. Fig. 8.43. Tx with different  0 = [0.5,0.7,0.9] and l = [0,5,10] under TE and TM mode. Panel (a) of Fig. 8.43 gives the trend of the spin torque component Tx acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different  0 for l = 10. For  0 = 0.5 (the blue line), the amplitude of Tx is equal to zero along z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at = 40 m with maximal magnitude of Tx = 11022 N.m. There is a trough about at z = 53 m with minor magnitude of 7 1023 N.m. There’s a region along the z -axis at region 60,90 m that’s symmetrically distributed along z = 70 m. After z > 85 m, Tx shows an upward trend, till the peak Tx = 8 1023 N.m. For  0 = 0.9 (the green line), Tx shows a downward trend till z = 28 m (the first trough with magnitude Tx = 11023 N.m). There is a first crest at region  40,70 m along z -axis. The second trough is about at z = 68 m. After z > 80 m, there is negative and positive value of Tx , with some unstable changes. The maximal magnitude of Tx is 1.5 1022 N.m, while the minimal 382 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) with 1.2 1017 N.m. There are few differences in two polarized mode. Panel (b) of Fig. 8.43 gives the trend of the torque component Tx acting on the sphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. The trends of Tx is decreasing as the increasing of l , and there is a few wave vibrating along z -axis. As has been noted, 0 > 0 represents the auto-focusing characteristic of the beam. ł (any real order) represents the bending characteristic. Whatever situation of optical torque components ( Fy , Fz and Fz < 0) and optical torque (Tx , Tx < 0), the scaling parameter  0 and the beam order l is sensitive to the raising or reducing. When the  0 is the same value, the width region of two bending arcs before focusing is wider and wider as the increasing of the beam order l. And the focusing spot location is further and further with the rising of the beam order. While the value of beam order is invariable, the width region of the two bending atcs before focusing is narrower and narrower, the focusing spot is closer and closer, with the increasing of the scaling parameter 0 . There is symmetrical characteristic of the optical force and torque along the y -axis. 8.4.3. Gaussian Light-sheet Gaussian light-sheet is the two-dimensional (2D) slice of a Gaussian beam which is a typical high-focused light. The optical scattering and mechanical effects when Gaussian light-sheet illuminates a dielectric sphere are paid attention to in this section. The effect of dimensionless waist radius k0 of the Gaussian light-sheet and ka is considered. Both TE-polarized and TM-polarized Gaussian light-sheet are studied. 8.4.3.1. Far-field Scattering Intensity Assuming the scatterer is located at the main lobe of incident light. Surrounding medium is vacuum (n0 = 1.0) and refractive index of scatterer is n1 = 1.59. The impact of k0 , ka and polarization type on I  is shown in Figs. 8.44-8.47. Fig. 8.44. I  (Log) of a TE-polarized Gaussian light-sheet scattered by a dielectric sphere. ka = 10. k0 varies as following list [6, 12, 18]. 383 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.45. The same as Fig. 8.44, but a TM-polarized Gaussian light-sheet is assumed. Fig. 8.46. The normalized far-field intensity of a TE-polarized Gaussian light-sheet scattered by a dielectric sphere. k0 = 12. ka varies as following list [2, 10, 20]. Fig. 8.47. The same as Fig. 8.46, but a TM-polarized Gaussian light-sheet is assumed. When a TE-polarized Gaussian light-sheet with varying k0 illuminates a dielectric sphere, I  is revealed in Fig. 8.44. k0 takes values as [6, 12, 18]. As the dimensionless waist radius k0 increases, the intensity of the main and side lobes increases. No matter how k0 varies, the distribution of I  is always symmetric about z -axis; the center and zone of the main and side lobes remain constant; the strongest intensity is located at  = 0 . When the incident beam becomes TM-polarized (Fig. 8.45), the intensity of lobes are weaker, and k0 affects amplitude of lobes more weakly. 384 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Fig. 8.46 (Fig. 8.47) exhibits the influence of ka on the far-field scattering intensitywith TE-polarized (TM-polarized) incident light. ka takes values as [2, 10, 20]. The distribution of I  still has perfect symmetry about z -axis. As ka increases, more side lobes appear, the intensity of the main lobe becomes stronger, and the width of the main lobe becomes smaller. When a TM-polarized Gaussian light-sheet illuminates, the amplitude of the main lobe becomes larger than the TE-polarized case. 8.4.3.2. Scattering, Extinction and Absorption Efficiencies The energy efficiency reflects the scatter’s ability to transform the incident energy to the scattering, extinction or absorption energy. We discuss the scattering, extinction and absorption efficiencies Qsca , Qext , Qabs in order. Suppose a Gaussian light-sheet shines on a gold particle (n1 = 0.26  7.0i) immersed in water (n0 =1.33). In this case, Figs. 8.48-8.50 show the variety of energy efficiencies with k0 and ka. k0 takes values as the list [6, 12, 18]. ka transforms from 0.01 to 60. The scatterer is also located at the main lobe of Airy light-sheet. Fig. 8.48. The Qsca of a Gaussian light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 60. k0 varies as following list [6, 12, 18]. Fig. 8.49. The Qext of a Gaussian light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 60. k0 varies as following list [6, 12, 18]. 385 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 8.50. The Qabs of a Gaussian light-sheet scattered by a dielectric sphere. ka changes in the range 0.01 < ka < 60. k0 varies as following list [6, 12, 18]. We discuss the scattering efficiency first. As shown in Fig. 8.48, whether the incident beam is TE-polarized or TM-polarized, Qsca rapidly increases to the maximum value in the range of 0.01< ka < 0.813 first; within the range of 0.813  ka  2.819, Qsca oscillates and has some peaks at ka = 0.813,1.414,2.217,2.819; in the range of 2.819 < ka < 60, Qsca decreases gradually and the decreasing speed declines with ka increases. Observing the three curves with different dimensionless waist radii, the larger k0 , the larger amplitudes of Qsca and peaks; with ka increases, the effect of k0 on the growth of Qsca increases first and then decreases. As for the impact of polarization, the curves of TE-polarization and TM-polarization are similar with the same parameters, but its value of the scattering efficiency is not equal. Within the range of ka < 57.994, the value of Qsca with TE-polarization is bigger than that with TM-polarization, however, the rule is contrary out of that boundary. As k0 increases, the point with the strongest effect of polarization on the scattering efficiency corresponds to a larger ka. Fig. 8.49 displays the impact of k0 and ka on the extinction efficiency Qext . The tendency of three curves representing Qext owns a striking similarity to that of Qsca . Due to the existence of absorption efficiency (the imaginary part of n1 is larger than zero), Qext is larger than Qsca throughout the computational zone. The absorption efficiency Qabs is also considered in Fig. 8.48. The trend of Qabs closely resembles that of Qsca . However, there are some differences between them. For Qabs , the amplitude is much smaller than that of Qsca ; the amplitude of peaks are located at ka = 0.612,1.414,2.016,3.421; the amplitude with TE-polarization is always larger than that with TM-polarization. 386 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) 8.4.3.3. Radiation Force and Torque Optical force acting on a Rayleigh (Figs. 8.51, 8.52) or Mie (Fig. 8.53) particle shined by a Gaussian light-sheet with different k0 is discussed. ka of the Rayleigh and Mie dielectric spheres are 0.1 and 10, respectively. We assign k0 value as the list of [6, 12, 18]. Both transverse and longitudinal force components Fy and Fz are considered. We pay more attention to negative longitudinal force Fz < 0. Assuming a Gaussian light-sheet illuminates on a sphere (n1 =1.59) in the vacuum (n0 =1.0). Fig. 8.51. The optical force acting on a dielectric sphere illuminated by a TE-polarized Gaussian light-sheet. ka = 0.1. k0 varies as following list [6, 12, 18]. We consider the Rayleigh scatters first. The calculation region is 20 < y < 20 and 0 < z <100 ( m). Due to the interaction between particle and light, Fy has two lobes that are symmetric about z -axis and have the opposite value, Fz has one negative main 387 Advances in Optics: Reviews. Book Series, Vol. 5 lobe near z -axis and two positive symmetric lobes on both sides of the negative lobe. A negative lobe of Fy is located in the positive y -axis. With k0 increasing, the distribution of main or side lobes of Fy and Fz enlarges, but the numbers of that are constant. When incident light-sheet is TM-polarized (Fig. 8.52), the same rule appears as TE-polarization. Fig. 8.52. The same as Fig. 8.51, but a TM-polarized Gaussian light-sheet is assumed. Changing the size parameter of scatterer to 10, Fig. 8.53 displays Fy and Fz with different k0 . The value of Fy and Fz with TE-polarization is equal to that with TM-polarization. Fy and Fz are symmetric about the origin and the line of y = 0, respectively. In panels (a,b), Fy has one negative lobe on the negative y -axis and one positive lobe on positive y -axis; as k0 increases, the main lobe has a smaller amplitude and larger zone, and the center of the main lobe move further away from the origin. Fz ranges more largely with k0 growing. 388 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Fig. 8.53. The optical force acting on a dielectric sphere illuminated by a Gaussian light-sheet. ka = 10. k0 varies as following list [6, 12, 18]. The radiation torque providing another mechanical degree of freedom is also studied as shown in Figs. 8.54, 8.55. Suppose n0 = 1.33 and n1 = 0.26  7.0i. We discuss Tx in this section. In Fig. 8.54, scatterer is a Rayleigh sphere (ka = 0.1). Tx owns one negative lobe ( y > 0) and one positive lobe ( y < 0) near z -axis. The larger k0 , the smaller the maximum value is, and the wider zone Tx takes.Finally, the case of a Mie sphere (ka = 10) is calculated (Fig. 8.55). Each curve of Tx is exactly symmetrical about region. With increasing k0 , the center of main lobe of Tx move towards the larger ka, and the amplitude of main lobe decreases. 8.5. Conclusions This chapter derives the equations for the light-sheet in the framework of GLMT. This part of the theory is suitable to any form of light-sheet and a spherical particle of arbitrary size. Then we deduced the scattering and mechanical properties of arbitrary light-sheet polarized by TE and TM illuminating a spherical particle of any size. When the incident light is an Airy light-sheet, the change of ys and ka has a huge influence on the far-field scattering intensity, the change of  has a little effect on it, and the polarization state has almost no influence on it. Qext , Qabs , and Qsca have roughly the same fluctuation trend, and the larger the ys , the smaller the ability of the particle to absorb and scatter the Airy 389 Advances in Optics: Reviews. Book Series, Vol. 5 light-sheet as ka increases.  has the opposite effect on this ability. The change of  and polarization state has little effect on the force. Torque is very sensitive to the variation of ys and  . In the case of a Gaussian incident light-sheet, as the dimensionless k0 increases, the intensity of main and side lobes increases. The distribution of intensity has perfect symmetry about the z -axis. As ka increases, more side lobes appear, the intensity of main lobe becomes stronger, and the width of main lobe decreases. Qext , Qabs , and Qsca have roughly the same fluctuation trend and the larger the k0 . The smaller the ability of the particle to absorb and scatter the Gaussian light-sheet as ka increases. Force and torque are very sensitive to the variation of k0 , but they are not affected by the polarization state. When a Bessel pincer light-sheet shines, the far-field scattering intensity is susceptible to the scaling parameter  0 and beam order l , and the dimensionless parameter ka. Whatever situation of optical force components ( Fy , Fz and Fz < 0) and optical torque (Tx , Tx < 0), their raising or reducing is susceptible to  0 and l. The research content of this chapter is expected to be applied to super-resolution imaging and optical control of light-sheet. Acknowledgements This work was supported by the National Natural Science Foundation of China [61901324, 62001345], the China Postdoctoral Science foundation [2019M653548, 2019M663928XB], the Project (B17035) and the CETC Key Laboratory of Data Link Technology (No: CLDL-20182418). Fig. 8.54. Tx acting on a dielectric sphere illuminated by a TM-polarized Gaussian light-sheet. ka = 0.1. k0 varies as following list [6, 12, 18]. 390 Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) Fig. 8.55. Tx acting on a dielectric sphere illuminated by a Gaussian TM-polarized light-sheet. ka = 10. k0 varies as following list [6, 12, 18]. References [1]. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, E. H. K. Stelzer, Optical sectioning deep inside live embryos by selective plane illumination microscopy, Science, Vol. 305, Issue 5686, 2004, pp. 1007-1009. [2]. R. Tomer, K. Khairy, F. Amat, P. 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Liu, Z. Lin, Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector Airy beam, Optics Express, Vol. 25, Issue 19, 2017, pp. 23238-23253. [46]. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, 1996. [47]. J. D. Jackson, Classical Electrodynamics, 3rd Ed., John Wiley & Sons, New York, 1999. [48]. H. Chen, N. Wang, W. Lu, S. Liu, Z. Lin, Tailoring azimuthal optical force on lossy chiral particles in Bessel beams, Physical Review A, Vol. 90, Issue 4, 2014, 043850. [49]. J. Schwinger, L. L. Deraad, K. A. Milton, W.-Y. Tsai, Classical Electrodynamics, Perseus Books, 1998. [50]. Y. Jiang, H. Chen, J. Chen, J. Ng, Z. Lin, Universal relationships between optical force/torque and orbital versus spin momentum/angular momentum of light, arXiv:Optics, 2015. [51]. Q. Ye, H. Lin, On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields, European Journal of Physics, Vol. 38, 2017, 045202. [52]. N. Ji, M. Liu, J. Zhou, Z. Lin, S. T. Chui, Radiation torque on a spherical birefringent particle in the long wave length limit: analytical calculation, Optics Express, Vol. 13, Issue 14, 2005, pp. 5192-204. [53]. M. Liu, N. Ji, Z. Lin, S. T. Chui, Radiation torque on a birefringent sphere caused by an electromagnetic wave, Physical Review E, Vol. 72, Issue 5, 2005, 056610. 393 Chapter 9. Laser Characterizations for Optical Wireless Communications Chapter 9 Laser Characterizations for Optical Wireless Communications Salam Nazhan1 9.1. Introduction to Optical Wireless Communications At the present time, optical communication technology, represented by optical wireless communication (OWC), which also known as free space optical (FSO) communication, or fibreless optical communications, that uses laser beam propagating through space, has substantially becomes a major leading topic in research articles. Laser-based OWC has basically given wide attention in research articles [1-3]. OWC is concerned with the transmission of information through the atmosphere from one point to another by using visual or laser beam to get optical communications, Fig. 9.1. A significant progress has been made since 1960s in OWC with the discovery of optical sources, particularly laser sources. Since then, the OWC has become one of the important fields in communications and has attracted a large number of researchers at a global level. The OWC first stated in the military application for covert communication because of inherent security compared with the radio frequency based technologies [4]. OWC technologies have a number of attractive features such as narrow optical beam width, thus avoiding potential interfere with the other beam, low costs system requirements, no fibre optic cables to lay (so no need for expensive rooftop connections equipment), very large bandwidth, excellent frequency re-use capabilities and compatibility with the exiting optical fibre communication networks [5, 6]. The optical sources that utilized for such systems are light emitted diodes or lasers. Both laser diode (LD) and light emitted diode are widely used in OWC systems. LDs are mainly used for outdoor and medium to long range applications, nevertheless the latter is ideal for high speed line of sight links, in both indoor and outdoor applications [7, 8]. Laser sources are one of the key components in the optical communication systems and network to enable high-speed data communications, therefore recent applications for the laser sources are highlighted in brief in the next sections. Salam Nazhan Department of Communication, College of Engineering, University of Diyala, Iraq 395 Advances in Optics: Reviews. Book Series, Vol. 5 Laser sources offer advantages in bandwidth and speed for free space applications over traditional systems which operated at shorter wavelength [2]. The laser beam increases the wireless capacity of 5G services and beyond for the future applications [1, 3]. Recently, high-speed underwater is also given wide interest in applications of OWC systems [9], as well as THz emitter based on laser chips [10]. The OWC technology is influenced by propagation of the laser beam through the space. Over the last decade, the OWC technology has to use applications such as satellite-to-satellite cross links and among mobile or stationary terminals to provide high bandwidth wireless communication links. OWC will be a powerful technique and one of the most unique tools to address issues of bandwidth limitation and security that have been created in high speed communication [11-13]. In addition, OWC based interconnect is the most promising scheme that could lead to increased speed, reduced size and compact packaging in future integrated circuits [14]. Normally, the free space communication systems use the wavelength range between 750 and 1600 nm because of the optical energy that travels through atmosphere have comparable properties at visible and near-IR wavelength [8]. However lower and higher wavelength are also being considered for specific applications. The wavelengths between 780-850 nm are the most popular because of readily available and inexpensive components, which have an attenuation of less than 0.2 dB/km [8]. The majority of OWC systems designed to operate at a transmission window located at a wavelength range of around 780-850 nm. Laser devices that emit 780 nm are available and inexpensive, but have short lifespan so must be considered during system design. While wavelength of 850 nm, high-performance transmitter, inexpensive and reliable, therefore they are commonly used in network and transmission systems. Furthermore, there are many devices that operate in this wavelength, such as avalanche photodiode (APD), highly sensitive silicon (Si) and vertical cavity surface emitting lasers (VCSELs). The latter, are extremely inserted in wide applications in such modern communication systems. In vast majority of the semiconductor lasers (SLs) diode applications, such as communication systems employing a high pump source and on optical fibre, the edge emitting lasers (EELs) are the dominant source and are widely used. However, EELs are too costly requiring optical fibre coupling, which results in additional power loss. Therefore, attention has been focused on VCSEL devices with potentially low manufacturing cost for various applications including optical communication systems. VCSEL is a SL, which has a resonant cavity that is vertically formed with the surfaces of the epitaxial layers. The lasing of VCSELs was first demonstrated in 1979 where a gallium indium arsenide phosphide grown on an indium phosphide GaInAsP/InP material was used for the active region, the emitted light had a wavelength of 1300 nm. VCSEL devices with optical and electrical properties offer a number of advantages such as a low threshold current, which enables these devices to be directly modulated at high frequencies, small beam divergence that allows good coupling efficiency with the optical fibre, symmetrical output beam profile, small switching transients, circular light-output mode, high packaging density, and very low power consumption compared to the conventional edgeemitting semiconductor laser devices [15]. VCSELs are the key and dominant source in local area networks thanks to their advantages such as wafer scale testing and ease of fabrication [16]. 396 Chapter 9. Laser Characterizations for Optical Wireless Communications Fig. 9.1. An alternative to traditional sources laser-based communications, or optic-based technologies, are prominent as another appliance for OWC systems. (AFCEA International, Airbus). The FSO communication systems typically uses wavelength between 750 and 1600 nm because of the optical energy that travels through atmosphere have similar properties at visible and near-IR wavelength rang [8]. However lower and higher wavelength are also being used for specific applications. The wavelengths between 780-850 nm are wide common owing to readily available and inexpensive components, which also has an attenuation of less than 0.2 dB/km [8]. The majority of OWC systems designed to operate at a transmission window located at 780–850 nm wavelengths. Lasers emit 780 nm are available and inexpensive, but have short lifespan so must be considered during designing such systems. While devices that emit 850 nm have high-performance, inexpensive and reliable, therefore they are generally used in network and transmission systems. Moreover, there are many devices that operate in this wavelength, such as avalanche photodiode, highly sensitive silicon and VCSELs. High security transmission data over a network is demand with the wide distributed of communication systems. OWC communications technology based on chaos becomes one of the most promising technique to avoid hacking information [11]. The message can be encoded with spectrum of the chaos and be sent through a free space channel. In the past few years, the chaotic dynamics of VCSELs has been the subject of great interest for researchers theoretically and experimentally [17, 18]; particularly, for secure optical communications. 9.2. Underwater Wireless Optical Communication Underwater wireless optical communication (UWOC) faces today challenges of absorption, scattering, and channel turbulence which still need an active solution, particularly for long distance wireless communication and high data rate transmission. These are real problems which could face many applications of UWOC channel. Optical wave offer high-data rate with power efficiency and short length data transmission compared with acoustic and radio frequency (RF). Latter suffer from limitation of high 397 Advances in Optics: Reviews. Book Series, Vol. 5 data transmission range owing to high water channel attenuation [19, 20]. Numbers of light sources, such as edge emitting laser diodes, super-luminescent diodes and VCSELs have been discussed in terms of modulation bandwidth and performance in the field of UWOC. High-speed underwater data transmission rate of about 3.4 Gbps was demonstrated with mixing lasers, with corresponding BERs of 3.5 × 10−3 where presented in [9]. An experimental results based on a green laser diode have achieved 34.5 m long distance UWOC with data rate of 2.70 Gbps and light attenuation coefficient of ~0.44 dB/m, which reported in [21]. For UWOC links the maximum attenuation length of 35.88 with symbol error rate of 6.31×10−4 at a distance of 120 m in clear ocean water at a bandwidth of 13.7 MHz have experimentally demonstrated in [22]. Based on both theoretical and experimental study, hundreds of meters long distance UWOC is quite possible in the future through optimization parameters of the optical links such as, power and beam divergence angle. Laser polarization modes, chaos and coherent beam are extremely attractive properties for the demand of high data rate transmission, power and security. 9.3. Chaos Based Secure Optical Communications Chaos-based optical communications using laser intensity have attracted intensive research interest due to its encryption capabilities [23]. The information signal is used intensity of modulated laser, which is set to operate in the chaotic oscillation of the polarization mode, and can be recovered at the receiver upon synchronization with an identical laser [17, 24]. Chaotic systems with a number of unique features including a noise-like shape with a broadband spectra, lower power implementations, and nonlinearity, have become appealing for modern secure communications applications. In such systems, a message is encoded into a noise like signal generated by a laser source with the chaotic behavior [25]. The chaotic communication system has three main parts, as depicted in Fig. 9.2, transmitter, channel, which including noise, and receiver [26]. The time delay between the two chaotic synchronization waveforms is the key to distinguish massage from these two signals of synchronization. It is important to match the corresponding parameters (internal and external) of transmitter and receiver to achieve a complete synchronization. External parameters such as bias current, temperature and so on of both transmitter and receiver can be controlled easily; however, the internal parameters such as optical loss, gain and quantum efficiency are difficult to be accurately controlled. Under the additive chaos modulation encryption scheme, the encoded messages can be successfully extracted based on a good synchronization oscillation between the transmitter and receiver signal. At the transmitter a chaotic carrier is employed to improve the overall security of the system. The modulated chaotic carrier signal is transmitted over the communication channel. At the receiver detection is carried out using exactly the same chaotic oscillator as in the transmitter to ensure successful recovery of the information signal. Successful message encoding/decoding in an unpredictability-enhanced chaotic VCSELs system is achieved numerically via the chaos-shift-keying technique. Using two VCSELs, which are subject to OF, and polarization-preserved optical injection, are adopted. 398 Chapter 9. Laser Characterizations for Optical Wireless Communications Improved decoding performance is achieved by choosing a proper polarizer angle, and the security is, to some extent, enhanced owing to the variple polarization OF [27]. Message encoding and decoding using the chaotic of VCSELs devices has been experimentally demonstrated in [28]. Signal + Noise Channel Synchronization technique Demodulator Chaotic signal Massage Transmitter Receiver Fig. 9.2. Block diagram of chaotic communication system. 9.4. Vertical-cavity Surface-emitting Lasers (VCSELs) VCSELs are relatively recent type of semiconductor laser devices, which are attractive for a number of applications from sensors to telecommunications. VCSEL is a semiconductor laser (SL), which has a resonant cavity that is vertically formed with the surfaces of the epitaxial layers. Typically SLs devices consist of semiconductor layers grown on top of each other on a substrate. Usually the growth processing is achieved in a molecular-beam-epitaxy (MBE) or metal-organic-chemical-vapor-deposition (MOCVD) growth reactor. Fig. 9.3 shows the oxide-confined VCSEL top-emitting structure [26]. The construction of VCSEL is particularly different from other lasers, an active layer sandwiched between highly reflectivity mirrors in range of about 99.5 to 99.9 %, which is placed at the bottom and top of the device structure as shown in the Fig. 9.3. These mirrors consist of distributed Bragg reflectors (DBRs), which is made up of several quarter-wavelength-thick layers of semiconductors. These layers have alternatively high and low refractive indexes. The VCSEL emission can be from the top or bottom of the device [29]. There are several categories of VCSELs based on the optical and electrical confinement techniques, active layer design and wavelength emission. The most important issue in fabrication such devices is the optical confinement factor, because of the conversion efficiency of the electrical-to-optical signal. VCSEL has become a very important source, which effectively displacing EELs for applications such as high speed data communication and chaos for local area networks [30]. VCSELs consist of a small active volume (few μm) and these have a very low threshold current (few μA) compared with EELs (~300 μm) [31]. This is one of the attractive features that give the higher reliability than edge-emitters devices. 399 Advances in Optics: Reviews. Book Series, Vol. 5 n-contact Light output Oxide Active Aperture Region DBRs p-contact Substrate Fig. 9.3. VCSEL with a selectively-oxidized top-emitting structure [26]. 9.4.1. Optical and Electrical Properties of VCSELs As discussed above the structure of VCSEL is completely different from other semiconductor structures devices that have been realised so far, such as distributed feedback laser (DFB) or EELs [32]. What makes VCSEL structure unique is that the emitting light is perpendicular to the surface of the laser. Consequently, this dramatic difference in the VCSEL structure makes a similar design technique impossible for convention facet emitting lasers. Because of the main concern in these devices is to achieve high-longitudinal side-mode suppression, which is completely disregarded in the VCSEL design [33]. VCSELs gained a reputation as a superior technology for applications such as fibre-channel, Gigabit Ethernet and intra-systems, free space optical communications, optical fibre communications and optical recording [34-36]. However, VCSELs have a number of problems based on large frequency chirp and polarization insensibility, which limit their performance in fibre-optic communication systems, as well as causing limitations in transmission distances and speed [37]. Such drawbacks are related to the laser noise properties and also depend on polarization mode fluctuation [38]. VCSELs are very sensitive to the effects of optical feedback (OF) and optical injection because of their high gain and very short cavity length [39]. 9.4.1.1. Light-current (L-I) Curve Characteristics VCSEL when operating near threshold current usually lases in a single polarization mode [40]. However, VCSEL can oscillate with the orthogonally polarized simultaneously, owing to their circular symmetry structure [41]. VCSEL emit mainly linearly polarized light. However, its orientation is not well distinguish because of the laser cavity and the gain medium are quasi isotropic in the active layer. In most sensing applications and data communications, a polarization stability of VCSEL and how it can be controlled is essential [42]. Consequently, the polarization mode linearity of VCSELs when modulated is of critical concern; particularly, in applications such as optical communications and 400 Chapter 9. Laser Characterizations for Optical Wireless Communications optical memory [43]. One of the essential parameter of a SL is light-current (L-I) curve, the laser efficiency can be predicted from the L-I curve properties [44]. The VCSEL lases in a fundamental mode with two orthogonal polarization modes for the entire range of bias current 𝐼𝑏 used as shwing in an experimental Fig. 9.5 (b). Fig. 9.4 displays the experimental setup for the free running L-I curve measurements of VCSELs. The laser is driven by laser diode driver (Newport, 505B) and is temperature controlled by temperature controlled with a thermoelectric temperature controller (TED 200C) to within 0.01 oC. The laser output is collimated using objective lens (Aspheric Lens, f = 4.51 mm). A half wave plate (HWP) (Zero-Order Half-Wave Plate) and a polarization beam splitter (PBS) (Cube, 620-1000 nm) are used to direct the orthogonal polarizations of the VCSEL to the photodetectors (PD) (New Focus Nanosecond photo detector, model No). 1621. The HWP and PBS are used only when measured the VCSEL polarizationresolved L-I curve using an optical power meter (Anritsu, ML9001A) and then removed when measuring the total output power (IT). The L-I plots were obtained using LabVIEW controlled by a personal computer. Power meter Temperature& current controller PD HWP Lens PD VCSEL PBS Fig. 9.4. Experimental setup to measure output power properties of VCSELs, half wave plates (HWP), polarization beam splitter (PBS) and photo detector (PD). Fig. 9.5 displays the L-I characteristics of VCSEL, where in Fig. 9.5(a) the total output power and in Fig. 9.5(b) the polarization-resolved output power of the VCSEL are presented. The first lasing mode in Fig. 9.5(b) with a full square black line refers to the X-polarization (XP) and the suppressed mode with a full dot red line corresponds to the Y-polarization (YP) mode. Fig. 9.6 shows the total output power (a) and the polarization resolved output power (b) of the standalone VCSEL combined with two polarization switching (PS) without any external perturbation. PS occurs between the orthogonal polarization modes of the VCSEL when 𝐼𝑏 increase from zero to 9 mA. The first type of switching (PSI) occurs at ~6.3 mA from the high-frequency mode (XP) to the mode with low frequency (YP). While the second type of switching (PSII) is observed at 𝐼𝑏 of ~7.6 mA corresponding to PS from the XP (low frequency mode) to the orthogonal mode (YP) with the high frequency mode, as depicted in Fig. 9.6(b). Both types of switching (i.e., PSI and PSII) are defined and studied in detail in [45]. 401 Advances in Optics: Reviews. Book Series, Vol. 5 0.6 Total Output power (mW) Output power (mW) 0.6 0.4 (a) 0.2 0.0 0 2 4 6 XP-Mode YP-Mode 0.4 (b) 0.2 0.0 0 8 2 4 6 8 Bias Current (mA) Bias Current (mA) Fig. 9.5. Light-current characteristics of free-running VCSELs under study (a) the total output power and (b) the polarization-resolved output power. Output power (mW) Output power (mW) XP-Mode YP-Mode 0.6 Total 0.6 0.5 0.4 0.3 (a) 0.2 0.1 0.0 0.5 0.4 0.3 (b) 0.2 0.1 0.0 0 2 4 6 Bias Current (mA) 8 0 2 4 6 8 Bias Current (mA) Fig. 9.6. The L-I characteristic of standalone VCSEL: (a) Total output power; (b) Polarization-resolved output power with two PS. 9.4.1.2. Polarization Switching Most VCSELs devices typically emit linearly polarized light. In fact, instability is a common features in VCSELs devices owing to weak material and cavity anisotropies [46], therefore polarization instability can happen without external perturbation. However, under some conditions - such as when the bias current is increased or an external OF is introduced - the linearly polarized state switches to the orthogonal linearly polarized state. This usually occurs due to changes in the gain and loss of the orthogonally polarized modes [47], and changes in the operating temperature as well as the magnitude and directionally of the bias current [48]. Based on the relevant studies in the literature, experimental and theoretical works on VCSELs have shown that increasing the injection current and the OF level can lead to increased polarization inversion and mode competition between the laser polarization modes. Mode competition is involving multiple switching, which destabilizes the laser emission and dynamic solutions. Interestingly, variable polarization based OF was recently proposed for external cavity feedback as an effective tool to control the polarization instability of VCSEL [46]. The PS features of SLs are of interest for a wide range of applications including optical switching, storage system and high capacity data processing. It has been observed that PS 402 Chapter 9. Laser Characterizations for Optical Wireless Communications can occur under the fixed bias current and OF conditions [49]. In VCSEL the position of PS is mainly determine by three factors; the net gain of the two polarization modes, the electric field and the injection current [50]. A critical issue in developing such laser devices is how to determine and control their polarization instability. Controllable PS has been investigated in [40, 51, 52] by considering the effects of a number of parameters including the OF strength, optical injection and frequency detuning. The first experimental work that demonstrated the PS in VCSELs was reported in 1993 [53], using optical injected. Since that time the polarization bi-stability of VCSELs has been the subject of extensive research from both theoretical and experimental perspective. However, the majority of previous studies have focused on influences of conventional OF on the dynamics and polarization characteristics of VCSELs [54-56]. The polarization properties of SLs subject to the OF with non-rotated [40, 41, 50-52, 54, 55, 57-61] and variable rotated polarization angles have been studied experimentally and theoretically in [49, 62-66]. It has shown that a laser under the OF effect can emit in certain polarization mode and increasing the level of dose lead to suppressed the polarization instability. Controllable PS of VCSEL has also been investigated in [40, 51, 52], where the effects of a number of parameters such as the OF strength, optical injection and frequency detuning were considered. Polarization mode hopping is also observed, which is associated with improvement of the antiphase synchronization dynamics [67]. Recently, numerical simulations results on the influence of polarization-rotated OF on the polarization properties of VCSEL have been reported in [49]. It has been observed that PS can be occur even for a fixed bias current and an OF level. Furthermore, PS properties are depending on the OF level and the polarization angle 𝜽𝒑 , where smaller 𝜽𝐩 is required to implement the PS when the OF level increase. In VCSELs it is possible to achieve PS by means of thermal effects [68], optical injection [69], strong enough OF [70] and rotated polarization angle of [65]. In the later, it was shown that by employing a polarization controller to obtain the variable polarization angle, VCSEL exhibit dominant PS for a fixed bias current and OF for the case of the selected polarization feedback, whereas in case of preserving the OF the dominant PS is not observed in the entire parameter space. 9.5. OWC Based on VCSELs Since thirty seven years ago, the VCSELs idea which was established in Japan is growing up rabidly worldwide [71]. After that VCSEL technology offers an exceptional advantage to high speed OWC links. Thanks to their unique features of low costs, power efficiency and low threshold current as well as desired output-power beam profile (circular and, single-longitudinal operation) with small divergence angle, VCSEL have introduced in wide ranges of applications including optical sensing, optical switching, short and long data transmission and networks, optical storage, optical HDMI cables, intra-chip communications, network security etc. [72-76]. Recently, VCSELs have been used as a promising candidate for photonic neuron networks. A neuron-like response dynamic based on polarization switching (PS) and dynamics nonlinearity, which have reported in VCSEL devices with applying polarization dynamic injection [47, 77, 78]. Underwater wireless communication (UWC) technologies have utilized VCSELs owing to desired 403 Advances in Optics: Reviews. Book Series, Vol. 5 modulation bandwidth range which past GHz range [79-81]. OWC based on VCSELs is becoming more common in wide variety of applications in optical communication including FSO communication [82], UWC [80], Light Fidelity or LiFi transmission based on VCSEL with injection technique, which reported in [82] and VCSELs’ chaos-based data transmission security as demonstrated in [67, 83]. VCSEL also adaptive for future metro applications, where it presented up to 50 Gb/s connections per flow over hundreds of km to address the needs of capacity and support multi-terabit connections networks [84]. Among wide light sources, VCSELs are recommended to be best candidate for both fiber-optics and wireless communication systems, in terms of their cost efficiency, higher data transmission and low power consumption [85, 86]. VCSELs are also considered to be ideal for gigabit Ethernet and optical interconnects [34, 87]. Lately, in [88, 89], high-speed and energy-efficient 850 nm VCSELs with direct modulation operated errorfree up to 57 Gbit/s. The VCSELs’ low power consumption minimizes the overall power dissipation of the optical sources, when it is used in high-speed data transmission systems. Conventional light sources, such as light-emitting diodes at a low-biasing current, experience an efficiency droop, which is a direct consequence of non-radiative recombination [90]. Both [91, 92] have shown that lowering the drive current of optical sources can achieve lower power consumption, as well as higher modulation speed. Indeed, reducing the lasing current below the threshold is essential, in order to decrease the optical device’s heat dissipation and increase its modulation bandwidth. Certainly VCSELs recorded very low threshold current of several micro ampere with high-efficient output power and performance [32, 93]. 9.6. Nonlinearity of VCSELs The nonlinear behaviour of SL with modulation signal is a subject of great interest in many researches [94] particularly for optical communication due to, for instance their effects to limit the RF dynamic range of optical devices. The nonlinear behaviour of optical devices such as EEL and VCSEL is a major limiting factor in analogue optical communications [95] and has been widely investigated [96-99]. The nonlinear dynamics of VCSELs as a result of optical injection and OF was investigated in [96]. Noise and different physical mechanisms, such as spatial-hole burning, phase coupling, gain anisotropy and birefringence effects can lead to rich nonlinearity properties in SL. In [100, 101] it was shown that nonlinear induced harmonic distortion can lead to decreased power efficiency of VCSEL. Recently, special attention has been paid to light polarization in VESCELs [96]. In the VCSELs under current modulation (CM) the nonlinear dynamics and chaos are easily realized due to the mode competition or polarization mode switching [102]. Furthermore, in semiconductors laser the nonlinear distortion appears when the laser is being driven near threshold current [101]. High nonlinearity in the L-I curve of VCSEL devices has been reported when the laser is biased near the threshold current at high modulating frequency [97], which has led to harmonic distortions. Nonlinear gain saturation of the lasing transition produce a nonlinear behaviour in the L-I characteristics 404 Chapter 9. Laser Characterizations for Optical Wireless Communications of VCSELs devices, which lead to increase the relaxation oscillation damping and eliminate the light intensity through the reduction of differential gain [103]. The result obtained in [104] demonstrated that the nonlinearity of VCSELs can entirely suppressed with orthogonal OF, and the 3-dB bandwidth of the devices is enhanced. As well as the harmonics distortions are significantly improved and reduced to the noise floor. This introduced OF as an effective tool to recover the nonlinearity behavior and improve the analogue performance of VCSELs. 9.7. VCSELs-based Chaos Dynamics In a number of applications such as communication systems, high pump sources and optical fiber systems, a potential light source that could be adopted is the VCSEL with unique features including high data rates [105], low power consumption and lower manufacturing costs [86]. VCSELs are sensitive to the OF and optical injection due to their higher mirror reflectivity inside the laser cavity [106, 107]. However, OF does lead to a number interesting complex dynamic behaviors in VCSELs including chaotic, time-period pulsing dynamics and PS [91, 108]. In the past few years, the chaotic dynamics of VCSELs with OF has been the subject of interest by researchers theoretically and experimentally [17, 18] particularly for secure optical-based communication systems. Recently, synchronization of the orthogonal polarization modes of VCSEL received wide attention as a chaotic source in secure communication systems [109, 110]. The anti-phase correlation of a semiconductor laser was experimentally observed in [7] for the case where the chaotic oscillation of the polarization modes was lower than the relaxation oscillation frequency. In VCSELs, detailed characteristics of chaos synchronization outlined in [15] showed that the anti-phase chaotic synchronization can be achieved between orthogonal polarization modes of the two mutually coupled VCSELs. An anti-phase synchronization phenomenon is a common feature in semiconductor lasers with polarization rotated OF technique. Furthermore, the time lag between the two synchronization signals can be zero in complete chaos synchronization [111]. A solitary VCSEL can exhibits strong anti-phase dynamics between orthogonal polarization modes [110]. Numerically and experimentally study in [11] outlined that the two polarization modes of VCSEL exhibit anticorrelated dynamics with OF. However, a study presented an anti-phase oscillation between transverse electric (TE) and transverse magnetic (TM) modes in a semiconductor laser with polarization-rotated OF [112]. Maximum values for the correlation coefficient of the synchronization modes of -0.68 and -0.99 obtained experimentally and theoretically, respectively with a zero time lag between the orthogonal modes under appropriate conditions have been reported [112]. In [113] demonstrated experimentally high anti-phase cross-correlation value of (0.99) between the polarization modes of a VCSEL with zero time delay between the synchronized signals. It shows that VCSEL can produce high-quality anti-phase polarization-resolved chaos synchronization between their polarisation modes under rotating polarization-preserved OF. Furthermore, its allow illustration that a different chaotic pattern can be achieved by means of rotated-polarization OF, which indicates the complex dynamic behaviour associated with the rotated polarization angle [114]. 405 Advances in Optics: Reviews. Book Series, Vol. 5 9.7.1. Chaos Synchronization in VCSEL In the recent years, chaos synchronization has become a hot topic due to their potential applications in optical communication systems where security is paramount. One of the important applications of the synchronized chaotic system is for encrypted communications, where the chaotic dynamic of an optical laser is controlled via synchronization phenomena using OF. Chaotic synchronization based on semiconductor lasers and their applications in secure communications have attracted considerable attention. The first demonstration of chaos synchronization was by Pecora and Carroll [115]. Recently, chaos synchronization in optical communications systems employing SL diodes has been a hot topic in applications where security is paramount. In such systems, a message is encoded into a noise like signal generated by laser source with chaotic behaviour [25]. In this context, chaos-based optical communications have attracted intensive research interest due to its encryption capabilities. A popular optical source adopted in such systems is the VCSELs because of their unique features especially when subjected to polarization OF. Furthermore, a different chaotic pattern and complex dynamics can be achieved under rotated-polarization angle of [17, 64, 114, 116]. Nowadays, we have seen a number of experimental and theoretical research activities on chaos, chaos synchronization and communication characteristics of VCSELs. Synchronization of the chaos is achieved experimentally in un-directionally coupled external-cavity vertical-cavity surface-emitting SLs operating in an open-loop regime. The polarization of the injected beam was perpendicular to that of the free-running receiver (x polarization). The injected beam and the y-polarized component of the receiver show good synchronization [17, 27, 28, 117-123]. It has been shown that two VCSELs can be synchronized under appropriate conditions. However, synchronization is lost at a higher mirror reflectivity (i.e., 50 %). Chaotic oscillations can be achieved with an external OF and optical injection [17, 94, 124], even more with a free running operation of semiconductor lasers [125]. Several theoretical and experimental studies have reported that the dynamics of polarization-rotated OF [126] in a semiconductor laser is quite different from the conventional OF [64, 114, 125]. Synchronization of chaos have attracted increasing attention especially in coupled VCSELs with a polarization-rotated OF and optical injection [17, 107]. The rotated polarization OF has employed to suppress the polarization nonlinearity [104] and polarization switching in VCSEL with the modulation signal [127]. In contrast, a higher external reflectivity leads to a more chaotic behaviour in the output power of VCSELs. VCSEL’s dynamic can be driven into the chaotic regime by means of an external OF. The synchronization has been observed over a range of values of the coupling parameter such as the drive current, external reflectivity and coupling coefficient [117]. Solitary VCSEL can exhibits strong anti-phase dynamics between own orthogonal polarization modes [110]. However, one study has presented an anti-phase oscillation between transverse electric (TE) and transverse magnetic (TM) modes in a SL with polarization-rotated OF. It has achieved a maximum value of correlation coefficient of the synchronization modes of -0.68 (-0.99) experimentally (theoretically) with zero time lag between the orthogonal modes under appropriate conditions [112]. The antiphase chaotic 406 Chapter 9. Laser Characterizations for Optical Wireless Communications synchronization is enhanced as the angle of orthogonal polarization of the OF is increased. Polarization modes are oscillated entirely in the chaotic regime in antiphase synchronization, with no time delay at low and high bias currents of 1.2 mA and 1.7 mA, respectively [67]. However, the synchronization quality of the two modes completely deteriorates when the bias current is increased to 1.7 mA at a polarization angle of 70o, where power mode differences are increased. 9.8. Theoretical Analysis of Lasers Dynamics In order to assess semiconductor lasers performance theoretically, it is imperative to utilise rate equations describing the time variation of the carrier and photon density. 9.8.1. Rate Equations Describing Carriers and Photons Density Dynamics These equations are shown below describing the rate equations for the carrier decay N and the photon decay S as a function of time t. These can be written as the subtracting carrier recombination rate via laser emission G νg S, and loss mechanisms N/τc, from the carrier generation rate ηi I/qV, by stimulated and spontaneous emissions [64, 97, 128]: 𝑑𝑁(𝑡) 𝑑𝑡 = 𝜂𝑖 𝐼 𝑞𝑉 − 𝑁(𝑡) − 𝜏𝑐 𝐺 (𝑡)𝜈𝑔 𝑆(𝑡) (9.1) The sources of photons are stimulated and spontaneous emissions. However, only a small fraction of the spontaneous emission is coupled into a given laser mode and this component may often be neglected. Losses of photons are governed by a photon lifetime τp , is given by: dS(t) dt = G (t)νg S(t) + Γβsp R sp − S(t) , τp (9.2) where I is the current density, ηi is the injection efficiency, V is the active volume, q is the unit charge, G is the gain coefficient, 𝜏𝑐 is the carrier lifetime, vg is the photon group velocity, Γ is the confinement factor, ßsp is the spontaneous emission factor, and R sp is the spontaneous recombination rate. The gain coefficient is assumed to be a linear function of carrier density and can be expressed as: G(t) = g ο N(t)−Nο 1+ϵS(t) (9.3) Here g ο is the linear gain coefficient, ϵ the gain saturation coefficient, and Nₒ is the carrier number at transparency. At the study-state condition, the gain of a laser above threshold should always equal the threshold gain, otherwise optical amplitude continues to increase which cannot happen in the steady-state. The same argument is valid for the carrier density in the steady-state because of both G and N are combine in the cavity of the laser, so [129]: 407 Advances in Optics: Reviews. Book Series, Vol. 5 G = Gth , (I > Ith ), (9.4) N = Nth , (I > Ith ) (9.5) At the study state condition, gain = loss and can be assumed to be constant, therefore we have: Gth = 1 , τp (9.6) where Ith is the threshold current. The average modal gain can be written as: 1 L 1 R < G >th = ΓGth = <ai > + ln ( ), (9.7) where L is the total cavity length, R is the mean mirror reflectivity, for simplicity the 1 1 1 mirror loss term ( L ln( R) ) is abbreviated as am and the photon loss ( τ ) is defined by: 1 τp = νg ( < ai > +am ) p (9.8) The internal cavity loss of the laser is ( ai + am ), ai is the photon losses, and am is the mirror loss parameter. ΓGth = 1 τp νg = < ai > +am (9.9) At the study state condition and from equation (9.1) the photon density above threshold can be written as: S = ηi (I−Ith ) , (I qVGth νg > Ith ) (9.10) To obtain the output power of the laser we have to extract the total optical energy inside the cavity Ein , which is defined in terms of the energy loss rate through the mirrors (νg am ) as: Ein = Shν Vc, Pout = Shν Vc νg am , (9.11) (9.12) where Vc is the cavity volume, and hv is the photon energy. Substituting from Eqs. (9.10) and (9.9) and using Γ = V/Vc in Eq. (9.12), now the laser out power can be written in a common expression as [130] a m Pout = ηi a +a i hν m q (I − Ith ) The differential quantum efficiency is given by: 408 (9.13) Chapter 9. Laser Characterizations for Optical Wireless Communications a m ηd = ηi a +a i We can simplify Eq. (9.13) to be as: Pout (t) = ηd m hν (I(t) − Ith ) q (9.14) (9.15) When a laser is subject to direct modulation current, the total injection current is given by: I(t) = Idc + Im (t), (9.16) Im (t) = Im eifmt , (9.17) Idc is the bias current (time-independent), and Im (t) is the modulating current (time varying), which could be given in a sinusoidal form as: where fm is the modulation frequency (Hz). 9.9. Conclusions In this chapter, an overview to optical wireless communication (OWC), underwater wireless optical communication (UWOC), Chaos-based optical communications and properties of one of the most important and commonly used semiconductor devices in communication systems (VCSEL), including of the optical chaos, optical and electrical properties, L-I characteristics, polarization switching (PS), and nonlinearity behaviours were presented. In addition, theoretical analyses for the dynamical properties of the semiconductor lasers (SLs) using the rate equations were described. It was known that OWC is a powerful and promising technique to increase the data transmission rat in communication systems. Laser devices are strongly introduced as alternative sources for OWC system due to their advantages in the bandwidth and speed properties over traditional sources. Although, properties of SLs such that VCSELs, for OWC have intensively been studied however, they are still under active investigation due to its potential applications as in secure communication and nonlinear optical systems. It was referring that the SLs properties are still under active study and not fully understood, especially when subjected to an external pertraptions. The dynamical properties of SLs were describe in a theoretical model and discussed in detail using rate equations to better understanding of the laser behaviours. Most OWC systems use a wavelength range between 780 to 850 nm. For this wavelength range, which has a low attenuation less than 0.2 dB/km and the optical devices are available and inexpensive. VCSELs devices can operated in instability polarization modes emitting owing to weak material and cavity anisotropies, therefore OF technics utalized to control the polarization instability of such devices. 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Woafo, Semiconductor lasers driven by self-sustained chaotic electronic oscillators and applications to optical chaos cryptography, Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 22, Issue 3, 2012, 033108. [122]. Y. Hong, et al., Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers, Optics Letters, Vol. 29, Issue 11, 2004, pp. 1215-1217. [123]. S. Priyadarshi, et al., Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization-angle of feedback, IEEE Journal of Selected Topics in Quantum Electronics, Vol. 19, Issue 4, 2013, 1700707. [124]. J. Ohtsubo, Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback, IEEE Journal of Quantum Electronics, Vol. 38, Issue 9, 2002, pp. 1141-1154. [125]. M. Virte, et al., Deterministic polarization chaos from a laser diode, Nature Photonics, Vol. 7, Issue 1, 2013, pp. 60-65. [126]. Y. Takeuchi, R. Shogenji, J. Ohtsubo, Chaos synchronization in semiconductor lasers with polarization-rotated optical feedback, Optical Review, Vol. 17, Issue 5, 2010, pp. 467-475. [127]. S. Nazhan, et al., Investigation of polarization switching of VCSEL subject to intensity modulated and optical feedback, Optics & Laser Technology, Vol. 75, 2015, pp. 240-245. [128]. L. A. Coldren, S. W. Corzine, M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits, Vol. 218, Wiley, 2012. [129]. L. A. Coldren, S. W. Corzine, Diode lasers and photonic integrated circuits, Optical Engineering, Vol. 36, Issue 2, 1997, pp. 616-617. [130]. S. Li, UWB Radio-over-Fiber System Using Direct Modulated VCSEL, UWSpace, 2007. 416 Chapter 10. Compact Solar-pumped Lasers Chapter 10 Compact Solar-pumped Lasers Hiroshi Ito, Kazuo Hasegawa, Shintaro Mizuno, Yasuhiko Takeda and Tomoyoshi Motohiro1 10.1. Introduction 10.1.1. Solar-pumped Lasers and Their Possible Applications A solar-pumped (or sun-pumped) laser (SPL) can be deemed as a variant of a conventional flash-lamp pumped laser. In SPLs, natural sunlight is used instead of flash-lamps for pumping the laser medium. Getting rid of electricity-consuming flash-lamps, SPLs are essentially renewable energy devices that convert incoherent sunlight with a wide spread of spectrum and low areal density into a monochromatic coherent laser beam with extremely high energy flux density. SPLs can be used to transmit the collected solar energy as laser beams across large distances wirelessly or via optical fibers. Some possible examples include power transmission between an orbiting space solar power station and the ground [1-5], power transmission between a lunar lander and a lunar rover exploring deep into permanently shadowed craters for frozen water [6, 7], or powering mobile objects such as drones, robots, and electric vehicles [4, 8, 9]. The transmitted laser beams can be received and converted into electricity by solar cells specially designed for the monochromatic light of the laser beams [8, 10-18]. SPLs have also been investigated to heat up materials to a higher temperature than the temperature that can be attained by simple solar concentration. A typical example includes a magnesium-based energy cycle in which solar energy is eventually stored and utilized by reduction of magnesium oxide using SPL irradiation and hydrogen production by reduction of water with magnesium [19, 20]. 10.1.2. Brief History of SPLs SPLs were first reported in the 1960s [21-23], soon after the discovery of laser, that is, stimulated optical radiation in ruby [24]. Since then, more than 600 papers publicly Tomoyoshi Motohiro Institute of Materials Innovation, Institutes of Innovation for Future Society, Nagoya University, Japan 417 Advances in Optics: Reviews. Book Series, Vol. 5 available papers have been published on SPLs. Iodine or iodides were the major laser mediums (LMs) intensively studied by research groups in USA (NASA and University of Florida, etc.) [25, 26], and Russia (Institute of Laser Physics, etc.) [27]. SPLs based on gas-phase dimer molecules have been actively studied by an Israeli research group (Weizmann Institute of Science) [28]. Various solid-state mediums, such as phosphate glasses [29], tellurite glass, silicate glass, fluoride glasses [22, 30-35], glass ceramics [36], borate crystals [37, 38], GdScGa-garnet crystals [39, 40], and YAlO3 [41], have also been studied for application as LMs. In one of the above-mentioned initial works on SPLs in the 1960s [23], Nd-doped yttrium aluminum garnet (Nd:YAG) single crystals were employed as LMs. Since then, Nd:YAG crystals have been the primary option for LMs [42-44]. Then, transparent Nd:YAG ceramic LMs were successfully fabricated and used widely in LMs [45, 46]. Further, transparent Cr-codoped Nd:YAG (Cr, Nd: YAG) ceramic LMs have been intensively studied by Institute for Laser Technology, Osaka and Institute of Laser Engineering, Osaka University, Japan [47]. Significant contributions have been made by Yogev et al. of Weizmann Institute of Science, Israel [48-55]. Active research on SPLs has also been ongoing in Academy of Sciences of Uzbekistan [56] and in NOVA University of Lisbon, Portugal [57-60]. 10.1.3. Stance and Orientation of This Work Typical sizes or diameters of solar concentrators in conventional SPL studies are in the range of 1-2 m [61, 62], such as parabolic mirrors [22, 23] and Fresnel lenses [61-64]. Some studies have employed larger solar concentrators, such as solar furnaces (mirrors) of 10 m aperture [42] and 54×48 m rectangle [65]. This trend is quite reasonable because it is necessary to harvest substantial amount of solar energy for practical solar energy utilization owing to the low areal density of sunlight (in the order of sub-kW/m2). However, for the next generation of SPLs that are more likely to be widely used for solar energy utilization (e.g., present solar cells), cost-effective and much smaller SPLs might be advantageous because they are suitable for mass production. Therefore, we explored the merits of compact SPLs and have developed prototype compact SPLs employing ⌀50.8 mm and ⌀76.2 mm off-axis parabolic mirrors (OAPs) for solar concentrators. An interim progress report of this research activity is provided in this chapter. 10.2. Background of the Concept Design 10.2.1. Preferable Oscillation Wavelength of SPLs Fig. 10.1(a) schematically shows the conversion of incoherent sunlight with a wide spread of spectrum into a monochromatic coherent laser beam by an SPL. Stimulated emission of photons from an SPL at an oscillation wavelength of λL is caused by the energy of the incident sunlight of wavelength λi shorter than λL unless a means of upconversion to utilize sunlight of wavelength longer than λL is employed [66-75]. If all the photon energies of the incident sunlight of λi < λL contribute to the laser-beam output of the SPL ideally, the conversion efficiency from solar energy to laser energy η, that is, the ratio of the output photon energy to the total incident photon energy is given by the following expression: 418 Chapter 10. Compact Solar-pumped Lasers 2   1 i  I  i  d i L 2  I  i  d i , (10.1) 1 where I(λi) stands for the areal power density of the incident spectral sunlight [J/(sec‧μm‧m2)]. The wavelengths λ1 and λ2 defining integral range are ideally 0 and ∞, respectively; however, in practice, they are 300 nm and 4000 nm if the main part of the intensity distribution of the solar spectrum is taken into consideration. The factor λi / λL in the integrand in the numerator in Eq. (10.1) comes from the ratio of the photon energies, (h‧c/λL)/ (h‧c/λi), representing the quantum defect caused by down-conversion. Variation of η as a function of λL is plotted in Fig. 10.1(b) together with the solar spectrum. Fig. 10.1(b) shows that η takes the optimal value in the range 1100 nm < λL < 1300 nm. Fig. 10.1(b) also shows that η of this single-wavelength emission type SPL cannot exceed 0.5 even in the ideal condition [76]. If an SPL has two emission wavelengths λL1 and λL2 (λL1 < λL2) employing two LMs, as shown in Fig. 10.2(a), η can be increased over 0.6 in the wavelength regions around 864 nm for λL1 and 1775 nm for λL2, as shown in Fig. 10.2(b). Fig. 10.1. (a) Function of a solar-pumped laser, (b) Variation of η as a function of the SPL emission wavelength λL displayed with the solar spectrum. Fig. 10.2. (a) Function of a two-wavelength emission type SPL, and (b) Variation of η as a function of SPL emission wavelengths λL1 and λL2. 419 Advances in Optics: Reviews. Book Series, Vol. 5 Solar-pumped lasers utilizing emissions from Nd ions such as a Nd-doped Y3Al5O12 (Nd:YAG) crystal, described in Section 10.1, are preferable practical systems in the case of single-wavelength emission type SPL because their emission wavelength of around 1.06 μm is near the optimal wavelength region between 1100 nm and 1300 nm shown in Fig. 10.1(b). In this case, η at a wavelength of 1.06 μm is obtained as 0.48 via Eq. (10.1). 10.2.2. Incident Power Density Required to Oscillate SPLs Here, we consider a case of an LM working on a four-level scheme, like the Nd:YAG transparent ceramic LMs shown in Fig. 10.3. When a four-level laser provides a constant output power under stable oscillation, the stable photon density in the LM Np [1/m3] is given by the following expression: N P   C  RP  1   L   c  , (10.2) where Rp [1/(sec•m3)] stands for the pumping efficiency, that is, the number of atoms excited per second per cubic meter; σ(λL) stands for the stimulated emission cross-section at the oscillation wavelength λL; c stands for the velocity of light; τ stands for the lifetime of spontaneous emission, which is equal to the reciprocal of the Einstein A coefficient (the rate of spontaneous emission, A32); and τc [sec] stands for the cavity photon lifetime, which is expressed as C  2  L , c    ln  R1  R2  (10.3) where L stands for the cavity length which is almost equal to the length of the LM, δ stands for the round-trip resonator loss, and R1 and R2 are the reflectivity of the cavity mirrors. Eq. (10.2) is schematically visualized in Fig. 10.4. Fig. 10.3. Four level laser scheme. 420 Chapter 10. Compact Solar-pumped Lasers Fig. 10.4. Stable photon density in the LM as a function of pumping efficiency. As elucidated in Fig. 10.4, the value of Rp when Np decreases to zero, Rpth is the minimum pumping efficiency, that is, the threshold pumping efficiency to cause laser oscillation. Using Eqs. (10.2) and (10.3), Rpth is expressed as RPth     ln  R1  R2  2  L    L (10.4) Here, we consider a rod type LM of diameter 2r and length L pumped by the light Ie [W/m2] of frequency νe from the left mainly on the left end facet and partly on the side surface, as schematically illustrated in Fig. 10.5. The total number of photons that contribute to pumping the rod is given by Ie   r 2  2  r   L  , h  e (10.5) where h stands for the Planck constant = 6.626×10-34 J‧sec. Here, the factor α (0 ≦ α ≦ 1) is used to express effective length αL of the side surface that receives photons. Dividing Eq. (10.5) by the volume of the rod, π‧r2‧L, the number of excited ions per second per volume Re is expressed by  I   1 2   Ie  1 Re   e        1  2  A  ,      h  e   L r   h  e  L (10.6) where A ( = L / r) is the aspect ratio of the rod, and η is the total pumping efficiency; it is expressed as follows:   QD QE abs m , (10.7) where ηQD stands for quantum defect representing the ratio of photon energy of the emitted laser to that of the pumping light, ηQE stands for quantum efficiency representing the ratio 421 Advances in Optics: Reviews. Book Series, Vol. 5 of number of photons contributing the excitation of ions to the number of absorbed photons, ηabs represents the ratio of the number of absorbed photons to the number of incident photons to the rod reflected by the degree of matching of the absorption spectrum of the rod material with the sunlight spectrum, ηm stands for mode-matching efficiency representing the ratio of number of exited ions distributing in the laser oscillation mode and contributing to the laser oscillation to the total excited ions in the rod. Laser oscillation takes place when Re in Eq. (10.6) exceeds RPth in Eq. (10.4). Combining Eq. (10.4) with Eq. (10.6), the threshold of the incident power density to start laser oscillation Ieth [W/m2] is expressed as follows:  h  e I eth      L       ln  R1  R2    1 1          A 2 1 2        QD QE  abs  m        (10.8) Fig. 10.5. A rod type LM pumped by light mainly on the left end facet and partly on the side surface. Eq. (10.8) shows that to decrease Ieth for ease of laser oscillation, lower νe, in other words, larger λe under the constraint of λe < λL (the lower quantum defect); larger σ(λL) ‧τ; lower cavity loss; higher reflectivity of the cavity mirrors; larger effective aspect ratio αA; and larger ηQD, ηQE, ηabs, and ηm are preferable. It should be noted that L is not related to Ieth in Eq. (10.8) explicitly because both Eqs. (10.4) and (10.6) contain L in their denominators. In a typical case of Nd:YAG laser with λL of 1064 nm, λe of 808 nm, σ(λL) of 2.8×10-19 cm2, τ of 230 μsec, R1 of 0.99, R2 of 0.95, ηQD of 0.61 ( = 650 nm / 1064 nm), ηQE of 1, ηabs of 0.3, ηabs of 0.29, A of 20 ( = 10 mm / 0.5 mm), and α of 0.03, Ieth was calculated to be 5427 kW/m2. Therefore, a solar concentration ratio of more than 5500 is required for 1 Sun solar insolation of 1 kW/m2 to start oscillation in this laser rod. 10.2.3. Preferable Shape of LMs for SPLs The volume V and the total surface area S of the typical LM shown in Fig. 10.5 are given by 422 V    r 2  L, (10.9) S  2  r  L  2  r 2 , (10.10) Chapter 10. Compact Solar-pumped Lasers respectively. Using Eq. (10.9), A = L / r, and normalizing S by the values of S when A is 2, SA = 2, under the fixed V, Eq. (10.10) can be modified as 2 S S A 2 23 1  1    A3  1   3  A (10.11) Fig. 10.6 shows the variation of S/SA = 2 as a function of A. Fig. 10.6 shows that the total surface area of the LM takes high values under fixed V in the left-hand side and the righthand side of the graph. Therefore, the total number of pumping photons expressed in Eq. (10.5) also takes a large value in the small A range and in the large A range in the fixed volume. This means that thin disk-type LMs, and thin fiber-type LMs are preferable for SPL because of the low areal energy density of sunlight. The high A values in the thin disk-type LMs and thin fiber-type LMs are also advantageous for thermal dissipation, as intuitively shown by the pink arrows in Fig. 10.6. In the case of a Nd:YAG crystal, for example, the theoretical upper limit of solar energy conversion efficiency into monochromatic laser at a wavelength of 1.06 μm is calculated to be 0.48, as described in Subsection 10.2.1. However, actual feasible conversion efficiency will be much lower [76]. Since most of the concentrated solar energy is converted to heat mainly in the LM, without rapid thermal dissipation, σ(λL) ‧ τ in Eq. (10.8) decreases and an inhomogeneous temperature increase in the LM can take place. This causes degradation of the laser-beam quality because of temperature dependent refractive index, including thermal lens effect, loss of the energy conversion efficiency, and even the breakage of LMs. Therefore, the advantage of thermal dissipation in thin disk-type, and thin fiber-type LMs is of key importance. Here, it is suggested that a glass LM is promising for SPLs because it can be easily molded into thin disks and thin fibers. High solubility of rare earth elements such as Nd is another attractive property of glass LMs. Further, the suitability of glass LMs for mass production ensures low production costs, which is a key factor for wide adoption of SPLs in the society. Fig. 10.6. Variation of S/SA = 2 as a function of A. 423 Advances in Optics: Reviews. Book Series, Vol. 5 10.2.4. Theoretical Limit of Solar Concentration Suppose the Sun is a perfect blackbody of radius RS (m) and surface temperature TS (K), the total radiant flux power from the Sun, PO (W), is given by P0  4  RS2    TS4 , (10.12) where σ stands for the Stefan-Boltzmann constant = 5.67×10-8 W/(m2K4), and σ‧Ts4 represents the radiant flux power density at the surface of the Sun based on the StefanBoltzmann law. At the average radius of Earth’s orbit around the Sun, that is, one astronomical unit Au (m), the radiant flux power density IO (W/m2) is called the solar constant and is given by IO  PO RS2     TS4 2 2 4    Au Au (10.13) IO is calculated to be 1367 W/m2. The solar concentration ratio C is defined as the ratio of the radiant flux power density at the most concentrated location near the focal point of the solar concentrator to the radiant flux power density of sunlight at the entrance of the concentrator. Thus, the radiant flux power density at the most concentrated location near the focal point of the solar concentrator I’ (W/m2) is given by I'C RS2    TS4    T '4 2 Au (10.14) Suppose the most concentrated location near the focal point of the solar concentrator is on a perfect blackbody surface, the surface temperature T K of this blackbody is required to be lower than TS based on the principles of thermodynamics. Then the following inequality expression can be deduced. Au 2 1 , C 2  RS (sin  S )2 (10.15) where θS is the apparent Sun semidiameter observed from the average Earth’s orbit around the Sun, as illustrated in Fig. 10.7. Fig. 10.7. Schematic geometry of the Sun, a solar concentrator on the average Earth’s orbit around the Sun and a blackbody absorber near its focal point. 424 Chapter 10. Compact Solar-pumped Lasers This is the theoretical limit of the solar concentration ratio CO deduced from the laws of thermodynamics. The same results can be obtained from the Lagrange-Helmholtz invariant or Luminance conservation law, naturally. By substituting the values RS = 6.96 ×108 m and Au = 1.496×1011 m into Eq. (10.15), we obtain C  CO  46200 (10.16) 10.2.5. Size of Images of the Sun and Concentration Ratios by Solar Concentrators 10.2.5.1. Convex Lens Fig. 10.8 illustrates the geometry of image formation by a convex lens comprised of an object, a lens, and an image of the object located graphically by means of geometrical optics [77]. The white arrow on the left side of the lens represents an object of length RS. The distance between the foot of this white arrow and the center of the lens O is supposed to be Au. Correspondingly, the smaller upside-down white arrow of length rS appears on the right side of the lens as an image of the white arrow on the left side of the lens. The distance between the foot of the image and O is expressed as f + x, where f stands for the focal length of the lens. A black arrow of length D, which is equal to the radius of the lens is marked on the white arrow on the left of the lens. Correspondingly, a black arrow of length d also appears upside-down on the image of the white arrow on the right side of the lens. From the two similar triangles with a common vertex F, the following Eq. (10.17) can be obtained: d D  x f (10.17) Fig. 10.8. Illustration of image formation by a convex lens tracing several characteristic rays. From the other two similar triangles with common vertex O, the following Eq. (10.18) can be obtained: d D  f  x Au (10.18) From Eqs. (10.17) and (10.18), rS is expressed as follows: 425 Advances in Optics: Reviews. Book Series, Vol. 5 rs  d  R S RS R f    Sf D Au 1  f Au Au (10.19) In Eq. (10.19), the denominator was partly approximated using the relation f /Au≪1. In a specific case of f = 50.8 mm, the radius of the image of the Sun is obtained to be 0.236 mm. The concentration ratio can be expressed as follows: A   D2 D2   u  2 2  rS    RS  R f   S   Au 1  f   Au   2 D    f  2 (10.20) As indicated in Eqs. (10.15) and (10.16), (Au/RS)2 is the theoretical upper limit of solar concentration ratio. In a typical case of f = 50.8 mm and 2D = 50.8 mm, the concentration ratio is 25 % of the theoretical upper limit. In practical aplanatic lenses that are nearly free of coma and spherical aberration, D/f does not exceed 0.7, usually. Therefore, solar concentration ratios are usually less than 50 % of the theoretical upper limit. In practice, a lens transmittance loss of between 0.25 and 0.9 ensures that solar concentration ratios are between 12.5 % and 45 % of the theoretical upper limit. 10.2.5.2. Parabolic Concave Mirror Fig. 10.9 illustrates the geometry of image formation by a parabolic concave mirror in the same manner as Fig. 10.8. Through similar geometrical consideration, the same relation as Eq. (10.18) is obtained. The equation corresponding to Eq. (10.17) can be obtained as Eq. (10.21). d  x D D2 f 4f Fig. 10.9. Illustration of image formation by a parabolic concave mirror tracing several characteristic rays. 426 (10.21) Chapter 10. Compact Solar-pumped Lasers Here, the factor D2/4f in the right-hand denominator corresponds to the displacement of the mirror surface to the left at its edge. From Eqs. (10.18) and (10.21), rS is expressed as follows: rS  RS R d  S  D Au R f  Sf 2 D  Au 1  1  f   Au  4f  (10.22) In Eq. (10.22), the denominator was partly approximated using the relation f /Au≪1. In a typical case of f = 50.8 mm, the radius of the image of the Sun is obtained to be 0.236 mm. The concentration ratio can be eventually expressed in the same manner as Eq. (10.20): 2 A   D 2  Au  D2    u    2 2  rs  Rs     Rs    f    2  1  1   f  D     Au  4 f   2 D    f  2 (10.23) Thus, for a typical case of f = 50.8 mm, and 2D = 50.8 mm, the concentration ratio is 25 % of the theoretical upper limit. 10.2.6. Merits of Miniaturization of SPLs Fig. 10.10 shows photos of Odeillo solar furnace at the Process, Materials and Solar Energy (PROMES)-CNRS laboratory. It is one of the largest solar furnaces in the world, situated in the Pyrenees mountain range in the south of France, and has been used as a solar concentrator for an SPL [65]. Its height, width, and focal length are 54 m, 48 m, and 18 m, respectively. With the aid of 63 flat plate heliostats, it attains a temperature of 3500 ºC and concentrates 1 MW at the focus. It should be noted that the expressions for the solar concentration ratio of a convex lens (Eq. (10.20)) and of a parabolic mirror (Eq. (10.23)) do not contain the absolute sizes of the solar concentrator, such as the focal length f and the diameter 2D, but contain the ratio D/f. Therefore, the same radiant flux power density as the Odeillo solar furnace can be attained by a much more compact solar concentrator, for example, of height, width, and focal length = 5.4 cm, 4.8 cm, and 1.8 cm, respectively, although the total amount of the concentrated solar power would be much less than 1 MW. As a typical example of a conventional SPL, a 1.4×1.05 m Fresnel lens with a focal length of 1.2 m and a Cr-codoped Nd:YAG ceramic LM was used for an SPL system and 18.7 W laser output was obtained [64]. The corresponding NA was 0.4-0.5. For simplicity, let us consider an SPL system composed of a ⌀1.37 m lens with a focal length of 1.2 m (NA = 0.5) and a ⌀10×100 mm cylindrical Cr-codoped Nd:YAG LM, as shown in Fig. 10.11(a). The sunlight-receiving area (1.37/2)2 × π is equal to 1.4×1.05 m2. Most of 427 Advances in Optics: Reviews. Book Series, Vol. 5 the focused sunlight is introduced into the rod because the size of the image of the Sun is calculated to be approximately ⌀8.2 mm according to Eq. (10.19). It is duly conjectured that this SPL can also oscillate and yield a similar amount of laser output as in reference [64]. Then, let us consider a miniaturized SPL system composed of ⌀76.2 mm with a focal length of 66 mm (NA = 0.5) and a cylindrical LM of ⌀1 × 10 mm as shown in Fig. 10.11(b). Most of the focused sunlight is introduced into the rod because the size of the image of the Sun is calculated to be approximately ⌀0.62 mm according to Eq. (10.19). This miniaturized SPL will be able to oscillate in natural sunlight because the focused radiant flux power densities are the same between the ⌀1.37 m lens and the miniaturized ⌀76.2 mm system because Eqs. (10.20) and (10.23) do not relate to the absolute size D or f but only D/f. However, the absolute value of the laser output power is approximately 1/323 in proportion to the ratio of the sunlight-receiving areas. Therefore, an array of 324 miniaturized SPLs will make more laser output power than the system of ⌀1.37 m lens. Here, the total volume of the ceramics LMs in this array is 810π mm3 (2.5π mm3 × 324), which is much less than the 2500π mm3 (25π × 100) in the ⌀1.37 m lens system. In addition to the advantage of having less volume, the ⌀1 × 10 mm Cr-codoped Nd:YAG LMs are easier to fabricate and more suitable for mass production compared to the ⌀10 ×100 mm LM. The ⌀1 × 10 mm micro laser rod is also advantageous in comparison with the conventional size laser rod, such as the one of ⌀10 ×100 mm, from the view point of rapid thermal dissipation owing to reasons explained earlier. As for the solar concentrator, compact solar concentrators together with the corresponding solar-tracking systems are cost-effective because they have already been mass produced and used in various commercial applications, such as the solar lighting system “Himawari” with a solar-tracking system by La Foret Engineering Company (Japan) [78]. This is in contrast to the large systems, such as ⌀1.37 m lens, which are still in the made-to-order stage. Fig. 10.10. Odeillo solar furnace of the Process, Materials and Solar Energy (PROMES)-CNRS laboratory in France (photograph taken on October 9th, 2018). 428 Chapter 10. Compact Solar-pumped Lasers Fig. 10.11. Concept of miniaturization of SPLs keeping laser output power unchanged. 10.2.7. Direct Solar Radiation and Diffuse Solar Radiation Fig. 10.12 shows the daytime variations in direct solar radiation and diffuse solar radiation on the horizontal ground plane simulated using the “SPECTRAL2” solar spectrum model by Bird and Riordan [79]. Detailed simulated conditions are as follows: longitude = 137°3’E and latitude = 35°10’N, August 1st, AM1.5, total column ozone = 0.34 cm, total precipitable water vapor = 1.34 cm, and aerosol optical depth at the wavelength of 500 nm = 0.27. It can be observed in the figure that the shape of the spectrum of the diffuse solar radiation does not vary but its intensity varies with time. In contrast to this, the spectra of the direct solar radiation shows a significant red shift with time in the afternoon. The effects of this spectrum shift must be seriously considered in the case of SPLs because they mainly work with direct solar radiation. If an aplanatic lens is used for solar concentration, the spectrum shift in the afternoon may affect the SPL oscillation in connection with the chromatic aberration of the lens. The spectrum shift may also affect the SPL oscillation in connection with the absorption and excitation spectra of the LM. The total radiant flux power density between wavelengths of 300 nm and 4 μm can be calculated as follows:  Global solar radiation: 1008 W/m2 (100 %);  Direct solar radiation: 787 W/m2 (78 %);  Diffuse solar radiation: 221 W/m2 (22 %). In a typical case in which a Cr,Nd:YAG transparent ceramic LM is employed, the total radiant flux density is considerably reduced because the wavelength range of the absorption spectrum is reduced to being between 300 nm and 900 nm, as shown below:  Global solar radiation: 696 W/m2 (100 %);  Direct solar radiation: 508 W/m2 (73 %);  Diffuse solar radiation: 188 W/m2 (27 %). 429 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 10.12. Daytime variations of solar spectrum for direct solar and diffuse solar radiations on the horizontal ground plane simulated using the “SPECTRAL2” solar spectrum model by Bird and Riordan [79] for longitude = 137°3’E and latitude = 35°10’N, August 1st, AM1.5, equivalent ozone depth = 0.34 cm, total precipitable water vapor = 1.34 cm, and aerosol optical depth = 0.27@500 nm. On an inclined plane or on a plane with its surface normal tracking the Sun, the total radiant flux density can be improved. Fig. 10.13 shows daytime variations of global solar radiation power density (1) on a horizontal plane, (2) on a plane inclined south at 37° to the horizontal plane, (3) on a plane with its surface normal tracking the Sun, and (4) direct solar radiation on a plane with its surface normal tracking the Sun, calculated using the “SPECTRAL2” solar spectrum model by Bird and Riordan [79] for longitude = 137°3’E and latitude = 35°10’N, under typical clear skies, on a Spring/Autumn day, with diffuse fraction = 25 %, and transmittance of the atmospheric layer = 75 %. Fig. 10.13 shows that sun-tracking can gain larger solar radiation both in the morning and in the afternoon. The following integrated solar radiation power densities throughout the day also show the advantages of sun-tracking:  Global, on the horizontal plane: 6462 Wh/m2;  Global, on the plane inclined south at 37° to the horizontal plane: 6530 Wh/m2;  Global, tracking the Sun: 8572 Wh/m2;  Direct, tracking the Sun: 6940 Wh/m2. In Subsection 10.2.2, a typical threshold power for oscillation of Nd:YAG laser was calculated to be approximately 5500 kW/m2. As described in Subsection 10.2.4, in actual practice, solar concentration ratios are between 12.5 % and 45 % of the theoretical upper limit of 46200. Therefore, the radiant flux power density is 5775-20790 kW/m2 when the direct solar radiation was 1 kW/m2. Therefore, the direct solar radiation utilized by Cr, Nd:YAG at noon (508 W/m2) can make the practical concentrated radiant flux power density = 2994-10561 kW/m2. This shows that it is important to use lenses or mirrors of larger D/f and higher transmittance or reflectivity for stable oscillation in the daytime – almost from sunrise to sunset – tracking the Sun. 430 Chapter 10. Compact Solar-pumped Lasers Fig. 10.13. Daytime variations of global solar radiation power density (1) on a horizontal plane (a black line), (2) on a plane inclined south at 37° to the horizontal plane (a green line), (3) on a plane with its surface normal tracking the Sun (a pink line) and (4) direct solar radiation on a plane with its surface normal tracking the Sun (a red line). Simulations were performed using the solar spectrum model “SPECTRAL2” by Bird and Riordan [79] for longitude = 137°3’E and latitude = 35°10’N, under typical clear skies, on a Spring/Autumn day, with the diffuse fraction = 25 %, and the transmittance of the atmospheric layer = 75 %. We also measured the direct solar radiation on Wednesday, August 9th, 2017 on the ground at 136°58’E longitude and 35°10’N latitude (Nagoya, Japan) using a pyrheliometer for the direct solar radiation of ⌀18 mm caliber mounted on an equatorial mounting that tracked the Sun. The power of the direct solar radiation in the view angle of 5° and in the wavelength range of 190 to 25000 nm captured with an aperture and a pinhole was measured with a THOLABS S302C thermal power meter sensor attached to the pyrheliometer, as shown in Fig. 10.14. The THOLABS S302C power meter sensor had been calibrated with an EKO INSTRUMENTS MS-53 (ISO 9060 First Class Pyrheliometer), which provided the value of the direct solar radiation in terms of W/m2. The measured value was 905 W/m2 for similar conditions in which 787 W/m2 was obtained via SPECTRAL2. Fig. 10.14. A THOLABS S302C thermal power meter sensor with a pinhole calibrated and attached to a pyrheliometer. 431 Advances in Optics: Reviews. Book Series, Vol. 5 We also monitored the daytime variations of the solar spectrum change using an Ocean Optics MAYA2000 solid-state optical spectrum analyzer. Fig. 10.15 shows variation in the measured solar spectrum as a function of the clock time. The shapes of the spectra qualitatively show a gradual red shift of the peak of the solar spectrum as the afternoon grew late, similar to the results obtained from the “SPECTRAL2” solar spectrum model shown in Fig. 10.12. Therefore, chromatic aberration must also be reduced in an aplanatic lens for solar concentration because of the wide-band range of the solar spectrum and its red shift with elapse of the time of the day. However, achromatic aplanatic lenses are not cost-effective. Therefore, parabolic mirrors are preferable as solar concentrators for SPLs. However, large-scale parabolic mirrors are also not cost-effective. Therefore, the preferable choice for the solar concentrator will be a small parabolic mirror. Fig. 10.15. Variation in the measured solar spectrum as a function of the clock time. 10.3. Compact Solar-Pumped Fiber Laser (SPFL) System 10.3.1. Choice of Laser Mediums (LMs) 10.3.1.1. Shape of LMs As described in Subsection 10.2.3, either disk-like and fiber-like shapes are advantageous for LMs from the point of efficient excitation and efficient thermal dissipation. As described in Subsection 10.2.1, solar-pumped lasers utilizing emissions from Nd ions are preferable practical systems in the case of a single wavelength emission type because their emission wavelength λL (approximately1.06 μm) is near the optimal wavelength region situated between 1100 and 1300 nm, as shown in Fig. 10.1(b). So far as the LMs utilizing emissions from Nd ions is concerned, the long optical path of a fiber-type LM enables absorption of almost all the pumping light propagating along its axial direction. In contrast, the short interaction length of a disk-type LM is insufficient for the same amount 432 Chapter 10. Compact Solar-pumped Lasers of pumping sunlight to be absorbed. Therefore, a fiber-type LM was chosen to develop a solar-pumped fiber laser (SPFL). 10.3.1.2. Glass LMs To realize a fiber-type LM as well as a disk-type LM, it was suggested in Subsection 10.2.3 that glass LMs are promising for SPLs because of their high moldability into thin disks/thin fibers, high solubility of rare-earth elements such as Nd, and suitability for mass production. Thus, TeO2-K2O (tellurite) glass, SiO2-B2O3-Na2O-Al2O3-CaO-ZrO2 (SBNACZ, borosilicate) glass, and ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAN, fluoride) glass were examined as host glasses [80]. The content of Nd was changed in the range of 0-5.0 mol %. TeO2-K2O (tellurite) glass has the advantage of having the largest integrated absorption cross-section of Nd3+ from 400 to 950 nm among the approximately 250 glass hosts. For example, it is about 1.5 times higher than those in borosilicate glass and fluoride glass [81, 82]. The effective phonon energy in tellurite glass is usually less than 800 cm-1, which is lower than that of the other oxides such as silicate (1000 cm-1) and phosphate (1200 cm-1) [83]. This enables efficient fluorescence from Nd3+ ions doped in the tellurite glass. SBNACZ borosilicate glass has the advantage of having excellent rare-earth solubility and chemical and thermal stability against crystallization because of its high glass transformation temperature. Crystallization behavior was not noticed for rare-earth doping less than 10 wt. %. Therefore, it was initially suggested as a host medium to immobilize high level nuclear waste [84]. SBNACZ borosilicate glass can hold up to 30 wt. % Nd2O3 without evidence of Nd clustering and has been proposed as an LM [81, 82]. ZBLAN fluoride glass, found in 1975 [85], has advantages in terms of its wide transparent window between 0.22 and about 8 μm, low minimum loss as low as 1 dB/km [86], low effective phonon energy of less than 600 cm-1 [87], and high allowable doping levels (up to 10 mol %) of rare-earth ions [34]. The internal quantum efficiencies under sunlight tracking the Sun using an altazimuth and Ti:sapphire laser excitation (at 807 nm for tellurite and borosilicate glasses and at 794 nm for fluoride glass) were measured directly using an integrating sphere method (cf. Fig. 10.22 (a)). The radiative quantum efficiency obtained by Judd-Ofelt analysis was almost 100 % for the tellurite and ZBLAN fluoride glasses and 50 % for the SBNACS borosilicate glasses. The quantum efficiency under laser excitation was respectively found to be 86 %, 34 %, and 88 % for the tellurite, SBNACS borosilicate, and ZBLAN fluoride glasses at a low Nd3+ content. Parasitic absorption in the SBNACZ borosilicate glass might have caused the low quantum efficiency. The quantum efficiency under sunlight excitation was up to 33 %, 21 %, and 70 % for the tellurite, SBNACS borosilicate, and ZBLAN fluoride glasses, respectively. Therefore, ZBLAN fluoride glass was chosen for the glass LM. The stimulated emission cross-section σ(λL) in the transition 4F3/2 to 4I11/2 corresponding to the 1064 nm emission for 0.5 mol. % Nd3+ doped in ZBLAN glass was calculated to be 2.95 × 10−20 cm2. With a fluorescence lifetime = 496 μsec for glass, σ(λL) ‧ τ was obtained to be 1.46 × 10−27 m2s, which is higher than 0.75 × 10−27 m2sec for Nd-doped silica glass codoped with Al [88]. 433 Advances in Optics: Reviews. Book Series, Vol. 5 10.3.2. Solar Concentrator An off-axis parabolic mirror (OAP) was chosen as the solar concentrator because it is free from chromatic aberration and because products of relatively short focal length, in comparison with the caliber diameter, can be commercially available. Specifically, a ⌀50.8 mm caliber aluminum OAP (Edmund Optics) with an aluminum coating layer (reflectance = 92 %) was employed for concentration of sunlight. Fig. 10.16 shows schematic drawings of the OAP, including typical traces of light rays emitted from certain points of an object and reflected at OAP. Fig. 10.16. Schematic drawings of an OAP of caliber diameter 2D = 50.8 mm with typical traces of light rays emitted from the head of the white arrow-shaped object of length RS and the head of a black arrow drawn on the white arrow-shaped object of length D. (a) A side view of a case in which the arrow-shaped object is pointing right in a side view of the system, (b) A case in which the arrow-shaped object is pointing left in a side view of the system, (c) A case in which the arrow-shaped object is pointing perpendicular to the plane of paper. A front view is also provided to facilitate understanding of the geometry. 434 Chapter 10. Compact Solar-pumped Lasers The offset angle of this OAP was 90º. The focal point F is located on the horizontal line, which penetrates the center point O of the OAP. The distance between F and O was also designed to be 50.8 mm. Among the commercially available products of ⌀50.8 mm caliber OAP, this was of the shortest FO length. Based on the geometrical consideration in Fig. 10.16 (a), the length of the inverted image of the black arrow d can be given as D 2D d     Au  1  12  D 5 Au (10.24) Supposing the ratio of the length of the image of the white arrow-shaped object rS to d is the same as the ratio RS to D, rS can be given as rS  d  R  RS  RS  2D      S   2D D  Au  1  12  D  Au  5 Au (10.25) Here, D/Au ≪1 is used for the approximation. The lateral location of the image is defined by the distance from F, x, which can be expressed as x 12  D  5  Au    2D  (10.26) Because D/Au ≪1, the image is located at x ≈ 0, that is, on the focal point F. Similarly, based on geometrical consideration in Fig. 10.16 (b), the length of the inverted image of the black arrow d can be expressed similar to Eq. (10.24). As for the lateral location of the image, x is obtained as follows: 4 D x   3  Au    2D  (10.27) Because D/Au ≪1, both the images in Figs. 10.16 (a) and (b) are located at x ≈ 0, that is, on the focal point F, as indicated in Eqs. (10.26) and (10.27). Figs. 10.16 (c) and (d) show the side view and front view for cases in which the arrow-shaped objects are pointing perpendicular to the plane of paper, respectively. Traces of two typical light rays emitted from the head of the black arrow drawn on the white arrow-shaped object are indicated with blue and red lines. It is shown that the two lines, blue and red, do not intersect with each other at a point three-dimensionally. More specifically, the red ray penetrates the focal point F but the blue ray does not. This is because the present OAP is not precisely an imaging system. However, the blue ray passes near the focal point. At the point of the nearest approach, the distance from F is (D/Au)‧2D. As for the light ray emitted from the head of the white arrow-shaped object corresponding to the blue line, the distance between the point of the nearest approach and F can be calculated as (RS/Au)‧2D. This is the same as Eq. (10.25). Therefore, all incident sunlight at the entrance of the OAP crosses inside 435 Advances in Optics: Reviews. Book Series, Vol. 5 the circle of radius rS at the focal point F. Therefore, 2 rS is not a diameter of the image of the Sun. It should be called the diameter of the pseudo image of the Sun. The solar concentration ratio can be expressed as follows:  D2   rs2  R S  A  u D2 2  2   4D  2 A  1  u    Rs  4 (10.28) Using 50.8 mm for 2D, 6.96 ×108 m for RS, and 1.496×1011 m for Au into Eqs. (10.25) and (10.28), the diameter of the pseudo image of the Sun and corresponding solar concentration ratio can be obtained as follows: 2rs  0.473 mm, (10.29)  D2  11550  rs2 (10.30) 10.3.3. SPL Resonator Based on a Double Cladding Optical Fiber Using the OAP in Fig. 10.16, the diameter of the image of the Sun near the focal point is 0.473 mm as shown in Eq. (10.29). A double-clad fiber (DCF) structure composed of a core (⌀5 μm), an inner cladding (⌀125 μm), and an outer cladding (⌀200 μm) was employed. Both end facets were coated with thin film mirrors having reflectivity of 98 % at 1050 mm [30-32]. A substantial part of the focused sunlight that hits out of the core can enter the inner cladding and propagates in it, after which it eventually enters the core to be absorbed by ions in the core as schematically shown in Fig. 10.17(a). This 10 m long Nd 0.5 at% doped ZBLAN DCF was spooled on a kidney-shaped spool to increase the chance of sunlight propagating in the inner cladding to enter the core. As a consequence, the mode-matching efficiency is as high as 80 %. The absorption spectrum, in comparison with the broadband solar spectrum, gives the spectrum matching efficiency of absorption of 13 %. Fig. 10.17 (b) shows a schematic diagram of this SPFL system. Fig. 10.18 shows photographs of this SPFL system mounted on an altazimuth-type solar-tracking system (Vixen SKYPOD). In this system, the factor in the first parenthesis in Eq. (10.8), h‧νe / (σ(λL) ‧τ), is 1.69×105 kW/m2. Supposing 0.8 for δ, and using 0.98 for R1 and R2, the factor in the second parenthesis in Eq. (10.31), (δ – ln (R1‧R2)/2, is 0.42. Supposing 0.62 (= 650 nm / 1050 nm) for ηQD, 1 for ηQE, 0.13 for ηabs, 0.8 for ηm, 4×106 ( = 10 m /2.5 μm) for A, and 2×10-4 ( = 2 mm / 10 m) for α (assuming 2 mm of cladding from the end facet functions effectively), the threshold power density is calculated to be 686 kW/m2. Supposing the incident direct solar radiation is 508 W/m2 as described in Subsection 10.2.7, and geometrical and reflectivity loss of the solar concentrator is 30 %, the required concentration ratio is 1930 (686 kW/m2 / (0.508 kW/m2×0.7)). Therefore, it was conjectured that the system shown in Figs. 10.17 and 10.18 has a good chance to realize oscillation. 436 Chapter 10. Compact Solar-pumped Lasers Fig. 10.17. Schematic diagrams of (a) double clad optical fiber (DCF), and (b) a prototype SPFL system composed of OAP and Nd 0.5 at% doped ZBLAN DCF laser medium. Fig. 10.18. Photographs of the prototype SPFL system comprising an OAP, a kidney-shaped spool for a 10 m long Nd 0.5 at% doped ZBLAN DCF, and an optical fiber for the SPFL output. The system is mounted on an altazimuth type solar-tracking system. 10.3.4. Outdoor Oscillation Performance of the Compact SPFL Lasing experiments using sunlight were performed at longitude 137°3’E and latitude 35°10’N under clear skies with occasional cumuli. Fig. 10.19 (a) shows the direct sunlight spectrum and the measured spectrum after passing through the Nd 0.5 at% -doped ZBLAN fluoride glass fiber together with the lasing spectrum. The direct sunlight spectrum was calculated according to the model by Bird and Riordan [79] under the following conditions: elevation angle = 37°, aerosol optical depth at 500 nm wavelength = 0.27, total 437 Advances in Optics: Reviews. Book Series, Vol. 5 column ozone = 0.34 cm, surface albedo = 0.20, total precipitable water vapor = 1.42 cm, longitude = 137° 3′ E, and latitude = 35° 10′ N. Absorption bands observed were indicated with the identified related excited electronic levels of Nd3+ ion. Sunlight was completely absorbed around the bands at wavelengths of 520 nm, 575 nm, 740 nm, 795 nm, and 867 nm. However, strong intensity transmissions, such as blue light with wavelengths between 410 nm and 510 nm and red light with wavelengths between 590 nm and 725 nm, still remain. The detailed lasing spectra around 1053.7 nm obtained at different times are shown in Fig. 10.19(b). Many peaks were observed between 1052 nm and 1054 nm. Typically, the full width of the half maximum of the peak at 1053.7 nm is approximately 0.01 nm. The spectra showed a complex and congested nature and fluctuated quickly. This was caused partly because of inhomogeneous broadening from Nd ions in glass and longitudinal-mode hopping. Fig. 10.19. (a) The direct sunlight spectrum (in red) and the measured spectrum after passing through the Nd 0.5 at% -doped ZBLAN fluoride glass fiber together with the lasing spectrum (in blue open circles). The locations of absorption bands are indicated with the identified related excited electronic levels of Nd3+ ion. (b) Detailed lasing spectra around 1053.7 nm obtained at different times. Fig. 10.20 shows the plots of the SPFL output power as a function of the input sunlight power captured in the inner cladding of the fiber. The input sunlight power was controlled by changing the flare angle of the bow-tie aperture placed at the sleeve for the parabolic 438 Chapter 10. Compact Solar-pumped Lasers mirror (cf. Fig. 10.26). A clear lasing threshold at 49.1 mW is confirmed by natural sunlight excitation. The slope efficiency and the total efficiency were 3.3 % and 0.88 %, respectively. The maximum laser output power was 0.57 mW. Because the reflectivity of both ends of the fiber are the same, adding the output powers from both ends, the slope efficiency and the total efficiency are doubled to be 6.6 % and 1.76 %, respectively [34]. This is “the first solar-pumped fibre laser” as picked up by Nature Materials [89]. The threshold power was derived to be 49.1 mW. Dividing the cross-sectional area within ⌀125 μm of the inner cladding and the core of the ZBLAN optical fiber, the threshold power density Ieth at the end facet of the fiber to start laser oscillation is approximately 4 MW/m2. Passing through the preposing optical system including a UV filter with a transmittance of 95 % and OAP with a reflectance of 92 %, the total transmittance of the preposing optical system was 71.5 %. Taking the solar concentration ratio (11550) of this OAP into consideration, the power density before the concentration is 484 W/m2. Similarly, the maximum sunlight power density in this measurement was obtained as 651 W/m2. In the example of a numerical estimate based on the Eq. (10.8) in Subsection 10.2.2, Ieth was estimated to be about 686 kW/m2, which is much less than the experimentally obtained value of 4 MW/m2. It is conjectured that a larger value of δ and smaller value of ηQD, ηQE, ηabs, ηm and α in practice may have caused this result. Fig. 10.20. Output laser power of SPFL as a function of input solar power. 10.4. Improvement of Spectral Matching Efficiency by Cr Codoping The spectral absorption range of the LM should cover a wide spectral range of sunlight – from UV to IR. This is referred to as “spectral matching”. It has been considered that an SPL using a single rare-earth element such as Nd cannot be efficiently driven because of the extremely narrow absorption bands of rare-earth elements [23]. A promising means for the improvement of spectrum matching is to introduce a sensitizer of another element that extends the absorption range considerably [90]. Particularly, it has been found that the transition metal Cr3+ is a promising sensitizer, which has broadband absorption characteristics and transfers the absorbed energy to Nd3+ [20, 42, 58, 63, 91-97]. In our ZBLAN fiber, the mode-matching efficiency ηm is as high as 80 % while the spectral 439 Advances in Optics: Reviews. Book Series, Vol. 5 matching efficiency ηabs is 13 %. Therefore, it is important to increase the spectral matching efficiency by Cr codoping in addition to Nd. Although Cr3+ codoping is a possible tactic to improve the performance of SPFL, it was eventually found difficult to codope Cr into Nd-doped ZBLAN fibers. Therefore, instead of taking advantage of the large aspect ratio A of fiber LMs shown in Fig. 10.6, we used Cr-codoped Nd:YAG transparent ceramic micro rods to make use of the advantages described in Subsection 10.2.6 and Fig. 10.11. Fig. 10.21 shows the measured spectral absorption coefficients of transparent ceramic mediums of (1) Nd (1.0 at%):YAG, (2) Cr (0.1 at%), Nd (1.0 at%):YAG, and (3) Cr (0.4 at%), Nd (1.0 at%): YAG. In contrast with the narrow adsorption peaks of (1) Nd (1.0 at%):YAG, (2) Cr (0.1 at%), Nd (1.0 at%):YAG and (3) Cr (0.4 at%), Nd (1.0 at%):YAG showed broad absorption peaks. It was also shown that the increase in Cr content from 0.1 at% to 0.4 at% caused an approximate 4 times increase in the broad absorption peaks. It is expected that spectral matching efficiency improves from 13 % to 35 % by the codoping of Cr ions. Then, introducing sunlight into an integrating sphere, spectral absorption, and emission properties of Nd:YAG and Cr-codoped Nd:YAG were examined. Fig. 10.22 (a) schematically shows a system to measure the spectral absorption and emission from the sample in an integrating sphere in the wavelength range of 300 nm to 1600 nm under natural sunlight irradiation (LUCIR FQEM-L1). Fig. 10.22 (b) shows the observed spectra with and without a sample. Subtracting the spectrum without a sample from the spectrum with a sample, a differential spectrum could be obtained that showed absorption of the solar spectrum and the spontaneous emission from the sample. Fig. 10.22 (c) shows a differential spectrum measured for the Nd (1 at%):YAG ceramic sample, which shows absorption of the solar spectrum and emission (1064 nm) from Nd3+ ions. Fig. 10.22 (d) shows a similar differential spectrum for Cr (0.5 at%)-codoped Nd (1 at%):YAG ceramics sample. In comparison with the spectrum in Fig. 10.22 (c), the absorption of the solar spectrum was enhanced, and the emission (1064 nm) increased 1.5 times in Fig. 10.22 (d). This indicates the photon energies absorbed by Cr3+ ions were transferred to Nd3+ ions and contributed the spontaneous photoemission. Thus, Cr doping to Nd:YAG transparent ceramic rod was experimentally reconfirmed to be a promising tactic to improve SPLs. Fig. 10.21. Measured spectral absorption coefficients of transparent ceramic mediums of (1) Nd (1.0 at%): YAG; (2) Cr (0.1 at%), Nd (1.0 at%): YAG; and (3) Cr (0.4 at%), Nd (1.0 at%): YAG. 440 Chapter 10. Compact Solar-pumped Lasers Fig. 10.22. Measurement of differential spectra showing absorption of sunlight and spontaneous emission from Nd:YAG ceramic samples and Cr-codoped Nd:YAG ceramic samples by guiding the natural sunlight into an integrating sphere using a solar-tracking system and optical fibers. 10.5. Compact Solar-Pumped Micro-Rod Laser (μSPL) System We replaced the Nd-doped ZBLAN glass fiber with a 1 × 1 × 5 mm quadrangular-prismshaped, Cr 0.1 at% codoped, Nd 1 at%:YAG transparent ceramic micro laser rod supplied by World Lab. Co. Ltd, Japan as shown in Fig. 10.23. Fig. 10.23. 1×1×5 mm, Cr 0.1 at% codoped, Nd 1 at%:YAG transparent ceramic micro laser rod. The solar concentrator was changed to a ⌀76.2 mm caliber OAP (Edmund Optics) keeping the distance between the focus F and the center of OAP, O, unchanged from that of the ⌀50.8 mm caliber OAP used in the SPFL described in Section 10.3, that is, 50.8 mm. Among the commercially available products of ⌀76.2 mm caliber OAP, this is of the shortest OF length. Expressing the caliber diameter 76.2 mm as 2D in this section, the distance between F and O is 4D/3 as shown in the schematic drawing Fig. 10.24, which is 441 Advances in Optics: Reviews. Book Series, Vol. 5 drawn in the same manner as Fig. 10.16. Based on the geometrical consideration in Fig. 10.24(a), the length of the inverted image of the white arrow-shaped object rS is obtained using a similar process as the derivation of Eq. (10.25): R rS   S  Au  4  D  3 (10.31) The same result is also obtained from Fig. 10.24(b). Then, it can be shown that all the incident sunlight at the entrance of the OAP crosses inside the circle of radius rS at the focal point F in the same manner as shown in Subsection 10.3.2. The solar concentration ratio can be expressed as follows: 2  Au  9  D2 D2      2 2 2  rS  RS   4   RS  16 2   D      Au   3  (10.32) Using values 36.1 mm for D, 6.96 ×108 m for RS, and 1.496×1011 m for Au into Eqs. (10.31) and (10.32): 2rs  0.448 mm, (10.33)  D2  25990  rS 2 (10.34) Fig. 10.24. Schematic drawings of an OAP of a caliber diameter 2D of 76.2 mm with typical traces of light rays emitted from the head of a white arrow-shaped object of a length Rs and the head of a black arrow drawn on the white arrow-shaped object of a length D. (a) A case in which the arrow-shaped object is pointing right in a side view of the system, (b) A case in which the arrow-shaped object is pointing left in a side view of the system. 442 Chapter 10. Compact Solar-pumped Lasers These results are more favorable than the results for the ⌀50.8 mm caliber OAP shown in Eqs. (10.29) and (10.30). Sunlight can be focused within the end facet of the ⌀1 × 10 mm rod. Fig. 10.25 schematically shows this μSPL system. Output coupler (OC) of different curvature radii were tested. In a typical case of Nd:YAG laser where λL was 1064 nm, λe was 808 nm, σ(λL) was 2.8×10-19 cm2, τ was 230 μsec [98], δ was 0.02, R1 was 0.99, R2 was 0.9995, ηQD was 0.61 ( = 650 nm / 1064 nm), ηQE was 1, ηabs was 0.35, ηm was 0.29, A was 10 (5 mm /0.5 mm) and α was 0.03, Ieth was calculated to be 5.9 MW/m2. Therefore, solar concentration ratio of more than 16244 is required for the direct solar radiation of 508 W/m2 to start oscillation of this laser rod, taking the transmittance of the preposing optical system to 71.5 % including a UV filter with a transmittance of 95 %, and OAP with a reflectance of 92 %. Fig. 10.26 shows a photograph of the nucleus of a μSPL. Bow-tie apertures of different flare angles were used to control the incident sunlight power to measure the input solar power dependence of the output laser power. Fig. 10.27 shows the μSPL system mounted on an altazimuth-type solar-tracking system (Vixen SKYPOD) in an outdoor oscillation experiment tracking the Sun. The μSPL is compact enough to use an altazimuth fabricated to mount an astronomical telescope for observation of celestial objects (Vixen SKYPOD) as a solar-tracking system. Fig. 10.25. Schematic diagrams of a prototype μSPL system comprised of an OAP and a Cr 0.1 at%, Nd 1 at% doped transparent YAG ceramic rod laser medium in an Al heat sink, an OC, and an optical fiber for laser output. Fig. 10.28 (a) shows the measured output laser power as a function of the input solar power into the rod shown in the abscissa at the bottom. An OC of a curvature radius = 100 mm was used. The input solar power into the OAP is also indicated in the abscissa at the top. The maximum input solar power into the OAP was 4.28 W, which corresponds to a direct solar radiation of 939 W/m2. Passing through the preposing optical system, including a UV filter with a transmittance of 95 % and OAP with a reflectance of 92 %, the maximum input solar power into the rod was 3.06 W. The total transmittance of the preposing optical system was 71.5 %. The threshold power at the end facet of the rod was 1.8 W. Dividing it by the area of the focused size, π‧rS2 of 1.58×10-7 m2, the threshold 443 Advances in Optics: Reviews. Book Series, Vol. 5 power density of 11.4 MW/m2 could be obtained. The slope efficiency against the input solar power at the rod and the total efficiency against the input solar power at the rod were 2.1 % and 0.9 %, respectively. The external slope efficiency and the external total solar conversion efficiency against the input solar power into the OAP were 1.5 % and 0.7 %, respectively. Fig. 10.28(b) shows optical transmittance spectra of sunlight before and after passing through the laser rod. The difference between the two spectra corresponds to the absorbed light by the rod. Fig. 10.28 (c) shows the laser emission spectrum peaking at 1064 nm. Fig. 10.26. Nucleus of the μSPL system composed of an OAP and a resonator comprising a Cr-codoped, Nd:YAG transparent ceramic micro laser rod and an optical coupler. Bow-tie apertures of different flare angles are used for controlling the incident sunlight power to measure the input solar power dependence of the output laser power. Fig. 10.27. Outdoor oscillation experiment of a μSPL on an altazimuth tracking the Sun. 444 Chapter 10. Compact Solar-pumped Lasers Fig. 10.28. (a) Output laser power of a 1×1×5 mm Cr (0.1 at%), Nd (1 %):YAG transparent ceramic rod type μSPL as a function of the input solar power; (b) Optical transmittance spectra of sunlight before and after passing through the laser rod; (c) Spectrum of the output laser. The inset figure shows a detailed peak profile between 1060 nm and 1068 nm. Fig. 10.29 shows the measured output laser power as a function of the input solar power into the ⌀1 × 10 mm, Cr (0.1 at%) codoped Nd (1.0 at%):YAG transparent ceramic rod (Konoshima Chemical Co. Ltd.) shown in the abscissa at the bottom using the same μSPL system shown in Fig. 10.25. An OC of a curvature radius = 100 mm was used. The input solar power into the OAP is also indicated in the abscissa at the top. The maximum input solar power into the OAP was 3.61 W, which corresponds to direct solar radiation of 792 W/m2. Passing through the preposing optical system of the total transmittance of 71.5 %, the maximum input solar power into the rod was 2.59 W. The threshold power at the end facet of the rod was 1.56 W, which was lower than that in Fig. 10.28. Dividing the value by the area of the focused size, π‧rS2 of 1.58×10-7 m2, threshold power density of 9.87 MW/m2 could be obtained. The slope efficiency and the total efficiency against the input solar power into the rod were 2.6 % and 1.0 %, respectively. The external slope efficiency and the external solar energy conversion efficiency against the input solar power at the OAP were 1.8 % and 0.73 %, respectively. The effects of different OC curvature radii, 500 and 1000 mm, were also examined. Further, two quadrangular-prism-shaped transparent ceramic rods of 1 ×1× 10 mm of Cr 0.1 at% codoped Nd 1 % doped YAG and Cr 0.4 at% codoped Nd 2 % doped YAG supplied by World Lab. Co. Ltd, were also tested in the same μSPL system shown in Fig. 10.25. Table 10.1 summarizes the results. Experiment (a) shown in Fig. 10.29 is the best data among the five results. 445 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 10.29. Output laser power of the μSPL as a function of the input solar power into the rod. The input solar powers into the OAP are indicated with italic letters to show the real soler energy conversion efficiency. Table 10.1. Slope efficiencies and threshold powers in the different experimental setups. Exp. No. (a) (b) (c) (d) (e) Size (mm) ⌀1 × 10 ⌀1 × 10 ⌀1 × 10 1 × 1 × 10 1 × 1 × 10 Dopants Cr0.1 %, Nd1 % Cr0.1 %, Nd1 % Cr0.1 %, Nd1 % Cr0.1 %, Nd1 % Cr0.4 %, Nd2 % Radius of OC (mm) 100 500 1000 100 100 Slope efficiency ( %) 2.60 1.68 1.73 0.78 1.40 Threshold power (W) 1.56 1.26 1.38 2.26 1.20 10.6. Continuous Oscillation of μSPL for Over 6.5 Hours Tracking the Sun Using the same condition as Experiment (a) in Table 10.1, continuous oscillation was tried outdoors tracking the Sun. Instead of the altazimuth (Vixen SKYPOD) in Fig. 10.27, a commercially available automatic equatorial mounting (VIXEN SXD2) with a tripod was employed seeking perfect solar tracking. The system was set at the roof terrace of Building 3 of Faculty of Engineering, Nagoya University located at latitude 35°10’N north and longitude 136°58’E, Nagoya, Japan. A continuous oscillation experiment was performed on a clear weather day, Wednesday, August 9th, 2017. The μSPL experimental rig is pictured in Fig. 10.30. Output power of the μSPL was measured with a power meter sensor every 0.3 seconds and the measured value was stored in the memory, coinstantaneously. A pyrheliometer shown in Fig. 10.14 and a solar telescope were fabricated and mounted on the same equatorial mounting with the μSPL and tracked the Sun simultaneously. The power of the direct solar radiation in 5 degrees in view angle and in the wavelength range of 190 nm to 25 μm captured with an aperture and a pinhole was measured with a THOLABS S302C thermal power meter sensor attached to the pyrheliometer. The THOLABS S302C power meter sensor had been calibrated with an EKO INSTRUMENTS MS-53 (ISO 9060 First Class Pyrheliometer) that provided the value of the direct solar radiation in terms of W/m2. These data were measured as well as recorded in memory every 0.3 seconds. The inset at the lower left in Fig. 10.30 is an image of the 446 Chapter 10. Compact Solar-pumped Lasers Sun on a computer display monitored by the solar telescope using a video-capturing attachment. As reference, global solar irradiance was measured with a digital illuminometer. After adjustment of the direction of the μSPL roughly using the image of the Sun on the PC display monitored by the solar telescope, precise adjustment of screws fixing the μSPL onto the mounting provided a sudden rise in the output power of the μSPL from several μW to several mW at 10:50, indicating that laser oscillation was taking place. After start of the μSPL oscillation, the solar telescope was replaced by the pyrheliometer, and the measurement of the direct solar radiation was performed in parallel with the measurement of the output power of the μSPL. Fig. 10.30. μSPL system tracking the Sun automatically on an equatorial mounting (VIXEN SXD2). The inset figure in the lower left is an image of the Sun on a PC display monitored by the solar telescope using a video-capturing attachment for a rough adjustment of direction of μSPL toward the Sun. Fig. 10.31 shows the variation in μSPL output power as a function of time in the lower blue line. The variation of the direct solar radiation in terms of kW/m2 is also shown in the upper red line. For reference, the global solar irradiances at three different clock times indicated on the illuminometer—109000 lx at 11:08, 114000 lx at 11:59, and 43600 lx – at 16:48 are also shown at the upper side of the figure. Simultaneous instantaneous falls both in the μSPL output and in the direct solar radiation around 13:22-13:36 and 17:17 were caused by passing clouds. With occasional manual readjustments of the equatorial mounting such as at 11:10, 11:23, 11:33, 11:43, 11:52, 12:00, 12:25, 12:55, 13:48, 14:46, 15:16, 15:37, 16:11, 16:29, and 17:00, the output power recovered 2-3 mW. Especially the readjustment at 12:00 provided an approximately 4.5 mW increase in the μSPL output. The maximum direct solar radiation over 898 W/m2 took place between 12:00 and 13:00 in which the μSPL output also reached its maximum of around 12 mW. The μSPL output kept going over 10 mW until 14:30. After continuous decrease, the output power of the μSPL suddenly fell from 0.2 mW to 0.0017 mW between 17:34 and 17:35, although instantaneous oscillations were observed several times beyond 17:35. After 17:35 until 17:51, almost all the measured output powers were around 0.0017 mW. Therefore, it is considered that the μSPL stopped continuous oscillation between 17:34 and 17:35. It was 447 Advances in Optics: Reviews. Book Series, Vol. 5 positive outcome that the μSPL kept continuous oscillation even if the direct solar radiation was lower than 550 W/m2 around 17:30, that is, 61 % of the maximum value. This indicates the possibility of continuous oscillation for 11 hours from 6:30-17:30, which is considerably longer than our expectation and favorable for solar energy utilization. Fig. 10.32 shows a plot of these synchronous 47012 data sets, that is, a plot of μSPL output against the incident direct sunlight power at the entrance of the OAP. The distribution of the data clearly shows the existence of a threshold incident direct sunlight power around 1.47 W (2.05 W) for laser oscillation and a linear increase in μSPL output power with incident direct sunlight power larger than 1.47 W (2.05 W). This plot was realized because the measurement and the data acquisition of μSPL output power were continued beyond 17:30 after the oscillation stopped because of shadowing of sunlight by clouds spreading flat to the western horizon. A more detailed inspection of Fig. 10.32 reveals existence of two straight lines: one is in the region under 2.7 W of the total incident power (under 5 mW of the μSPL output), and the other is in the region over 2.7 W of that (over 5 mW of the μSPL output), with an inflection point around 2.7 W. Since the μSPL output was over 5 mW in the major day time before 16:30 in this experiment as shown in Fig. 10.31, we estimated the slope efficiency (external slope efficiency) to be 1.15 % (0.82 %) from the straight line over 2.7 W. The slope efficiency based on the straight line below 2.7 W is discussed next using the inset figure in Fig. 10.32. Fig. 10.31. Variation of μSPL output power as a function of time with an occasional manual readjustment of equatorial mount during automatic solar tracking. Variation of the direct solar radiation in terms of kW/m2 measured by a pyrheliometer is also shown. For reference, the measured global solar irradiance in terms of lx at three different clock times are shown at the upper side. To understand whether the red shift as the afternoon grew late (as displayed in Figs. 10.12 and 10.15) influenced the oscillation properties of the μSPL or not, a replot is shown in the inset figure in Fig. 10.32 in which the 8000 data points measured in the last 40 minutes after 16:50 are indicated using red markers while the others are indicated using gray markers. Those red-marked data were measured under insolation with the spectrum almost equal to or more redshifted than that at 16:50 indicated in Fig. 10.15. The distribution of 448 Chapter 10. Compact Solar-pumped Lasers the red markers is consistent with the second straight line below 2.7 W of the total incident power described above. The slope efficiency deduced from this second straight line is evidently lower than that deduced from the total data in the main figure. To compare the slope efficiency quantitatively, the slope efficiency (external slope efficiency) of 1.08 % (0.77 %) was deduced from the second straight line while that deduced from the first line was 1.15 % (0.82 %). The two straight lines do not positively indicate the existence of two different threshold incident direct sunlight powers for laser oscillation but rather the same one around 1.47 W (2.05 W). Fig. 10.32. Plots of μSPL output power against incident direct sunlight power at the entrance of the OAP using the 47012 data sets from those shown in Fig. 10.31. The inset at the upper left is a replot emphasizing the 8000 data points that correspond to the last 40 minutes after 16:50 indicated using red markers while others are marked using gray markers. The results in Figs. 10.29 and 10.32 were obtained employing the same laser rod together with the same solar concentrator of ⌀76.2 mm caliber OAP. Although there is no remarkable difference in the threshold power between the two experiments, the slope efficiency and the total efficiency are more than halved in the result in Fig. 10.32 in comparison with that in Fig. 10.29. Almost two years had passed when the experiment in Fig. 10.32 was performed since the experiment in Fig. 10.29. Between the two experiments, repeated test experiments were done using the same laser rod. It is known that there is a degradation of Nd:YAG transparent ceramics, known as solarization, which is thought to be caused by sunlight exposure. However, the origin of solarization has not been clarified yet because the degradation is difficult to detect via instrumental analyses. Although a UV filter (Kenko Tokina Zeta UV L41) was always set at the entrance of the OAP in our experiments, degradation of the laser rod is a possible cause of the decrease in the efficiencies. The prevention of degradation of the laser rod will be one of key challenges before commercial adoption of SPLs. In previous study [23], a water-cooled Nd:YAG crystal rod was pumped with a ⌀61 cm, solar tracking, equatorial mount solar collector to give 1 W of continuous wave (CW) laser output. Although the study claims that “Operation over many hours was obtained with no evidence of reduction of output,” no evidential data of continuous oscillation was provided. An evidential record of continuous oscillation for 11 min was reported employing a water-cooled Nd:YAG rod and a 10 m aperture solar collector [42], in which 449 Advances in Optics: Reviews. Book Series, Vol. 5 tracking error was corrected manually 8-10 times during this 11 min continuous operation. A record of 60 sec stable continuous oscillation was also reported employing a watercooled Nd:YAG ceramic rod and 2 m × 2 m Fresnel lens to generate 110 W CW laser output [63]. More recently, records of continuous oscillation of solar-pumped 1064 nm laser emission with a simple Gaussian intensity distribution around the center axis of the beam (TEM00-mode) for 4 minutes have been reported [99]. Therefore, the result shown in Fig. 10.31 is a record long continuous laser oscillation of SPL tracking the Sun [100]. This is an indispensable experimental step if SPLs are to be applied to terrestrial solar energy utilization. 10.7. Effect of Cr Content in a Cr-Codoped Nd:YAG Transparent Ceramic Laser Rod for μSPLs Direct experimental comparison of the effect of Cr, Nd:YAG ceramic LMs with different Cr contents on the SPL output was performed. Quadrangular-prism-shaped LMs of transparent Cr-codoped Nd:YAG ceramics (1 × 1 × 10 mm, and 1 × 1 ×20 mm) were prepared with Cr contents of 0.0, 0.4, 0.7, and 1.0 at% with a fixed Nd content of 1.0 at%. These LMs were fabricated by World Lab. Co. Ltd., Atsuta, Nagoya, Japan [45, 46]. The μSPL system is the same as shown in Figs. 10.25 and 10.26. Fig. 10.33 shows a photograph of the prepared LMs; the greenish color is attributed to Cr3+. The end facet of an LM was coated with a high-reflectivity (HR) coating consisting of a multilayered thin film optical filter with 95 % transmittance for focused sunlight coming from the OAP and a reflectivity (RHR) of 99.95 % for the 1064 nm monochromatic light emitted by Nd3+ in the LM. The other side of the LM was coated with a broadband antireflection (BBAR) coating consisting of a multilayered thin film optical filter with more than 95 % transmittance of sunlight with wavelengths between 780 and 1064 nm and almost 100 % transmittance of 1064 nm monochromatic light. The laser cavity contained the HR coating on the end facet of the LM and the concave mirror of OC with a curvature radius r of 100 mm and a diameter of 10 mm. Fig. 10.33. Photograph of the prepared Cr-doped Nd (1.0 at%): YAG rods. Cr content is indicated on the left. Length of the rod is indicated at the top. The black points near the left ends of the rods indicate that the left end facets of the rods were coated with HR coatings, whereas the right end facets were coated with BBAR coatings. 450 Chapter 10. Compact Solar-pumped Lasers An outdoor μSPL oscillation experiment was performed on the same location as shown in Figs. 10.27 and 10.30. Figs. 10.34(a) and (b) show plots of measured μSPL outputs against the incident direct sunlight power into the OAP (IOAP) for the 10 mm and 20 mm LMs, respectively [101]. The incident sunlight power at the edge of LM, ILM, is 71.5 % of IOAP in this light collection system employing the OAP, as was reported in Section 10.3. Here, the OC had a reflectively Rc of 99 % for laser emission (1064 nm). For both the LMs, the output power of the μSPL took the maximum at a Cr content of 0.4 at%. The output power at a Cr content of 0.4 at% was approximately 8.5 times that at a Cr content of 0.0 at%. The output power decreased as the Cr content increased beyond 0.4 at%. At a Cr content of 1.0 at% for the 20 mm LM, the output power was even lower than that for the LM without Cr doping. It is not easy to define the portion of linear increase of μSPL output with IOAP. However, we can compare the threshold input powers and slope efficiencies of the LMs using the values of the IOAP intercept Ieth with the line connecting the point of the highest μSPL output and the point of the second highest μSPL output obtained in this experiment for each Cr content and the values of the corresponding slope ηslope. Table 10.2 lists the obtained values of Ieth and ηslope. Fig. 10.34. Plots of measured μSPL output power against the incident direct sunlight power into the OAP, IOAP for (a) 10 mm and (b) 20 mm transparent Cr-doped Nd (1.0 at%) YAG LMs with different Cr contents. In addition to the outdoor laser oscillation experiments pumped by direct sunlight, laser oscillation experiments in which Nd3+ was pumped directly using an 808 nm LD were performed in an indoor laboratory. To couple the 808 nm excitation beam, a long-working-distance objective lens (focal length = 10 mm) was used. Here, OCs of r = 100 mm, diameter = 20 mm, and reflectivity Rc = 90 %, 95 %, and 99 % for laser emission (1064 nm) were employed. Fig. 10.35 (a) shows a typical oscillation property of the 10 mm LM of Cr (0.4 at%), Nd (1.0 at%):YAG ceramic obtained in the laser oscillation experiments by pumping Nd3+ directly using an 808 nm LD in an indoor laboratory. Here, an OC with a reflectivity Rc of 95 % for the laser emission (1064 nm) was employed. To evaluate the round-trip loss δ808 in the laser cavity, the oscillation properties were also obtained using OCs with Rc of 90 % and 99 %. Fig. 10.35 (b) shows a plot of Ieth (808) for the three measurements at different values of Rc against loge Rc. The 451 Advances in Optics: Reviews. Book Series, Vol. 5 extrapolation of a line determined by least-square fitting of the three data points intercepts the horizontal loge Rc axis at -0.026. Therefore, δ808 was found to be 0.026 [102]. Similarly, the δ808 values of the eight LMs were obtained and are listed in Table 10.2, together with the ηslope (808) and Ieth (808) values obtained in these indoor experiments. The results in Table 10.2 show that δ808 increased with increasing Cr content. The value of ηslope (808) did not increase as dramatically as ηslope but did show a slight increase between 0.0 at% Cr and 0.4 at% Cr. At Cr contents exceeding 0.4 at%, ηslope (808) decreased gradually in both the 10 mm and 20 mm LMs. The increase in Ieth (808) was like that of δ808 for both the 10 mm and 20 mm LMs. These results clearly show that the presence of Cr3+ does not affect the oscillation process because of direct excitation of Nd3+ by the LD. In contrast, for both the 10 and 20 mm LMs, Ieth showed a steep decrease between 0.0 at% Cr and 0.4 at% Cr, reaching a minimum between 0.4 at% Cr and 0.7 at% Cr and increased again toward 1.0 at% Cr. According to Eq. (10.8), the increase in Cr content is thought to contribute to the increase in Ieth via the increase in δ. However, the increase in the Cr content also contribute to the decrease in Ieth via the increase in ηabs in Eq. (10.8). The minimum value of Ieth at a Cr content of 0.4 at% is attributed to these two conflicting contributions. Because the increase in Ieth was observed even at low Cr contents between 0.4 % and 0.7 %, the possible effect of concentration quenching is thought to have little effect on the increase in Ieth in this experiment. Recall that ηslope increased dramatically between 0.0 at% Cr and 0.4 at% Cr and decreased gradually beyond 0.4 at% Cr. The maximum ηslope at 0.4 at% Cr was 2.43 % for the 10 mm LM and 3.19 % for the 20 mm LM. These values are converted to 3.40 % and 4.46 %, respectively, if the μSPL output is plotted against not IOAP but ILM. These values are competitive with the current state-of-the-art values [60]. Table 10.2. Measured values: Ieth and ηslope. of the eight LMs obtained outdoors under sunlight, and the round-trip loss L808, Ieth (808) and ηslope (808) obtained indoors under 808 nm LD pumping. Cr3+ I e th (at%) (W) 0 2.98 0.4 2.41 0.7 2.49 1 2.52 η slope (%) 0.36 2.43 2.00 1.15 10 mm LM δ 808 η slope(808) (%) (%) 2.5 47.2 2.6 49.2 3.0 37.4 4.5 26.6 I e th(808) (mW) 45.5 50.6 70.9 85.3 I e th (W) 3.03 2.64 2.52 3.26 η slope (%) 0.67 3.19 1.44 0.34 20 mm LM δ 808 η slope(808) I e th(808) (%) (%) (mW) 2.8 37.9 58.6 2.6 43.5 59.7 3.8 32.3 79.3 5.8 18.1 93.8 These results indicate that Cr doping is effective for decreasing Ith and increasing ηslope because of increased optical absorption by Cr3+. Although it was demonstrated that Cr doping increases δ808, it was also found that the increase in δ808 between 0.0 at% Cr and 0.4 at% Cr was small compared with the increase between 0.4 at% Cr and 1.0 at% Cr. Therefore, the μSPL output showed an outstanding maximum at 0.4 at% Cr, because it is determined by competition between the positive effect of increased energy transfer from Cr3+ to Nd3+ and the negative effect of increased δ with increase in Cr content. It appears that these conditions were the same for both 10 and 20 mm LMs. 452 Chapter 10. Compact Solar-pumped Lasers Fig. 10.35. (a) Plots of μSPL output against the incident 808 nm laser power I(808) into the 10 mm LM of Cr (0.4 at%),Nd (1.0 at%):YAG, where an OC with r of 100 mm, a diameter of 20 mm, and a reflectivity Rc of 95 % at 1064 nm was employed. (b) Plot of Ieth (808) for Rc of 90 %, 95 %, and 99 % against loge Rc. 10.8. For Improvement of Mode-matching Efficiency In this work, it has been established that sunlight can be concentrated to exceed Ieth at the end facet of LMs by using OAPs. However, this raises a serious issue, known as the “mode-matching problem,” which means that there is sufficient overlap between the laser mode profile and the propagation mode of the pumping light is difficult to achieve. Fig. 10.36 schematically shows how the concentrated direct sunlight propagates in an LM. The thin, red cylindrical region stands for a simplified laser mode profile. The overlap is maximum when the length of the LM is small, that is, in the case of disk-shaped LM. With increase in the length of LM, the overlap decreases toward the constant value because the converged sunlight at the end facet of the LM diverges again in the LM. Pumping the region not overlapping with the laser mode does not contribute to the laser output but only leads to energy loss. In Figs. 10.16 and 10.24, several typical traces of light rays are indicated to show the image formations. Here, more detailed tracing of sunlight was performed using a commercially available simulation software ZEMAX. Fig. 10.37(a) shows the result for traces of sunlight reflected by an OAP and converged on an end facet of ⌀1 × 10 mm LM. Fig. 10.37(b) shows a distribution of the absorbed power of the pumping light propagating in the LM. The distribution reflects asymmetric nature of the converged light by OAP in the first 1 mm and extends widely in the LM, implying significant loss in the peripheral region out of the laser mode. Combining a laser cavity analysis software LASCAD, the output laser powers were simulated as a function of the input solar power into the rod assuming 2.8×10-19 cm2 for σ(λL), 230 μsec for τ [98], 0.02 for δ, and 0.8 for ηQE. The results are shown in Fig. 10.38. The simulated results give a threshold power of 1.44 W and a slope efficiency of 2.8 % while the corresponding experimental values are 1.56 W and 2.6 %. Although the simulated results approximately reproduced the experimental results in Fig. 10.29, the slightly better threshold power and the slope efficiency imply ηQE lower than 0.8, thereby reflecting poor energy transfer efficiency from Cr3+ to Nd3+. If the doping region in ⌀1 × 10 mm rod could be reduced, 453 Advances in Optics: Reviews. Book Series, Vol. 5 for example, within ⌀0.4 mm, 0.6 mm, or 0.8 mm as shown in Fig. 10.39(a), the absorption of the light would take place only within the doped region as shown in a simulated result in Fig. 10.39(b). Here, we call this structured rod shown in Fig. 10.39(a) as a composite rod. Simulated output laser powers plotted against the input solar power into the rods by ZEMAX and LASCAD shown in Fig. 10.39 (c) indicate a decrease in the threshold power with decrease in the diameters of the doped regions. This is possibly caused by the increase in mode-matching, that is, the decrease of the absorption out of the laser mode that does not contribute the laser output. However, reduction of the doped region less than ⌀0.8 mm led to decrease in the slope efficiency, which is possibly caused by the decrease in the absorbed light in the laser mode region because of the excessive reduction in the doped region. Fig. 10.36. Schematic representation of a laser mode profile and pumping light propagation in a laser medium. Fig. 10.37. (a) Ray tracing of the sunlight concentration onto ⌀1 × 10 mm, Cr (0.1 at%) codoped Nd (1.0 at%):YAG transparent ceramic rod by an OAP; (b) Absorbed power distribution in the laser medium (optical simulation code: ZEMAX). 454 Chapter 10. Compact Solar-pumped Lasers Fig. 10.38. Output laser power of the μSPL as a function of the input solar power into the rod simulated combining ZEMAX and LASCAD. The corresponding experimental data shown in Fig. 10.29 were also replotted for reference. Fig. 10.39. Effects of reduction of the doped region in a YAG micro rod. (a) A schematic composite rod structure; (b) Absorbed power distribution in the composite rod simulated using ZEMAX; (c) Output laser power of the composite rod SPL as a function of the input solar power into the rod simulated combining ZEMAX and LASCAD. Although the reduction in the doped region to form a composite rod is effective for improving mode-matching, the difference in the refractive indices between the core and the cladding causes another loss, that is, diffraction loss between the exit end facet of the rod and a resonator mirror as shown in Fig. 10.40 (a). By doping 5 at% non-absorbing Gd, the refractive index of the cladding was matched with that of core (Fig. 10.40 (c)), and the diffraction loss could be eliminated as shown in Fig. 10.40(b). For ease of production, quadrangular-prism-shaped composite rods in place of concentric cylinders, as shown in Fig. 10.39 (a), were chosen. Fig. 10.41 shows photographs of (a) a colorless Gd 5 at% doped YAG transparent ceramic LM, (b) a greenish Cr 0.4 at%, Nd 1 at% doped transparent ceramic LM, (c) a cross-sectional view of 2D composite LM, (d) a 455 Advances in Optics: Reviews. Book Series, Vol. 5 cross-sectional view and part side three-dimensional (3D) composite LM. view of a quadrangular-prism-shaped Fig. 10.40. Refractive index matching between the Cr, Nd: YAG core and the Cr, Nd non-doped YAG cladding by non-absorbing Gd in the cladding to reduce the diffraction loss between the exit end facet of the rod and a resonator mirror. Fig. 10.41. Photographs of (a) a colorless Gd 5 at% doped YAG transparent ceramic LM, (b) a greenish Cr 0.4 at%, Nd 1 at% doped transparent ceramic LM, (c) a cross-sectional view of a quadrangular-prism-shaped two-dimensional(2D) composite LM, (d) a cross-sectional view, and a part of a side view of a quadrangular-prism-shaped three-dimensional(3D) composite LM. Using a convergent monochromatic light of NA = 0.4 formed from an 808 nm diode laser that directly excites Nd3+ ions (Fig. 10.42 (a)), preliminary evaluations of composites rods shown in Fig. 10.41 (c) and (d) were performed. The results shown in Fig. 10.42 (b) elucidate a lower threshold power and a higher slope efficiency for 2D composite rod in comparison with those for a conventional homogeneous rod, and even better results for 3D composite rod than that for 2D composite rod [103]. Although an experimental proof of the advantages of 3D composite rods by outdoor oscillation tracking the Sun remains a challenge for the future, the slope efficiency in a 3D composite rod with a 0.4×0.4 mm core in a 1×1 ×10 mm LM under white light pumping is simulated to be 18 %, as shown in Fig. 10.43 whereas that in a 2D composite rod is 3.1 % and that in a homogeneous (conventional) rod is 1.3 %. Therefore, more precise adjustment of core size may bring better mode-matching and better efficiencies. 456 Chapter 10. Compact Solar-pumped Lasers Fig. 10.42. (a) Monochromatic convergent beam optics for preliminary evaluation of 2D and 3D rods; (b) Results of comparative evaluations of a conventional homogeneous rod, 2D composite rod, and 3D composite rod. Fig. 10.43. Simulated oscillation property of a 3D composite rod under white light pumping. 10.9. Evaluation of Energy Transfer Efficiency from Cr3+ to Nd3+ in μSPL in Outdoor Operation As shown in Fig. 10.34, Cr codoping is effective in increasing the output laser power at 0.4 at% in comparison with no codoping. However, in the cases of codoping of 0.7 at% Cr and 1 at% Cr, the demerit of the increase in the round-trip loss δ offset the merit and the output laser power decreased. There is a previous study that reports that the performance of Cr, Nd:YAG ceramics was inferior to that of undoped Nd:YAG crystals in an SPL where a 3 × 9 × 100 mm transparent Cr, Nd:YAG ceramic LM was tested in a specially designed SPL employing end-pumping and side-pumping simultaneously using 457 Advances in Optics: Reviews. Book Series, Vol. 5 a liquid light-guide lens [91]. The reason suggested was as follows: (1) Cr codoping increased the scattering coefficient of transparent Nd:YAG ceramic rods, (2) the increase in the scattering coefficient of the rods leads to an increase in the round-trip loss δ in the laser cavity, (3) the increase in δ leads to a decrease in ηslope, (4) the decrease in ηslope offsets the advantage of the decrease in the saturation gain caused by increased optical absorption by Cr3+ followed by energy transfer from Cr3+ to Nd3+. This agrees with the results observed in Fig. 10.34. Since then, undoped transparent Nd:YAG LMs have been used in SPLs [58, 59]. However, using ⌀4.5 mm × 35 mm Cr (0.1 at%)-doped Nd (1.0 at%):YAG ceramic rod, the same authors recently reported a slope efficiency 1.28 times their own previous record [58, 60]. Therefore, the advantage of using transparent Cr, Nd:YAG ceramic LMs instead of Nd:YAG single-crystal LMs in SPLs is not conclusive. It was also suggested from the simulated result in Fig. 10.38 that ηQE might be less than 0.8, reflecting poor energy transfer efficiency from Cr3+ to Nd3+. Fig. 10.44 schematically shows the energy diagram of Cr3+ and Nd3+ in YAG crystal including energy transfer process from Cr3+ to Nd3+. Cr3+ ions excited by sunlight relax to 4T2 state or 2E state. Through the transition of Cr3+ ions from these excited states (4T2 or 2E) to the ground state (4A2), it is considered that energy is transferred from Cr3+ to Nd3+ ions to put Nd3+ ions eventually into the metastable excited state 4F3/2. It is considered that this process increases the pumping efficiency and decreases the threshold power for laser oscillation. However, the direct excitation of Nd3+ ions by sunlight may increase the number of Nd3+ ions at the metastable excited state 4F3/2 so much that Cr3+ ions excited by sunlight cannot find enough Nd3+ ions at the ground state to which energy is transferred. This may cause poor energy transfer efficiency from Cr3+ to Nd3+. To simulate this situation experimentally, laser oscillation experiments with simultaneous direct excitation of Nd 3+ by an 808 nm laser and the direct excitation of Cr3+ by 561 nm that does not excite Nd3+ directly were performed as shown in Fig. 10.45 (a). Fig. 10.45 (b) shows output laser powers as a function of 808 nm pumping power in cases with and without 50 mW pumping at 561 nm. With 50 mW pumping at 561 nm, the output laser power increased in comparison with the case without 50 mW pumping at 561 nm. The increment can be attributed to the Cr to Nd energy transfer. Fig. 10.45(c) shows the decrease in the threshold power with increase in the input 561 nm power. From these results, the energy transfer efficiency was estimated to be 65.3 % [97]. A more realistic approach to access the Cr to Nd energy transfer efficiency was also tried using natural sunlight pumping outdoors. For the samples, Cr-codoped Nd (1.0 at%):YAG transparent ceramic LMs of 1 × 1 × 10 mm in size with Cr contents of 0.0 at%, 0.4 at%, 0.7 at%, and 1.0 at% shown in Fig. 10.33 were used again. Apart from the measurement of the stimulated emission, as shown in Fig. 10.25, the spontaneous emission spectra by sunlight excitation were also measured in a similar geometry as shown in Fig. 10.46 (a). Figs. 10.46 (b) and (c) show measured spontaneous emission spectra in the cases of sunlight intensity of 1.12 W and 3.07 W, respectively. Using the decrease in the threshold power caused by the energy transfer from Cr3+ ions to Nd3+ ions, the values of Cr3+ to Nd3+ energy transfer efficiency ηCr→Nd were estimated as shown in Fig. 10.47 [104]. In the 458 Chapter 10. Compact Solar-pumped Lasers figure, the values estimated are not for ηCr→Nd but for (ηCr/ ηNd) ⋅ηCr→Nd. Here, ηCr is the quantum efficiency of the formation of the excited Cr3+ ions in the energy level 2E and 4T2 from the number of absorbed photons by Cr3+ ions, and ηNd is the quantum efficiency of the formation of the excited Nd3+ ions in the metastable energy level of 4F3/2 from the number of absorbed photons by Nd3+ ions. In other words, ηCr stands for the efficiency of relaxation of the excited electrons in higher energy levels of Cr3+ to their 4T2 or 2E levels, and ηNd stands for the efficiency of relaxation of the excited electrons in higher energy levels of Nd3+ to the metastable 4F3/2 levels. The ratio γ ( = ηCr/ηNd) is not known. For the first-order approximation, γ can be assumed to be unity because both ηCr and ηNd represent similar properties of the isolated metal ions doped in YAG ceramics. For the second-order approximation, γ can be assumed to be a value slightly less than unity, γ ≾1. Because the energy levels of rare-earth ions such as Nd3+ are not significantly influenced by the surrounding electronic structure of YAG matrix atoms, ηNd is considered to be higher than ηCr of a transition metal ion, Cr3+. Since γ, which is nearly equal to unity, does not majorly influence the present discussion, the values listed in Fig. 10.47 are deemed to represent ηCr→Nd. When IOAP was increased to 3.07 W, ηCr→Nd in spontaneous emissions decreased to 0.675γ (0.4 at%), 0.623γ (0.7 at%), and 0.625γ (1.0 at%). In stimulated emissions, ηCr→Nd further decreased to 0.448γ (0.4 at%), 0.213γ (0.7 at%), and 0.166γ (1.0 at%). The decrease in ηCr→Nd in the increased IOAP in spontaneous emissions, and further decrease in ηCr→Nd in stimulated emissions may be attributed to the fact that the energy of the Cr3+ ions excited by sunlight cannot transfer to Nd3+ ions because the number of Nd3+ ions directly excited by sunlight and already occupying the metastable excited state, 4F3/2, increased with increasing IOAP in spontaneous emissions, and more seriously so in the population inversion in stimulated emissions. As for the dependence on the Cr content, the highest values of ηCr→Nd were obtained at 0.4 at% in line with the Cr content dependence of output powers of SPL shown in Fig. 10.34. Fig. 10.44. Energy diagrams of Cr3+ and Nd3+ in YAG crystal under the sunlight excitation. 459 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 10.45. (a) Experimental set-up of the Nd/Cr: YAG pumped by 808 nm and 561 nm lasers. (b) Laser output power property by two-wavelength pumping. (c) Laser oscillation threshold vs. the 561 nm pumping power. Fig. 10.46. (a) Geometry to monitor solar-pumped spontaneous emission spectra. The optical filter transmits 70 % for λ of 1064±2 nm; (b) and (c) Measured spontaneous emission spectra in 1060 nm<λ<1070 nm at two different sunlight intensities of 1.12 W and 3.07 W, respectively. 460 Chapter 10. Compact Solar-pumped Lasers Fig. 10.47. Results of estimated ηCr→Nd·γ displayed in bar charts to show (1) the dependence on the sunlight power in spontaneous emissions, (2) the difference between spontaneous emissions and stimulated emissions and (3) dependence on the Cr content. The Cr content dependence of the output power of μSPL at the pumping power of sunlight = 3.07 W is also attached. γ is the ratio ηCr/ηNd ≾ 1, ηCr is the quantum efficiency of the formation of the excited Cr 3+ ions in the energy level 2E and 4T2 from the number of absorbed photons by Cr3+ ions, and ηNd is the quantum efficiency of the formation of the excited Nd 3+ ions in the metastable energy level of 4F3/2 from the number of absorbed photons by Nd 3+ ions. Deeming γ to be unity, the values of ηCr→Nd obtained here in stimulated emissions are also much less than the values of ηCr→Nd in spontaneous emission. This indicates that Cr-codoped Nd:YAG is not an ideal LM. Improvement of ηCr→Nd is awaited for SPL. This is consistent with the motivation to develop novel LMs such as in Cr, Nd-doped garnet crystal families for SPLs in place of Cr, Nd:YAG [105, 106]. However, the advantages of Cr, Nd:YAG ceramics over novel LMs, such as grain sizes of a few micrometers resulting in suppressed scattering on the grain boundary, controllability of Cr and Nd concentrations, and thermal conductivity, have been still unchallenged. As briefly explained earlier, the problem of degradation due to solarization also needs to be solved in Cr-codoped Nd:YAG transparent ceramic LMs. To improve the spectral matching efficiency, we believe that it is promising to employ a vertical cavity surface emitting semiconductor laser (VECSEL) that absorbs all the solar photons with their energy larger than the bandgap of a semiconductor as a future challenge [107-109]. A VECSEL is essentially a disk-type LMs shown as an ideal form of LM in the top left corner in Fig. 10.6, in contrast to the current Nd:YAG rod-like micro LMs in our μSPL. 10.10. Coordinated Solar Tracking of an Array of μSPLs to Harvest Larger Amount of Solar Energy In contrast to a conventional SPL employing a large Fresnel lens or a large converging mirror with typical sizes or diameters in the range of 1-2 m and a water-cooled thick LM 461 Advances in Optics: Reviews. Book Series, Vol. 5 of a typical size of ⌀10×100 mm [63-65], a μSPL can be mounted on a commercially available solar-tracking system for a telescope for amateur astronomers, as shown in Fig. 10.30, for all-day continuous lasing tracking the Sun only with natural air convection. This is enabled by rapid thermal dissipation from a thin micro-LM of a typical size of ⌀1 ×10 mm. However, a ⌀76.2 mm OAP can harvest only 4.1 W solar energy even in the relatively high direct solar radiation of 905 W/m2 we observed, using the apparatus shown in Fig. 10.14. The improved sunlight-to-laser energy conversion efficiency of 14 % shown in Fig. 10.43, and the thermodynamical limit deduced from a detailed valance theory 31 % [76] gives the output laser power of only 0.6 W and 1.3 W, respectively. For meaningful amount of solar energy utilization, arrays of large numbers of μSPLs should be used. Mass production and cost-reduction of μSPLs is indeed feasible, as demonstrated by the commercially available “Himawari” solar lighting system composed of compact concentrators and a solar-tracking system [78]. Large-sized solar concentrators with a solar-tracking system and solar-tracking type solar panels cannot be operated on windy days because they suffer from wind forces. Taking this fact into account, a coordinated solar tracking system of an array of compact solar-pumped lasers was designed, as shown in Fig. 10.48 (a). In Fig. 10.48 (a), parallel sliding motion of an upper white panel transmits motions of a single solar-tracking engine to 25 μSPLs to realize coordinated solar tracking. The upper white panel is always kept horizontal during its motion to keep the wind pressure minimum. The distances between neighboring μSPLs are determined in the array to avoid spatial interventions between them during the coordinated solar tracking. In the outdoor performance test shown in Fig. 10.48(b), thermal expansion of the upper panel by solar irradiation could disturb precise coordinated tracking. Application of super Invar alloy, whose thermal expansion coefficient between 20 and 90 ℃ is lower than 0.1×10-6 K-1 (which is as low as 1/100 of that of the popular stainless steel), could overcome this problem. However, application of more cost-effective materials or structures to realize low thermal expansion remains a future challenge [110]. Fig. 10.48. (a) Schematic rendering of a coordinated solar tracking system of 5×5 compact SPLs and (b) a photograph of outdoor solar tracking operation test. 462 Chapter 10. Compact Solar-pumped Lasers 10.11. SPL-PV Combined System and Its Application to Optical Wireless Power Transmission From the spectral quantum efficiency measured on a Si solar cell of energy conversion efficiency η = 24.0 % [111], the spectral energy conversion efficiency η(λ) can be calculated as shown in Fig. 10.49, where λ stands for wavelength [8]. η(λ) takes a maximum value of over 40 % at λ just below the optical absorption band edge of Si = 1117 nm. The solar spectrum, which is also illustrated with its intensity in arbitrary unit for comparison, has a maximum at around 500 nm. Obviously, the solar spectrum does not make a good fit with η(λ). On average in the range of the solar spectrum below 1117 nm, η for crystalline Si solar cells decrease to a level as low as 25-30 % as those in the market. Fig. 10.49. Spectral conversion efficiency of a crystalline Si solar cell under monochromatic illumination of around 50 mW/cm2. The solar spectrum is also illustrated. The lasing wavelength of the Nd:glass laser is indicated by an arrow. The residual 70-75 % of the solar energy contributes only to the temperature rise of the cell. In contrast, the lasing wavelength of a Nd:glass laser is located just below 1117 nm and closely fits the peak of η(λ). This means that the photon energy can be converted to electricity with minimal heat loss. Therefore, it will be reasonable to convert the output of μSPL into electricity via specially designed Si photovoltaic (PV) cells [10-17]. Optical wireless power transmission [109] to mobile objects may find applications in near future such as power feeding to drones and electric vehicles [4-9]. The extraterrestrial AM0 solar radiation on the Earth’s orbit is 1.36 W/m2, which is 36 % larger than the terrestrial AM1.5 solar radiation. Owing to a lack of interrupters such as clouds, the concept of space solar power stations, on which electricity is generated using solar cells and transmitted to the earth via microwaves or laser beams, has been proposed [1-5]. More recently, electricity generated by solar cells at a lunar lander has been planned for transmission via laser beam to a lunar rover exploring deep into permanently shadowed craters for frozen water [6, 7]. Because all the 1.36 W/m2 solar radiation in space is direct solar radiation, SPLs can be also applied to generate laser beam for power feeding. The 463 Advances in Optics: Reviews. Book Series, Vol. 5 combined loss of solar energy via energy conversion from sunlight to electricity by solar cells and that from photovoltaic electricity to laser beam by of a laser diode can be higher than the single loss due to conversion from sunlight to laser beam by an SPL [8]. In the terrestrial application of SPLs to laser-beam power feeding, the high concentration ratios of OAPs such as 11550 in Eq. (10.30) and 25990 in Eq. (10.34) exclude the chance to utilize diffuse solar radiation. The reason why the high concentration is requested is to realize high radiation flux density that surpasses Ieth in Eq. (10.8). According to Eq. (10.8), Ieth can be lowered if the aspect ratio A ( = L/r) of an LM is large, and the factor α, the effective utilization rate of the length L of the LM is large. Disk-shaped LMs such as VECSELs discussed in Section 10.9 and shown as an ideal form of an LM in the top left corner of Fig. 10.6, show good promise in the future. As is also shown in the top right corner of Fig. 10.6, a fiber-type LM is another approach to attain large A and a large α by side-pumping in Eq. (10.8). As a promising approach, transverse excitation has been introduced into a solar pumped fiber laser system to decrease the threshold excitation power for oscillation using a low sunlight concentration at the ratio of 1/15 [112-115]. 10.12. Conclusion We have developed compact solar-pumped lasers (SPFL and μSPL) employing an off-axis parabolic mirror with an aperture of 50.8 mm or 76.2 mm diameter and a Nd-doped ZBLAN fiber or a Cr-doped Nd (0.1 at%):YAG transparent ceramic rod of ⌀1 × 10 mm as the LM. Here, the stimulated emission takes place via the transition of Nd3+ in YAG from its excited state to lower energy states. Cr was doped to absorb sunlight in broad wavelength ranges and to transfer the absorbed solar energy to Nd3+ because Nd3+ absorbs sunlight only in the limited narrow wavelength ranges. The laser oscillation wavelength of 1.05-1.06 μm, just below the optical absorption edge of Si solar cells, is suitable for photoelectric conversion with minimal thermal loss after optical wireless (laser) power transmission to distant places. The small LMs and solar concentrators realize more stable oscillation by rapid natural air convection cooling and increased mechanical stability during wind exposure in contrast to the conventional large SPLs typically employing a 2 m size solar concentrator. Outdoor operation tracking the Sun yielded continuous oscillation exceeding 6.5 h, considerably improving upon the previously reported 11 min. This shows the applicability of SPLs to whole-day operation and terrestrial and extraterrestrial solar energy utilization. The measured incident sunlight-to-laser output total energy conversion efficiencies are 0.700.9 %. (slope efficiencies against the input solar power to LMs are 1.15-3.3 %.). However, several promising data have been obtained for future improvements. The SPL output increased more than eightfold between an LM with Cr content of 0.0 and that of 0.4 at%. The possibility of 14 % total energy conversion efficiency or more was shown in preliminary experimental and simulated results by the improvement of mode-matching efficiency introducing a refractive index matched core-cladding 3D composite structure into an LM. In contrast, there was evidence of degradation of Cr-codoped Nd:YAG transparent ceramic LMs, possibly by the accumulated sunlight irradiations, and relatively low Cr3+ to Nd3+ energy transfer efficiency under broadband sunlight pumping. To 464 Chapter 10. Compact Solar-pumped Lasers overcome these problems, future studies need to explore novel LMs or to investigate alternative choices of LMs including side-pumped fiber LMs and VECSELs of high spectral and mode-matching efficiency. Acknowledgments This chapter contains research results supported by Advanced Low Carbon Technology Research and Development Program (ALCA), Japan Science and Technology Agency, and Toyota central R&D Laboratories Inc. The authors acknowledge Dr. A. Ikesue for his essential contribution in fabrication of laser mediums. The authors also acknowledge Dr. K. Higuchi, Dr. T. Ito, Dr. T. Kajino, Dr. A. Ichiki, Dr. H. N. Luitel, Dr. D. Inoue, Mr. T. Ichikawa, Mr. Y. Suzuki, Mr. H. Terazawa, Mr. T. Kato, Ms. N. Hara, Mr. K. Watanabe, Mr. S. Takimoto, Mr. D. Kano, Ms. L. T. A. Phuc, Dr. H. Iizuka, Dr. T. Tani, and Dr. M. Umehara for their helpful contributions in this work. The authors also acknowledge Prof. M. Yamaga (Gifu University), and Prof. Y. Ohishi and Assoc. Prof. T. Suzuki (Toyota Technological Institute) for their suggestions and valuable discussions regarding this work. The authors would like to thank Editage (www.editage.com) for English language editing. 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Si is transparent in telecommunication wavelength (1550 nm). It has high index of refraction in mid-infrared range which makes it suitable for creating photonic crystal structures with complete bandgaps. It is also resistant to the radiation damage. The most significantly, it has been practiced for decades in electronics and nano structuring. Therefore, there is a solid background in Si research [1]. Surface and subsurface modification of the silicon in electronics and photonics applications have become one of the most popular research areas during the last decades [2-5]. Optical switches [6], waveguides [7, 8], photonic crystals [6, 9], modulators [10], and periodic structures [11-13] have been reported during the last two decades. Processing of Si surfaces in micron and submicron level is an important element to enhance the performance of photonic devices. Lithography techniques; electron ion-beam [14] and focused ion-beam [15, 16]; are known methods, which are used to pattern Si in these levels. However, these methods are very time consuming and not cost effective. Processing large scale of samples (> 1 cm) is the main difficulty on these methods. To solve this, the direct writing method is an alternative approach to processing of semiconductors. The use of pulsed lasers in order to form structures having photonic properties on the surfaces of semiconductor crystal materials is quite common. The high peak power and the controllable pulse duration within the designed limits open the gates of new design techniques and the future technology, and enable the past ideas to put into practice in Fırat İdikut, Burcu Karagöz Department of Physics, Middle East Technical University, Ankara, Turkey 473 Advances in Optics: Reviews. Book Series, Vol. 5 semiconductor fields. In last decades, the ultrafast lasers (femtosecond, picosecond and nanosecond) have started to be used for processing the material surfaces. Surface and subsurface modification of semiconductors such as silicon became possible with ultrafast lasers [8, 17]. The focal sizes formed in far-field processing techniques remain 10 µm and above due to the limited numerical aperture (NA) [18]. In silicon, the spot size is enlarged due to high index of refraction of the silicon [17]. This causes the wavelength bands in which the photonic structures that can be produced can work effectively to be limited to mid- and far-infrared wavelengths. Therefore, in order to reach the sub-micron processing alternative methods are needed. Reducing the processing pixel sizes to 1 µm and below will be able to expand the potential photonic application on Si. Femtosecond laser pulses have been used for precise processing and modification of a range of materials including silicon at micrometer scales [19] through the ultrashort pulse duration and high intensity of these ultrafast pulses. Therefore, two-photon absorption (TPA) is higher in femtosecond lasers for low pulse energy. Ablation precision of femtosecond laser pulses is also high through reduced heat affected zone and small ablation threshold. Taking advantages of these characteristics, well-defined micro structuring is possible with femtosecond pulses. The smallest size of features that can be achieved by laser irradiation is limited due to diffraction of laser beams. The far-field resolution is therefore limited to approximately half of the wavelength of illuminating beam. Direct laser writing cannot be used for nanoscale structuring due to the diffraction limitation of far-field patterning [20]. To overcome this limitation in patterning of substrate surfaces, contact particle lens array (CLPA) technique where these particles serve as near-field focusing elements has been extensively employed [21-25]. In this method, self-assembled monolayer of micrometer sized spheres on a substrate subjected to a laser irradiation acts as a micro lens array that focuses the laser beam to substrate. By scanning the laser beam through this microsphere lattice, surface patterning beyond the diffraction limit can be observed on substrates [26]. The mechanism of this patterning technique is based on the intensity enhancement at contact regions between spheres and substrate [27]. This enhanced field causes ablation on substrate regions in contact with spheres and a patterned surface is obtained upon the removal of ablated material. In this chapter, the uniformly distributed microspheres on silicon (Si) substrate which is functioned as focusing micro lenses having capability to achieve the ablation of Si under the two-photon absorption regime with 1.55 µm ultrafast laser will be discussed. In addition to the direct writing with microspheres, periodic structuring inside the processed region is reported. 11.2. Laser-material Interactions 11.2.1. Gaussian Beam Propagation The beam shape is outcome of combination of the cavity modes generated in the laser. In Fig. 11.1, several modes and their intensity profiles are represented. Depending on the 474 Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser laser design for the usage of its application field, single mode or multimode beam profiles are generated. In our case, TEM00 mode is considered as the output beam shape. This mode is generally known as Gaussian beam [28]. Fig. 11.1. Representation of the intensity distribution of the transverse electromagnetic modes (TEMij). The Gaussian beam has an exponential decaying intensity distribution profile along x-y plane which is perpendicular to the propagation direction z. In equation (11.1), the intensity distribution of the Gaussian mode is indicated in cylindrical coordinates. The distribution has dependencies along radial r and z. w(z) is the beam waist at the measurement distance where the irradiance drops to 1/e2 factor, which is calculated with respect to the beam waist (w0) as referred in equation (11.2). 𝑤 2 −2𝑟 2 (11.1) 2 (11.2) 0 𝐼(𝑟, 𝑧) = 𝐼0 (𝑤(𝑧) ) 𝑒𝑥𝑝 (𝑤(𝑧)2 ), 𝜆𝑧 𝑤(𝑧) = 𝑤0 √1 + (𝜋𝑤 2 ) , 0 The equation (11.2) can be rearranged in terms of the Rayleigh range (ZR) and the axis distance from the center. The Rayleigh range is the length at which the spot size increases to 1.414w0. 𝑧𝑅 = 𝜋𝑤02 𝜆 (11.3) Combining the equations (11.2) and (11.3), the beam waist is described by the parameter, z. 𝑧 2 𝑤(𝑧) = 𝑤0 √1 + (𝑧 ) 𝑅 (11.4) The beam width with varying z- distance is represented in Fig. 11.2. The equation (11.4) and Fig. 11.2 are the general picture of the Gaussian beam spreading. The spot sizes in specific distances (z- values) are 𝑤(𝑧 = 0) = 𝑤0 , (11.5) 475 Advances in Optics: Reviews. Book Series, Vol. 5 𝑤(𝑧 = 𝑧𝑅 ) = √2𝑤0 , 𝑤(𝑧 ≫ 𝑧0 ) = 𝑤0 𝑧 𝑧𝑅 = 𝜆 𝑧 𝜋𝑤0 (11.6) (11.7) The equation (11.5) is the spot size at the beam waist. At z = zR, the spot size is defined by the equation (11.6). ZR is Rayleigh distance, which is related to the focal depth of the Gaussian beam focusing. zR is also taken as a reliable distance for measuring the focal depth in near-field zone. In far-field approximation (z >> zR), the beam waist can be calculated by the equation (11.7) [29]. Fig. 11.2. The illustration of Gaussian beam. The ideal Gaussian beam intensity distribution and the beam shape are defined by the equation provided above. However, in reality, the output beam shape may not match with the ideal beam shape. It has some defects due to the quality of the equipment and environmental conditions. For this reason, the beam quality factor (M2) is introduced. It provides the value that implies how close the ideal beam shape is obtained. The ideal value is 1 for the ideal case. If the M value is introduced to equations (11.5), (11.6) and (11.7), the spot sizes can be modified to the equations (11.8), (11.9) and (11.10), respectively [28]. 𝑤0,𝑒𝑓𝑓 = 𝑀𝑤0 , 𝑤𝑒𝑓𝑓 = 𝑀𝑤(𝑧), 𝑤𝑒𝑓𝑓 ≅ 𝑀2 476 𝜆 𝑧 𝜋𝑤0,𝑒𝑓𝑓 (11.8) (11.9) (11.10) Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser 11.2.2. Ablation Mechanisms In interaction of laser pulses with solids, there are different kinds of processes depending on the material type and laser properties. These processes can be thermal or nonthermal according to laser wavelength. In a thermal process, laser light energy absorbed by electrons in materials is converted into heat and this transference of energy results in melting and vaporization of the material [30]. For cases of photon energies sufficient to break atomic lattice, ablation of the material can occur without traditional heating processes (nonthermal process) [31]. Laser pulse duration is also a crucial parameter in ablation process. Vapor pressure and recoil pressure of laser light induce removal of materials through melting when lasers in long pulse regime are used [32]. Ablation governed by a continuous wave (CW) and a nanosecond wave is a thermal process. Because pulse durations of these ranges are longer than thermalization time of materials, much of the incident beam energy is conducted to and deposited in exposed regions. Heat affected zone for ns pulses is smaller than those of CW lasers [33]. Material removal using lasers emitting ultrashort pulses can be thermal or non-thermal [34]. Pulse durations of femtosecond and picosecond laser pulses are shorter than the necessary time for energy transference between the electrons and lattice. This mechanism results in very small heat affected zones and low thermal effect enhances micromachining quality of materials [19, 35]. Material ablation takes place when necessary energy is accumulated in processing region. When a beam with intensity below ablation threshold illuminates the material, the temperature of illuminated region will not increase sufficiently for ablation of the material. The ablation threshold of a material depends on different parameters including pulse duration. Ablation threshold for ultrashort pulses is smaller due to rapid pulse energy deposition in laser spot. When a laser beam interacts with a material, the energy of irradiating pulse may be absorbed. Absorption mechanisms are varied depending on the material type and incident photon energy. In semiconductors, incident light is strongly absorbed provided that total energy of incident photons exceeds the energy gap of the material. When photons energy is smaller than the bandgap energy, light passes through the semiconductor as if it is transparent and the interaction of light with the material is weak. At high laser intensities, nonlinear absorption becomes apparent. The bandgap energy of crystalline silicon is 1.11 eV at 300 K [36] corresponding to a wavelength value of 1.1 µm in vacuum. Absorption mechanism in silicon for photons of wavelength longer than 1.1 µm is multiphoton absorption which is a nonlinear process. The simplest form of multiphoton absorption is two-photon absorption (TPA) where simultaneous absorption of two photons occurs. In a two photon absorbing material, light intensity change along the propagation direction is 𝐼(𝑧) = 𝐼0 , 1+𝛽𝑧𝐼0 (11.11) 477 Advances in Optics: Reviews. Book Series, Vol. 5 where 𝑧 is the propagation direction, 𝐼0 is the incident light intensity and 𝛽 is the TPA coefficient of material. The two-photon absorption is defined by the equation 𝑑𝐼 − 𝑑𝑧 = 𝛽𝐼 2 (11.12 Because absorption probability is proportional to the square of intensity, TPA is higher in femtosecond lasers for low pulse energy. Silicon is also transparent to wavelengths longer than 1.1 µm. This characteristic enables the use of lasers operating above this wavelength value in processing of both the surface and bulk of the silicon. For a pulsed laser, optical intensity is 𝐼 = 𝐸 , 𝛥𝑡×𝐴 (11.13) where E is the energy per pulse, ∆t is the pulse duration and A is the beam spot area. A higher energy value is provided as laser pulse duration gets shorter at a specific pulse energy. Required pulse energy for ablation is therefore lower in femtosecond regime as compared to nanosecond and picosecond regimes. The strong intensity dependence of absorption increases applicability of lasers operating in femtosecond regime in multiphoton processing. 11.2.3. Simulations The simulation programs have been started to use during the last decade in order to predict the possible outcomes of the ideas before performing the experiments. In this part, the light intensity distribution inside and vicinity of two different diameter spheres, which are placed on smooth Si surface, under the laser exposure will be discussed. Two software (MNPBEM and Lumerical-FDTD) were used for our purpose. The first one is MNPBEM. It is a MATLAB toolbox, which was developed by F. J. Garcia de Abajo and A. Howie [37, 38]. By using the approach of the boundary element method (BEM), it solves Maxwell`s equations for arbitrary shaped (homogeneous or inhomogeneous) dielectric mediums which are expressed as surface integral equations. A second alternative approach to calculate the intensity distribution of the model is the Finite Difference Time Domain (FDTD) method. The method was introduced by Kane Yee [39] in 1966. Similar to MNPBEM, Lumerical-FDTD also uses Maxwell’s equations in order to calculate electric field distribution in and underlying region of models by integrating over lines, surfaces or volumes. In this technique, field distributions can be modeled in near- and far-field regions as well [40-42]. 478 Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser 11.3. Experimental Details Our previous direct writing study on Si by using microspheres has been reported in OPAL`2019 [43]. The custom design femtosecond laser having 1.55 µm central wavelength, and 1 W average power at 1.0 MHz repetition rate was used in the experiment. The experimental setup was quite similar to the updated version which will be covered in detail. In first trials, the initial step was to determine the limits of the processing parameters by observing the hot spot under the surface of the silicon in order to eliminate surface damaging. To do so, a piece of c-Si (p-type) double side polished having 1 mm thickness, 2 mm wide (mechanically diced and polished) and 3 cm length was prepared. In Fig. 11.3a, the beam was focused by a lens having f = 8 mm and NA = 0.5. To monitor the heated zone, which is created by the focused beam, a modified Si based camera was used as an artificial IR camera (IR filter removed). The thermal radiation of the heated zone at the focus generates photons having enough energy (>1.11 eV) for the detection by Si based chips. The second step was to place the microspheres on the surface. In Fig. 11.3b, scanning electron microscope (SEM) image shows that the spheres were distributed by drop casting method and the single layer microsphere clusters containing varying number of spheres was observed on the surface. In Fig. 11.3c (region 1), the ablation threshold of c-Si with TPA was reached by using 4.5 µm diameter spheres. In Fig. 11.3c (region 2), the misaligned sample with an amount of Rayleigh length along the optical axis also created the linear pattern on c-Si surface. According to the first results in Fig. 11.3, it was shown that the microspheres can be used as a micro lens array for processing c-Si with femtosecond laser at 1.55 µm wavelength by two photon absorption. Fig. 11.3. (a) The plasma tail is observed by the camera; (b) SEM image of the distributed microspheres having 4.5 µm in diameter; (c) In region 1, the ablation under the spheres was observed. In region 2, the ablation due to the direct writing was observed. 479 Advances in Optics: Reviews. Book Series, Vol. 5 In order to process the large array structures by focusing the beam with microspheres in different diameters, it is needed to run the simulation for estimating the intensity distribution on the surface by microlensing and to enlarge the area of c-Si which is covered by the sphere clusters, also required to improve the experiment setup for pattering larger areas (>1 cm). To do so, we first run some simulations for estimating intensity distribution of the focused beam on the surface by MNPBEM and Lumerical-FDTD. The spheres, which are 1.5 µm and 4.5 µm in diameters, have an index of refraction 1.565 at 1.55 µm. Si substrate has an index of refraction 3.5 at the same wavelength. A microsphere was placed on Si substrate and illuminated by a linearly polarized plane wave. Under these conditions, the field distribution around the sphere is shown in Fig. 11.4. The outcomes of simulations indicate that both spheres in different diameters focus the incoming plane wave to spot, which is close to the interface. Both simulation approaches gave the similar results. Fig. 11.4. (a, b) The intensity distribution of focused laser beam with 1.5 µm and 4.5 µm diameter spheres by MNP- BEM, respectively. (c, d) The intensity distribution of focused laser beam with 1.5 µm and 4.5 µm diameter spheres by Lumerical- FDTD, respectively. Secondly, the spin coating method was used for coating microspheres having 4.5 µm and 1.5 µm in diameter. Certain amounts of aqueous suspension of these microspheres were deposited by spin coating on silicon wafers which were made hydrophilic with an oxygen plasma treatment before the deposition process. After a 24 h drying at room temperature, a self-assembled microsphere layer was generated on silicon substrate. In Fig. 11.7, the optical microscope image of the microspheres (1.5 µm in diameter) layers is shown. Thirdly, the optical path was changed to the periscopic configuration via integrating the 480 Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser new 3-axis stage in Fig. 11.5a. The stage has resolution ~20 nm in z-axis and ~10 nm in horizontal axes. The maximum scanning speeds are 20 mm/s in z-axis and 10 mm/s in horizontal axes. Fig. 11.5b illustrates processing of c-Si by means of microspheres. Further details about the experiment including spheres, processing and scanning parameters are listed in Table 11.1. Fig. 11.5. (a) Position of the laser spot is controlled with a 3-axis stage. The polarization direction of the beam on sample is shown. (b) The laser beam is focused on multiple areas by microspheres. Table 11.1. Experimental parameters of Fig. 11.6 and Fig. 11.7, respectively. Spheres 4.5 µm Laser Average Output Power 300 mW Repetition Frequency 1 MHz Pulse Duration 480 fs Stage Parameters Step Distance 50 µm Scan Speed 1.5 mm/s Diameter 1.5 µm 470 mW 1 MHz 480 fs 50 µm 2.5 mm/s In Fig. 11.6, optical microscope and SEM images obtained using 4.5 µm spheres are shown. After processing c-Si, the microspheres were cleaned and the patterned structure was revealed. In Fig. 11.6 (c), SEM image shows that the line thickness of the processed zone is around 13.6 µm and the damaged zones were only observed in that region. Therefore, the diameter of the spot on the surface can be taken the same as the line thickness (13.6 µm). 481 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 11.6. (a) The optical microscope image of the processed area with 50 µm step distance. It is shown that 4.5 µm diameter spheres were burned under the 1.55 µm laser pulses. The spheres were evaporated in an instant of time under the laser exposure. (b) After the acetone cleaning and chemical etching of the sample, the modified area was observed by optical microscope. (c) The SEM image of the region is shown in (b). Three regions (A, B, C) are indicated in Fig. 11.7. These are corresponding to multilayer, monolayer and uncoated regions on c-Si, respectively. In region A, multilayer of spheres (1.5 µm in diameter) causes the beam to defocus and the ablation threshold of c-Si cannot be reached. The borders of the regions show that the direct writing is mostly functional on monolayer area (region B). Intersection of the regions B and C is an evidence for TPA processing of c-Si triggered by the microsphere focusing. Between the region A and B, the surface pattering is also stopped because of the defocusing. Different from SEM image given in Fig. 11.6c, a periodic structuring, which was created by 1.5 µm spheres, is observed with SEM image in Fig. 11.8. The periodicity of the structures inside the processed region is approximately ~300 nm and the simulation could not predict this outcome. 11.5. Conclusions In literature, microlensing with microspheres has been already demonstrated for the surface structuring of c-Si via ultrafast lasers having central wavelength which has enough photon energy for direct absorption. However, in this work, we showed that submicron processing of c-Si in TPA regime was achieved by focusing the femtosecond laser at 482 Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser 1.55 µm wavelength with microspheres. c-Si was successfully processed by using two types of microspheres (1.5 µm and 4.5 µm in diameters). The localized spot by the microspheres overcame the diffraction limits of the optical system. The focal depth of the system is limited by Rayleigh length, which covers 1.5 µm microspheres as well. It is observed that the spot was defocused by multiple scatterings inside of the spheres in multilayer region and the monolayer region was the only processed area among 3-regions. An unexpected outcome of this work was to observe the periodic structuring on Si under the damaged spots by focusing the beam with 1.5 µm microspheres. The spatial period of the structures was calculated around ~300 nm (~λ/5). The orientation of the periodic structures depends on the polarization direction of the electric field. The structures aligned perpendicular to the electric field polarization in our results. In the future, the triggering mechanism of the periodic structuring under the ablated regions by TPA will be investigated in detail. Fig. 11.7. Three regions which are shown as A, B and C are corresponding microsphere (1.5 µm in diameter) layers on Si surface. Region A includes multilayer microsphere coated area, B is the monolayer coated region and C is Si substrate. Fig. 11.8. SEM images of submicron periodic structure patterning with 1.5 µm microspheres shown in Fig. 11.7. 483 Advances in Optics: Reviews. Book Series, Vol. 5 Acknowledgements The authors acknowledge the support and contributions of Mona Zolfaghari Borra and Nasim Seyedpour Esmaeilzad at Micro and Nano Technology (MNT) program of the Middle East Technical University in the initial phase of this study. This work is supported by TÜBİTAK under grant nr. 118F132 and the Middle East Technical University under grant nr. BAP-105-2018-2798. We thank Zafer Artvin for the simulations. References [1]. B. Cowan, Optical damage threshold of silicon for ultrafast infrared pulses, AIP Conference Proceedings, Vol. 877, 2006, 837. [2]. F. Priolo, T. Gregorkiewicz, M. Galli, T. F. Krauss, Silicon nanostructures for photonics and photovoltaics, Nature Nanotechnology, Vol. 9, 2014, pp. 19-32. [3]. A. E. Lim, J. Song, Q. Fang, C. Li, X. Tu, N. Duan, K. K. Chen, R. P. Tern, T. 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W. Razali, N. N. M. Rashid, S. R. Hosman, N. K. Aid, Surface patterning of silicon and germanium using focused ion beam for the development of FinFET structure, in Proceedings of the IEEE Regional Symposium on Micro and Nanoelectronics (RSM’19), 2019, pp. 116-118. [17]. D. Grojo, A. Mouskeftaras, P. Delaporte, S. Lei, Limitations to laser machining of silicon using femtosecond micro-Bessel beams in the infrared, Journal of Applied Physics, Vol. 117, 2015, 153105. [18]. R. Gattass, R. Mazur, Femtosecond laser micromachining in transparent materials, Nature Photonics, Vol. 2, 2008, pp. 219-225. [19]. X. Liu, D. Du, G. Mourou, Laser ablation and micromachining with ultrashort laser pulses, IEEE Journal of Quantum Electronics, Vol. 33, Issue 10, 1997, pp. 1706-1716. [20]. L. Li, M. Hong, M. Schmidt, M. Zhong, A. Malshe, B. l. Huis, V. Kovalenko, Laser nano-manufacturing – State of the art and challenges, CIRP Annals – Manufacturing Technology, Vol. 60, Issue 2, 2011, pp. 735-755. [21]. Y. 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B, Vol. 65, Issue 11, 2002, 115418. 485 Advances in Optics: Reviews. Book Series, Vol. 5 [38]. U. Hohenester, A. Trügler, MNPBEM – A MATLAB toolbox for the simulation of plasmonic nanoparticles, Comp. Phys. Commun., Vol. 183, 2012, 370. [39]. K. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation, Vol. 14, Issue 3, 1966, pp. 302-307. [40]. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press Series, New York, 2000. [41]. A. Taflove, Computational Electromagnetics: The Finite-Difference Time-Domain Method, Artech House, Boston, 2005. [42]. S. D. Gedney, Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromagnetics, Morgan & Claypool Publishers, 2011. [43]. F. İdikut, M. Borra, N. Esmaeilzad, A. Bek, Subsurface silicon processing by microsphere focusing of ultrafast infrared laser, in Proceedings of the 2nd International Conference on Optics, Photonics and Lasers (OPAL’19), Amsterdam, Netherland, 24-26 April 2019, pp. 69-70. [44]. A. Dhiman, Silicon photonics: A review, IOSR Journal of Applied Physics (IOSR-JAP), Vol. 3, Issue 5, 2013, pp. 67-79. [45]. X. Liu, D. Du, G. Mourou, Laser ablation and micromachining with ultrashort laser pulses, IEEE J. Quantum Electron., Vol. 33, Issue 10, 1997, pp. 1706-1716. 486 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence Chapter 12 Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence Annette M. Colón-Mercado, Karla M. Vázquez-Vélez, Francheska M. Colón-González, John R. Castro-Suárez, Nataly J. Galán-Freyle, Vladimir Villanueva-López, Edwin Caballero-Agosto, Leonardo C. Pacheco-Londoño, and Samuel P. Hernandez-Rivera1 12.1. Introduction Quantum cascade lasers (QCLs) have emerged as a promising technology for developing applications in defense, security, and pharmaceutical processes. QCLs possess clear advantages over conventional techniques due to their sensitivity, high signal-to-noise ratios (S/N), rapid response, and remote detection capabilities. The high output power of these mid-infrared (MIR) sources allows the detection of analytes in the nanogram to the picogram mass regime. These fast-scanning systems can achieve responses in less than five seconds, offer portable detection at room temperature, and are commercially available. QCLs are semiconductor lasers that undergo sub-interband transitions that are precisely engineered in the solid-state device. Modifying the heterostructure in the lasing device allows multiple quantum well transitions. There is a constant effort to continue improving techniques to detect analytes based on taking advantage of the QCLs' capabilities. Most previous research has focused on detection using infrared spectroscopy (IRS) primary modalities: absorption, transmission, emission, reflection, and attenuated total reflection. In this contribution, we describe a novel optical setup that couples a QCL source to a compact grazing angle (GA) probe mount that enables a double-pass beam trajectory that allows a higher sensitivity. At the same time, measurements at the GA warrants a higher coverage area for detection and enhances the S/N, particularly when Annette M. Colón-Mercado Department of Chemistry, University of Puerto Rico, Mayagüez 487 Advances in Optics: Reviews. Book Series, Vol. 5 using highly reflective surfaces as sample substrates. By adapting the QCL’s vibrational response to multivariate analysis (MVA) and artificial intelligence (AI) algorithms, a better representation of the data is obtained. Therefore, the QCL-GAP represents a promising new path for detecting analytes in a wide range of applications. 12.2. Quantum Cascade Lasers The invention of QCLs originated in 1994 at Bell Labs by Faist, Capasso, and collaborators [1]. QCLs are semiconductor injection lasers that differ from ordinary diode lasers in the transition band process [2]. For the lasing process, diode lasers depend on a direct transition between the valence band and the conduction band, and the population inversion is achieved by optical or electrical pumping. On the other hand, QCLs the lasing process is based on interband transitions in which the population inversion is distributed over several overlapping bands, resulting in a relatively broad spectrum. One of the significant advances of QCLs is that modifications to the quantum confinements allow tailoring the emission wavelength. Therefore, a wide spectral range from MIR to the submillimeter-wave region of the electromagnetic (EM) spectrum can be achieved. In addition, wide bandgap materials can be used without the necessity of incrementing the temperature. In a QCL, the conduction band acquires a staircase shape where electrons stream down, and photons are emitted at each step of the process. Population inversion occurs in each step where a discrete transition is achieved by electron tunneling. The resulting emission has a higher density, thus enhancing the laser power. Fig. 12.1 shows the conduction band diagram, where a discrete transition between a higher energy level n = 3 to n = 2 occurs. Consequently, n = 2 and n = 1. In each transition, a photon is emitted while the tunneling escape time decreases. A final transition between this first active region to the following occurs from n = 1 to E3, where the photon is injected to the next level. Fig. 12.1. Schematic of the conduction band structure of a QCL laser. 488 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence 12.3. Grazing Angle Reflection-Absorption Infrared Spectroscopy (RAIRS) 12.3.1. Design of the Optical Setup One of the most sensitive techniques of infrared spectroscopy (IRS) is MIR spectroscopy operating at the GA since this arrangement enables the optical effect known as reflectionabsorption infrared spectroscopy (RAIRS). The technique requires coupling a MIR excitation source to an optical probe mount at an incident angle 80 with respect to the surface normal). In the present case, a QCL based pre-dispersive spectrometer (model LaserScan, Block Engineering, Southborough, MA) was coupled to a compact GA probe (GAP) consisting of two compact gold-coated plane mirrors fixed near the grazing angle (~82) as illustrated in Fig. 12.2). The LaserScan contains a thermoelectrically cooled mercury-cadmium-telluride (MCT) detector and a 3-in (diam) zinc selenide (ZnSe) lens that focuses the laser beam onto the right-hand mirror. Reflecting the laser beam expands the laser beam. Gold (Au) coated mirrors were positioned as illustrated in Fig. 12.2. The first gold-coated mirror was placed at 49 with respect to the surface normal, deflecting the light at an angle of 8 with respect to the surface, hence 82 with respect to the surface normal as shown in Fig. 12.2 [3]. Fig. 12.2. Setup for QCL-GAP for enabling RAIRS. The LaserScan system has an output beam size of approximately 2×4 mm2. The GAP optical setup expands the laser spot on the sample plane, forming an elliptical illuminated image for a wider sampling area. Thermal measurements show that the first pass of the laser beam creates an ellipse of 40×4 mm2, and the returning beam produces an even larger spot on the sample of 50×5 mm2. A comparison between a directly reflected beam from a metal substrate and the GA probe setup is shown in Fig. 12.3. 489 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 12.3. Thermal images of the laser beam incident on an aluminum substrate. a) QCL directly reflected beam b) QCL GAP reflected beam. 12.3.2. Advantages over Conventional Methods Developing techniques for detecting emerging defense and security threats and organic residues after cleaning sessions in pharma industries has been a top priority of this research group for several years. Various methods for detecting and characterizing active pharmaceutical ingredients (APIs) and high explosives (HEs) traces have been devised [4]. Chromatographic techniques have been the leading choice for analysis due to their robustness and low detection limits (DL) [5]. In spectroscopic applications in these critical tasks, Raman scattering (RS) and IRS have been the two main branches used to characterize the vibrational signatures of targets. Modalities based on these techniques are non-invasive, non-destructive, and require minimal sample preparation, making them ideal for rapid characterization. To improve the performance, investigators have started to couple existing techniques. In the MIR region, fiber optic probes have been coupled to Fourier transform infrared (FT-IR) technology, achieving a higher sensitivity when operating at the grazing angle [6-8]. The mentioned techniques still lack essential features that are needed when detecting traces. High-performance liquid chromatography (HPCL) and mass spectrometry coupled to gas chromatography (GC-MS) are time-consuming and require sample preparation. Laser excitation of RS in the visible region of the EM spectrum can be plagued with fluorescence from the target analytes or impurities present in the samples. Another possibility is sample thermal decomposition due to the high laser powers required in the near-infrared region arising from the ~4 dependence of the inelastic scattering in a Raman event [9]. IRS also presents limitations due to the radiation density of a thermal source which results in low absorptions/reflectances [10]. The necessity of a fast and reliable technology without sample preparation to detect residues at trace level was evident. QCLs tunable lasers have emerged as a promising new source of MIR radiation capable of overcoming the challenges discussed. A laser source meets portability requirements and adds stability, high absorptions/reflectances, low spectral noise. It is a high brightness source that can be adapted to any optical setting to enhance the S/N. QCLs have the advantage of working under ambient conditions, having minimal interference from external sources [11, 12]. Furthermore, the high output power makes the system an excellent choice for the standoff detection of threats [13]. Our approach using the QCL-GAP optical setting goes one step further: coupling data acquisition with powerful 490 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence statistical models for data analysis to demonstrate how the technique surpasses common challenges by understanding the behavior of the vibrational patterns in the spectra obtained. This approach is critical when analyzing RAIRS data. 12.4. Sample Preparation for Trace Detection Analyses 12.4.1. Deposition Techniques Quality assurance of an analyte on a sample surface is directly related to the deposition technique used for the analysis. Sample homogeneity and the time of residence of the analyte on the substrate are two factors that significantly affect the detection hence, the vibrational response of the instrument. There are complex deposition techniques used in the field of materials science and engineering for manufacturing devices, particularly in the areas of computer chips and electronic devices. Chemical vapor deposition (CVD), molecular beam epitaxy (MBE), and ion beam deposition (IBD) have distinguished throughout the years. The need for practical analyte deposition methods for preparing samples and standards has led to new alternate deposition techniques such as spin coating, inkjet printing, air spray, sample immersion (partial or total) meniscus-guided coating as shown in Fig. 12.4. Spin coating deposition uses the centrifugal force caused by the rotation of the surface to spread the liquid into a film. The thickness of the uniform film ranges from 1-10 μm [14].. Regarding printing methods used by our research group, thermal inkjet printing deposits samples on surfaces with high homogeneity, providing a thin film deposition. The selection of the proper parameters ensures a deposition that simulates real-world samples. In the meniscus-guided coating deposition, the smearing method uses a Teflon stub to spread the analyte on the surface of the substrate with the advantage of minimal loss of the product. The properties of Teflon assist in minimizing analyte aggregation; therefore, the sample is deposited on the surface nearly homogeneously. When using absorbent materials as substrates, a dip-coating approach is recommended. Although, dip coating presents the challenge of not easily being quantifiable. Sample sieving is also of great interest. Fig. 12.4. Deposition methods: (a) Smearing; (b) Inkjet printer; (c) Smearing using micropipette); (d) Spin coating. 491 Advances in Optics: Reviews. Book Series, Vol. 5 12.4.2. Effect of the Substrate: Specular and Diffuse Reflectance Measurements When analyzing the influence of the substrate, its chemical and physical characteristics should be considered. Understanding the substrate role in the deposition process and the final product obtained helps determine the results obtained in the measurements conducted. This is when analyzing the reflectance signals of the analytes and the substrates (background) on the spectra collected. The roughness and reflectivity of the substrate affect the laser beam trajectory. Hence, two different optical effects are present: specular reflectance and diffuse reflectance. Specular reflectance results in the reflection of light occurring at the same angle as the incident angle (with respect to the surface normal) for highly reflective smooth substrates [15]. In diffuse reflectance, the light is reflected by rough surfaces in all directions. Most of the light reflected from rough materials comes from the surface, but some come from internal reflections. Diffuse reflectance is mainly observed in non-reflective and opaque substrates. 12.4.3. Sample Preparation of HEs and APIs We have recently established sample deposition protocols that improve the validation of the detection process of the sensors developed. We have worked with all the mentioned deposition methods. Here, we present the validation of the QCL-GAP system by using reflective and non-reflective standards obtained from external laboratories (for cross-validation procedures) or prepared in our laboratories. HE standards were obtained from the Navy Research Lab at Washington, DC. These were prepared as part of a Department of Homeland Security (DHS) program that focuses on standoff explosives detection on vehicles (SED-V) and also as part of DHS research program entitled Methods for Optical Detection of Explosives (MoDEx). Sample coupons were prepared by sieving deposition of HEs on acrylonitrile butadiene styrene (ABS) plastic substrates and aluminum (Al) substrates [5]. The HE included as primary standards were RDX, Tetryl, and PETN. These 1 in. × 1 in. substrates (coupons) were dosed with well-known surface concentrations of the HEs and used for validating our QCL-GAP methodology. Fig. 12.5 shows the white light micrographs of the coupons. In addition, the Applied Physics Laboratory of the Johns Hopkins University (Baltimore, MD) also participated in the MoDEx program, also prepared and sent primary standards in the form of brushed AL coupons loaded with HEs. These were used to characterize and compare two QCL acquisition modes: direct back-reflectance and GAP diffuse reflectance. Thermal inkjet printing was used to deposit the HEs. The HE selected for the study was C-4, which is a formulation of the nitramine RDX (91.0 %), dioctyl sebacate (5.3 %), polyisobutylene (2.1 %), and mineral/motor oil (1.6 %) [16] HPLC analysis confirmed the mass deposited on the substrates, this being in a 0.1-10.0 g/cm2 range. Fig. 12.6 shows white light micrographs of the coupons. Besides HE detection experiments at near trace level, experiments designed for pharmaceutical cleaning validation applications were conducted to expand the applicability of the methodology based on QCL-GAP. Sample preparation was carried 492 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence using direct smearing deposition with a micropipette tip, minimizing the contact with the surface. In this case, the Teflon stub was not required. We analyzed stainless steel plates loaded with an API to look for similar substrates that represent surfaces inside a typical pharmaceutical reactor contaminated with organic residues. The analytes under study were [2-n-butyl-3-((2'-(1H-tetrazol-5-yl) biphenyl-4-yl) methyl)-1,3-diazaspiro (4,4) non-1-en-4-one; the API] commonly known as irbesartan (IRBS) and RDX. Our goal was to develop and analyze standards that represent real-life surfaces to generate a robust technology capable of detecting trace amounts of analytes. Fig. 12.5. White-light micrographs of ABS and Al substrates loaded with HEs used as standards for QCL-GAP validation. (a) Set of ABS substrates with HEs at various surface concentrations; (b) the second set of Al substrates loaded with the HEs. Fig. 12.6. White light micrographs of JHU standards. Al brushed substrates are loaded with C-4 HE in a 0.1 - 10.0 ug/cm2 range. 493 Advances in Optics: Reviews. Book Series, Vol. 5 12.5. Trace Detection Analyses Using QCL-GAP 12.5.1. Detection of HEs When detecting HEs and APIs, the vibrational response from the QCL-GAP is used to achieve a functional group characterization of the target molecules. There are several steps to follow: instrument performance check, spectra acquisition of the substrate, and spectra acquisition of the analyte. A spectrum of solid potassium bromide (KBr) is performed to check the system stability determining the variation in the noise across the entire spectrum. This process is denoted as acquiring the blank or the 100 % line (100 PC). To obtain the substrate spectrum without the analytes, a KBr spectrum was used as a reference. Fig. 12.7 shows typical spectra of the substrates without HE. Fig. 12.7. 100 % line and reflectance spectra for Al and ABS substrates without the HE. A total of 24 spectra were acquired for each sample. The obtained spectra were characterized by identifying the vibrational signals of the HEs by comparing them with the ab initio computed MIR spectrum (in the case of Tetryl) or with the measured MIR spectra for the neat substances (RDX and PETN). The IR spectrum of Tetryl was simulated using OPT+FREQ-job setup in Gaussian-09 (Gaussian, Inc., Wallingford, Connecticut, USA). The spectrum was generated using Density Functional Theory 6-311+G(2d,p) basis set and B3LYP hybrid functional. The aliphatic explosives RDX and PETN were synthesized in the lab, and the spectra were acquired using a small sample to compare with the primary standards from the MoDEx program. Fig. 12.8 shows an average of the QCL-GAP reflectance spectra obtained for each of the HEs deposited on ABS and Al. As mentioned before, signals can be affected by substrate roughness (ABS), but this must be taken into account. Also, RDX and PETN were synthesized in our facilities impurities could be present, affecting the results. Tables 12.1 and 12.2 show the identification of the characteristic signals for each HEs. 494 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence Fig. 12.8. Normalized MIR reflectance spectra for (a) RDX (b) Tetryl and (c) PETN. Table 12.1. QCL reflectance signals for the HEs deposited on ABS substrates. HE/ABS Tetryl RDX PETN Experimental band position (cm-1) 954 1312 1348 1149 1359 1080 942 1370 1171 Reference band position (QCL, cm-1) 954 1287 1342 1148 1349 1093 954 1369 1176 Vibrational mode Sym. ring str. N-NO2 str. N-NO2 sym. str. CNstr./CH3 bend NO2 sym. str. CN str./ring str. CH2 + CCC def. CH2 wag + C5 sk. + NO2 rock The study was extended by including the APL-JHU standards to compare two QCL acquisition modes. The first is denoted as point detection. The second approach was based on the QCL-GAP arrangement, where the optical setup allowed a larger sampling area and a faster and more accurate analysis. Fig. 12.9 shows the two modes studied. 495 Advances in Optics: Reviews. Book Series, Vol. 5 Table 12.2. QCL reflectance signals for the HEs deposited on Al substrates. HE/Al Tetryl RDX PETN Experimental band position (cm-1) 1144 1337 1075 954 1310 1351 955 1372 1175 Reference band position (QCL, cm-1) 1148 1349 1093 954 1287 1342 954 1369 1176 Vibrational mode CN str./CH3 bend NO2 sym. Str. CN str./ring str. Sym. ring str. N-NO2 str. N-NO2 sym. str. CH2 + CCC def. CH2 wag + C5 + NO2 rock. Fig. 12.9. QCL acquisition modes: (A) Single-path optical mount with Mini-QCLTM system, (B) QCL-GAP double-path optical mount. The sample coupons from JHU were loaded on the center of the substrates with C-4. The averaged normalized spectra from the samples using both systems are shown in Fig. 12.10. To identify C-4, it was necessary to use RDX QCL reflectance bands as a reference, which are located at: 1220 cm-1 (-C-N stretch), 1261 cm-1 (-NO2 stretch), 1310 cm-1 (-N-N stretch), and 1370 cm-1 (-NO2 stretch). In the analysis, the most prominent signal of RDX, -NO2, located at 1261 cm-1, was used as evidence of the presence of the explosive in C-4. The low surface loadings on the JHU samples required a confirmatory technique to understand how C-4 crystals were distributed. Raman microspectroscopy (RmS) allowed 496 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence the identification of the HE by focusing the particles at the sub-micron scale. Micrographs and spectra confirmed the presence of the crystals on the Al substrate for the 0.1 g/cm2 and 10 g/cm2 surface concentrations as shown in Fig. 12.11. The identification of the NO2 vibrational signatures were found as (vs) symmetric and (vas) antisymmetric modes in 1309 cm-1 for C-4, RDX at 1306, and 1304 cm-1. An excellent correlation between the peak location for vs-NO2 of RDX and C-4 can be observed when comparing the RmS spectra of RDX and C-4 in Fig. 12.12. Antisymmetric vibrations of -NO2 are shown at 1536 cm-1 for RDX and 1527 cm-1 for C-4. RmS confirmed the presence of this analyte on the surface, which was previously detected using both QCL setups. Fig. 12.10. Normalized spectra of C4 deposited on Al substrate. (A) System 1; (B) System 2. Fig. 12.11. Micrographs of C-4 on Al: (a) 0.107 μg/cm2; (b) 1.074 μg/cm2 and (c) 11.096 μg/cm2. 497 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 12.12. Raman spectra of C-4 deposited on Al substrates. RDX and substrate. 12.5.2. Detection of API The QCL-GAP back reflection setup was also used to detect irbesartan (IRBS, API, deposited on stainless steel substrates. RDX was also detected on the same substrate to compare the methodology's performance, but our primary focus in this section will be IRBS. As part of the data analysis, six (6) vertical regions were used following the elliptical illumination image formed by the GAP on the sample plane. Figs. 12.13a and 12.13b illustrate the diffuse reflectance spectra for the concentration loadings of both analytes. Band assignment for functional group characterization was achieved by reducing the spectral range to 1200 to 1428 cm-1. IRBS showed significant signals at 1236 cm-1 tentatively assigned to δNNH + νNN + νC-C bridge bond in biphenyl and at 1337.11 cm-1 corresponding to CH2 wagging were observed. Interference patterns present in this region are shown in the clean substrates and on the samples. These patterns will be further discussed in Section 12.6.3. 498 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence Fig. 12.13. (a) Comparison of average spectra of clean SS substrate RDX/SS (~16 ng/cm2), IRBS/SS (~60 ng/cm2) and their reference QCL spectra; (b) magnification of spectra (1200-1428 cm-1). 12.6. Improving Detection Performance 12.6.1. Development of Spectral Library Instrumentation for vibrational spectroscopy, such as infrared and Raman, has evolved in recent years. IRS has been used in academic, research, and industrial applications and has become an alternative analytical technique in fieldwork. These developments are due to technological advances and the miniaturization of equipment. The latter has led to smaller, easier to use, smarter devices, and faster data processing. A good example of these are the QCL based portable and handled spectrometers. However, to achieve quick and timely decisions during field analyses, a suitable data processing analysis of the acquired spectra is necessary. To answer some of the following questions: is a target analyte present? (i.e. detection) Is this material what I think it is ? (i.e. confirmation) What is this? (i.e. identification) To which group does it belong? (i.e. classification) How much of this is present? (i.e. quantification) [17]. Various algorithms that allow detection, confirmation, classification, identification, and quantitation of a target are used to carry out this task. All these algorithms perform a comparison between the target spectra (test) 499 Advances in Optics: Reviews. Book Series, Vol. 5 and the spectra of known compounds stored in a spectroscopic library. The simplest spectral comparison is based on calculating the hit quality index (HQI) values [18]. The HQI is a numerical quantity that indicates the correlation between two spectra and has been widely used in spectroscopy to indicate the degree of spectral matching in library searches [19-24]. In the process, a spectrum of an unknown compound is compared with all spectra of known compounds in a library, and the best match is determined based on the calculated HQI values. HQI values are calculated using different algorithms, but the most commonly used are spectral correlation algorithms. A spectral correlation algorithm utilizes the Pearson product-moment correlation coefficient, rxy, which is a measure of the strength and direction (Euclidean distance) of the linear relationship between two variables. This parameter is defined as the covariance of the variables divided by the product of their standard deviations. In this case, the spectral correlation algorithm is applied between two spectra: a reference (spectra form library) and an unknown spectrum to be identified [20-23]. The rxy values are calculated using Eq. (12.1): 𝑟𝑥𝑦 = 1 𝑛 ∑ 𝑥𝑖 𝑦𝑖 − (∑ 𝑥𝑖 )(∑ 𝑦𝑖 ) 1 2 1 2 1⁄2 [{∑ 𝑥𝑖2 − (∑ 𝑥𝑖 ) }{∑ 𝑦𝑖2 − (∑ 𝑦𝑖 ) } ] 𝑛 𝑛 , (12.1) where 𝑥 and 𝑦 represent the spectral responses of the reference spectrum and an unknown spectrum, respectively, measured at the ith wavenumber for a set of n corresponding wavenumber points; in this case, the HQI assumes values between +1 and -1. An HQI value of +1 is obtained when the spectral similarities are maximum (the unknown spectrum is identical to a library spectrum), -1 when the spectral similarities are inverted (the unknown spectrum is identical to a library spectrum but with peaks inverted with respect to the library spectrum), for example, when the transmittance spectrum is compared to an absorbance spectrum, and 0 when there is no spectral similarity. Often, the values are rescaled to values between 0 and 1, taking the square of r (r2) [22, 23] to yield positive values between 0 and 1. The spectral correlation method can be used in algorithms for identification or verification. The identification mode compares a test spectrum to all of the spectra in a reference library and ranks the library spectra using an HQI value. In identification mode, the library spectrum that gives the highest HQI value identifies the most probable composition of the test material. The verification mode is more common in the pharmaceutical industry (also in national defense and forensic science applications). It is widely used to test incoming pharmaceutical raw materials and excipient to verify if a compound is an explosive or a controlled substance. The identification mode compares each sample spectrum acquired to all of the library spectra. The verification mode operates by comparing the spectrum of each sample to a predetermined library reference spectrum. Verification tests use a minimum HQI threshold to assign a “Pass” or “Fail” determination to the sample under study. A typical minimum HQI threshold for a “Pass” is 0.85 [22]. This section illustrates the usefulness of the spectral correlation method, calculating the HQI value for the identification of HE deposited on metallic surfaces. The explosives were deposited on an Al plate, using the smearing technique (already described). Then, the QCL reflectance spectra were recorded using back-reflectance (point detection setup). The spectra were divided into two groups: calibration spectra and test spectra. The respective 500 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence HQI value for each explosive tested was calculated. Ten explosives were analyzed, as shown in Table 12.3. The analyzed compounds correspond to aromatic explosives (TNT, 2,4-DNT, p-nitrotoluene (p-NT), 3,4-dinitrotoluene (3,4-DNT), and 2,4-dinitrophenol (2,4-DNP)), aliphatic explosives (PETN and RDX), and peroxide explosives (TMDD, HMTD, and TATP), Fig. 12.14 shows the spectra of HEs used. Table 12.3 shows the spectral correlation coefficients for unknown spectra of HE deposited on Al substrates. Before calculating the r values, a spectral normalization was performed on the entire spectral range of 1000-1600 cm-1 for both the unknown (measured) and library spectra (measured on Al substrates). Normalization of a spectrum for library searching is a two-step process, as recommended by ASTM E2310-04 (2009): “Standard Guide for Use of Spectral Searching by Curve Matching Algorithms with Data Recorded Using MIR Spectroscopy.” [18] First, the minimum spectral response value in the selected spectral range is subtracted from the complete spectral response in the same range. The resulting values are then scaled by dividing by the maximum value in that range. The net result is a spectrum in which the minimum intensity value is zero (0) and the maximum value is one (1). It is generally accepted to consider a spectrum of an unknown compound similar to a spectrum stored in the library when the spectral correlation coefficient is larger than ~0.85. However, the spectral resolution of IR spectra (reference and target), the shift of baseline, and the sample preparation method's difference may be considered factors affecting correlation coefficients. In these cases, an HQI equal to 0.8 may be sufficient to confirm a questioned spectrum. As is shown in Table 12.3, spectra were correctly identified when QCL spectroscopy coupled with spectral correlation algorithms were used. In general, all the calculated HQIs presented high values, allowing to confirm that the questioned spectrum belongs to an HE. Table 12.3. Values of spectral correlation coefficients (HQI) for spectra of HEs using QCL. Library spectra TNT 2,4 DNT pNT 3,4 DNT 2,4 DNP PETN RDX TMDD HMTD TATP TNT 1.0 0.9 0.7 0.6 0.4 0.0 0.1 0.0 0.7 -0.2 2,4 DNT 0.9 1.0 0.6 0.6 0.4 0.0 0.0 0.1 0.6 -0.2 pNT 0.7 0.7 0.9 0.3 0.4 0.2 0.4 0.1 0.6 0.1 HQI/ Spectral Correlation Spectra Questioned 3,4 2,4 PETN RDX TMDD DNT DNP 0.8 0.6 0.0 0.1 0.2 0.8 0.6 0.0 0.0 0.2 0.5 0.6 0.1 0.3 0.5 0.8 0.2 -0.1 -0.2 0.1 0.2 0.8 -0.1 0.1 0.3 -0.1 0.0 1.0 0.4 0.4 -0.1 0.2 0.5 1.0 0.6 0.0 0.1 0.3 0.3 0.8 0.7 0.5 0.0 0.2 0.3 -0.5 0.3 -0.2 0.2 0.2 HMTD TATP 0.6 0.5 0.6 0.1 0.2 -0.1 0.2 0.1 0.8 0.2 -0.1 0.0 0.4 -0.3 0.2 0.9 0.2 0.1 0.1 0.9 501 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 12.14. IR Spectra of HEs tested for calculating HQI values using spectral correlation. 502 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence 12.6.2. MVA Routines MVA offers new possibilities for data treatment in many fields of science, engineering, agriculture, or even in the arts for understanding and helping to express the dependence and independence of the variables in a data set. When conducting a study, 30 % effort is directed to mathematical and statistical analysis to evaluate the data. How MVA plays a vital role in detecting chemical threats and cleaning validation of vessels in pharmaceutical industries? In the detection arena, the research has migrated from a univariate analysis to consider all the obtained data to decode a pattern fully. In cleaning validation, it is essential to have a panorama of the possible impurities that may appear when conducting this procedure and MVA permits to control it. In both situations, MVA can be used as a qualitative analysis tool or for quantitative analysis, which are different processes of treating the data depending on the need in the study. Applying MVA with data obtained from QCL helps us elucidate behaviors that we could not observe with the naked eye. This analysis provides more relevant chemical information from a data set, and it can help us reduce the dimensions in a data set to identify patterns in the data. Today several statistical programs allow better interpretation of the data on a larger scale, making the process more user-friendly. This approach provides extensive research of vibrational peak analysis of molecules in the MIR region, seeing where there is more significant variation, and identifying patterns that discriminate the data. 12.6.2.1. Data Preprocessing How can we be sure that we are analyzing the chemical effects of the data? It is crucial that when the vibrational data is obtained, before performing the MVA, some pretreatments (scatter corrections) are applied. These must be performed to ensure that the variations observed in the mathematical models are from the chemical effects of the analyte and not from a physical influence that may appear. Baseline distortion, dispersion, and noise are typical physical effects that can affect spectral data. These effects are particle size, shape, surface roughness, and others may mask the samples' vibrations' chemical aspects [25]. These effects can be dominant in the spectral data, which can invalidate subsequent steps of the MVA models. For this reason, one should consider using some of the following preprocessing steps to reduce these effects: a) Normalization Normalization helps us reduce the baseline effect in the spectra, enabling all data sets to be on the same scale. It allows seeing the interactions that would lead to spectral variations not related to the composition of the samples. This pretreatment places the data on a scale of 0 to 1, according to their respective intensities. If one desires to take the data to a quantitative MVA analysis, special care must be taken since the proportions in data can change. This can lead to having concentration values different from what they are. 503 Advances in Optics: Reviews. Book Series, Vol. 5 b) Savitzky-Golay Algorithm for Data Smoothing Noise can be a factor that interferes with the variations that we need to observe in the vibrational data. This algorithm allows estimating the spectrum's shape using a polynomial fit of a specific rank and degree. c) Multiplicative Scatter Correction (MSC) and Standard Normal Variate (SNV) These two also help with the baseline distortion and data dispersion that appear in some of the spectra. MSC can estimate the coefficient by regressing the spectrum to correct with a reference. SNV is a scaling algorithm, where the average of a data set is subtracted, and each spectrum is divided by the standard deviation of the data set. It should be considered that these should only be applied to spectra that present physical information influenced by these multiplicative effects. If misused, they can provide erroneous results or when performing predictive MVA models [25]. 12.6.2.2. Data Exploration Once the spectral data is prepared, what would be the next step? When conducting an MVA, it is essential to visualize the behavior of the spectral data to obtain a correct interpretation of the variation that the study is trying to explain. With the help of what is known as principal component analysis (PCA), an exploratory data analysis (EDA) can be done. EPA is usually used for qualitative data analysis. PCA models provide information based on the score and the loadings plots. The variation of the data is decomposed in the form of principal components. The selection of the principal components falls into the direction that has the most variation, and this is the principal component 1 (PC 1). PC 2 is selected orthogonal or perpendicular to PC 1. This procedure is carried out for all the PCs required to explain all the variations in the data. This is represented on a score plot as the distance from the mean along a PC for a sample. The loadings plot provides information on the variables of the data. In spectral data, the variables will be the wavenumber (cm-1) to see how much they contribute to the PC by correlation. Before applying a PCA model, the data must be subjected to other treatments that help visualize them, as seen in Table 12.4. This is done to determine how the spectral data behaves and see which preprocessing step provides more information to interpret. The best pretreatments of the spectral data are usually the first (1SG) and second (2SG) Savizky-Golay derivatives. Derivatives are the most common treatments applied to spectral data to resolve peak overlapping (thus enhancing spectral resolution) and to eliminate constant and linear baseline drifts between samples. One disadvantage is that application of derivatives could increase noise, but this can be evaluated using the loadings plots to determine which variables affect the variation on the data. How do we know which of the pretreatments is ideal for my data? Selection of the optimum preprocessing step (or combination of them) can be based on the patterns observed in the data when PCA is performed with that pretreatment (s). The objective is 504 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence to place the data in clusters with similarities but differences compared to other clusters. There are variations in the clusters’ characteristics. This helps to find an ideal model that explains the topic of interest in the research. One also constantly seeks to obtain the most significant interpretation of the data with the least number of PCs. Table 12.4. Set of pretreatments that can be used in chemometric routines. SNV MSC 1SG 2SG Pretreatments SNV + 1SG SNV + 2SG MSC + 1SG MSC + 2SG 1SG + SNV 1SG +MSC 2SG + SNV 2SG + MSC How do the PCs correlate with spectral data? Once the characterization of the data is carried out and the ideal preprocessing step for the PCA is identified, one can further explore what is happening with this data. An example of this can be seen in the PCAs shown in Fig. 12.15, where the treatment selected was 1SG + SNV. In these PCAs, the objective was to differentiate between the HEs deposited on ABS and Al, where the main components PC 1 and PC 2 showed a higher percentage of variation for both substrates. As seen in both PCAs, the clusters are from the differences between the vibrational signatures of PETN, RDX, and Tetryl. Another way to understand where the most remarkable variation occurs in the spectral data is by viewing the loadings plots. The loadings are the map of the variables; in this case, the wavenumber (cm-1). The vibrational signals from each HE can be evaluated on the loading plot to fully corroborate that the variation analyzed is the one from each one of the vibrations of the nitro groups. The correlations are shown in Fig. 12.16. Fig. 12.15. Scores plot for PETN, RDX, and Tetryl on (A) ABS and (B) Al substrate in terms of PC-2 versus PC-1. The 95 % confidence level used in the analysis is also shown (dotted trace). 505 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 12.16. Scores plots for PC-1 and PC-2 for HEs/ABS performed for data collected in the range of 833 to 1428 cm−1. 12.6.2.3. Data Discrimination QCL spectral data can be further investigated by using MVA prediction and classification models. Prediction models are known as Soft independent modeling by class (SIMCA) or PCA projections. SIMCA allows objects to display intrinsic individualities as well as common pattern characteristics. However, only the common properties of the classes are modeled by PCA [26]. This model takes advantage of similarities among members of the same class where a PCA is built for each class where samples well described by the model are "accepted," and others are "rejected" or not classified. The model generates a classification table where the samples from each of the model's classes are listed. Can a quantitative analysis be done using spectral data from QCL? Partial Least Squares (PLS) is an excellent tool to analyzed quantitative data. As seen in Fig. 12.17, models like this can detect explosives at low concentrations when using these multivariate routines. For this reason, regardless of how the sample is created, the combination of PLS models with data obtained from QCL helps the detection of explosives, making it have much lower detection limits. Likewise, there are other mathematical models such as support vector machines (SVM or VM), k-nearest neighbors (KNN), partial least-squarediscriminant analysis (PLS-DA), linear discrimination analysis (LDA), among others that can be applied to generate classification models to assist in the detection analysis. 506 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence Fig. 12.17. PLS regression plots of predicted vs. measured surface concentrations for C-4 deposited on Al substrate: (A) System 1. (B) System 2. Spectral range, 950 -1374 cm-1. 12.6.3. Fourier Transform Data Preprocessing Algorithm When the QCL-GAP system is well aligned, the interaction of the MIR light with the sample can produce interference patterns. At first, these patterns were considered a problem in the acquired vibrational response, but it was noticed that they displayed changes in the patterns observed when the analyte was deposited on the surface. The preprocessing step to determine if these patterns could be used for identification and quantification analysis using chemometric methods is fast Fourier transform (FFT). FFT is a tool used to extract robust and accurate information from reflectance spectra. This algorithm transforms a function from the time domain to the frequency domain, and conversely. The result of applying the FFT is a complex function that consists of an imaginary part [Im(n)] and a real part [Re(n)]. Its magnitude is expressed as the absolute value of lz(n)l, and “n” represents the number of points used. This information was used for discrimination and quantification using PLS-DA and PLS. Figs. 12.18A and 12.18B show the FFT preprocessing steps applied to the clean SS substrate when the GAP mirrors are aligned and misaligned. Fig. 12.18. (a) Representation of interference patterns; (b) applying FFT preprocessing, and representative spectral patterns obtained after applying FFT preprocessing. 507 Advances in Optics: Reviews. Book Series, Vol. 5 12.7. Conclusions A QCL-GAP system was presented in this chapter as a novel technique for detecting APIs and HEs. Our approach demonstrated the system's feasibility in detecting HEs for national defense and security applications and cleaning validation in pharmaceutical batch reactors applications. The GAP setup presents many advantages that surpass common challenges posed by conventional point detection systems: a double-pass (double reflectance) system and the RAIRS advantage, known as the most sensitive MIR technique. Detection quality can be affected by the deposition technique selected and the substrate properties. Deposition techniques can affect the vibrational response in the analysis, mainly due to the lack of homogeneity and the time of residence of the analyte. In terms of the substrate, non-reflective and opaque surfaces produce a diffuse reflectance signal, while reflective substrates with high homogeneity produce specular reflectance. Coupling chemometric routines (MVA) to the QCL-GAP detection system improves the performance of the analysis. FFT-derived functions using MATLAB were tested to evaluate the complete spectral range, including the regions where interference patterns were present. These patterns determined the alignment of the system. The QCL-GAP described herein will have a far-reaching impact due to its high specificity and high sensitivity results when analyzing low concentrations of analytes. Acknowledgments Support from the US Department of Homeland Security under Award Number 2013-ST061-ED0001 is acknowledged. However, this document's views and conclusions are those of the authors. They should not be considered a representation of the official policies, either expressed or implied, of the US Department of Homeland Security. The US Naval Research Lab prepared the sieved sample coupons mentioned in Section 12.4.3 under Department of Homeland Security Inter-Agency Agreement HSHQPM-15-X-00166. References [1]. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, Quantum cascade laser, Science, Vol. 264, Issue 5158, 1994, pp. 553-556. [2]. Y. Yao, A. J. Hoffman, C. F. Gmachl, Mid-infrared quantum cascade lasers, Nature Photonics, Vol. 6, Issue 7, Jul. 28, 2012, pp. 432-439. [3]. S. P. Hernández-Rivera, L. C. Pacheco, Grazing Angle Probe Mount for Quantum Cascade Lasers, Patent, Ser. #62/587,557, 2017, submitted. [4]. W. Zhou, D. Wu, R. McClintock, S. Slivken, M. Razeghi, High performance monolithic, broadly tunable mid-infrared quantum cascade lasers, Optica, Vol. 4, Issue 10, Oct. 2017, pp. 1228-1231. [5]. A. M. Colón-Mercado, K. M. Vázquez-Vélez, E. Caballero-Agosto, V. Villanueva-López, R. Infante-Castillo, S. P. Hernández-Rivera, Detection of high explosives samples deposited on reflective and matte substrates using mid-infrared laser spectroscopy at the grazing angle of incidence assisted by multivariate analysis, Opt. Eng., Vol. 59, Issue 9, Aug. 2020, 092011. [6]. O. M. Primera-Pedrozo, Y. M. Soto-Feliciano, L. C. Pacheco-Londoño, S. P. HernándezRivera, Detection of high explosives using reflection absorption infrared spectroscopy with fiber coupled grazing angle probe/FTIR, Sens. Imaging, Vol. 10, Issues 1-2, 2009, pp. 1-13. 508 Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy at the Grazing Angle Incidence [7]. O. M. Primera-Pedrozo, L. C. Pacheco-Londono, L. F. De la Torre-Quintana, S. P. Hernandez-Rivera, R. T. Chamberlain, R. T. Lareau, Use of fiber optic coupled FT-IR in detection of explosives on surfaces, Proceedings of SPIE, Vol. 5403, 2004. [8]. M. L. Ramírez-Cedeño, N. Gaensbauer, H. Félix-Rivera, W. Ortiz-Rivera, L. PachecoLondoño, S. P. Hernández-Rivera, Fiber optic coupled raman based detection of hazardous liquids concealed in commercial products, Int. J. Spectrosc., Vol. 2012, 2012, 463731. [9]. T. Vankeirsbilck, et al., Applications of Raman spectroscopy in pharmaceutical analysis, TrAC – Trends Anal. Chem., Vol. 21, Issue 12, 2002, pp. 869-877. [10]. D. T. D. Childs, R. A. Hogg, D. G. Revin, I. U. Rehman, J. W. Cockburn, S. J. Matcher, Sensitivity advantage of QCL tunable-laser mid-infrared spectroscopy over FTIR spectroscopy, Appl. Spectrosc. Rev., Vol. 50, Issue 10, 2015, pp. 822-839. [11]. C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, Recent progress in quantum cascade lasers and applications, Reports Prog. Phys., Vol. 64, Issue 11, Oct. 2001, pp. 1533-1601. [12]. J. Faist, et al., High power mid-infrared (λ∼5 μm) quantum cascade lasers operating above room temperature, Appl. Phys. Lett., Vol. 68, Issue 26, Jun. 1996, pp. 3680-3682. [13]. F. Capasso et al., New frontiers in quantum cascade lasers and applications, IEEE J. Sel. Top. Quantum Electron., Vol. 6, Issue 6, 2000, pp. 931-946. [14]. R. Smith, C. Peters, H. Inomata, Historical background and applications, in Supercritical Fluid Science and Technology, Vol. 4, Elsevier B. V., 2013, pp. 175-273. [15]. A. Kumar, Object appearance and colour, Book Chapter in Principles of Colour and Appearance Measurement Object Appearance, Colour Perception and Instrumental Measurement, Elsevier, 2014, pp.53-102. [16]. S. Chong, B. Long, J. K. Maddry, V. S. Bebarta, P. C. Ng, Acute C4 ingestion and toxicity: Presentation and management, Cureus, Vol. 12, Issue 3, Mar. 2020, e7294. [17]. C. Gardner, R. L. Green, Identification and confirmation algorithms for handheld spectrometers, in Encyclopedia of Analytical Chemistry, John Wiley & Sons Ltd, 2000, pp. 1-18. [18]. Standard Guide for Use of Spectral Searching by Curve Matching Algorithms with Data Recorded Using Mid-Infrared Spectroscopy, https://www.astm.org/DATABASE.CART/ HISTORICAL/E2310-04R09.htm [19]. K. Banas, et al., Multivariate analysis techniques in the forensics investigation of the postblast residues by means of Fourier transform-infrared spectroscopy, Anal. Chem., Vol. 82, Issue 7, Apr. 2010, pp. 3038-3044. [20]. E. C. Wong, J. C. Reid, Data-reduction and -search system for digital absorbance spectra, Appl. Spectrosc., Vol. 20, Issue 5, Sep. 1966, pp. 320-325. [21]. K. Tanabe, S. Saëki, Computer retrieval of infrared spectra by a correlation coefficient method, Anal. Chem., Vol. 47, Issue 1, Jan. 1975, pp. 118-122. [22]. J. D. Rodriguez, B. J. Westenberger, L. F. Buhse, J. F. Kauffman, Quantitative evaluation of the sensitivity of library-based Raman spectral correlation methods, Anal. Chem., Vol. 83, Issue 11, Jun. 2011, pp. 4061-4067. [23]. S. Lee, H. Lee, H. Chung, New discrimination method combining hit quality index based spectral matching and voting, Anal. Chim. Acta, Vol. 758, Jan. 2013, pp. 58-65. [24]. FT-IR Search Algorithm – Assessing the Quality of a Match, https://www.spectroscopyonline.com/view/ft-ir-search-algorithm-assessing-quality-match [25]. J. W. Jin, et al., Quantitative spectroscopic analysis of heterogeneous mixtures: The correction of multiplicative effects caused by variations in physical properties of samples, Anal. Chem., Vol. 84, Issue 1, Jan. 2012, pp. 320-326. [26]. K. H. Esbensen, An introduction to Multivariate Data Analysis, including Process Analytical Technology (PAT) and Quality by Design (QbD), Camo, Oslo, 2018. 509 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers Chapter 13 Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers Paul Ikechukwu Iroegbu, Min Liu and Tao Zhu1 13.1. Introduction Light is vital to the human race and our persistent attempt to understand the world around us. Our earliest efforts to comprehend light dates far back to the dawn of civilization. If we characterize the existence and evolution of light ranging from antiquity until the middle of the 19th century, its classification is appropriately recognized as incoherent light source. Such a phrase in classification is adopted because it consists of the sun, candlelight, sodium lamp, or light bulb. In the middle of the 19th century, a new light source called Laser (Light Amplification by Stimulated Emission of Radiation), which is a coherent light source, was invented and ultimately ended up as one of the greatest inventions of the 20th century [1]. The concept of laser acting as a light source, as introduced in 1958 by Townes and Schawlow, has revolutionized the world we live in by actively participating in a vast range of modern applications. These applications are ultrahigh geographical monitoring and surveying with detection ranges as high as pico-meters, highly delicate eye surgery, security, and surveillance devices capable of measuring vibrations. This chapter presents the fundamental dynamics of lasers, their classification, and their applications to optical fiber communication and optical fiber sensing systems. Furthermore, we present the advances done on lasers regarding linewidth compression, linewidth measurements, noise characterization, wavelength switching, and wavelength tuning. Lastly, we present our predictions on the future directions and development of lasers. Paul Ikechukwu Iroegbu School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, China 511 Advances in Optics: Reviews. Book Series, Vol. 5 13.2. Laser Dynamics The term laser is an acronym that stands for (Light Amplification by Stimulated Emission of Radiation). This is because the output light signal from a laser is amplified in a cavity regarded as the gain medium, which is fundamentally saturated by atoms or molecules. The amplification of light signal in the gain medium is referred to as excitation, whereas the reverse process, referred to as de-excitation, leads to photons' release. A propagational encounter between these released photons and other excited atoms inevitably stimulates those excited atoms/molecules sufficient enough to release their stored energy in a chain reaction manner that creates an avalanche emission of radiation in the form of photons with equivalent wavelength. In the knowledge that these released photons are of equivalent wavelength and monochromatic (i.e., stimulated by the same photons propagating nearby), it is further understood that these emitted photons are as well in phase with each other. This phenomenon establishes what is globally regarded and accepted as a coherent light source and consistent across all laser types. Lasers generally consist of three core elements; pump, gain medium, and resonant cavity. The difference in types of lasers is mainly associated with the difference in the gain medium. Since light sources are classified into coherent light sources and incoherent light source, candle lights and sunlight are practical examples of incoherent light source [2]. 13.3. Classification of Lasers As a result of Townes and Schawlow's laser concept, various lasers have sprouted out over the years with a broad range of physical and operating parameters. Provided lasers are classified according to their gain medium's physical state, the description of lasers into various classes such as solid-state lasers, liquid state lasers, and gas lasers is adopted. However, besides these renowned classifications of lasers, other rather special cases exist to classify lasers, such as free-electron lasers whose gain medium consists of free electrons propagating past a periodic magnetic field relativistic velocity. In addition, more special classifications of lasers are infrared lasers, visible lasers, UV (Ultra-Violet) lasers, and X-ray lasers which are resultants of cases where lasers are characterized by the wavelength of their emitted radiation [3]. Furthermore, a semiconductor laser and fiber laser are both classified as independent classes of their own. An excellent example of a semiconductor laser is a Distributed Feedback (DFB) laser. On the other hand, a perfect example of a rare-earth-doped fiber laser is an EDF (erbium-doped fiber). 13.4. Lasers and Systems In this section, we present semiconductor laser and fiber laser. We also present how the laser parameters of semiconductor lasers and fiber lasers influence their applications in communication and sensing systems. Various types of lasers are used as a light source in a large array of modern-day system applications. However, we will emphasize semiconductor lasers and fiber lasers and their advances and applications in optical communication and optical sensing systems. Our distinct choice and focus on the advances relating to semiconductor lasers, and fiber lasers are centered on the fact that 512 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers there has ever remained an essential device in communication and sensing systems. This is because lasers lasing at single longitudinal mode (SLM), whose laser linewidth is compressed, low phase or frequency noise, low relative intensity noise (RIN), and wavelength-tunable or switchable, are critical to the performance level of these systems. For example, the quality of the light source employed in terms of its compressed spectral linewidth, phase or frequency noise, RIN, and wavelength-tunable range or switching has significant impacts on an optical fiber communication system, such as its bandwidth (i.e., channel capacity), signal-to-noise ratio (SNR), and sensitivity. Likewise, it also affects the performance of optical fiber sensing systems, such as their detection range, sensitivity, and accuracy. Hence, the development of these systems is tremendously accelerated by the advances in these lasers. 13.4.1. Semiconductor Lasers DFB lasers, which were simultaneously invented at three different industrial labs, MIT Lincoln Laboratory, IBM, and General Electric, were inspired by the desire to investigate a relatively new laser gain medium. These lasers have been highly desirable due to their small size, low cost, stable single-longitudinal mode operation, and easy compatibility with other devices [4]. Semiconductor optical amplifier (SOA), on the other hand, was discovered around the early 1960s, wherein a gallium arsenide laser diode was found to be a suitable laser diode. It was also discovered to be a good amplifier of infrared light, provided its reflectivity at the facets is decreased with an anti-reflection coating material. These lasers have been highly desirable due to their fast gain dynamics and simultaneous wavelength operation. The gain medium commonly used in an SOA and a DFB are indium, gallium, arsenide, phosphide (InGaAsP). SOA is often employed to amplify transmitted optical light signals by stimulated emission in a fiber waveguide, which attenuates long fiber length spans. Depending on the need, an SOA can be adopted as a post-amplifier, in-line amplifier, pre-amplifier, or a wavelength converter [4-6]. 13.4.2. Fiber Lasers The emergence and continuous advancement of fiber lasers whose active gain medium is a piece of optical fiber that is doped with a rare-earth element such as thulium, ytterbium, erbium, etc., are associated with the progressive development of various manufacturing technologies. For example, low-loss erbium-doped fiber (EDF) with a commendable monochromaticity was successfully prepared by the SB Poole team at the University of Southampton, UK, in the 1980s [7]. Since then, it has received wide-spread applications due to its overall merits, such as high conversion efficiency, low insertion loss, hightemperature resistance, high output power, and strong anti-electromagnetic interference capabilities. 13.4.3. Lasers in Optical Fiber Communication Systems A typical optical fiber communication system consists of a transmitter (laser light source), medium (optical fiber), and a receiver (optical detectors). Semiconductor lasers and fiber 513 Advances in Optics: Reviews. Book Series, Vol. 5 lasers are good examples of light sources in optical fiber communication systems. A laser's linewidth, wavelength, phase or frequency noise, and relative intensity noise, have often been of particular interest and high demand in academia and industry for optical communication systems. This is because the laser light source's linewidth often affects the bandwidth of the fiber. In addition, the wavelength determines the number of accessible channels. The phase or frequency noise determines the long-term stability of the laser and the detection accuracy of local oscillators (LO) in an optical receiver. Likewise, the relative intensity noise influences the long-term stability of the laser and the sensitivity of the optical receiver. Hence, stringent requirements on the parameters of the laser light source employed have ever remained increasingly dominant over the years. These requirements are inspired by the advent of emerging and promising optical communication technologies that strive to resolve the growing capacity demands of optical networks in long-haul, metro, and local area networks. Some of these technologies are wavelength division multiplexing (WDM), dense wavelength division multiplexing (DWDM), digital coherent technology, multi-level modulation formats, and spatial division multiplexing (SDM). 13.4.3.1. Lasers and Optical Communication System Technological Evolution Over the years, considering the growing demand for large transmission capacity, academia and industry are actively seeking to develop a system whose transmission capacity spans beyond 10-Gbits/s. These same systems will be capable of adopting advanced modulation formats such as differential phase-shift keying (DPSK), differential quadrature phase-shift keying (DQPSK) as opposed to an intensity modulation (IM) format. In order to meet fastgrowing internet traffic, a combination of advanced multi-level modulation formats in a coherent WDM/DWDM system that utilizes a wavelength-tunable or switchable laser is required. For example, a wavelength tunable range of 100 nm, can guarantee a high-speed fiber optic transmission system with bitrates that span over 100 Gbits/s in today's optical networks [8]. Regardless of the emergence of various technologies to resolve issues relating to high capacity demand, it is essential to note that a broad array, if not all of today's optical networks, adopts a combination of these technologies to resolve the same problem. This is practically evident in today's 100 Gb/s optical transmission systems that adopt a combination of advanced modulation formats such as quadrature phase-shift keying (QPSK), which work together with polarization division multiplexing (PDM) [9]. Like optical transmitters, the most current focus is to actualize digital coherent receivers that can handle bit rates as high as 400 Gb/s in the nearest future. The trade-off in requirements between an optical transmitter and an optical receiver is this. In optical receivers, phase signals are detected or identified with a local oscillator, which practically suggests that a local oscillator's detection accuracy in an optical receiver is much dependent on the phase noise of the laser. Furthermore, the feasibility of attaining high transmission capacity typically requires a laser of low relative intensity noise. This is because the RIN noise level of the laser affects the SNR of the system. Hence, the need to obtain a tunable laser with a low phase or frequency noise and low RIN is paramount [10]. 514 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers In regards to the latest advancement towards attaining massive data rate increase and breaking the Shannon capacity limit, SDM is today's world-leading technology to achieve this tremendous progress in a submarine network system. This is because SDM is the last unexplored physical dimension in optical communication. It is merely the act of adopting multi-core fiber (MCF) or a few-mode fiber (FMF) to exceed the expected capacity crunch. However, little attention was paid to this promising novel technology over the years due to the effectiveness of other emerging technologies such as WDM, multi-level modulation formats, and digital coherent technology. These technologies collectively and individually played a critical role in resolving issues relating to capacity demand. More so, as the increase in capacity demand continued to rise exponentially in the past, more technologies to mitigate this issue were developed and employed, such as Erbium-doped fiber amplifier (EDFA) [11] and PDM [12]. This ultimately limited the need to adopt SDM. More importantly, it is anticipated that the massive increase in demand for higher data rates will directly supersede today's optical network's bandwidth capacity, which will create that which is globally coined as 'capacity crunch' [13]. In the era of capacity crunch (i.e., around the year 2018), global commercial systems are expected to deliver bandwidth capacity as high as 80 Tbit/s, which is way beyond the limit of today's single-mode fiber (SMF) based optical networks [14]. Consequentially, SDM, as an unexplored optical domain, is regarded as a promising technology and a monumental leap forward towards resolving these capacity demands. In retrospect to the previous discussion on modulation formats that modulates phase and the amplitude to increase data rates significantly in an optical communication system that adopts a wavelength-tunable or switchable laser, another spectral characteristic of the laser in terms of a narrow linewidth is likewise essential. The need to modulate the phase of the light source directly places stringent requirements on the laser linewidth. For example, a communication system that adopts a multi-level advanced modulation format such as a 16-state quadrature amplitude modulation (16-QAM) requires a laser linewidth below 100 kHz. However, recent research proves that higher modulation formats in contrast to 16-QAM, such as 32-QAM and 64-QAM, need the MHz laser linewidth of a laser to be deeply compressed down to the single kHz level [15]. This is because narrowlinewidth lasers with suitable phase or frequency noise are vital in optical communication systems. Such systems seek a light source of long coherence length or superior phase sensitivity to expand the capacity of long-haul, metro, and short-distance optical networks. 13.4.4. Lasers in Optical Fiber Sensing Systems A typical optical fiber sensing system consists of a laser, transmission fiber, detectors, and other parts. Once again, semiconductor lasers and fiber lasers are good examples of light sources in optical fiber sensing systems. The quality of the laser employed has often been of particular interest and high demand in academia and industry for sensing systems. This is because the laser light source's linewidth often determines the measurement distance. In addition, the wavelength determines the accuracy and precision of the detection 515 Advances in Optics: Reviews. Book Series, Vol. 5 process. The phase or frequency noise determines the stability and sensitivity of the system. Likewise, the relative intensity noise determines the accuracy of the sensing system towards detecting and demodulating small signals. Hence, stringent requirements on the quality of the laser light source employed have ever remained increasingly dominant over the years. These requirements often influence the performance of various emerging and promising optical sensing technologies such as distributed acoustic sensing (DAS), distributed vibration sensing (DVS), distributed temperature sensing (DTS), and distributed strain sensing (DSS). These technologies find versatile applications in the oil and gas industries, roads, bridges, borders, perimeter security, etc. In order to realize the sensitivity, resolution, accuracy, and sensing range of these technologies used in a distributed optical fiber sensing system, these technologies, such as DAS, DVS, DTS, and DSS, functions alongside various optical techniques. Some of these techniques are phase-sensitive optical time-domain reflectometry (φ-OTDR), optical frequency domain reflectometry (OFDR), Brillouin optical time-domain reflectometry (BOTDR), Brillouin optical time-domain analysis (BOTDA), and Raman optical time-domain reflectometry (ROTDR). Furthermore, other sensing technologies such as light detection and ranging (LIDAR), which is also regarded as a sub-field (i.e., optical remote sensing) within the broader field of optical sensing, also places stringent requirements on the quality of the laser light source employed for various applications. Some of these applications are found in unmanned driving, law enforcement, environmental monitoring, etc. 13.4.4.1. Lasers and Optical Sensing System Technological Evolution LIDAR supports autonomous (self-driving) cars that adopt automotive coherent LIDAR systems during the cruise as a technology for object recognition such as pedestrians and other vehicles. Object recognition is feasible when the receiver of the LIDAR system characterizes the intensity or frequency information corresponding to the reflected laser signal [16, 17]. Narrow-linewidth wavelength-tunable lasers are essential to extend the dynamic sensing range and sensitivity of coherent LIDAR systems in autonomous vehicles to over 400 m. Linewidths of 100 kHz are sufficient for LIDAR applications [18]. When a tunable laser is used in LIDAR, its wavelength needs to be continuously linearly scanned to obtain beat frequency signals related to the state to be measured. The linearity of laser wavelength scanning affects the positioning accuracy of the object to be measured. Suppose we categorize the desired sensing parameters that distributed sensing systems intend to detect. In that case, we can appropriately obtain four categories of optical sensing technologies corresponding to each desired parameter. These categories of sensing technology include DAS [19], DVS [20], DTS [21], and DSS [22]. These sensing technologies of DAS, DVS, DTS, and DSS corresponds to the desired sensing parameter of acoustic, vibration, temperature, and strain, respectively. In order to realize these desired sensing parameters, we need to utilize at least either one of the three types of scatterings in the optical fiber of the distributed sensing system, namely Rayleigh, Brillouin, or Raman scattering. 516 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers The Rayleigh scattering effect in an optical fiber of a distributed sensing system is mainly associated with two techniques. These techniques are φ-OTDR [23-26] and OFDR [27-29]. Consequently, the φ-OTDR is used to detect the dynamic change in vibration and acoustic sensing. In contrast, OFDR detects the slow change in strain and temperature. The Brillouin scattering effect is mainly associated with two techniques, namely BOTDR [30-32] and BOTDA [33-35]. Consequently, the BOTDR and BOTDA are used for detecting strain and temperature. The Raman scattering effect is mainly associated with the ROTDR [36-37], which is used for detecting temperature. Therefore, a distributed sensing system's principle of operation or technique mainly includes the above-listed φ-OTDR, OFDR, BOTDR, BOTDA, and ROTDR. These techniques are used to realize the various sensing technologies listed above by monitoring the physical parameter that changes with time and space within an optical fiber. In fiber-based sensing systems, the fundamental characteristics of an optical signal launched from a laser and utilized to conduct sensing during transmission within the fiber are subjected to changes induced by various parameters. These parameters. are pressure, temperature, polarization, rotation, displacement, frequency, strain, bending, and vibration, which affect its light path. Hence, academia and industry have stringent requirements on the quality of laser sources used in distributed sensing systems. For example, in OFDR based sensing systems, these requirements now demand laser linewidth less than 100 Hz to be adopted in scenarios where the sensing fiber length is 80 km [38]. In addition, the coherence of the laser linewidth directly characterizes the impact of phase or frequency noise of these systems, which hampers the linearity of frequency sweeps. Therefore, low phase or frequency noise lasers are desirable for such systems [21]. OFDR based sensing systems also strive to use a tunable light source to tune the frequency wherein aspects relating to frequency tuning range, frequency tuning accuracy, and frequency tuning time of the tunable light source are considered. Therefore, to achieve high accuracy coherent detection and demodulation of the optical signals in light frequency domain-based systems, it is essential to ensure that the laser possesses fast-periodic sweep frequency attributes. This suggests that the modulation rate of the laser should be higher, with a sweep speed of approximately 10 pm/μs [39]. Similar requirements are demanded as well in BOTDR based sensing systems [40]. Furthermore, we generally require a low RIN laser when demodulating small signals in the φ-OTDR based sensing system [20]. Likewise, suppose we intend to significantly increase the sensing range of a BOTDA based sensing system by employing Raman amplification. In that case, a low RIN laser is desirable [41]. The ROTDR technique can comfortably and independently conduct distributed temperature sensing. However, for case scenarios where a multi-parameter hybrid-based sensing system that can conduct simultaneous temperature and acoustic sensing with the ROTDR technique and the φ-OTDR technique, respectively, a laser linewidth less than 1 kHz is often required. Such a hybrid sensing system utilizes an MCF fiber that contains SDM reflectometers for sensing [42]. 517 Advances in Optics: Reviews. Book Series, Vol. 5 13.5. Linewidth Compression This section presents laser linewidth and various methods used to compress the linewidth of semiconductor lasers and fiber lasers. The laser linewidth is a measurement of the spectral qualities of a laser light source which is predominantly related to the temporal coherence often characterized by a coherence length and time. In other words, it is the full width at half maximum (FWHM) of its optical spectral. In most situations, the linewidth of a laser results from quantum fluctuations generated from the excited spontaneous emission of ions and atoms that propagates without a specific direction and phase. Thus, the photons of each spontaneous emission event randomly propagate in a manner that ensures that an addition to the optical field in terms of a random phase is experienced. Sometimes, the effect of the spontaneous emission ensures that some radiation is inseparable from the stimulated emission, which is centered on the fact that they propagate along an approximate direction. Thus, the core aspect of spontaneous emission noise is none other than the desire to ensure that the laser emits a finite spectral width [43]. In typical practice, the value of a spectral linewidth often varies across different orders of magnitude and is predominantly consistent with the type of laser. The Schawlow-Townes expression, which contributes to the linewidth as a result of phase changes initiated by spontaneous emission, is an expression that is globally accepted and adopted as a fundamental value to characterize a laser’s spectral linewidth. It also suggests that the width of the Lorentzian is inversely proportional to the laser’s output power. The Schawlow-Townes equation aimed to calculate the quantum limit of a laser’s linewidth was put forward even before the emergence of the first laser. This Lorentzian spectral line shape equation can be expressed as:  laser   h   c  / Pout , 2 (13.1) where hv is the photon energy, vc is the bandwidth of the cavity (FWHM), and Pout is the output power of the laser. The Schawlow-Townes linewidth is proportional to the square of the cavity bandwidth divided by the output power. It can also be written as:  wST P2 hw 2  [ ] c , P2  P1 Pout (13.2) where P2 and P1 represents the steady-state lower-level and upper-level population, respectively.  c  (c / 2d )ln( R1R2 ) is the power damping rate of the laser cavity. c represents the speed of light in a vacuum, and d is the laser’s resonator length. R1 and R2 are used to represent the reflectivity of the mirror [44]. 518 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers 13.5.1. Whispering Gallery Mode (WGM) Whispering gallery mode resonator with high quality (Q) factor in the order of 1010, mainly found in calcium fluoride (CaF2) crystal-line disk and characterized by their relatively smaller structure size, is a rather promising candidate for filtering and stability in a laser cavity. For example, Sprenger et al. [45] demonstrated a ring laser linewidth compression system that adopts a calcium fluoride (CaF2) with a diameter of 5 mm, the passive linewidth of 15 MHz, and Q factor of 107 acting as a whispering gallery mode resonator for linewidth compression of an SOA as shown in Fig. 13.1 (a). Compared to other research, the authors further enhanced the stability of their system by adopting a dual prism structure attached to both sides of the CaF2, which ensures that light is coupled to and from the SOA in numerous cyclic roundtrips. Even more, a single longitudinal mode operation is guaranteed by mainly adjusting the coupling on either side of the resonator. Furthermore, the authors characterized their detected compressed laser linewidth by adopting the three-cornered-hat method to ensure absolute accuracy, as shown in Fig. 13.1 (b). To achieve this, in the linewidth detection method, the authors utilized two other lasers alongside the ongoing investigated CaF2 disk laser to beat the output of the lasers and detect them by three photodetectors whose results are shown in Fig. 13.1 (c). These two other lasers are MP 100-Littro configuration grating stabilized laser and Toptica DL Pro laser. According to the authors, by utilizing the renowned Allan deviation, it appears to be that the spectral linewidth of the MP 100-Littro configuration grating stabilized laser is relatively larger in comparison to the other two corresponding lasers, as shown in Fig. 13.1 (d). In other words, after numerous averaging times, the spectral linewidth of the CaF2 disk laser is enhanced by a factor of over 103 order of magnitude. The authors verified their claims by further characterizing the detected spectral linewidth with a traditional self-heterodyne method which adopted a long delay fiber of 45 km, whose results are shown in Fig. 13.1 (e). Remarkably, the authors compressed the passive 15 MHz linewidths of the CaF2 disk laser to 13 kHz for a Q-factor of 103. This linewidth compression method appears to be effective towards deep linewidth compression and ensuring stability to an extent. Utilizing a WGMR (Whispering gallery mode resonator) is arguably an effective approach to compress the spectral linewidth of any commercial semiconductor laser while still maintaining its well-renowned desirable fundamental qualities and further guarantee stability. The reason being, it is an alternative approach to circumvent the drawbacks associated with other linewidth compression methods tailored for semiconductor lasers, which, when adopted, undesirably diminishes the fundamental qualities of semiconductor lasers (i.e., size, weight, and power). Typically, the host material of any WGMR usually contains, for example, some crystallike material. This is because such crystal-like materials are highly transparent and can guarantee that material absorption of the laser light is mitigated. More so, crystals are arguably the most transparent material in the entire electromagnetic spectral. 519 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (c) (b) (d) (e) Fig. 13.1. (a) Stabilized laser setup. Gain from a SOA passes through polarization control (PC) and is coupled through a CaF2 disc, using two prisms and GRIN lenses. 1 % of the 1550 nm emission is coupled out, and unidirectional lasing is ensured by an isolator. (b) Schematic of the frequency mixing for the three-cornered-hat measurement. (c) Allan deviations of simultaneous beat signals between a Toptica DL Pro, a home-built grating-stabilized diode laser (MP 100), and the CaF2 disk stabilized laser. A common 1:5 kHz harmonic due to vibration is removed from the beats with the MP 100 before the Allan deviation calculation. Inset, frequencies of the beats as a function of time (10 ms). MP 100 and CaF2, DL Pro and CaF2, then DL Pro and MP 100 from top to bottom. (d) Allan deviations of the three individual lasers, calculated using the three-cornered-hat method. (e) Self-heterodyne technique measurements. Beats are at the AOM frequency of 100 MHz. From left to right, recording spans of the MP 100, DL Pro, and CaF 2 stabilized laser are 5 MHz, 1 MHz, and 100 kHz, respectively. 3 dB linewidths are 550 kHz, 60 kHz, and 18 kHz. The delay is 45 km [45]. In 2015, Liang et al. [46] demonstrated a diode laser that is self-injection locked to a WGM micro-resonator that consists of a crystalline host material which guaranteed the stability of the heterogeneously integrated laser. Furthermore, the authors ensured that issues relating to size are resolved by meticulously placing the entire component (laser, optics, and resonator) into a single package, as shown in Figs. 13.2 (a), (b), (c). Through the self-injection locking technique, Rayleigh scattering, an effect of surface and/or volumetric inhomogeneities, is backscattered into the laser when the frequency of the emitted propagating light signal interferes with the frequency of the selected resonator mode. Hence, the resultant linewidth is gradually compressed until it achieves dynamic equilibrium. In other words, the 2 MHz free-running laser linewidth was compressed to seven orders of magnitude, as shown in Figs. 13.3 (a), (b). Such a deep linewidth compression was feasible simply because, with the aid of a prism, the authors coupled the light signal from a diode laser into a WGMR whose host material is MgF2 (Magnesium fluoride). The loaded quality factor is ~6 ×108, and its unloaded resonator Q-factor is ~6 ×109. The authors reported that the frequency noise of their laser is on the order of 0.3 Hz Hz-1/2 above 10 kHz, as shown in Fig. 13.4 (a), (b). The 520 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers uniqueness of the self-injection locking technique also improves stability by reducing ambient acoustic noise which affects the laser frequency. (a) (b) (c) Fig. 13.2. Schematic of the experimental set-up. Light from the pump laser (a semiconductor DFB laser) enters the whispering gallery mode resonator (WGMR) through the prism. Part of light is reflected back to the laser because of Rayleigh scattering in the resonator. The light exiting the prism is collimated and used for processing. Insets show power distribution in the resonator mode (a), a picture of the actual resonator (b) as well as a picture of the packaged laser (c) [46]. (a) (b) Fig. 13.3. Power spectral of the RF signal generated by beating two self-injection locked DFB lasers on a fast photodiode. RF carrier frequency is kept at 8.7 GHz, resolution bandwidth is 300 kHz and video bandwidth is 3 kHz. The noise floor is determined by Johnson–Nyquist noise of the photodiode. Inset (a): linewidth measurement performed with 30-Hz resolution bandwidth. Points stand for the experimental data. Continuous red line is a 60-Hz Lorentzian fit of the data. Inset (b): comparison of the RF spectra generated by beating of two self-injection locked lasers (curve (2)) and one injection locked and one free running lasers (curve (1)) [46]. 521 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (b) Fig. 13.4. Spectral purity and stability characteristics of the laser. (a) Linear frequency noise of the RF beat note of the lasers, (1) compared with the noise determined by conversion of the laser power fluctuations to the frequency fluctuations, (2) as well as fundamental thermorefractive, (3) and thermoexpansive (4) noise. (b) Allan deviation interpolated using the frequency noise data (1) and actual measurement of the Allan deviation (2). The 500-s peak in curve (2) results from the air conditioner cycle in the laboratory. This systematic frequency shift exceeds the internal laser noise rather significantly [46]. 13.5.2. Parallel Feedback Mechanism Wang et al. [47] demonstrated a linewidth compression mechanism based on an external parallel feedback cavity. This structure was established by mainly inserting several pieces of MMF (Multimode fiber) whose core diameter is 62.5 µm, length is 15 m, into a linear single-mode fiber cavity as shown in Fig. 13.5 (a). The authors coined this external parallel feedback cavity as a single-mode-multimode-single-mode fiber (SMS) structure which was used in the experimental scheme shown in Fig. 13.5 (b), (c). According to the authors, the parallel feedback mechanism generated by this external parallel feedback cavity is similar to the Weiner effect, alternatively known as the cursor effect. In principle, compared to the propagation of light signals only within the main mode as consistent with an SMF, the propagation of the light signal in an MMF is very different, resulting in a parallel traverse light signal with different propagation constant. Such a phenomenon is present within the authors' external parallel feedback cavity structure, which renders a compression in laser linewidth since the internal cavity of the DFB laser and that of the external structure correspond to a compound resonator cavity. In other words, the propagating light signal within such a compound cavity must conform to every restriction placed by various light paths within the cavity, which ultimately leads to compression in spectral linewidth. The authors compressed the free-running 200 kHz spectral linewidth of a normatively designed DFB laser without an internal isolator down to 1.25 kHz and 430 kHz, as shown in Fig. 13.6 (a), (b), and in Fig. 13.7 (a), (b). This was feasible by optimizing their laser cavity wherein an EDFA is absent and present, respectively, for the same structure at different compression scenarios. Firstly, the reported value of the compressed spectral linewidth (i.e., 1.25 kHz) was characterized by a traditional delayed self-heterodyne 522 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers interferometer (DSHI) system were at different detection scenarios, a delay fiber of 100 m and a single acoustic optic modulator (AOM) of 70 MHz was used in the detection structure at a particular time. Likewise, two different AOM of 70 MHz and 75 MHz frequency shifts alongside the delay fiber of 100 m were used in the interferometer structure at another particular time. (a) (b) (c) Fig. 13.5. (a) The single-mode–multimode–single-mode fiber (SMS) Structure; (b) Experiment setup for compressing the line-width of a DFB Laser with a passive External Parallel Feedback Cavity, and (c) Experiment setup for compressing the line-width of a DFB Laser with an active External Parallel Feedback Cavity [47]. (a) (b) Fig. 13.6. Measurement of laser line-width compressed by the external cavity: (a) The whole span view, and (b) An image zoomed in [47]. 523 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (b) Fig. 13.7. Measurement of laser line-width compressed by the external cavity with an EDFA: (a) is the whole span view, and (b) is an image zoomed in [47]. Secondly, the authors argued that provided the narrow linewidth laser's coherence length is longer than the length of the delay fiber, accuracy is questionable. Therefore, the external parallel feedback cavity, whose parallel feedback mechanism was optimized by attaching an EDFA within the compound structure for loss compensation, had its spectral linewidth rather characterized by a loss-compensated recirculating delayed selfheterodyne interferometer (LC-RDHSI). This interferometer structure consisted of an adjustable gain EDFA, an AOM of 500 kHz frequency shift, and a delay fiber of 100 m. Through the LC-RDHSI, the authors reported a compressed spectral linewidth of 430 Hz. Based on the authors findings, this is indeed in many ways, a commendable approach to attain linewidth compression of such magnitude. 13.5.3. Optical Self-Injection Feedback In 2018, Kasal et al. [48] demonstrated a 1.5 µm distributed feedback-laser diode (DFB-LD) array system with the characteristics of a highly compressed, high output power, and wavelength-tunable merits by adopting two different types of self-optical feedback circuits. According to their findings, in the linewidth compression structure that adopts a monolithically integrated 12 DFB-LD and an optical circulator-based feedback circuit as shown in Fig. 13.8 (a), the authors reported a compressed spectral linewidth of 8 kHz as shown in Fig. 13.8 (b). Similarly, in the linewidth compression structure that adopts a monolithically integrated 12 DFB-LD, a Multimode interferometer (MMI), an SOA, and a partial reflection mirror as shown in Fig. 13.8(c), the authors resolved issues relating to compactness and obtain a spectral linewidth of 11 kHz as shown in Fig. 13.8 (d). In other words, both systems generally adopted the well renowned optical self-injection feedback method for such a large laser diode array with a wavelength oscillation separation of 3.2 nm and lasing at the 1.5 µm wavelength. 524 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers (a) (c) (b) (d) Fig. 13.8. (a) Configuration of tunable DFB LD array with circulator-based self-optical feedback circuit; (b) Delayed self-heterodyne spectrum of LD 9; (c) Configuration of tunable DFB LD array with partial reflection mirror-based self-optical feedback circuit; (d) Delayed self-heterodyne spectrum of LD 9 [48]. A minor drawback to this laser structure that adopts this compression method is the inability to operate all twelve LD simultaneously. Therefore, the performance test of one single LD lasing at 1538.8 nm (i.e., hereinafter referred to as LD 9) was driven at an LD current of 250 mA, whereas the SOA was driven at 400 mA. The stability of the laser structure is enhanced by packaging the LD array, MMI, SOA, and two lenses in a temperature-controlled butterfly-type pigtail module. Even more, the authors packaged the self-optical feedback circuit into a metal structure of reasonable dimensions. The authors ensured that the intrinsic loss of the MMI coupler situated in between the LD array and the SOA is compensated for by the SOA. As a typical self-optical feedback circuit, light signals are fed back into the DFB-LD array after crossing through the polarization maintaining (PM) circulator, whose short loop length of 1.4 m has its propagating optical feedback power finely controlled by a variable optical attenuator attached within the loop. The length of the short loop plays a decisive role in the linewidth compression effect. This is because the Q-factor of the cavity increases provided the length of the loop is meticulously increased, thereby resulting in spectral linewidth compression. The authors adopted the delayed self-heterodyne interferometer (DSHI) method with a delay fiber of 25 km to characterize the compressed spectral linewidth of 8 kHz. Such attention and design is focused on the laser structure that adopts the optical circulator-based feedback circuit. On the other hand, for the same DFB LD array module packaged with its previously mentioned components, the authors replaced the previously used self-optical feedback 525 Advances in Optics: Reviews. Book Series, Vol. 5 circuit with a partial reflection mirror acting as the self-optical feedback mechanism. Such optimization enabled the authors to utilize a PM partial reflection mirror in a linear cavity length of 30 cm to feedback 10 % of optical light signals into the DFB-LD array. In this configuration, the authors studied the linewidth compression behavior of their laser system by fixing the drive current of the LD array to 250 mA and gradually alternating the SOA drive current where they discovered an increase in SOA drive current leads to a decrement in linewidth. In other words, the optical feedback power also increases sufficient to obtain a compressed linewidth that is less than 11 kHz at an optical feedback power above 1.5 mW. For this optimized system, the reported output power was mainly as high as 18 mW. Thus, it demonstrates a slight difference in spectral linewidth and output power compared to the first linewidth compression structure (i.e., the optical circulator-based feedback circuit). The authors claimed that such a difference in investigated performance and reported results for the same LD 9 was induced by the short feedback length of the partial reflection mirror-based linewidth compression structure. This linewidth compression method is a significant improvement in laser stability. 13.5.4. Microfibers Fan, et al. [49] demonstrated a microfiber ring laser whose spectral linewidth was compressed to 2 kHz, and the side mode suppression ratio (SMSR) is as high as 38 dB with an output power higher than 0.95 µW. Structurally wise, the microfiber employed as the gain medium in this laser configuration lasing at the 1536.1 nm wavelength is made up of a 125 µm Er3+/Yb3+ co-doped phosphate glass fiber which is 1.88 µm in diameter. The authors ascertain SLM operation in their laser structure by spooling two fiber knot with the diameter of 206 and 351 µm, thereby creating a high-Q double-knot resonator whose stability was further enhanced by placing the resonator on an MgF2 crystal substrate as shown in Fig. 13.9 (a), (b), (c). The characteristics of a non-flat gain profile of the gain medium, which is ideal for an easier single-mode generation in this laser configuration and its corresponding validation of a satisfactory single longitudinal mode operation without mode hopping or mode competition, are shown in Fig. 13.10 (a), and in Fig. 13.10 (b), respectively. The authors paid particular attention to the joining point between the double-knot resonator and the taper-drawn fiber to ensure that light is evanescently coupled to and from the resonator. This joint point was established by electrostatic and van der Waals force. As a result, the traversing optical light signal within the resonator tends to propagate in an anti-clockwise direction of the double knot. Furthermore, the authors claimed that the inter-twisted part of the double knot ensures that a part of the light signal is recoupled back into the double knot using the evanescent coupling effect. In other words, there is a somewhat interference between the injected pump light signal from the semiconductor LD and the luminescent recirculated light signal, thereby rendering the resonator cavity into a lasing operation with the merits of a 2 kHz compressed laser linewidth as shown in Fig. 13.10 (c). This linewidth compression scheme and method appears to be a costeffective and portable method. 526 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers (a) (b) (c) Fig. 13.9. Configuration of the microfiber laser: (a) Schematic diagram; (b) Optical microscope image of the double-knot resonator with green upconverted photoluminescence, and (c) The scanning electron microscope image of the coupling region of the double-knot structure [49]. (a) (b) (c) Fig. 13.10. (a) Gain characteristics of the Er3+∕Yb3+ co-doped phosphate glass fiber; (b) Single frequency characteristics of the microfiber laser investigated by the scanning Fabry-Perot interferometer; (c) Line shape of the homodyne signal measured with 48.8 km fiber delay [49]. 13.5.5. Electrical Feedback Control In principle, spectral linewidth compression by an electrical feedback method is regarded as converting frequency fluctuations into intensity variations through a frequency discriminator that depends on a transfer function that creates a situation where a fast transmission variation is plotted against its corresponding optical frequency. The transfer 527 Advances in Optics: Reviews. Book Series, Vol. 5 function is created due to either an interference effect of optical wave(s) or the effect of absorption at a specific wavelength. Such a frequency discriminator could be either a π-phase-shifted FBG or a high finesse FP resonator. To that end, Poulin et al. [50] put forward the design choices and the electrical feedback method adopted by TeraXion for laser linewidth compression, as shown in Fig. 13.11 (a). As a linewidth compression method, a typical electrical feedback method adopts a reference photodetector to limit the impact of the RIN of a laser pose towards the error signal. The efficiency of the frequency discriminator is dependent on its isolation against vibrations. Regarding utilizing a feedback loop to correct fast frequency fluctuations that can further reduce frequency noise at high frequency, the authors claimed that TeraXion is currently conducting technological advances to attain it. The findings of their numerical analysis are shown in Fig. 13.11 (b). According to the authors, TeraXion reported using a 34 mm long π-phase-shifted fiber Bragg grating (FBG) of 15 MHz widths serving as a frequency discriminator in their laser configuration whose transmission and reflection spectra are shown in Fig. 13.11 (c), and Fig. 13.11 (d). In order to resolve issues relating to the compactness of a laser module adopting this suppression method, the authors put forward the technological prospects by TeraXion, which aims to reduce the size of the employed FBG for a superior withstand against vibration effects. For any laser that adopts this compression method, the spectral linewidth will be compressed by a servo feedback control system that is proportional to the error signal generated by the intensity variations. (a) (b) (c) (d) Fig. 13.11. (a) Schematic illustration of the electrical feedback method. (b) Simulations showing the power spectral density of frequency noise (PSDFN) for a free-running semiconductor laser (dark blue), for the same laser with electrical feedback for linewidth reduction (light blue) and width an additional fast loop (red). (c), (d) Theoretical reflection (red) and transmission (blue) spectrum of a 34 mm long π-phase shifted FBG providing a 15 MHz wide peak [50]. 528 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers As shown in Fig. 13.12 (a) and Fig. 13.12 (b), the authors reported a linewidth compression of less than 5 kHz (approx. 4.4 kHz) of a 365 kHz free-running semiconductor DFB laser that adopts this compression method in a compact laser module. This laser module consists of a temperature controller, current driver, and other optical and electronic components are integrated into a 10×15×2.5 cm3 sized module. For case scenarios that will adopt the well-renowned heterodyne configuration, in this work, the authors emphasized that an additional LD with the ability to serve as an offset-frequency will be phased locked to the laser under test (LUT). This is done to synthesize pure RF signals attached along the optical parts created to induce the electrical feedback effect. With this electrical feedback method, linewidth compression from a higher order of magnitude to lower orders of magnitude is very much feasible. However, to some extent, the structure of an electrical feedback control method appears to be a somewhat complex electrical system. (a) (b) Fig. 13.12. (a), (b) Measured power spectral density of frequency noise (left) and calculated optical spectrum (right) for the Narrow Linewidth Laser (red) and for the free-running laser (blue) [50]. 13.5.6. Rayleigh Backscattering Our research group put forward a new linewidth compression mechanism centered on Rayleigh backscattering (RBS) with a great potential to be utilized in any waveguide structure operating at any wavebands. For the first time, Zhu et al. [51] simulated and experimentally demonstrated a Rayleigh backscattering linewidth compression model 529 Advances in Optics: Reviews. Book Series, Vol. 5 (RBSLCM). In principle, Rayleigh backscattering has a strong compression effect on the linewidth of a laser. This is because the linewidth of a propagating laser signal after scattering in a high Rayleigh scattering structure is narrower than the linewidth of the incident laser light signal. The authors claimed that when an incident light signal is injected into an optical fiber, the RBS signal centered at the central wavelength of the scattering intensity profile decreases faster only at a region beyond the central wavelength. This is feasible since RBS is regarded to be inversely proportional to the fourth power of the incident light signal. This same RBS signal undergoes a process of amplification, thereby generating an RBS signal but with a relatively higher Rayleigh coefficient when it encounters an active fiber or gain medium such as the EDF employed by the authors. In principle, the light signal is usually scattered at every scattering point during propagation which then serves as an input for the next scattering point. In other words, the amplified light signal recreates the RBS signal with a spectral linewidth that is comparably narrower than the incident light signal. Consequently, the spectral linewidth of a laser will be compressed for every excitation and amplification of cyclic roundtrips within the laser cavity. The numerical analysis of the author's RBSLCM model utilized an incident Lorentzian linewidth of 5.03 × 10-5 nm at a fixed number of 5000 iterations in a simulated waveguide of 100 scattering points, as shown in Fig. 13.13 (a), (b), (c). Upon successful 5000 iterations, the authors discovered that the laser linewidth was compressed by ~ 99.89 %. Furthermore, the authors discovered that the scattering medium's length and damping rate play a decisive role in the linewidth compression. (b) (a) (c) Fig. 13.13. Simulation of RBS-based linewidth compression mechanism: (a) Rayleigh backscattering spectrum; (b) Power spectra with different iteration number; (c) Power spectra under different scattering factors after 5,000 iterations with one scattering point [51]. 530 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers To further validate this novel mechanism, the authors utilized and investigated this compression mechanism in a fiber ring laser system which consists of a 12 m long EDF serving as the gain medium that is a fusion spliced and connected to an array of other optical components, as shown in Fig. 13.14 (a). The authors ensured that the two points (point A and point B) presented a laser structure wherein the RBS effect can be applied to or isolated from the ring laser system at any given time. Therefore, another part of their laser structure was established by connecting another optical circulator (OC) to a high Rayleigh backscattering structure (HRBSS). The far end of this structure was accompanied by a variable optical attenuator (VOA) and faraday rotation mirror (FRM). Such a configuration was made for the solemn purpose of loss and gain balancing in the cavity. This HRBSS serving as a unique material to induce the desired RBS signal of high RBS coefficient, low optical attenuation, strong optical confinement, and high stimulated Brillouin scattering (SBS) threshold, was obtained from a slightly tapered 110 m long SMF with characteristics of ~112 m waist diameter, 2 cm tapering length, and a total number of 21 tapers. (a) (b) (d) (c) (e) Fig. 13.14. (a) Schematic diagram of fiber ring laser combing RBS and self-injection feedback. HRBSS high RBS structure. The coupling ratios are C1: 80/20; C2: 50/50; C3: 50/50; C4: 90/10, (b) The longitudinal mode characteristics of the output laser without and with the self-injection feedback based on RBS, respectively, (c) The output spectra measured by the self-heterodyne method with the change in feedback power, (d) The relationship between the laser linewidth and the feedback power under the condition of pump power of ~165 mW, (e) Lorentz fitting linewidth for the narrowest laser linewidth [51]. 531 Advances in Optics: Reviews. Book Series, Vol. 5 Two couplers (C1 and C2) were employed in the fiber ring laser system to induce the effect of optical self-injection feedback, whose feedback power was intuitively controlled by the VOA2. Thus, issues relating to enhanced linewidth and laser operational stability are guaranteed. In a way, there is a trade-off between the desired compressed linewidth and the obtained SMSR. This is because the authors were compelled to adjust the VOA1 attached to the optical pathway wherein the HRBSS was situated, which ultimately lead to an increase in reported SMSR and a detected linewidth compression of ~ 300 Hz at the first instance of injection into the unique material as shown in Fig. 13.14 (b). The authors also discovered that an increase in feedback pump power at a controllable range of 0.2 to 900 µm by mainly gradually tuning the VOA2 attached to the self-injection feedback scheme ensures an SLM operation (characterized by a broadened FSR) is guaranteed. It is apparent that the side modes of the output laser gradually disappear at a tuned feedback injection power of 20 µm, as shown in Fig. 13.14 (c). In this way, the spectral linewidth was further compressed to 130 Hz at a reported controllable feedback power of ~600 µm, as shown in Fig. 13.14 (d) and Fig. 13.14 (e). Utilizing this novel compression mechanism, the authors obtained an ultra-narrow linewidth laser output of ~130 Hz and SMSR as high as 75 dB. The output power of the laser is considerably high, and the stability of the laser structure in terms of its output power and spectral linewidth fluctuation is commendable for such a laser structure. Since it has been predicted that Rayleigh backscattering could be collected in any waveguide medium, member(s) of our research group [52] investigated the characteristics of this weak signal in various types of optical fibers. They employed a heterodyne detection scheme to isolate the Brillion scattering within a fiber laser to achieve this. These fibers are in polarization-maintaining fiber (PMF), large effective area fiber (LEAF), and yet again in the well renowned single-mode fiber (SMF-28e). The stimulated Rayleigh scattering (STRS) investigated characteristics consist of its linewidth, threshold, and frequency shift. A unique experimental system that consisted of an array of optical devices, as shown in Fig. 13.15, was employed to investigate aspects relating to linewidth and threshold. Fig. 13.15. Measurement of Rayleigh scattering in optical fibers. Laser: Fiber laser; EDFA: erbium-doped fiber amplifier; PC1 and PC2: polarization controller, PBS: polarization beam splitter; PM Circulator: polarization maintaining circulator; AOM: acoustic-optic modulator; PD: photon detector; ESA: electrical spectrum analyzer [52]. 532 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers The authors' findings on the three employed types of fibers suggest that the amount of applied input power plays a decisive role in the level from which STRS occurs alone and that the SBS is initiated henceforth. Even more, when the power of the SMF-28e, LEAF, and PMF is within a controllable range of 11 dBm to 17 dBm, 4.5 dBm to 11.6 dBm, and 16.5 dBm to 26 dBm, respectively, it is believed by the authors that the bandwidth of the STRS is somewhat little in fluctuation. In other words, the authors believed and claimed that the effect of STRS is evident in the region of an approximate 6 dB, excluding that of the PMF, which exists at a power range of 10 dB. The spectral linewidth whose evolutionary state of the spectra, contrast, and bandwidth variation is shown in Fig. 13.16(a)-(f). According to the authors, the reported bandwidth of the three types of fibers under test are 9 kHz, 10 kHz, and 11 kHz for the SMF-28e, LEAF, and PMF, respectively. Fig. 13.16. Evolution of spectra, 3 dB bandwidth changing, and contrast of Rayleigh scattering signal for 2 km SMF-28e, 7 km LEAF, and 100 m PMF. (a), (c) and (e): Rayleigh spectra evolution for SMF-28e, LEAF and PMF, respectively. (b), (d) and (f): 3 dB bandwidth and contrast of Rayleigh scattering for SMF-28e, LEAF and PMF, respectively [52]. The preceding (i.e., minimum) power value in those mentioned above 'controllable range' represents the threshold power values of the STRS in each of the three different types of fibers. Hence, any slight increase beyond the maximum power values of each of those mentioned above 'controllable range' for the different fibers under test will instead inject power into the SBS process since the gain factor of SBS is more significant than that of SBS the STRS. In addition, the authors discovered that the linewidth of STRS in the three 533 Advances in Optics: Reviews. Book Series, Vol. 5 different fibers under test was compressed in comparison to the 11.2 kHz free-running linewidth of the laser. What is more, a measured compressed linewidth of ~9 kHz for the 2 km SMF was the most compressed of them all. In regards to the reported threshold power values of 10.2 dBm, 3.4 dBm, and 9.4 dBm for the 2 km SMF, 7 km LEAF, and 100 m PMF, respectively, as shown in Fig. 13.16(a)-(f), the authors put forward two core reasons surrounding the difference in the measured values. Firstly, these reasons are the dependency of the threshold power on the polarization, fiber length, and effective core area of the three different types of fibers under test. Secondly, the efficiency of STRS in a PMF fiber is arguably higher than that of traditional optical fibers such as a LEAF and an SMF-28se. Hence, the approximate corresponding Rayleigh gain coefficient of the three employed fibers are 2×10 -13 m/W, 2.8×10-13 m/W, and 1.0×10-12 m/W, for the 2 km SMF, 7 km LEAF, and 100 m PMF, respectively. Likewise, investigating the characteristics of frequency shift, the authors adopted the same experimental structure in a modified manner, as shown in Fig. 13.17 (a). Regarding the frequency shift, the authors claimed that the fundamental reason surrounding the reported findings, as shown in Fig. 13.17 (b), is centered on the fact that the electrostriction and absorption process possesses an equivalent magnitude when the applied input power is low. In other words, the input power plays a decisive role in observing a frequency shift wherein a substantial increment in its value ensures that the electrostriction effect becomes rather dominant. The authors further reported that when the input power is increased beyond 25 dBm, the SBS effect is initiated within the 100 m PMF since the gain coefficient of STRS is two orders of magnitude lower than that of the SBS. In light of this, a frequency shift measurement is arguably impossible in a fiber of longer lengths such as the 2 km SMF and the 7 km LEAF, except if only the threshold values of SBS are suppressed within such fibers in general. Fig. 13.17. (a) Measurement system of frequency shift of Rayleigh signal in optical fibers, Laser: fiber laser; EDFA: erbium-doped fiber amplifier; PM Circulator: polarization maintaining circulator; PC: polarization controller, PBS: polarization beam splitter; AOM: acoustic-optic modulator; PD: photon detector; ESA: electrical spectrum analyzer. (b) Frequency shift experimental results [52]. Besides the classical discoveries of STRS in liquids and gases during its earliest days, the monumental findings of this work happen to be a significant addition to the literature of scattering effects. The reason being, over the years, researches have been particularly 534 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers tailored to discoveries on stimulated Brillion scattering (SBS) and stimulated Raman scattering (SRS), which are renowned members of the non-linear scattering group. Thus, the findings of this work on STRS as a linear scattering effect that focused on observing the dynamic change from spontaneous to stimulated Rayleigh scattering by characterizing its frequency shift, threshold, and linewidth, is an improvement on the literature. Furthermore, the findings of this work distinctively proved that STRS, characterized by its low threshold power and narrow bandwidth, is indeed a weak effect in fibers and can be employed for spectral linewidth compression purposes. Member(s) of our research group [53] demonstrated ultra-narrow linewidth compression in an erbium-doped fiber ring laser system which consists of a tapered optical fiber. The linewidth suppression of this work was induced by the RBS effect from within a ~110 m short tapered optical fiber. The SBS effect is suppressed because the shape of the altered fiber core and cladding (i.e., the tapered fiber) influences the propagation of acoustic waves (photons and phonons). The fabrication process of the tapered fiber (SMF-28, Corning, Inc.) was done on a translation stage where an oxyhydrogen flame is exposed onto the fiber's jacked-off sections to induce heat into the fiber to create the uniform tapering process, as shown in Fig. 13.18 (a), and Fig. 13.18 (b). Consequently, the authors emerged with a total number of 21 tapered zones along an ~110 m tapered fiber, thereby increasing the SBS threshold value by approximately 6 dB. (a) (b) Fig. 13.18. (a) Schematic of the fiber taper manufacture; (b) Schematic illustration of the tapered fiber with two tapers. The left inset shows a side view of the waist and the right inset shows the waist of the tapered fiber cross-section. The diameter of the waist is ∼112.09 µm, which is slightly different from single-mode fiber (D = 125 µm) [53]. The RBS effect in this laser configuration is evident within the tapered fiber, where multiple scattering occurs to present multiple reflections similar to a distributed mirror. The most potent potion of these reflections, which gradually builds up with Rayleigh scattering of other nearby scattering points, is typically situated at the front end of the tapered fiber. To further facilitate the feasibility of linewidth compression by the circulating RBS propagating in nanoseconds, the employed EDF with characteristics of a millisecond-scale fluorescence relaxation time permits the RBS to undergo numerous cyclic round trips within the ring laser. Thus, leading to laser performance modification (characterized by SLM operation). Spectral linewidth compression occurs in the laser system as shown in Fig. 13.19 (a), when the laser system's cavity loss is increased by adjusting the VOA situated in between 535 Advances in Optics: Reviews. Book Series, Vol. 5 an FRM and the spool of tapered fiber. When there is a significant cavity loss, the RBS effect situated at the far end of the ~110 m tapered fiber plays a dominant role. It appears to be relatively easier in amplification as compared to the reflected signal originating from the FRM. Hence, a cyclic round trip of numerous times inevitably creates a high gain, thereby compressing the spectral linewidth of the laser to an ultra-narrow level of ~200 Hz, as shown in Fig. 13.19 (b), Moreover, obtaining an SMSR as high as 50 dB. The authors further demonstrated the sensitivity of the laser linewidth evolution to a continuously varied power by gradually adjusting the VOA, which directly increases the cavity loss of the laser, as shown in Fig. 13.19 (c). In so doing, the minimum linewidth of ~200 Hz was discovered because by adjusting the VOA, the RBS signal dominates the front end position of the fiber and can be easily amplified for deep linewidth compression compared to the reflected pump signal. The authors also ensured that the growing demand for ultra-narrow seed laser lasing at various wavelengths is met in their work. They achieved this by ensuring that their system can be used at any desired wavelength due to the wavelength selective filter (FBG) impact upon which the wavelength selection is dependent. (a) (b) (c) Fig. 13.19. (a) The configuration of the experimental setup. WDM: wavelength division multiplexing device; EDF: erbium-doped fiber; FBG: fiber Bragg grating; OC1, OC2: optical circulators; C1, C2, C3: 50:50 couplers; PC: polarization controller; VOA: variable optical attenuator; FRM: Faraday rotating mirror; AOM: acoustic optic modulator; PD: photodetector; ESA: electrical spectrum analyzer. (b) The photocurrent power spectrum of the delayed self-heterodyne signal of the laser output and the Lorentz fitting results. (c) The minimum linewidth the laser can achieve: its mean linewidth and its standard deviation [53]. In some situations, it appears to be that SBS and RBS could be simultaneously utilized in a laser system for spectral linewidth compression, provided the threshold value of SBS is reduced intuitively. To that end, members of our group research; Huang et al. [54], 536 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers established a laser system, as shown in Fig. 13.20 (a) that utilizes multi-linewidth compression methods to compress the spectral linewidth of their free-running laser to an ultra-narrow level. These mechanisms include a dual cavity feedback structure (DCFC), the SBS effect, and the RBS effect. Firstly, the authors utilized a DCFS whose principle of operation has been explained in their previous work [55] to compress the original MHz linewidth of a commercial butterfly DFB laser to ~3 kHz to serve as the pump laser to induce the SBS effect. Two spools of SMF fiber of 25 m and 17 m length were coupled alongside two couplers and two optical circulators to establish the DCFS. This was done simply because the theoretical analysis of the authors predicted that a narrow linewidth of the pump laser in an SBS configuration would obtain a relatively narrower bandwidth of the SBS spectral for situations where the spectral linewidth of the pump laser is less than 10 kHz. (a) (b) (c) (d) Fig. 13.20. (a) Laser linewidth compression system based on SBS and RBS systems. (b) Frequency domain characteristics of the final output laser by SCELD with a 50 km delaying fiber. (c) Different frequency noise under different conditions: free-running DFB laser (blue curve), DFB laser with a DCFS (red curve), and the final output laser after a DCFS, SBS, and RBS (purple curve). (d) RIN of the final output laser [54]. Hence, the SBS effect further compresses the linewidth to ~200 Hz as the second compression method. Reason being that a low loss fiber with a high gain nonlinear scattering factor (i.e., the employed ~20 m HNLF) was coupled in a ring cavity configuration with a polarization controller (PC), and an optical circulator. Thus, this configuration was used to sufficiently accumulate the SBS effect and further guarantee SLM operation. A C-band EDFA firstly amplified the pre-compressed 3 kHz linewidth before being injected into the SBS ring resonator. To further validate their notion on the pump laser linewidth and SBS effect, the authors investigated a series of situations (output characteristics of the SBS ring resonator) wherein the pump laser linewidth of the 537 Advances in Optics: Reviews. Book Series, Vol. 5 resonator was changed. Hence, it appears to be that the gradual decrement in pump laser linewidth (from 1.5 MHz to 150 Hz) leads to a gradual increment in SMSR (from 25 dB to 62 dB), and likewise, the linewidth of the SBS ring resonator (from ~9 kHz to ~200 Hz). Finally, an ultra-narrow compressed linewidth of ~75 Hz with an SMSR as high as 70 dB was achieved by an RBS configuration which consists of two pairs of FBG whose reflectivity are ~90 % and ~50 % each, with a spool of 50 m SMF situated in between them as shown in Fig. 13.20 (b). The authors deliberately choose a mismatch in the percentage of grating reflection (particularly the second FBG) to ensure that the SLM operation of the laser achieved by the SBS effect is not altered. Similarly, this was done to ensure that the randomly distributed Rayleighscattering gain is established by the SMF and the second FBG precisely. Consequently, RBS is sufficiently accumulated in this chamber whose linewidth is narrower than the incident laser (SBS ring resonator). Hence, this signal originating from the RBS structure is then feedback into the SBS chamber wherein numerous cyclic round trips and amplification occur to further compress the linewidth to ~75 Hz. Furthermore, the parameter of the output laser in terms of its frequency noise which is another representation of its compressed linewidth, is shown in Fig. 13.20 (c). Likewise, the RIN was found to be −120 dB/Hz for frequency ranges beyond 1 MHz, as shown in Fig. 13.20(d). The authors demonstrated a commendable laser system with the merits of an ultra-narrow linewidth and stable frequency noise that seeks wide-spread applications in communication and sensing systems. 13.6. Linewidth Measurement This section present various linewidth measurement methods. The fundamental linewidth of a laser tends to be arduous whenever the need to measure it arises. This difficulty is associated with the fact that all methods developed to determine the ideal linewidth of a laser are often subjected to a duration of time coined by academia and industry as a ‘finite measurement time.’ In this instance, the compressed linewidth of a laser light source is impacted by frequency jitter arising from multiple aspects such as vibrations, pump laser noise, acoustic noise, etc. Therefore, for a compressed laser linewidth, the value of the detected linewidth can be regarded as an integrated frequency jitter resulting from technical noise originating across the integration duration of the linewidth detection system [56]. 13.6.1. Heterodyne Detection Besides the fundamental reason for obtaining data from a laser linewidth measurement, this particular method transcends its primary detection purpose and into a feasible method to provide data of an optical power spectral. Likewise, it is feasible enough to characterize a non-symmetrical linewidth line shape of a laser. This method's effectiveness depends on the quality of the reference laser light source employed in the detection scheme in terms of its narrow linewidth value and laser operational stability. A traditional heterodyne detection scheme typically requires two laser light sources where one serves as the signal 538 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers laser. In contrast, the corresponding laser serves as the reference laser, often referred to as an LO, as shown in Fig. 13.21. Propagating light signals of the signal laser and the LO laser are combined and mixed by a 3 dB optical coupler (OC). This mixed optical signal is further connected to a photodetector, where the interference beat note is detected and converted into an electrical signal. The measurement accuracy of this method is dependent on the fact that two laser light sources are required in this configuration and that the linewidth of the LO needs to be narrower or approximately comparable to the signal laser [57, 58]. Fig. 13.21. Schematic of the setup for optical heterodyne detection [58]. In typical practice, in order for the two lasers to undergo a signal mixture or impairment and fall within the bandwidth of the photodetector, the central frequency of the reference signal needs to be finely tuned to nearly coincide with the central frequency of the signal laser. In this way, the photocurrent of the mixed frequency signal that is detected by the photodetector can be expressed as [58]: i  t   R  Ps  t   PLO  t   2 P2  t  PLO cos  2  vs  vLO  t   (t )   ,   (13.3) where Ps(t) and PLO(t) represents the power of the laser under test and the reference laser, respectively. vs and vLO represents the central frequencies of the two laser signals. Furthermore, Δϕ(t) and R represents the phase difference of the two lasers and responsivity of the photodetector employed, respectively. 13.6.2. Delayed Self-Heterodyne Detection In contrast to a traditional heterodyne detection method, the DSHI method is a relatively more straightforward method to measure the spectral linewidth [57, 58]. This is because it practically requires a single laser light source (signal laser) in its configuration instead of having two laser light sources (i.e., the usual LO). It adopts a sizeable optical delay fiber situated on one arm of the laser, which induces a delay of the optical signal, and an acoustic optic modulator (AOM) situated on the second arm of the MZI, which induces a frequency shift of the optical signal. By adopting a directional optical coupler in this traditional detection method, the optical light signal injected into the MZI structure and propagating across, is split into two equal paths wherein one potion is delayed, and the corresponding signal is frequency shifted as shown in Fig. 13.22. 539 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 13.22. Schematic setup for optical delayed self-heterodyne detection [58]. In this way, the two optical signals from the signal laser are mixed with themselves by another directional optical coupler situated at another end of the MZI leading up to the photodetector, where the interference beat note is detected and converted into an electrical signal. Hence, the electric field that is detected by the photodetector can be expressed as [58, 59]: ET  t   P  cos  2 f 0t    t     P0  cos  2  f 0  f1    t   d     t   d   , (13.4) where P and P0 represents the power of the upper and lower arms, respectively. f0 and f1 represents the frequency shift of the laser and AOM, respectively. τd represents the delay difference induced by the optical path imbalance of the interferometer. Therefore, the relationship of the photocurrent I(t) and the light intensity at the PD can be expressed as:   I  t   ET  t  ET  t   2  exp i   t     t   d  exp  it   c.c, (13.5) where c.c represents the complex conjugate of the output light field. More so, it is understood that the photocurrent characterized by a constant amplitude that is equivalent to the phase jump of the laser light field, has a quasi-monochromatic signal at an angular frequency of Ω = 2πf1. 13.6.3. Delayed Self-Homodyne Detection In a typical coherent detection method, the frequency of the optical light signal injected into the measurement system is converted into the RF (Radio-Frequency) domain by using a signal mixture or impairment with a LO (i.e., a reference laser signal). However, in a traditional delayed self-homodyne detection method [57], this is not the case. The reason being that a delayed self-homodyne detection method does not require an LO. In a delayed self-homodyne detection method, the optical light signal from the laser light source practically mixes with what is rather referred to as a delayed version of itself originating from the same laser after propagating across an extended length of delayed optical fiber. 540 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers This delay fiber is usually situated on one arm, which causes an optical path difference compared with its corresponding second arm without a delay fiber, as shown in Fig. 13.23. Fig. 13.23. Typical experimental set-up for linewidth measurement using self-homodyne method [57]. It is understandable that since the two mixed signals possess an equal mean optical frequency, the signal to be detected is mainly dependent on the precise length of the optical path imbalance in the interferometer. Thus, in principle, by employing this MZI structure to measure the spectral linewidth of the LUT, the overall concept is fixated on the fact that we intend to convert the laser’s optical phase or frequency fluctuations into variations of intensity. 13.6.4. Amplitude Difference Comparison of Coherent Envelope The desire to obtain laser linewidth measurement, which is less than 1 kHz, has propelled the numerical analysis and experimental demonstration of a new linewidth measurement method. This new method was coined as amplitude difference comparison of coherent envelope (ADCCE) by members of our research group [60]. This linewidth measurement method reveals the actual value of a signal laser less than 1 kHz by comparing the contrast difference between the second peak and the second trough (CDSPST) of the coherent envelope of the power spectral. Technically, this method follows the traditional DSHI linewidth detection method. However, compared to a long delay optical path imbalance adopted in a DSHI scheme, the ADCCE method utilizes a short delay fiber. In other words, it is somewhat analogous to a short delayed-self heterodyne interferometer (SDSHI). This method is not only new but has proved to be stable and effective in characterizing the spectral linewidth of a signal laser less than 1 kHz (i.e., 100 Hz ~ 1 kHz), thereby rendering it a practical approach for narrow or ultra-narrow linewidth detection. This is centered on the fact that the typical structure of this method, based on an SDSHI, ensures that the impact of the broadening spectral created by the 1/f frequency noise is approximately ignored. Simply put, a short delay fiber is arguably promising to mitigate this drawback. In principle, provided the linewidth is further compressed, and the Lorentzian line shape is used to fit the power spectral in the DSHI, a delayed fiber of hundreds of kilometers is required to be utilized to measure the spectral laser linewidth of 541 Advances in Optics: Reviews. Book Series, Vol. 5 sub-kilohertz level. A typical structure of the SDSHI scheme is shown in Fig. 13.24, consists of a signal laser, VOA, isolator (ISO), 1×2 optical couplers, AOM, delay fiber, PD, and an ESA. Fig. 13.24. Schematic of the delayed self-heterodyne interferometer to measure laser linewidth [60]. The propagation of the optical signal within this detection scheme can be explained as follows. Firstly, a signal laser injects a single longitudinal mode beam via the VOA and ISO, arriving at a typical MZI. The MZI consists of two optical couplers that ensure that the light signal is split into two paths (i.e., an equal path where a delay is induced and another equal path where the signal is frequency shifted). Secondly, at the output arm of the MZI, which holds another optical coupler, the light signal is therein recombined and propagated towards the PD and ESA for power spectral light detection and analysis, respectively. In this way, the output power spectrum of the SDSHI can be expressed as [60]: S  f , f   S1S2  S3 (13.6) In the above equation, S1, S2 and S3 are respectively expressed as, S1  P02 f 2 , 4 f 2  f 2  sin  2  f  f1  d   S2  1  exp  2f  d  cos  2  f  f1  d   f , f  f1   S3   P02 2 (13.7) exp  2f  d    f  f1  , where f represents the frequency, f1 represents the frequency shift of the AOM, the item τd (τd = L/c) which refers to the time delay of an optical path with respect to the another path, consists of L which represents the length of the delayed fiber; c is the speed of light, and Δf is the FWHM of the power spectral. The shorter the length of the delayed fiber, the narrower the laser linewidth, a more distinct coherent envelope of the power spectral is bound to appear. Ultimately, the accuracy of 542 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers this measurement method is directly dependent on the length of the delay fiber, the resolution of the ESA, and the narrow spectral linewidth of the signal laser under test. Theoretically, the ADCCE remains unchanged once the length of the delayed fiber and spectral linewidth is determined. This ensures that the laser linewidth is revealed from the power spectral of the coherent envelope by estimating the value of contrast difference between the second peak and the second trough (CDPT). The second frequency is chosen to evaluate the contrast difference between the peak and trough because it holds a more distinct advantage in minor error detection and higher stability than the center frequency. Furthermore, members of our research group established an equation as shown below to express the value of the CDSPST (ΔS) in their subsequent work [61]: S  f   10log10 S peak  10log10 Strough   10log10 S peak Sthough  2l  1 c   S  f1  , f  2 nL   10log10  c   S  f1  m , f  nL     2c 2   nfL    1      1  exp  2 c      nfL    (l  2, m  2)  10log10   3 c 2   nfL    1      1  exp  -2 c      2 nfL    (13.8) Similarly, f represents the frequency, f1 represents the frequency shift of the AOM, and n is the refractive index of fiber. L represents the length of the delayed fiber, c is the speed of light, and Δf is the FWHM of the power spectral (Lorentzian linewidth). The position of peaks and troughs can be represented by the item l and m whose values are within the range of 2,3,4, and 1,2,3, respectively. In principle, the Lorentzian linewidth is induced by the white frequency noise, while the Gaussian broadening is induced by the 1/f frequency noise of a long delay fiber. Voight fitting is often adopted to separate these two effects. However, it is difficult to determine the true spectral linewidth since the Lorentzian linewidth is enveloped by the 1/f frequency noise of the long delay fiber. With the desire to resolve this drawback and ensure that an ultra-narrow linewidth laser is accurately determined (i.e., Lorentzian linewidth), the Gaussian broadening effect needs to be eliminated. Hence, the amplitude difference comparison of the coherent envelope method our research group puts forward by mainly estimating the contrast difference between the second peak and second trough of the second frequency in an SDSHI scheme is feasible and effective. Since the accuracy of this method depends on the spectral linewidth, ESA resolution, and the length of the delay fiber, it is essential to know an exact choice of short delay fiber length for the SDSHI. 543 Advances in Optics: Reviews. Book Series, Vol. 5 Such emphasis is made because a suitable length can effectively filter out the 1/f frequency noise and is capable enough to characterize the actual value (Lorentzian linewidth) of an ultra-narrow linewidth signal laser of different spectral magnitude. Such a length decision can be determined by combining the estimated spectral linewidth and a specific value of CDSPST. 13.7. Noise Characterization This section present various phase or frequency noise, and relative intensity noise characterization methods. Phase noise is regarded as the phase fluctuations occurring within a signal which are random, rapid, and short duration. Similarly, frequency noise is considered to be the fluctuations of the instantaneous frequency of a signal. The phase and frequency noise of a laser are related because they are parts of the same noise and are mainly different ways of characterizing the same phenomenon. In fact, the measurement method adopts the same setup. Some authors often prefer to present the power spectral density (PSD) of the noise carrier in frequency due to the advantageous technicalities in demodulation. A typical example of a frequency noise spectrum which is easily identified as series of jitter that equally spread across either side of a signal and manifesting itself in the form of sidebands across the signal, is shown and discussed in some parts of the preceding section (see Fig. 13.20. (c)). Relative intensity noise is regarded as the fluctuations in the optical output power of a laser. The RIN noise of a laser can be induced from aspects relating to the fluctuations of the laser gain medium, vibrations of the external cavity, or the intensity noise effect originating from a laser light source. An ideal level of RIN for high-performance lasers to meet communication or sensing system application demands is lower than −155 dB/Hz [62]. A typical example of a relative intensity noise spectrum is shown and discussed in some parts of the preceding section (see Fig. 13.20. (d)). 13.7.1. Digital Cross Correlation The phase noise or frequency noise characterization of a narrow linewidth laser is vital to evaluate the stability of a laser. In order to actualize this method, a digital cross-correlation scheme can be established where one laser is the signal laser. In contrast, the corresponding two lasers whose linewidths are relatively lower than the signal laser under test act as the reference laser [63]. A typical structure of this characterization method consists of chambers, as shown in Fig. 13.25. To be precise, the three-laser light source is required to be in a chamber of theirs while its light signal is transported over tens of meter fiber lengths to the detection chamber. The detection chamber consists of two photodiodes. In addition, an optical amplifier is often attached immediately after the PD to amplify the beat note signal, whose signal is first amplified and then frequency down-converted by a frequency synthesizer. Hence, the signal launched into the digital cross-correlator is power amplified, and a low pass filtered signal. The first procedure undergone by this correlator is analog to digital conversion 544 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers (ADC) before analysis. A cascade of these three lasers ensures that the two reference lasers beat with the signal laser to generate two RF electrical signals after photo-detection. The photodiode requires that the frequency difference between the LUT and the reference laser needs to be within the bandwidth of the PD in order for proper conversion from an optical signal to an electrical signal. A digital cross correlator can analyze the detected signal. The noise features of the laser under test can be revealed by averaging the statistical estimator of the cross PSD of the phases of these two beat-note signals. Such an approach is feasible since the noise level from the two other light sources is independent and does not contribute to the noise level in the signal laser. Furthermore, this method does not require the linewidth of the reference signal lasers to have a superior performance level to the linewidth of the laser under test. Fig. 13.25. Experimental setup of the laser phase noise characterization. Three separate ultra-stable 1542 nm lasers are used in this scheme. Laser B is beat with the reference Laser A and Laser C to get two electrical signals. These two electrical signals are first mixed down to a lower frequency and then analyzed by a home-designed digital cross correlator to reveal the phase noise PSD of Laser B [63]. 13.7.2. Michelson Interferometer The emergence of the Michelson Interferometer (MI) method for phase or frequency noise measurement was inspired by the desire to resolve drawbacks associated with measurement methods such as the digital cross-correlation method, which requires a reference laser source of high coherence. The Michelson Interferometer method is an efficient and promising method that conveniently characterizes the phase and frequency noise conveniently. This is feasible by demodulating an accumulated laser’s differential phase in a delay time and then employing fundamental mathematical relations, which involve the laser’s phase noise and differential phase. Such mathematical relations ensure that the actual value of a laser’s phase and frequency noise is revealed [64]. Noise measurement by this method involves a typical structure that consists of a signal laser, OC, 3×3 optical coupler, short delay fiber, etc., as shown in Fig. 13.26. The signal laser generates the optical signal injected and traversing across the optical circulator and optical coupler before being reflected by the FRM. This reflection ensures the optical signal propagates backward to be firstly mixed within the OC with different 545 Advances in Optics: Reviews. Book Series, Vol. 5 delay times. Furthermore, it is detected by the photodetectors before being analyzed and demodulated. The 3×3 optical coupler with two FRM attached at the far end of only two coupler ports guarantees the restriction of polarization fading of the interferometer arising from surrounding noise. Such a configuration is adopted in this unbalanced MI simply because of its unique ability to accommodate a laser injected propagating light signal and demodulate its corresponding differential phase. Fig. 13.26. Experimental setup used to measure the laser phase and frequency noise, and the output interference fringe of the PD1, PD2, PD3 (inset). LUT: laser under test, C: circulator, OC: optical fiber coupler, FRM: Faraday rotation mirror, PD: photodetector, DAC: data acquisition board [64]. Even more, the 3×3 optical coupler holds superior performance simply because it does not require any form of an active controlling operation. Thus, the unbalanced MI structure does not suffer from the impact of polarization, nor does it require any adjustment. In this way, the signal laser’s differential phase fluctuation generated by the MI is demodulated, thereby paving the way to reveal the noise features of a compressed laser linewidth utilizing PSD estimation and calculation of the differential phase. Although the previously discussed delayed self-heterodyne detection method and the delayed self-homodyne detection method can characterize the phase or frequency noise [65], most authors prefer to adopt the beat-note cross correlation method and the Michelson interferometer detection method as discussed above. In addition, the delayed self-heterodyne detection method and the delayed self-homodyne detection method adhere to stringent requirements when employing these methods to characterize the phase or frequency noise. These rigorous requirements seek a long delay fiber and the need to control quadrature point using special, well-calibrated active feedback methods [63, 64]. Hence, the beat-note cross correlation and Michelson interferometer detection methods are robust methods to characterize the phase or frequency noise accurately. 13.7.3. Cross-Spectrum Method The cross-spectrum method is used to characterize the PSD of the relative intensity noise of a laser [66]. The system configuration for this method possesses an upper and lower arm that consists of a high-speed photodetector of low noise qualities used to detect the varying propagating optical light signal. In some cases, a compromise can be made on the choice of amplifier attached to this detection scheme, such as a PD with an internally 546 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers packaged amplifier or a typical external stand-alone amplifier connected to the electrical path. Considering a possible large flicker typically induced by a PD with an internally packaged voltage amplifier, a low noise non-inverting configuration-placed amplifier is preferred to be attached in front of the PD to amplify the detected signal in this scheme shown in Fig. 13.27. In addition, a dual-channel Fourier frequency transform (FFT) analyzer is utilized to measure and analyze the photo-detected and amplified signal by conducting FFT on these two independent signals, which will then reveal the calculated cross-spectrum of the RIN in the frequency domain. It is assumed that for this detection configuration, the optical coupler is reasonably stable, Hanbury-Brown Twiss effect free, and does not induce any partition noise on either arm while operating with the intensity. Furthermore, it is also assumed that any environmental impact does not influence the two channels. This cross-spectrum detection scheme to characterize the RIN seems to be of significant advantage compared to a conventional direct detection scheme. A conventional direct detection scheme consisting of one channel is often limited by the noise within the single channel consisting of a single PD and a single external amplifier. Fig. 13.27. Basic RIN measurement method [66]. 13.8. Wavelength Switching and Wavelength Tuning This section presents various methods for wavelength switching or tuning in semiconductor and fiber laser, either from the perspective of a single-wavelength generation, dual-wavelength generation, or multi-wavelength generation. Wavelength switching is simply the act of reproducing the peak spectra of a particular wavelength band into a different position within the wavelength band. In contrast, wavelength tuning is the act of gradually shifting the peak spectra of a specific wavelength band over a considerable range of the wavelength band. Furthermore, wavelength converted lasers aimed to avoid wavelength blocking is another aspect of a laser wavelength that has witnessed a sizable amount of widespread research. 13.8.1. Mechanical Switching/Tuning In typical practice, various types of gain mediums such as ytterbium and erbium used independently or simultaneously with each other can be exploited when it comes to 547 Advances in Optics: Reviews. Book Series, Vol. 5 establishing a fiber-based laser system with attributes of wavelength switching or tuning. However, for erbium-doped fiber lasers, the fundamental step towards attaining an extended wavelength laser is to primarily adopt the approach of suppressing the amplified spontaneous emission (ASE) and reducing the reabsorbing effect within the EDF. The concept of ASE suppression is achievable by either utilizing an EDF with the parametric fiber structure of a depressed cladding or by basically employing a multistage scheme that consists of filters situated between each of the two stages. On the other hand, the reabsorbing effect can be eliminated by ensuring that a fiber of a shorter length is employed to accommodate sufficient pumping that can guarantee a relatively high inversion level. In light of this, Chen et al. [67] demonstrated an Erbium-doped fiber ring laser with a wavelength switching phenomenon. The authors obtained a short wavelength band and wavelength switchable dual laser by the principle of bending-dependent loss. This was feasible because the authors adopted a 12 m length of EDF with a depressed cladding scheme. By spooling the EDF to a coil with a diameter of 6.25 cm, the authors were able to induce the bending dependent loss effect on the EDF by basically pressing the fiber from either direction at the same time (i.e., from the top or the bottom) as shown in Fig. 13.28(a). At a fixed diameter of 6.25 cm, it appears to be that the laser system is somewhat susceptible to the increment in pump power. The authors varied the pump power of their laser from 25 mW to 100 mW and discovered that the peak wavelength of their grating (i.e., 1488.9 nm) is less susceptible to the change in pump power. Even more, a constant increase in pump power instead leads to the emergence of a new and unexpected peak that progressively grows with the increment of pump power, as shown in Fig. 13.28(b). In this work, the authors claimed that the dual-wavelength peaks' emergences are mainly due to the FBG central wavelength (i.e., 1488.9 nm) and the strong reabsorbing gain effect of the EDF, which is believed to correspond to the peak wavelength of 1533.9 nm according to the authors on an approximate explanation. When the fiber coil is reduced from its initial diameter of 6.25 cm by a factor of 0.75 cm from the bottom up as induced by the bending effect, the wavelength peak of 1533.9 nm completely switches to a rather dominant wavelength peak of 1488.9 nm, as shown in Fig. 13.28 (c). However, the peak wavelength at 1533.9 nm is rather dominant than at 1488.9 nm when the diameter was at its initial length. As the diameter decreases, this gradually switching phenomenon progresses before reaching its lowest length of 5.50 cm, whose spectra phenomena were discussed above. The authors emphasized that any further decrement of D beyond 5.50 cm will lead to a fall in power of the emitting peak. Also, the authors emphasized that the efficiency of the laser is influenced by the impurities incurred from the doped EDF. Even more, a dual FBG should be considered as an effort to control the lasing power. On the brighter side, this method is a cost-effective and portable approach to attain wavelength switching. In 2007, Chen, et al. [68], with a length of 6.4 m EDF, proposed a ring cavity laser that is wavelength switchable and tunable, as shown in Fig. 13.29 (a). Compared to the consistent usage of wavelength-selective components in laser systems for wavelength switching or tuning, such as the researches mentioned above, this structure, for the first time, instead 548 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers adopts a birefringent and high nonlinear Photonic crystal fiber (PCF). With this PCF, single-wavelength lasing and dual-wavelength lasing are achieved. The dual-wavelength peak of the laser is generated by the inclusion of the PCF, which directly creates two cavities of different lengths whose polarization state is different for each cavity. In this work, it is understandable that the operational mode of single and dual lasing of the laser is switched by mainly adjusting the PC attached behind the PCF, as shown in Fig. 13.29 (b) and (c), respectively. This laser configuration's wavelength tunability and range depend on the birefringence of the nonlinear PCF. The nonlinear coefficient of the PCF was reported to be as high as 11 W-1 km-1 for the fiber length of 25 m. (a) (b) (c) Fig. 13.28. (a) Experimental setup of the switchable dual-wavelength erbium-doped fiber ring laser; (b) Output spectrum variation with pump power; (c) Output spectrum variation with D. Inset schematically shows the shape of the EDF loop and the definition of D [67]. Suppose we leverage the already established theory surrounding the feasibility to extend the wavelength range of an erbium-doped fiber laser. In that case, it is not far-fetched to suggest that the short length of the EDF will, in fact, alongside a highly birefringent PCF, might attain a substantially wider tunable region beyond the range of 1563 nm to 1569 nm, as shown in Fig. 13.30 (a). Another compelling reason surrounding the remarkable stability of the single lasing wavelength of the laser detected over some time, as shown in Fig. 13.30 (b), is somewhat dependent on the fact that the gain medium is considerably short even for such a high pump power. 549 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (b) (c) Fig. 13.29. (a) Schematic configuration of the proposed fiber ring laser. LD: laser diode. WDM: wavelength division multiplexer. OC: optical coupler. ISO: isolator. PC: polarization controller. EDF: Erbium-doped fiber. PCF: photonic crystal fiber; (b) Output spectrum of the proposed fiber laser with single-wavelength lasing; (c) Output spectrum of the proposed fiber laser with dual-wavelength lasing [68]. (a) (b) Fig. 13.30. (a) Repeated scanning spectrum each 5 minutes. Inset shows the lasing wavelength can be tuned by adjusting the polarization controller, and (b) Repeated scanning spectrum each 5 minutes [68]. Liu et al. [69] leveraged the multi-wavelength generation qualities of a PCF and the wavelength switching merits of a high birefringence (Hi-Bi) fiber grating to establish a multi-wavelength, stable channel spacing, and wavelength switchable EDF laser system as shown in Fig. 13.31 (a). Quite similar to Chen, et al. [68], the authors adopted a high 550 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers nonlinear coefficient PCF with a relatively long length of about 70 m. The PCF is inserted into a loop to establish a loop mirror consisting of a rotated PC at an angle sufficient to attain multi-wavelength lasing with suitable channel spacing of 0.4 nm to 0.8 nm, as shown in Fig. 13.31 (b) and (c). The stability of the multi-wavelength with an amplitude variation of 0.2 dB, as shown in Fig. 13.31 (d), represents an equalization of the EDF mode competition induced by the PCF loop mirror. By rotating the PC, this loop induces an intensity-dependent cavity loss which is somewhat similar to that reported by Chen et al. [67]. However, slightly different since no bending effect was induced on the PCF but instead intensity effect by the PC. The authors demonstrated and obtained a switchable wavelength laser whose channel spacing functions at a three to four wavelength lasing operation with the aid of the Hi-Bi and adjustment of the PC. It is understandable that provided the PC1 is fixed, and the PC2 is finely adjusted to an angle of 45 degrees, the channel spacing grows even smaller (i.e., approx. 0.8 nm) with the lasing operation of four orders. (a) (c) (b) (d) Fig. 13.31. (a) Schematic diagram of the proposed laser; (b), (c) Output spectra of the channelspacing and wavelength switchable multiwavelength EDF laser by adjusting the PC2; (d) Output spectra of the stable multiwavelength lasing oscillation with 16 times repeated scans [69]. Indeed, a gain medium doped with a rare-earth element can provide high saturation power and a significant gain that is also low polarization-dependent. Nevertheless, laser structures that adopt gain mediums such as EDF lasing at room temperature are subjected to drawbacks such as instability and intense mode competition. These drawbacks primarily account for the homogeneous line broadening effect of the gain medium. Provided we ignore a few aspects such as complexity and cost, it appears to be that quite a few solutions have been put forward to mitigate these drawbacks, such as adopting a 551 Advances in Optics: Reviews. Book Series, Vol. 5 four-wave mixing effect, utilizing a high-nonlinear fiber, and so on. Suppose we instead leverage the effectiveness of the approaches mentioned above. In that case, it is regarded that, to some extent, there have served to be of substantial impact in resolving the drawback. More importantly, an ideal approach is to circumvent this drawback by adopting an SOA whose core attributes of inhomogeneous broadening can also facilitate other aspects of a lasing laser’s wavelength, such as the generation and simultaneous oscillations of multi-wavelengths. In 2009, Zhang et al. [70] demonstrated an SOA-based fiber ring laser with multiwavelength generation and fine-tuning attributes. Based on the nonlinear polarization rotation of an established polarizer consisting of two polarization controllers, a polarization-dependent isolator, and of course, an SOA, a multi-wavelength is easily generated. The multi-wavelength generation is feasible by mainly adjusting the two polarization controllers that function to adjust the polarization of the input light signal traversing across the SOA and PDI. These two polarization controllers are placed in a nonlinear direction to each other, as shown in Fig. 13.32 (a). If the polarization controllers are appropriately adjusted, and of course in consideration of the phase difference between the two modes (i.e., transverse electric and transverse magnetic) generated by the SOA, the two adjusted PCs inevitably serve to guarantee that the output light signal traversing across the PDI is decreased with respect to an increase of the light intensity. In other words, mode competition is avoided, leading to the ease generation and lasing of multiwavelength. Furthermore, as shown in Figs. 13.32 (b) and (c), the authors demonstrated a multi-wavelength comb laser and wavelength spacing as high as 126 and 0.08 nm, respectively. (a) (b) (c) Fig. 13.32. (a) Experimental setup of our proposed multiwavelength SOA fiber laser. The part surrounded by dashed line is a general configuration of nonlinear polarization rotation based on SOA; (b) Multiwavelength output spectrum with wavelength spacing of 0.08 nm; (c) Zoom-in of the part surrounded by dashed lines in (b) [70]. Besides evaluating the stability of the multi-wavelength comb by changing the polarization-dependent isolator to a polarization-insensitive isolator as shown in Fig. 13.33 (a), the authors failed to demonstrate the stability of the multi-wavelength comb over an extended duration of time. Even more, it appears to be that the length and fiber characteristics of the polarization-maintaining fiber inserted in the Sagnac loop should also play some decisive role alongside the polarization-dependent isolator towards the stability of the wavelength comb. More importantly, with this unique and commendable 552 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers laser configuration, a fine wavelength tuning range as high as 20.2 nm has been achieved, as shown in Figs. 13.33 (b)-(e). (a) (b) (c) (d) (e) Fig. 13.33. (a) Output spectrum from the SOA fiber laser with 0.08 nm Sagnac loop filter when the PDI is replaced by a polarization-insensitive isolator. (b); (c); (d); (e); Tunable multiwavelength generation through adjusting polarization [70]. In 2020, Hao, et al. [71] demonstrated a multi-wavelength tunable SOA fiber laser with a considerably ultra-narrow wavelength spacing of 0.081 nm and a wide tunable wavelength range that spans from 168 nm to 250 nm. Similar to Zhang, et al. [70] the authors mitigated the mode competition effect induced by the SOA by mainly establishing a linear polarizer consisting of two PCs, an SOA, and a polarizer, as shown in Fig. 13.34 (a). This linear polarizer structure generates the intensity-dependent loss effect for the mode competition suppression. It is also used to adjust the wavelength of the laser by mainly tuning the PC, which directly controls the intensity of the independent-dependent loss. In principle, the orientation and components of the linear polarizer structure suggest that the direction of propagating light signals ought to be in the forward direction. However, by adjusting the two PCs inserted in this structure, the authors could change the direction of the light signal. Thus, creating a non-linear polarized light signal whose polarization state of the output light signal is directly proportional and dependent upon the intensity of the input light signal. This change in orientation of the polarized light signal is only needed for ultranarrow wavelength spacing. Likewise, the corresponding tuning of the lasing lines is achieved by the induced effect of the intensity-dependent loss, whose intensity is controlled by adjusting the PCs to different settings. Of course, the orientation of the propagating light signal, which ought to be a forward light signal traversing across the linear polarizer, has its orientation changed with the aid of the polarizer attached within the structure. The multi-wavelength comb-like spectral of the laser is generated by constructing an MZI consisting of two equally bidirectional optical couplers with an SMF situated in between them. In this way, light traverses across the interferometer in a double pass with a phase difference, creating a comb as shown in Fig. 13.34 (b) and filtered by this same double pass MZI. The authors argued that the stability of their multi-wavelength laser, as shown in Fig. 13.34 (c), was mainly induced by the presence of a Sagnac loop created and consisting of a considerably long fiber length of 600 m and a PC. Even more, the Sagnac loop also serves as a power equalizer. If we 553 Advances in Optics: Reviews. Book Series, Vol. 5 consider the length of the SMF the authors used in their Sagnac loop, compared to that reported by Zhang et al. [70], the cost of a traditional telecommunication fiber is relatively cheaper. In other words, a compromise between cost and stability needs to be considered. The stability of this fiber laser is characterized by the progressive 3 min time interval measurement over an entire duration or course of 15 min, which demonstrated little or no apparent power fluctuation, as shown in Fig. 13.34 (d). (a) (b) (c) (d) Fig. 13.34. (a) Experimental setup of wavelength-number-tunable multi-wavelength SOA fiber laser based on NPR effect; (b) Stable multiwavelength output with ultra-narrow wavelength spacing and different number of laser lines; (c) Multiwavelength output spectrum without Polarizer. (d) Repeated scans of the output optical spectra from1535 nm to1540 nm every 3 min over 15 min [71]. Yeh, et al. [72] demonstrated a hybrid fiber ring laser with a broad wavelength tuning that spans from the C to L-band. This hybrid fiber ring laser was established by encompassing a C-band SOA and a C-band EDFA of 3 m EDF aligned serially (i.e., first stage and second stage), as shown in Fig. 13.35 (a). This configuration was made to extend the effective amplification range. Furthermore, the authors adopted an intracavity C-and L-band tunable bandpass filter (TBF) with tuning ranges spanning from 1520 nm to 1560 nm and 1560 nm to 1610 nm to lase and tune, respectively. In order to obtain a hybrid and wide ASE source amplifier (i.e., consisting of both the first stage and second stage amplification), the authors verified the threshold and maximum ASE state of the adopted SOA. This verification was feasible by mainly varying the drive current of the first stage SOA within the range of 80 to 200 mA. According to their findings, it appears to be that the wavelength of the SOA shifts to a lower wavelength region just as the current is progressively increased. Thus, a hybrid amplifier (cascaded with the EDFA) is formed. In other words, a bias current of 200 mA and a pump power of 25 mW on the SOA and EDF, respectively, is sufficient for proper cascading, as shown in Fig. 13.35 (b). 554 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers (a) (c) (b) (d) Fig. 13.35. (a) Experimental setup of CW tuning fiber ring laser scheme covering both Cand L-bands; (b) Output ASE spectral of SOA at the driving current of 80, 140, and 200 mA respectively. And red line is the ASE profile of proposed hybrid amplifier; (c) Output wavelength spectra of the proposed fiber ring laser in the wavelengths of 1518.5–1610.0 nm, at the driving current and pumping power of 200 mA and 25 mW, respectively; (d) Output power and SMSR of the proposed fiber ring laser under different lasing wavelength range of 1518.5–1610.0 nm [72]. With the aid of the TBF, a wavelength tuning range of 91.5 nm (i.e., from 1518.5 to 1610.0 nm), as shown in Fig. 13.35 (c), is commendable for such a simple and costefficient system architecture. Furthermore, for such a hybrid fiber laser, the reported maximum output power stability (characterized by its detected fluctuation) and wavelength variation in the order of 0.9 dB and 0.2 nm, respectively, is a massive leap in laser system designs. However, although the authors demonstrated a laser system with the characteristics of a 62.5 dB high SMSR, there still exists room for further development in the laser system to advance the reported 5.9 dBm in maximum output power as shown in Fig. 13.35 (d). Concisely and insightfully, Mao et al. [73] demonstrated an optical fiber short ring cavity SOA laser system with a broad and continuous wavelength tuning range spanning from 1498 nm to 1623 nm. The authors adopted a traveling wave SOA that acted as the gain medium in the laser cavity, consisting of a fiber Fabry-Perot tunable filter (FFP-TF) serving as the wavelength selective element and filter to attain a remarkable stable wideband tunable range. Compared to other authors, the authors mitigated the penalty of ASE noise by brilliantly placing their FFP-TF with characteristics of 128 nm FSR and finesse of 850 nm directly in front of the output coupler, as shown in Fig. 13.36 (a). Moreover, for such a widely tuned wavelength range of 125 nm, as shown in Fig. 13.36 (b), the laser exhibited a relatively stable output power with fluctuations 555 Advances in Optics: Reviews. Book Series, Vol. 5 accounted for approximately less than 2 dB and lasing at a single longitudinal mode operational state. It appears to be that the presence of the polarization-independent isolator functioning alongside the PC serves as a contributing factor towards profound stability reported by the authors. (a) (c) (b) (d) Fig. 13.36. (a) Configuration of the semiconductor fiber-ring laser; (b) The tunable semiconductor fiber-ring laser spectrum over the 125 nm wavelength range; (c) The laser output power plotted against the laser emission wavelength for different driver currents. Driver currents: 250 mW, 200 mW, 150 mA, and 100 mA; (d) The measured laser output powers at different output ratios of 0.5, 0.3, and 0.1 [73]. To further validate their laser system's effectiveness and performance level, the authors investigated the drive current of their SOA and its corresponding output coupling ratio. The authors discovered that at an SOA drive current of 250 mA (herein, we only considered the maximum current investigated), the output power of their laser was found to be 4.9 mW at an output coupling ratio of 30 %, shown in Fig. 13.36 (c). Besides the output coupler of 70:30 (characterized by an output coupling ratio of 0.3) used in their laser structure, the authors further investigated the feasibility of attaining and maintaining such a wide tunable range. This was achieved by mainly leveraging various output couplers such as 90:10, 50:50 which correspond to output coupling ratios of 0.1 (10 %) and 0.5 (50 %), respectively. According to the authors, it appears to be that a lower output coupling ratio (i.e., either 0.1 or 0.3) is rather suitable if we intend to achieve an increment in the tunable range. The reason is that cavity loss arising from a higher output coupling ratio of 0.5 (i.e., a decrement in light signal injected back into the ring laser cavity) induces substantial loss, which significantly impacts the cavity stability. On the other hand, a relatively lower output coupling ratio of 0.1 demonstrated a more widely tunable range of 556 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers 127 nm, as shown in Fig. 13.36 (d). For such a low-cost and compact ring laser, the reported output power is commendable. However, further optimization is needed if the authors intend to meet the rapid and high demand for higher output power tunable lasers. As much as the application needs of dual-wavelength lasers are in great demand, it is difficult to achieve a dual-wavelength laser lasing at SLM operation because they both adopt different methods to achieve the two output aspects of a laser simultaneously. Therefore, various methods have been put forward to achieve these simultaneous aspects of a laser. However, yet these methods suffer from drawbacks such as low output power and somewhat operational complications associated with the laser's cavity. In 2017, Shen et al. [74] demonstrated a long ring cavity double SOA-based laser system with tunable dual-wavelength lasing qualities in SLM operation. A conventional SOA laser (i.e., SOA-S) is primarily utilized as the laser system's gain medium. In addition, for the sake of attaining SLM operation based on spectral narrowing effect, the authors creatively adopted a nonlinear semiconductor optical amplifier (NL-SOA) into their laser cavity to induce the effect of inverse four-wave mixing (FWM), as shown in Fig. 13.37 (a). (a) (b) (c) (d) Fig. 13.37. (a) Experimental setup of the proposed dual-wavelength SLM fiber laser; (b) Individual transmission spectrum of the OCF (black solid line) and the OTF (red dotted line), and (c) overall transmission spectrum of the OCF and OTF; (d) Output optical spectrum of the proposed dual-wavelength SLM fiber laser [74]. Naturally, such a laser structure is expected to generate a comb-like spectral similar to a multi-wavelength laser. However, for mainly obtaining a dual-wavelength and tunable laser, the authors endeavor to attach two key components (i.e., the TOF and the OCF) that function basically in this manner. Firstly, the authors demonstrated a laser system with a wavelength tuning range of approximately 56 nm by basically tuning the central wavelength of the TOF, which in turn, filters out the various distinct number of channels 557 Advances in Optics: Reviews. Book Series, Vol. 5 stacked and adjacent to each other within the OCF, as shown in Fig. 13.37 (b). Secondly, for the tuned wavelength, a channel spacing of 0.4 nm (in other words, wavelength spacing) was achieved and dependent upon the wavelength separation of the tunable FSR of the OCF adopted in the laser system, as shown in Figs. 13.37 (c) and (d). Ultimately, the previously reported tunable wavelength ranges at SLM operation and dualwavelength selection over a wide and narrow wavelength range, as shown in Fig. 13.38 (a) and (b), was achieved by utilizing a tunable optical filter (TOF) and an optical comb filter (OCF) within the laser cavity. The insertion loss and cost of this laser system should be considered as we predict a compromise of these will not be inevitable. Nevertheless, the authors demonstrated a relatively stable dual laser system that tremendously mitigates the drawbacks of attaining dual-wavelength laser lasing at an SLM through other methods. Furthermore, the reported tuned wavelength ranges and wavelength spacing for such a long ring cavity laser system are commendable. (a) (b) Fig. 13.38. Wavelength tunability over (a) a wide wavelength range, and (b) a narrow wavelength range [74]. In 2012, Ummy, et al. [75] proposed and demonstrated an SOA-based dual-wavelength switchable and continuous wavelength-tunable fiber laser. As shown in Fig. 13.39 (a), the authors established a relatively cost-effective laser system whose structure consists of three main components. These components include two Sagnac loop interferometers acting as a broadband reflection mirror, a C-band SOA acting as the gain medium whose ASE propagates in both directions. It also consisted of wavelength selection components such as an FBG (0.28 nm bandwidth) and a tunable filter (0.3 nm bandwidth) to switch and tune the wavelength. Structure-wise, the second Sagnac loop is a lot similar to the first Sagnac loop. However, the slight difference in loop emerges from the absence of a wavelength selection element and filter in the second loop whose reflectivity is deliberately kept at 99.9 % in order to reflect, and adjust with a PC, a substantial amount of optical light signal towards the direction of the first loop. The authors demonstrated a fiber laser with a single or dual-wavelength operation using the two filters in the laser cavity. Thus, the wavelength reflected by the fiber gratings (reflection peak wavelength) and the wavelength selected by the thin film tunable filter (passband wavelength) establish the operation lasing modes. To be more precise, within 558 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers the first Sagnac loop, a series of counterclockwise and clockwise propagation of light signals undergoes reflectance and transmittance. The ASE injected into the first Sagnac loop encounters the FBG within the loop wherein the propagating light signal that falls upon the resonance bandwidth of the grating (i.e., 1544.32 nm) is reflected in the same direction of its initial propagation. In other words, the reflected signal becomes the selected wavelength as induced by the FBG. Acting as a standard mirror (i.e., the FBG), the transmitted light signal effortlessly traverses across the grating in a propagating manner headed towards the thin film tunable filter. (a) (b) Fig. 13.39. (a) Experimental setup of the SOA based dual loop mirror fiber laser; (b) Dual wavelength operation of the laser when the SOA is driven by a bias current of 100 mA [75]. In the counterclockwise direction, the first wavelength, also regarded as the selected wavelength, survives a continuous circulation within the first Sagnac loop. However, since it is typically a reflected light signal (i.e., the wavelength signal reflected by the FBG), it recombines with another reflected light signal induced by the almost total reflection (i.e., 99.9 % reflected signal from the second Sagnac loop). This combination produces a Michelson-like interference at the first output region of the laser cavity. Regarding the second wavelength generation, it is essential to note that the ASE that encounters the thin-film tunable filter mainly permits a wavelength signal (i.e., 1549.5 nm) that is selected by the tunable filter to propagate further. This selected wavelength signal from the tunable filter effortlessly traverses across the FBG due to the difference in resonance bandwidth of the fiber grating. A continuous circulation within the loop ensures that either in the clockwise or counterclockwise direction, the selected wavelength signal by the tunable filter is effortlessly recombined at the 3-dB splitter. The PC within the first Sagnac loop is used to control the birefringence of the optical fiber, which directly ensures that there is a control of the Sagnac loop’s reflectivity. In other words, a compromise in control of gain and cavity loss is feasible by adjusting the PC. The authors were able to switch the wavelength operation of their laser system from a single to a dual-wavelength and vice-versa by mainly adjusting the PC1 as shown in Fig. 13.39 (b) and Figs. 13.40 (a), (b). In this way, the authors generated a dualwavelength laser system whose wavelength was induced due to the reflected wavelength by the FBG and the selected wavelength by the tunable filter. The wavelength tunability over the entire C-band was feasible by fixing the wavelength of the FBG and finely tuning the band-pass filter's wavelength, which increases or decreases the wavelength separation between the two adjacent generated wavelengths as shown in Figs. 13.41 (a)-(c). The separation between the dual adjacent wavelength is 559 Advances in Optics: Reviews. Book Series, Vol. 5 dependent on the bandwidth of the filters. Provided a narrower bandwidth filter is adopted, the minimum separation can be further reduced. In other words, a broader tunability is arguably feasible. The stability of the total output power of 6.9 dBm (characterized by a value of 0. 33 dB) and wavelength stability (characterized by a value of 0.014 nm) is commendable for such a distinguished laser configuration. This laser provides a rather substantial cost-efficient and compact laser system with multiple functionalities. (a) (b) Fig. 13.40. Single-wavelength operation of the fiber laser (a) wavelength selected by the FBG; (b) wavelength selected by the thin film based tunable filter [75]. (a) (b) (c) Fig. 13.41. Variable wavelength difference Δλ obtained by tuning the tunable thin film filter. (a) 5.3 nm, (b) 10.28 nm, (c) 15.28 nm [75]. 13.8.2. Acousto-optic Switching Delgado-Pinar et al. [76] developed a fiber laser system with the capacity of achieving wavelength switching by adopting acoustic waves, as shown in Fig. 13.42. The authors coined this method as AOSLM, an acronym that stands for acousto-optic superlattice modulator. The wavelength is switched by controlling the relative peak reflectivity between the sidebands originating from the AOSLM. Using this method, the authors predict the feasibility of adjusting the wavelength spacing between adjacent lasing wavelengths by mainly tuning the frequency of the RF signal. Undoubtedly, the acoustic wave is based on a Bragg grating upon which the propagation occurs. The authors employed a tapered boron codoped germanosilicate fiber. According to the authors, the fundamental essence of the tapering that is characterized by a 55 mm uniform waist, 100 µm diameter, and a transition length of 12 mm, is to ensure that acousto-optic interaction is enhanced in the FBG. The grating length, bandwidth, and reflectivity are 5 m, 0.1 nm, and >30 dB, respectively. 560 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers Fig. 13.42. Experimental setup [76]. In the fiber ring laser system, the authors endeavored to employ a piezoelectric disk driven by an RF signal meticulously connected to the fiber grating through a silica horn to generate a longitudinal acoustic wave that eventually propagates across the tapered FBG. The authors argued that the acoustic reflection at the Bragg-based AOSLM system's transitions is significantly reduced because the diameter of the tapered transitions gradually decreases. Ultimately, sufficient acoustic power from the excitation system (characterized by the PZ-disk, silica fiber ‘Horn,’ and RF supply) effectively traverses into the FBG. The performance and findings of this acousto-optic superlattice modulator were investigated firstly, as shown in Fig. 13.43 (a), (b), and (c). According to the authors, the reflection band of the FBG and fourth-order sidebands, as shown in Fig. 13.43 (a), is reduced to 16 dB and detectable, respectively. The appropriate minimal voltage needed to ensure reflectivity of the bands within this fiber laser configuration was explored, as shown in Fig. 13.43 (b), to guarantee the laser's wavelength switching process. The authors discovered that 7 V is suitable for the central band, from 7 to 17 V is suitable for the first-order sidebands, and from 17 to 24 V is suitable for the second-order sideband. What is more, the authors demonstrated the tunability of the first-order sidebands and the second-order sidebands, as shown in Fig. 13.43 (c). The linearity between the wavelength shift from the central band and the RF frequency exists to reveal rates of 0.142 and 0.285 nm/MHz for the first-order sideband and the second-order sideband, respectively. With the change in applied voltage, the authors demonstrated a switchable wavelength laser. A single wavelength peak corresponding to the central band exists for situations wherein no voltage is applied, as shown in Fig. 13.44 (a). However, an increase in voltage inevitably generates another wavelength peak regarded as the first-order right-hand sideband, which creates a dual lasing wavelength laser in addition to the existing central band, as shown in Fig. 13.44 (b). The authors claimed that the gain effect of the EDFA ensured that the first order left-hand sideband does not exist since more gain is given to the right-hand sideband only. Nevertheless, their laser system could lase a first-order lefthand sideband only when an additional FBG was cascaded into the ring laser cavity and diminishes the right-hand sideband based on reflection and elimination by the circulator. A switch between the first-order sideband, first and second-order sidebands, and a single peak corresponding to the second-order sideband demonstrated by the authors is feasible with the increase in voltage amplitude as shown in Fig. 13.44 (c), (d), and (e), respectively. 561 Advances in Optics: Reviews. Book Series, Vol. 5 Furthermore, the authors demonstrated the transient regime that occurs when a switch from the central band to the second-order sideband of their laser exists, as shown in the inset of Fig. 13.44. (a) (b) (c) Fig. 13.43. (a) Reflection spectra when an RF signal of frequency 1.017 MHz and amplitude 15 V is applied to the piezoelectric (solid line). The reflection spectrum of the FBG is also shown (dashed line); (b) Peak reflectivity of the “central” band (triangles), and the first- (open circles) and second-order (squares) sidebands against voltage applied to the piezoelectric, at a constant frequency of 1.017 MHz; (c) Wavelength shift of the first- (open circles) and second-order (squares) sidebands as a function of frequency [76]. (a) (b) (c) (d) (e) Fig. 13.44. Laser emission spectra for different voltages applied to the piezoelectric. Inset shows the transient regime when the laser emission is switched from (a) to (e) [76]. 562 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers 13.8.3. All Optical Tuning Members of our research group, Li, et al. [77], established an all-fiber optical-controllable laser assisted by graphene for wavelength tuning in a Q-switched mode-locking state laser. The employed element for wavelength tuning in this work was mainly fabricated by attaching graphene onto the surface of a micro-FBG (MFBG) which was firstly a commercial Bragg grating whose reflectivity is 90 %, bandwidth is 0.2 nm, and lasing at a wavelength of 1550 nm as shown in Fig. 13.45 (a). In order to avoid evanescent waves from escaping the grating, a part of the cladding is etched in a hydrofluoric acid of 8 % concentration. In this way, with the aid of the graphene micro-FBG (GMFBG), the wavelength of the laser system, as shown in Fig. 13.45 (b), was easily tuned by adjusting the intensity of the 1530 nm pump laser. In addition, the pump power was enhanced by a 1530 nm EDFA to interact with the graphene appropriately. (a) (b) (c) (d) Fig. 13.45. (a) Diagram schematic of the GMFBG, and the inset is the photograph of junction region of the GMFB and MFBG; (b) Experimental schematic of the GMFBG-based wavelengthtunable passively QML fiber laser; (c) Output laser spectra under pump powers with a range from 6.5 mW to 99.7 mW; (d) Peak wavelengths at different controlling pump powers with two tuning processes and the experimental data is linearly fitted by curve 1 and 2, respectively [77]. Furthermore, the authors discovered that the etching process altered the grating's typical internal structure (characterized by single-mode propagation), thereby rendering the grating as a mix-waveguide (characterized by the propagation of single and multi-mode) and somewhat induced a loss of ~2 dB. Nevertheless, the authors validated the impact of this fabricated element towards wavelength tuning by attaching it to a fiber ring laser system mode-locked with a saturable absorber (SA) and a 1 m length of EDF serving as the gain medium. Furthermore, the authors emphasized that envelopes of mode-locked pulses are easily generated in their laser system. This was achieved with the aid of a 563 Advances in Optics: Reviews. Book Series, Vol. 5 fabricated single-wall carbon nanotubes saturable absorber (SWCNT-SA) coupled alongside an OC, which modulated the laser system's net cavity loss. In order to achieve the process of optical-controlled wavelength tuning, the authors maintained the pump power of the 980 nm pump laser at 162 mW while tuning the intensity (pump power) of the 1530 nm pump laser from 6.5 to 99.7 mW. Hence, a wavelength tuned range spanning from 1550.348 nm to 1552.520 nm was obtained as shown in Fig. 13.45 (c), with a relatively high wavelength tuning sensitivity of 23.5 pm/mW, which is dependent on the GMFBG's response time in the order of a few milliseconds as shown in Fig. 13.45(d). The authors further mentioned that the filter situated in front of the GMFBG is used to mainly tune the wavelength of the laser lasing at a Q-switched mode-locking (QML) state. As a result, the stability of their laser system had a fluctuation in the order of ∼±0.04 dB, which is rather commendable even for a lab-based setting. Li, et al. [78] employed graphene-coated micro FBG (GCMFBG) as shown in Fig. 13.46 (a) to achieve a light-controlled wavelength-tunable fiber laser. An infrared continuous-wave light source continuously drives this fabricated device (GCMFBG). This light source leverages the graphene wideband absorption property. In other words, the evanescent field of the controlling infrared light source which interacts with the graphene ensures that the central reflected wavelength is tuned when the photo-thermal effect of graphene is stimulated. This is mainly feasible by etching a standard FBG in hydrofluoric acid to decrease the FBG's cladding, thereby creating a micro-FBG with a diameter of ~15.6 µm, ensuring that the graphene is conveniently wrapped unto the surface of the etched FBG. Therefore, the evanescent field of the controlling light can adequately interact with graphene, which is coated on a micro FBG, whose reflection spectrum and energy distribution of the controlling light within the FBG are shown in Fig. 13.46 (b) and (c), respectively. (a) (b) (c) Fig. 13.46. (a) Experimental schematic of the fabricated GCMFBG and (b) its reflection spectrum; (c) Circular symmetrical energy distribution of the controlling light on the cross section [78]. 564 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers The fabricated GCMFBG in this work is attached to a fiber laser system similar to the experimental scheme used by the same authors(s) in their previous work [77]. Compared to their previous work, the modified experimental scheme of this present work, as shown in Fig. 13.47, employs an EDFA to amplify the laser and provide sufficient gain. It also consists of a 20 m length of polarization maintaining fiber and two polarizationmaintaining devices (i.e., optical circulator 2 and optical coupler 4), forming a Brillouin laser cavity for this laser configuration. Fig. 13.47. Experimental setup of the light-controlled wavelength-tunable narrow linewidth fiber laser [78]. The principle of operation in terms of wavelength tuning is somewhat similar, wherein the controlling pump power is increased from 0 mW to 275.4 mW. In this way, the wavelength of the laser is optically tuned from 1549.936 nm to 1553.608 nm with wellpreserved linearity, as shown in Fig. 13.48 (a) and (b), respectively. This laser's wavelength tuning speed depends on the switch time of graphene, which is ~ 10 ms. Excluding the imperfections induced by EDFA and other environmental factors, this fabricated device does not in any way alter the SLM operation and frequency stability of the laser but rather enhances it, as shown in Fig. 13.48 (c) and (d). 13.8.4. Thermal Tuning In 2017, Sun et al. [79] demonstrated a 1.5 µm narrow-linewidth fiber ring laser that consists of a π-phase-shifted fiber Bragg grating (π-PSFBG) which serves as a narrow band filter, and a wavelength tuning component whose central wavelength is thermally tuned by a thermo-electric cooler (TEC). The fiber ring laser system presented in this work consists of C-band active and passive fiber components, as shown in Fig. 13.49 (a). Like a traditional FBG, the π-PSFBG employed in this work possesses a typical periodic refractive index modulation (i.e., grating period) along the fiber axis with a π-phase shift whose length is odd times half of the grating period as shown in Fig. 13.49 (b). 565 Advances in Optics: Reviews. Book Series, Vol. 5 (a) (c) (b) (d) Fig. 13.48. (a) Tuned spectra at output 1 under different controlling pump powers; (b) Corresponding wavelength positions with different controlling pump powers; (c) Spectra at output 1 and 2; (d) RF spectrum at output 2 with a frequency span of 1 GHz measured by the delayed self-heterodyne system [78]. (a) (b) Fig. 13.49. (a) Experimental setup of the tunable narrow linewidth fiber laser. EDFA, erbiumdoped fiber amplifier; TEC, thermo-electric cooler; OSA, optical spectrum analyzer; PM, power meter; HRFBG, highly reflective fiber Bragg grating; π-PSFBG, π-phase-shifted fiber Bragg grating; (b) Schematic diagram of a π-PSFBG [79]. Such a grating structure creates an ultra-short resonant cavity along the fiber axis, which induces an ultra-narrow transmission band in the central band of the π-PSFBG, whose 3-dB bandwidth is ~2.5 pm. According to the authors, the shape of this transmission band spectrum is unchanged irrespective of the increase or decrease in temperature-induced on it since the relationship between the length of the phase shift and grating period is constant. Furthermore, the authors emphasized that the renowned and acceptable temperature coefficient of the refractive index (α ≈1.1 × 10-5/°C) and the thermal expansion coefficient of silica fibers (β≈5.2 × 10-7/°C) are believed to be constant for the investigated temperature range in their laser configuration. In order words, the π-PSFBG employed in this work is not damaged by the thermal effect, which presents the excellent durability of this laser system. However, when heat is induced on the grating with the aid of the thermoelectric cooler, which increases temperature, the shape of the transmission 566 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers spectrum will undergo a red or blue shift along the fiber axis and inevitably result in a shift of the central wavelength. In essence, the central wavelength shift is feasible simply because the grating period and the effective refractive index of the fiber core, which determines the central wavelength, change in response to the induced thermal effect. The authors employed two fiber gratings alongside an optical circulator to determine the lasing wavelength. The light signal is initially reflected and filtered by a high reflective FBG (HR-FBG) with a bandwidth of ~0.18 nm. In contrast, the π-PSFBG is employed to serve as a narrow band filter and wavelength tuning component. The authors emphasized that a narrow linewidth laser whose wavelength is tunable is feasible only when the reflection band of the HR-FBG coincides with the narrow transmission band of the π-PSFBG as shown in Fig. 13.50 (a) and (b). (a) (b) Fig. 13.50. Spectra of (a) a π-PSFBG, where the blue solid line shows it measured with a tunable laser with a resolution of 1 pm. Pink dashed line: measured with an OSA; (b) HR-FBG measured with an OSA with a resolution of 0.02 nm [79]. Hence, these two gratings are separately placed on two refrigerator ceramic wafers, somewhat like a substrate whose temperatures are controlled by two TECs. This renders the simultaneous control of the central wavelength of these two gratings whose wavelength shift demonstrated a linear bathochromic shift with a wavelength shift coefficient of ~12 pm/°C when the temperature values of the TECs are adjusted as shown in Fig. 13.51 (a) and (b). The detected output spectral and wavelength tuning as a function of the temperature rise is shown in Fig. 13.52 (a) and (b) wherein the authors demonstrated that a gradual increment in temperature of the π-PSFBG from 5 °C to 95 °C obtained a maximum tunable range of 1.29 nm with a sensitivity of approximately 14.33 pm/°C. The authors further emphasized that for the previously mentioned temperature tuned range, the laser system required several seconds to obtain a stable laser output for each temperature change of 10 °C. The authors attached a second π-PSFBG with a transmission bandwidth of ~4 pm, as shown in Fig. 13.53 (a) into their laser scheme, which serves as a reference narrow bandpass filter. This reference grating was inserted in between the two couplers. Like the 567 Advances in Optics: Reviews. Book Series, Vol. 5 other two initial gratings, it likewise has a TEC whose temperature value was adjusted to ensure that the central wavelength coincides with the laser for each wavelength stability observation process when the pump power is varied. In order words, the wavelength stability can be characterized by observing the laser output power. Fig. 13.51. Measured central wavelength of (a) a π-PSFBG and (b) a HR-FBG with temperature [79]. (a) (b) Fig. 13.52. (a) Measured laser spectra as the temperature rises; (b) The laser wavelength as a function of the temperature of the π-PSFBG [79]. As shown in Fig. 13.53 (b) and (c), it is observable that the maximum power fluctuation is approximately 5%, suggesting that the maximum wavelength fluctuation is approximately 0.2 pm (~25 MHz) and consistent with the transmission spectrum of the reference π-PSFBG. In this way, we can deduce that the wavelength of the laser system is very stable. More work is to be done to expand the wavelength-tunable range. However, other commendable features of this laser are its high output power of approximately 1.1 W and low power fluctuation of < 3%, which is an impressive representation of the watt-level output power stability even for such a low-cost laser configuration. Furthermore, this laser tuning method is more straightforward and poses minor damage when objectively compared with other wavelength tuning methods. 568 Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization, Wavelength Switching, and Wavelength Tuning of Lasers (a) (b) (c) Fig. 13.53. (a) Measured transmission spectrum of the reference π-PSFBG. The measured laser power stabilities after passing through the reference π-PSFBG: (b) three power stages with same wavelength, inset: zoom in near the power of 740 mW; (c) three different wavelengths with near powers. All of the measurement durations are about 44 min [79]. 13.9. Conclusion We reviewed the advances on the laser parameters of semiconductor and fiber lasers by considering a few superior parameters such as linewidth, phase or frequency noise, relative intensity noise, and wavelength. Improving a laser parameter is highly desirable in modern-day systems such as optical communication and optical sensing systems, whose performance level depends significantly on the parameters of the laser employed. Hence, various methods used to improve the performance of laser parameters for widespread applications were presented in this work. Therefore, we predict that the future development directions of lasers will focus on the persistent advances in material and structural design, linewidth compression, high output power, low phase or frequency noise, low relative intensity noise, wavelength switching, wavelength conversion, and wavelength tuning. Acknowledgements The first author will like to acknowledge his loving parents, Rev. Engr. Joseph Chidozie Iroegbu and Bishop Agnes Benedicta Iroegbu for their loving support and unwavering encouragement throughout this project. His siblings, Dr. Engr. Udochukwu Frank Iroegbu, Chukwuemeka Daniel Iroegbu MD., Ph.D., and Pharm. Chioma Phoebe Iroegbu was a tremendous source of inspiration and motivation to him. 569 Advances in Optics: Reviews. Book Series, Vol. 5 This work was supported by the Ministry of Science and Technology (2016YFC0801202); Natural Science Foundation of China (61635004, 61825501, 61705024); Fundamental Research Funds for the Central Universities (106112017CDJZRPY0005). References [1]. M. S. Zubairy, A Very Brief History of Light, Optics in Our Time (M. D. Al-Amiri, M. M. El-Gomati, M. S. Zubairy, Eds.), Springer Nature, 2016. [2]. S. Lee, Introduction to Laser Physics with Applications in Fiber Optics, Lulu.com. [3]. O. Svelto, Principles of Lasers, 5th Ed., Springer Science + Business Media LLC, New York, 2010. [4]. G. P. 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Dorsinville, Switchable dual-wavelength SOA-based fiber laser with continuous tunability over the C-band at room-temperature, Optics Express, Vol. 20, Issue 21, 2012, pp. 23367-23373. [76]. M. Delgado-Pinar, J. Mora, A. Diez, J. L. Cruz, M. V. Andres, Wavelength-switchable fiber laser using acoustic waves, IEEE Photonics Technology Letters, Vol. 17, Issue 3, 2005, pp. 552-554. [77]. Y. Li, L. Gao, T. Zhu, Y. Cao, M. Liu, D. Qu, F. Qiu, X. Huang, Graphene-assisted all-fiber optical-controllable laser, IEEE Journal of Selected Topics in Quantum Electronics, Vol. 24, Issue 3, 2018, pp. 1-9. [78]. Y. Li, L. Huang, S. Huang, L. Gao, G. Yin, T. Lan, Y. Cao, I. P. Ikechukwu, T. Zhu, Tunable narrow-linewidth fiber laser based on light-controlled graphene, Journal of Lightwave Technology, Vol. 37, Issue 4, 2019, pp. 1338-1344. [79]. J. Sun, Z. Wang, M. Wang, Z. Zhou, N. Tang, J. Chen, and X. Gu, Watt-level tunable 1.5 μm narrow linewidth fiber ring laser based on a temperature tuning π-phase-shifted fiber Bragg grating, Applied Optics, Vol. 56, Issue 32, 2017, pp. 9114-9118. 574 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers Chapter 14 Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers Meriem Kemel, Mohamed Salhi, Charles Ciret, Georges Semaan, Ahmed Nady and François Sanchez1 14.1. Introduction Over the past decades, high-energy lasers attract considerable attention in various fields including medicine [1-4], security and defense [5, 6], and industry [7]. Until recently, the chirped pulse amplification (CPA) [8], which won the Nobel Prize of physics in 2018, was the most used technique to generate ultrashort high-energy optical pulses in mode-locked oscillators. Despite the effectiveness of the CPA technique, its implementation requires several amplifiers at the oscillator output. To make a step forward, in 2004, Fernandez et al. has reported a new technique enabling the generation of high-energy femtosecond pulses without having recourse to external amplifiers [9]. The principle was to use a chirped multilayer mirrors in the oscillator to produce a net positive dispersion over a broad spectral range. Then, the output chirped pulses are compressed, using a dispersive delay-line at the output of the laser cavity. Since then, several studies have emerged, offering a multitude of solutions for increasingly compact, efficient, and high-energy lasers. In fiber lasers, Chong et al. studied the key parameters (nonlinear phase shift, spectral filter bandwidth, and group-velocity dispersion) to determine the behavior and properties of the laser [10], thus permitting a better comprehension of the fiber laser functioning. Meanwhile, many investigations have been conducted in this regard and the concept of dissipative soliton resonance (DSR) was introduced by Chang et al. in 2008. The authors have numerically studied a new regime in which the energy of pulses increases indefinitely for particular values of the system parameters of the cubic-quintic complex Ginzburg-Landau equation (CGLE) [11]. They Meriem Kemel Laboratoire de Photonique d’Angers, Université d’Angers, Angers, France 575 Advances in Optics: Reviews. Book Series, Vol. 5 have also shown that this regime can occur in both normal and anomalous dispersion regimes [12]. Based on these first reports, the generation of high-energy square pulses has been extensively investigated numerically [13-18], and experimentally [19-29], in various configurations of passively mode-locked fiber lasers. In the experimental literature of the DSR regime, square pulses are generally assumed to be a manifestation of the DSR when they reveal some features versus the pump power. These features correspond to a linear increase of the pulse width, a linear increase of the energy per pulse, constant peak power, flat-top pulse shape, and invariant optical spectrum. This behavior occurs without experiencing any wave-breaking. Besides, to be considered as DSR, the square pulses must be temporally coherent, i.e., there have to be no fine structures within the square envelope. Despite the importance to verify the temporal coherence to affirm whether the square pulses are DSR or not, only a few authors have investigated it by providing the autocorrelation traces [13, 15, 29]. It should be noted that they mainly reported relatively short pulse durations (few hundreds of picoseconds) with relatively high repetition rates. However, for higher energy pulses, the laser cavities are adjusted to obtain much larger pulses. So, most of the reported square pulses in the DSR regime are in the range of a few ns to few hundreds of ns of pulse width, and with low repetition rates (in kHz). However, the usual autocorrelators do not always permit a correct measurement when the laser's repetition rate is lower than a certain value. In our case, the autocorrelation trace is not accurate for repetition rates lower than 1 MHz. Moreover, even when the repetition rate is compatible with the autocorrelator characteristics, the resulting autocorrelation trace is not enough to conclude about the coherence of the pulse. In theory, if the pulse is not coherent, a peak appears in the autocorrelation trace whereas it is absent in the case of a coherent pulse. The problem is that in the experiment, when a polarization controller (PC) is inserted at the input of the autocorrelator to optimize the signal, the presence of a peak in the autocorrelation trace depends on the adjustment of the PC. Therefore, it is impossible to conclude whether the square pulses are coherent or not since the absence of a peak in autocorrelation trace can have several origins. To circumvent these problems, we successfully implemented a new experimental setup to investigate the coherence of large square-shaped pulses in passively mode-locked Er:Yb co-doped double-clad fiber laser in anomalous dispersion regime. The setup consists of two different methods of measurement: Mach-Zehnder interferometer (MZI) and Dispersive Fourier Transformation (DFT). The MZI method is widely used for many applications [30-35]. It is the same concerning the DFT method which is extensively used in fundamental research, for example, to study pulse buildup dynamics [36-40], soliton breathing phenomena [41-43], laser instabilities [44, 45], and multipulse regimes [46-48]. In this work, we use these two methods to study the coherence of large nanosecond square pulses [49]. The chapter is organized as follows: Section 14.2 shows the experimental setup, including the laser cavity and both MZI and DFT techniques for coherence characterization. The experimental results of DSR-like square pulses are presented in Section 14.3. The results of coherence measurements performed with the MZI and DFT techniques are 576 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers presented in Section 14.4 and Section 14.5, respectively. A general conclusion is made in Section 14.6. 14.2. Experimental Setup The experimental setup, implemented for the coherence characterization of DSR-like square pulses is illustrated in Fig. 14.1. It consists of three different parts: (a) the laser cavity, (b) the MZI, and (c) the DFT line. The laser cavity is an all-fibered unidirectional ring (UR) enabling the generation of mode-locked square pulses using the nonlinear polarization rotation (NPR) mechanism [50]. The cavity includes a C-band double-clad co-doped Er:Yb 30 dBm fiber amplifier (EYDFA), a polarizer and two polarization controllers (PC1 and PC2), a polarization-insensitive isolator (PI-ISO) to ensure the unidirectional propagation of the light and an additional coil of single-mode fiber (SMF 28). The total cavity length is about 290 m (which corresponds to a round-trip time of 1.45 µs), including 5 m of double-clad fiber with second-order dispersion β2 = −0.021 ps²m−1 and 285 m of SMF 28 with β2 = −0.022 ps²m−1. The net cavity dispersion is about −6.375 ps². The formed signal is then extracted by a 1×2 optical coupler (OC) from the output port with an extracting ratio of 20 %. The laser is adjusted to generate square pulses in the nanosecond range (few tens of ns). Thus, the spatial extension of the pulse is few meters, so it is not possible to scan it entirely by standard interferometric methods. The coherence is then characterized by two different methods. Fig. 14.1. Experimental setup. EYDFA, Erbium-Ytterbium co-doped fiber Amplifier; OC, optical coupler; PC, polarization controller; SMF, single-mode fiber; PI-ISO, polarization-insensitive isolator; DFT, dispersive Fourier transformation; PZT, Piezoelectric translator; MZI, Mach-Zehnder interferometer; L, lens; TS, translation stage. The first method is based on a Mach-Zehnder interferometer (MZI), as illustrated in part (b) of Fig. 14.1. A 50 % coupler splits the beam into two equivalent signals. One part propagates through one arm, which includes two collimating lenses fixed on a translation stage (TS). The second part of the signal propagates through a piezoelectric (PZT) fiber stretcher which introduces a modulated delay δ(t) around the fixed path difference ∆. The 577 Advances in Optics: Reviews. Book Series, Vol. 5 total path difference is then ∆ + δ(t), as illustrated with the two pulses in the inset of part (b) of Fig. 14.1. The oval form PZT-based fiber stretcher consists of two U-shaped caps made of polyethylene resin designed for winding the optical fiber around it. One of the caps is fixed and made immobile while the other is mounted on a PZT translation stage driven by a function generator with a triangular function at 200 Hz. The free-space propagation made by the two lenses is crucial for a precise adjustment of the delay between the two arms. It also permits to easily suppress the delay if needed, which is tedious to achieve with all-fibered arms because of the eventual errors of the fiber length introduced by the fusion splicing. Finally, both signals are recombined with a 50 % OC and local temporal coherence of the square pulses is studied. The second method is based on the dispersive Fourier transform (DFT) technique as displayed in part (c) of Fig. 14.1. It consists of a simple delay-line of 30 km of SMF 28 through which the pulse propagates. DFT is a real-time measurement technique. In fiber lasers, it aims either to measure the dispersion D (GVD per unit length) of optical fiber or the spectrum waveform of a pulse [51]. This latter is performed by using the mapping relation between time and wavelength, given by Δτ = L. D. Δλ, where L is the length of the fiber, Δλ is the bandwidth of the laser, and Δ𝜏 is the time duration into which the laser spectrum is mapped [52]. For ultrashort pulses of few tens of picoseconds or less, the effect of DFT is to map the optical spectrum into a temporal waveform, i.e., the temporal intensity trace of the pulses mimics the spectrum. In our work, we use the DFT technique in a non-standard way to study the coherence of large nanosecond square pulses to characterize their coherence. These two methods (MZI and DFT) are completely independent but each one permit to confirm results of the other thus concluding whether the studied pulses are coherent or not. The laser output characteristics and the coherence measurements are monitored by a 13-GHz oscilloscope, combined with two 12-GHz photo-detector, an optical spectrum analyzer, and an electronic spectrum analyzer to investigate the temporal trace, the optical spectrum, and the radio-frequency signal, respectively. At last, a high-power integrating sphere is utilized to measure the laser average power. 14.3. DSR-like Square Pulses In this section, the characteristics of square-shaped pulses are studied at the reference laser cavity output (see the reference output in Fig. 14.1, part (a)). This allows verifying some of the DSR features of the pulses. The pulse temporal trace is given in Fig. 14.2 (a), obtained for the pump power of 2 W. There is one pulse per cavity round-trip, with a pulse width of 35.4 ns, as shown in the inset of Fig. 14.2 (a). The corresponding optical spectrum is centered at a wavelength of 1612 nm as shown in Fig. 14.2 (b). The spectral bandwidth measured at -3 dB is equal to 9.2 nm, and the spectral waveform is typical for square pulses in the DSR regime. Fig. 14.2 (c) represents the RF spectrum trace with a signal-tonoise ratio of 68 dB, indicating high signal stability. This is confirmed by the stability of the regime for pump power ranging from 0.8 W to 4.02 W, without any polarization controller (PC) readjustment. As illustrated in Fig. 14.3 (a), the temporal shape remains perfectly square and there is one square pulse per cavity round-trip. So while increasing 578 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers the pump power there is no wave-breaking occurred, which is an important feature of the dissipative soliton resonance (DSR) regime. The same observation is made concerning the optical spectrum, which also keeps its initial shape, as shown in Fig. 14.3 (b), by holding the spectral bandwidth at -3 dB around the same value in function of the pump power, as depicted in Fig. 14.3 (c). Fig. 14.2. (a) Temporal trace of square pulses, (b) the corresponding optical spectrum, and (c) the radio-frequency trace. Additional characteristics are required to affirm the eventual DSR nature of these square pulses. As mentioned before, the most common characteristics of the DSR regime to verify are the linear increase of the pulse width and the pulse energy in the function of the pump power, and the peak power clamping effect, i.e., the peak power has to be constant while the pulse width increases with the increase of the pump power. Fig. 14.4 (a) represents a linear increase of the pulse width (red stars fitted with a linear red curve), and the blue dots represent a nearly constant peak power while increasing the pump power. Pulse energy also follows the linear evolution with the pump power, as shown in Fig. 14.4 (b). These three characteristics are as expected for a DSR pulse, and in many publications, these criteria are enough to conclude that the obtained square pulses are in the DSR regime. As specified before, the autocorrelation trace was provided in few studies but for relatively short duration pulses. Herein, the study is completed by verifying the temporal coherence of the obtained nanosecond pulses, and only then we can conclude whether it is DSR pulses or not. 579 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 14.3. (a) Temporal trace of square pulses for pump power ranging from 0.8 W to 4.02 W; (b) The corresponding optical spectrum for the same range of pump power; (c) 3-dB bandwidth of the spectrum as a function of the pump power. Fig. 14.4. Characteristics of square pulses. Evolution of the (a) pulse width (red stars linearly fitted by red curve), the peak power (blue dots fitted by linear blue curve), and (b) the pulse energy (black dots with a linear fit curve) with pump power ranging from 0.8 to 4.02 W. 580 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers 14.4. Coherence Characterization of Square Pulses with MZI Technique Square pulses in passively mode-locked fiber laser are a complex regime. Before studying their coherence, the MZI is tested and validated with a continuous wave laser source. We used a Distributed Feed-Back (DFB) laser source emitting at a wavelength of 1556.5 nm. The optical spectrum of the DFB laser is represented in Fig. 14.5 (a). Fig. 14.5 (b) displays the temporal trace of the signal at the output of the MZI. Fig. 14.5. (a) Optical spectrum trace of the DFB laser source. (b) Temporal trace of interferometric fringes of the DFB laser at the output of the MZI. The temporal trace exhibits the interferometric fringes with high contrast. The modulated delay which is performed by the PZT fiber stretcher driven by a 200 Hz triangular function generator (see part (b) of Fig. 14.1), corresponds to the period T = 5 ms. One can note that the curve is not a perfect sine as expected for a continuous phase variation, but there exist phase jumps around 1 ms, 3.5 ms, and 6 ms. These phase jumps are located at every minimum and maximum of the triangular function which causes an inhomogeneous response of the fiber to the PZT fiber stretcher. The frequency of the triangular function is chosen at 200 Hz, according to the characteristics of the PZT fiber stretcher, the fiber response to the mechanical stretching, and also for better visualization of the interferometric pattern on the oscilloscope. Finally, except for the phase jumps, the MZI is operational and could be used to measure the coherence of more complex laser regimes. From Fig. 14.5 (b) we deduce that there are about 33 fringes for one modulation period of the PZT so that a 2𝜋-phase delay between the two arms of the interferometer corresponds to about 150 µs. The period of the interferometric fringes is about 100 times greater than the round-trip time of the cavity (1.45 µs). Consequently, the interferometric fringes will not be visible along one square pulse but will modulate a large number of pulses. Fig. 14.6 is a schematic representation of the expected temporal intensity evolution of the square pulses at the MZI output. In the case of coherent pulses, the intensity of interfering beams is expected to experience a sinusoidal evolution corresponding to consecutive destructive and constructive interference (as observed in the temporal trace of interferometric fringes of the DFB laser). So, if we measure the intensity of the top of the center of each two interfering pulses (see red crosses in Fig. 14.6 (a)), the resulting curve would be sinusoidal, as shown in the same 581 Advances in Optics: Reviews. Book Series, Vol. 5 figure (red curve representing the envelope versus time). However, in the case of incoherent pulses, the intensity reached by the interfering part of each pulse remains constant, i.e., as the pulses are not coherent, the resulting intensity is an incoherent addition of intensities of the interfering pulses (see Fig. 14.6 (b)). The interpretation of the interferometry results of the square pulses is made based on this reasoning. Fig. 14.6. Temporal evolution of the pulse intensity, and the envelope, in case of (a) coherent pulses, and (b) incoherent pulses. Fig. 14.7 (a) represents the interferometric fringes of the square pulses with 26.5 ns pulse width, obtained at a pump power of 1.67 W, for different fixed optical delays ∆ (~0, ~100, ~200, and ~300 µm). For a fixed delay of ∆ ~0 µm, i.e., when the path difference is nearly zero, the interferometric fringes are visible with relatively high contrast. When ∆ increases to 100 µm and 200 µm the contrast of the fringes decreases and disappears for ∆ ~300 µm. Fig. 14.7 (b) represents the corresponding peak intensity (envelope) traces for each value of path difference, in which the evolution process of fringe visibility is much clear. Taking into account the discontinuities introduced by the fiber response to the stretching, we can see that for ∆ ~0 µm, the contrast of the fringes is high and mimics the sine function as it is expected in the schematic of Fig. 14.6 (a). When ∆ increases, the contrast of interferometric fringes decreases. For ∆ ~300 µm, the fringes completely disappear, and we find the case of incoherent pulses of Fig. 14.6 (b), where the peak intensity is constant versus time. The same measurements are conducted for different pulse durations. In Fig. 14.8, we represent the results for square pulses with a duration of 69 ns, obtained at a pump power of 4.02 W. As in the previous figure, one can observe a relatively high contrast interferometric fringes for ∆ ~0 µm, but the coherence pattern disappears around the same value of the path difference as previously (∆ ~300 µm). We conclude from Fig. 14.7 and Fig. 14.8, that when the path difference is equal or higher than ∆ ~300 µm, corresponding to a time delay of 𝜏 = 1 𝑝𝑠, the coherence completely disappears. This means that the square pulses are not coherent and could consist of many shorter pulses within the square envelope. The duration of these pulses is estimated to be about 1 𝑝𝑠. 582 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers Fig. 14.7. (a) Temporal evolution of the interferometry fringes of the 26.5 ns pulses at the pump power of 1.67 W, for different optical delays ∆ (ranging from 0 µm – 300 µm), at the output of MZI; (b) The corresponding peak intensity evolution. Fig. 14.8. (a) Temporal evolution of the interferometry fringes of the 69 ns pulses at the pump power of 4.02 W, for different optical delays ∆ (ranging from 0 µm – 300 µm), at the output of MZI; (b) Intensity envelopes of the interferometry fringes. In conclusion, by using an MZI, we prove that although the studied square pulses fulfill almost all characteristics of the DSR pulses (such as linear increase of the pulse width and pulse energy, constant peak power, and invariant optical spectrum, versus the pump power), they are not necessarily DSR but may be packets of very short pulses (which can be difficult to detect by usual means). These results are confirmed by other measurements conducted with the DFT technique. Indeed, with the MZI, we can scan a short part of a square pulse so the coherence measurement is local. It is possible to adjust the delay ∆ with the free-space collimators and scan different parts of the pulse as shown previously for different path differences ∆ (~0, ~100, ~200, and ~300 µm). However, the studied square pulses have a spatial extension of few meters and it is not possible to create such delays with collimators. The DFT technique allows more global coherence characterization. By studying the shape evolution of the whole pulse after propagating a long fiber coil, it is possible to discriminate between coherent and incoherent square pulses. 583 Advances in Optics: Reviews. Book Series, Vol. 5 14.5. Coherence Characterization of Square Pulses with DFT Technique In this section, the DFT technique is used to study the coherence of square pulses. As explained in Section 14.2, the usual DFT technique allows performing real-time measurement of the spectral profile. This occurs if the fiber length 𝐿 satisfies the condition 𝐿 ≫ 𝐿𝐷 [53], where 𝐿𝐷 is the dispersion length. Experimentally, this condition can be easily fulfilled with femtosecond or picosecond pulses while it is impossible in the nanosecond range. Considering a Gaussian pulse with a duration of 𝜏0 = 10 𝑛𝑠, the required fiber length to perform an efficient DFT exceeds 109 𝑘𝑚. Despite this limitation for large pulses, the DFT allows to easily discriminate between a coherent and incoherent square pulse. Indeed, let us consider a square pulse in the nanosecond range and a dispersive line of few tens of km of fiber. If the pulse is coherent, such fiber length is not enough to modify the pulse shape since the dispersion has no time to spread the pulse. In contrast, if the square pulse experiences considerable modifications in the temporal profile and imitates the spectrum waveform, we can conclude that the pulse is incoherent and assume that it is composed of subsequent fine structures. In our experiment, the studied square pulses propagate through 30 km of SMF (see part (c) Fig. 14.1). Fig. 14.9 (a) shows the temporal trace of the initial square pulses at the laser output (at the reference point in part (a) of Fig. 14.1), while Fig. 14.9 (b) represents the temporal trace of the pulses at the output of the DFT line (part (c) of Fig. 14.1). Fig. 14.9. Temporal traces of the square pulses for different pump powers (a) at the reference output, and (b) at the output of the DFT-line. (c) The rise time of the initial pulses (red squares fitted with the linear red curve) and after the DFT-line (blue dots fitted with the linear blue curve). 584 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers Comparing the initial pulses and those after propagating through the DFT-line, the pulses are no more square-shaped after 30 km of fiber. This shape-change is more pronounced for the shortest pulses. We can see that for the pulses at the pump power of 0.46 W and 0.8 W, corresponding to initial duration of 6.5 ns and 12.2 ns, respectively, the temporal trace of the pulses after propagation through the DFT-line are more likely to mimic the spectral waveform as in standard DFT technique. For higher pump powers and larger pulses, the DFT pulses remain flat at the center. However, the rise time of the initial square pulses is around 0.1 ns while the rise time of these pulses after the DFT line is around 10 ns as depicted in Fig. 14.9 (c). This means that if the available fiber is twice longer for example (60 km instead of 30 km), but still much lower than the dispersion length, the pulse depicted in blue in Fig. 14.9 (a) would experience the same shape change as in the case of the pulse depicted in red. These observations permit us to conclude that the studied square pulses are not temporally coherent but consist of many shorter pulses that allow the envelope to be stretched by only 30 km of fiber. Indeed, as explained above, if the square pulses were coherent with a nanosecond range duration, they would not experience any change of the temporal profile because of the considerable length of the fiber that would be necessary to observe the DFT effect. For further investigation of the DFT-line effect on the pulses, additional numerical simulations have been performed. In the simulations, we have considered the optical fiber as a purely second-order dispersive medium. Additionally, we consider the initial peak power of the pulse relatively low so that the nonlinear effects can be neglected, which is the fundamental assumption for DFT to be performed. In the reference frame moving at the group velocity, the electric field envelope at the abscissa 𝑧 is related to the incident electric field envelope through the integral formula [33]: 1 1 (𝑡−𝑡′)2 ) 𝑑𝑡 ′ , 𝛽2 𝑧 𝐴(𝑧, 𝑡) = √2𝑖𝜋𝛽 𝑧 ∫ 𝐴(0, 𝑡′) exp (− 𝑖 2 2 (14.1) where 𝐴(0, 𝑡) is the initial pulse profile, and 𝐴(𝑧, 𝑡) is the pulse profile after propagation through the fiber of length z. The coefficient 𝛽2 is the group velocity dispersion. In these simulations, we consider a super-Gaussian pulse profile given by: 𝑡 𝑚 𝐴(0, 𝑡) = exp (−(1 + 2𝑖𝐶) ∙ (𝑡 ) ) 0 (14.2) To start, the linear chirp parameter C is chosen randomly as 𝐶 = 15. The order of the super-Gaussian is 𝑚 = 100 to better fit the square shape of the experimental pulses. We take 𝑡0 = 2.5 ns leading to an initial pulse duration of 5 ns, defined as the full width at half maximum. The group velocity dispersion value of silica single-mode fiber at the wavelength of 1.55 µm is 𝛽2 = −0.022 ps 2 m−1. The results are shown in Fig. 14.10, where the initial square pulse and its spectrum are represented in the blue curve in Figs. 14.10 (a) and (b), respectively. The red and black 585 Advances in Optics: Reviews. Book Series, Vol. 5 curves in Fig. 14.10 (a) represent the pulse after propagating 30 km and 2.5 × 108 km of fiber, respectively. One can note that the pulse shape after the 30 km of DFT line remains almost square. The rise time difference is very low, increasing from 0.42 ns for the initial pulse to 0.57 ns for the DFT pulse. However, after propagation in 2.5 × 108 km of fiber, the temporal trace of the pulse is no more square but mimics the optical spectrum of the pulse. These observations demonstrate that to perform an efficient DFT on a coherent pulse with an initial duration as large as 5 ns, the needed fiber length is at least equals to 2.5 × 108 km and the 30 km are far from being enough. Fig. 14.10. (a) Temporal trace of the square pulses representing the initial pulse in blue, the pulse propagation after 30 km in red, and the pulse propagated in 2.5 × 108 km in black; (b) The corresponding optical waveform for the initial square pulse. Thus, in the case of the experimental results, as the 30 km of fiber is enough to strongly modify the shape of the pulses and mimic the optical spectrum, the pulses are not coherent. In the previous calculations, the chirp was fixed to C = 15, so further study of the effect of this parameter is required. Since DSR square pulses are highly linearly chirped [54], many values have been tested in the simulations considering two pulses with pulse widths of 10 ps and 5 ns. The range of chirp values is estimated based on the duration and spectral bandwidth of the pulse. The 10 ps short pulse requires around 1 km of fiber to perform an observable DFT effect. As shown in Fig. 14.11 (a), for an initial pulse width of 10 ps, the temporal trace after 1 km of fiber has mapped the optical spectrum shown in Fig. 14.11 (b). For the same 1 km propagation, different values of the chirp are tested such as C = 0.1, 1, 10, 20. The curves are represented in a different color for each chirp value. One can note that for an initial pulse with a duration as short as 10 ps, the chirp has no significant effect on the DFT process even with the highest chirp C = 20. It is worth mentioning that 1 km of fiber is a limit fiber length for an observable time-wavelength mapping, so if the chirp has any influence on the DFT process, it would be visible in the simulations. 586 Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked Fiber Lasers Fig. 14.11. (a) Temporal trace of the square pulses representing the initial pulse in blue with a pulse width of 10 ps, and the other curves represent the pulse after propagating 1 km of fiber with different values of the initial chirp (C = 0.1, 1, 10, 20); (b) The corresponding optical waveform for the initial square pulse. Other chirp values (C = 15, 500, 5000, 50000) are applied for a pulse with an initial duration of 5 ns. The result is shown in Fig. 14.12. We observe in Fig. 14.12 (a) that even if a very high linear chirp is applied, the temporal trace doesn’t mimic the optical spectrum shown in Fig. 14.12 (b). Fig. 14.12. (a) The temporal trace of the square pulses representing the initial pulse in blue with a duration of 5 ns, and the other curves represent the pulse after propagating 30 km of fiber with different values of the initial chirp (C = 15, 500, 5000, 50000); (b) The corresponding optical waveform for the initial square pulse. 587 Advances in Optics: Reviews. Book Series, Vol. 5 In conclusion, the initial chirp of the square pulses has no considerable impact on the DFT process. So, the value of the chirp of experimental pulses, shown in Fig. 14.9 (b) (especially whose duration is about 5 ns), is not the reason why the pulses tend to mimic the optical spectrum and perform an efficient DFT. As the DSR pulses are characterized by a high chirp, these simulation results allow understanding the impact of the chirp while the pulses are propagating in the fiber. Thus, both MZI and DFT methods confirm that the studied square pulses are not temporally coherent. Therefore, the square pulses are not DSR pulses. 14.6. Conclusion In conclusion, using both MZI and DFT techniques, we studied the temporal coherence of square pulses in a passively mode-locked fiber laser. In the case of MZI, we were able to circumvent the limitations due to the repetition rate and the pulse’s duration, to visualize in real-time the interferometric fringes on the oscilloscope. We observed that, as soon as the interferometer’s time delay reaches 1 ps, the interferometric fringes disappear. This proves that the square pulses contain shorter pulses within the square envelope. In addition, through the DFT technique, we confirm that these pulses cannot be coherent as the temporal shape of the pulses changes after only 30 km consisting of the DFT line. The result of the DFT is more significant for the pulses with a duration of 6.5 ns and 12.2 ns as they tend to mimic the optical spectrum. This would be impossible to occur in the case of coherent pulses because of the length of the fiber needed to complete the DFT effect on these large duration pulses. The simulation of the DFT on a coherent square pulse of a few ns confirms that the 30 km available are far from being enough to produce any shape change to mimic the optical spectrum. We also studied the effect of the linear chirp on the DFT process, and confirm that its value doesn’t affect the shape-change of the pulses. Finally, with both methods, we conclude that even if the square pulses verify most of the characteristics of DSR as a function of pump power (a linear increase of the energy and width of the pulses, clamped peak power, and invariant optical spectrum), they are not necessarily DSR as long as they are not temporally coherent. We assume that these square pulses may be noise-like pulses with fine structures that we are not able to detect with the usual acquisition system. References [1]. A. A. Boni, D. I. Rosen, S. J. Davis, L. A. Popper, High-power laser applications to medicine, J. Quant. Spectrosc. Radiat. Transfer, Vol. 40, Issue 3, 1988, pp. 449-467. [2]. R. W. 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New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers Chapter 15 New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers Do Tan Si1 15.1. Introduction Every physicist and engineer know that a periodic function f (x) of period P may be expanded into an infinite series of cosine, sine and exponential functions following the formulae f (x)    (n) cos(2 n x ) P a  x  b  a  P, (15.1) f (x)    (n)sin(2 n x ) P a  x  b  a  P, (15.2) nZ nZ f (x)   c(n) exp(i 2 n  n nZ x ) P a  x  b  a  P, (15.3) This revolutionary idea was introduced by Fourier in 1807 at the Academie Française under the title “Treatise on the propagation of heat in solid bodies” [1]. Formula (15.3) means that if the variable x represents a time then “A periodic movement t  t0 ) with frequencies of period P of an object creates a series of waves exp(i 2 n P n  2n / P each has relative amplitude equal to c(n) ”. Do Tan Si Ho Chi Minh-city Physical Association, Vietnam 593 Advances in Optics: Reviews. Book Series, Vol. 5 For example the vibration of a guitar string generates octaves notes of music; rain water creates wavelets on a pool, change of electron orbits creates light, etc. Many works on Fourier series has been done afterward and may be found on the net for examples in [4, 5]. Nevertheless looking at the integrations utilized until now to calculate the coefficients  (n),  (n), c(n) we notice that these integrations are rather complicated to perform, i.e. there is not yet a general formula easy to apply for obtaining them. This work proposes such a formula where lengthy integrations are replaced with short derivations. Moreover, from a simple property of the exponential function we obtain the equivalence between each term of a Fourier series of a function f ( x) with a polynomial. Based on this property of the exponential function, we obtain also the recurrence formula for calculating the Bernoulli polynomial Bm (x) from and only from Bm1 (x) which open the way for obtaining the Lucas symbolic formula Bm (x) : ( B  x)m and a simple matrix formula for calculating easily Bernoulli numbers Bm . Corollary the powers of pi, the values of positive even Riemann zeta functions may also be obtained. Details of calculations are exposed in the hereafter paragraphs. 15.2. Fourier Series 15.2.1. Fourier Series Coefficients Let f (x) be a periodic function defined on an interval a  x  b and has the period P  b  a. We want to have the expansion of f (x) into a series of exponentials e f (x)   c(n)e i 2 n x P a  x  b  a  P,  a e  i 2 n 0 x P x P (15.4) nZ b i 2 n x b   i 2 n0 x dx P f ( x)d ( )  c(0)  e  a P P   c(n) (e i 2 ( n  n0 ) nZ n 0 b P i 2 ( n  n0 ) e i 2 (n  n 0 ) a P ) (15.5) If n 0  0, the second term is equal to zero and we have  b a f ( x)dx  c(0) P If n 0  0, the first term is zero and we have 594 (15.6) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers  b a e  i 2 n 0 x P x e f ( x)d ( )   c(n) P nZ i  ( n  n0 ) ba P i  ( n  n0 ) n  n0   c(n ) nZ n n0 e i  ( n  n0 ) 0 b-a P (e e i 2 (n  n 0 )  i  ( n  n0 ) b-a P )  ba P (15.7) 2i sin  ( n  n0 )  0  c( n0 ) i 2 (n  n0 ) Combining (15.6) and (15.7) we get the formula for calculating the coefficients in a Fourier series c ( n)  x  i 2 n 1 b P f ( x )e dx  P a (15.8) For continuous functions, in order to simplify the calculation in (15.8) we utilize the method of integrations by parts x  i 2 n 1 b P f x e dx  ( ) P a b a  x  i 2 n  i 2 n  i 2 n i b i   P P P f x e dx, )  '( )   f(a)e (f(b )e a 2 n   2 n c ( n)  i c ( n)  2 n (15.9) b x x  i 2 n   i 2 n  i b P P  f( ) '( ) x e f x e dx,   a   a 2 n (15.10) where we have introduced the notation b  x b a  i 2 n   i 2 n  i 2 n   P P P  f(a)e f(x)e   f(b )e  a (15.11) In the cases where the function has successive derivatives one may continue successively integrations by parts beginning with  b a f '( x)e  i 2 n x P b b  e  i 2 nx / P  e i 2 nx / P  "( ) dx  P  f '( x) P f x dx,  a (i 2 n)  a ( i 2 n)  (15.12) in order to obtain an interesting new formula following our knowledge for calculating the Fourier series coefficients b x  i 2 n  1  P k  ( k 1) P c ( n)    ( ) f ( x )e  ,n0 P k 1 2i n  a (15.13) 595 Advances in Optics: Reviews. Book Series, Vol. 5 Remarks: o In actual literature about Fourier series it is considered integrations of  b a  b a f ( x)dx instead f ( x)dx. We think that this is not convenient for periodic functions f ( x) which are discontinuous at b such as rectangular, Heaviside functions and so all; o The great advantage of Eq. (15.13) is that we don’t have to calculate integrations as in Eq. (15.9) but only derivatives. 15.2.2. Formula for Obtaining Fourier Series of Functions From (15.11), (15.13) we get the formula for obtaining Fourier series of functions b x m b  i 2 n   P k i 2 n Px 1 1 P f ( x)    f ( x)dx     f ( k 1) ( x)e ( ) e ,  a P P nZ k 1   a 2i n (15.14) n0 where m may be infinite. Remarking that e  i 2 n b P e  i 2 n a P (15.15) , we may also write f ( x)  c(0)  b 1 m P k i 2 n x-aP ( k 1)   f ( x ) (   a  2i n ) e P k 1  nZ (15.16) n0 Eqs (15.14) and (15.16) may be written under the form of sum of sine and cosine Fourier series, for example m   b x  i 2 n  1  2 x P 2 k 1 P ) 2i sin(2 n )  f ( x)  c(0)    f (2 k ) ( x)e  ( P P n 1 k 0   a 2i n  m 2    1  (  f (2 k 1) ( x)e  P n 1 k 1  x  i 2 n P (15.17) b  x P 2k ) 2cos(2 n )  ( P  a 2i n b From (15.17) we see that a function f ( x) such that  f (2 k 1) ( x)   0, k is a pure Fourier a b sine series and such that  f (2 k ) ( x)   0, k a pure Fourier cosine series. a 596 Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers 15.2.3. Fourier Series of Even and Odd Functions We have for even functions f ( x)  f ( x),  a  x  a a a x x  i 2 n   i 2 n    (2k) P P  f( x ) e f ( x ) e     0  a  a (15.18) According to (15.18) we see that an even function has pure Fourier cosine series. However a pure Fourier cosine series may also came from functions such as Bernoulli polynomials f ( x)  B2m ( x), a  0, b  1 as we can see in a hereinafter paragraph. Similarly an odd function f ( x)   f ( x),  a  x  a a a x x  i 2 n   i 2 n   (1)  (2k 1) P P  f ( x ) e f ( x ) e     0   a   a (15.19) is a pure Fourier sine series but functions which are not odd such as Bernoulli polynomial f ( x)  B2m1 ( x), a  0, b  1 has also pure Fourier sine series. 15.2.4. From Fourier Series to Fourier Transform Consider a periodic functions f (x) defined on the interval a  x  b  a  P we have following (15.4) and (15.8) f (x)  b 1 eikx  f ( x)e  ikx dx  a P nZ Making the not forbidden extension a  , b   and putting k  (15.20) 2 n we get the P formula f (x)   1 eikx  f ( x)e  ikx dx   P kZ (15.21) which in a 3D space becomes  1 f (r)   (  ei k.r f (r )dr ) ei k.r , k R3 P  (15.22) where P is determined within a multiplicative constant. 597 Advances in Optics: Reviews. Book Series, Vol. 5 We may conclude then that the Fourier transform is the limit when a  , b   of the Fourier series and that the Fourier transform of a function, if it exists, is the sum of an infinity of plane waves exp(i k.r) each has wavelength   2 / k and relative amplitude  f (k )    ei k.r f (r ) dr , (15.23)  where kr is dimensionless and f (k) is the Fourier transform of f (r). It is not superfluous to inform readers that the formula (15.22) was found to be very important because it alone may explain the principles in quantum mechanics and laws of optics [2, 3]. For examples concerning laws of optics, we have deduced the theorem: “In the diffraction of a plane wave k0 by a 3D object represented by a function f D (r ) equal to unity for r  D and to zero for r  D under Fraunhofer conditions the diffracted wave is composed of plane waves k each having amplitude equal to the Fourier transform of the object function calculated for the deviation k  k  k0 ”. 15.2.5. Obtaining Fourier Series of Current Functions 15.2.5.1. Periodic Function Built from Delta-like Function  a ( x)   ( x),  a  x  a, P  2a We simply have c ( n)  1 a  i 2 n Px 1 e  ( x)dx  ,   a P 2a so that formally x 2 i n 1 1 1 1  n x 2a  a ( x)    e    cos( ) 2a 2a nZ 2a a n 1 a n0 This is not a convergent series. 598 (15.24) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers 15.2.5.2. The Saw-tooth Function Built from f ( x)  x, 0  x  a The saw-tooth function is described by the graph shown in Fig. 15.1. Fig. 15.1. Graph of f ( x)  x, 0  x  a. We have a a  xdx   0 a x x a 2  xe  i 2 n P   a, 1.ei 2 n P   0,  ,    2  0  0 so that from the formula (15.14) for calculating Fourier series we deduce that x a  P i 2 n Px  ( )e , 0  x  a, P  a, 2 nZ 2i n (15.25) n0 i.e.  1 x 1 x  n sin(2 n a )   ( 2  a ), 0  x  a, P  a (15.26) n 1 Formula (15.26) permits to calculate the value of pi by the formula by putting 4x  a 1 3 1 5 1 7   4(1     ....), (15.27) which is the historical formula found by James Gregory (1638-1675) then by Euler in 1734 by a famous method very well explained by Ayoub in Ref. [6]. More generally we may prove that b b a a  i 2 n  i 2 n  i 2 n  i 2 n Px  P P P  xdx   b  a ,  xe    be ae Pe ,   a 2  a b 2 2 b  i 2 n Px  1.e   0,  a 599 Advances in Optics: Reviews. Book Series, Vol. 5 x x-a  i 2 n ab P P   (  )e 2 nZ 2i n (15.28) n0  P x-a   ( )sin(2 n ) a  x  b, P  b  a P n 1  n 15.2.5.3. Periodic Function Defined from the Heaviside Function Consider the function described by the graph shown in the following Fig. 15.2. f ( x)  H a ( x)  0 if  a  x  0, H a (0)  1 , 2 (15.29) H a ( x)  1 if 0  x  a, P  2a Fig. 15.2. Graph of H a ( x). We have a  H a ( x )   a,  a a 0 0 a x 1 x x  i 2 n   i 2 n   i 2 n   i 2 n      2a 2a 2a 2a H ( x ) e H ( x ) e H ( x ) e H ( x ) e     a   a   a   a  ,  a   a   0   0 so that by (15.14) H a ( x)  a  i 2 n 1 1 1 1 2a i 2 n 2ax 2a )e ,    (0  0   (  )  (e  1))( 2 2a nZ 2 2 2i n n0 600 Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers H a ( x)  1  2 2   n sin( n 0 n odd n x ) a (15.30) Formula (15.30) is conformed with the result given by Spiegel in Ref [4]. 15.2.5.4. The Square Wave Function SW ( x) This is the function described by the graph in the following Fig. 15.3. Fig. 15.3. Graph of SW ( x). and related to the function H a ( x) by the relation SW ( x)  2H a ( x)  1, (15.31) so that according to (15.30) SW ( x)  4   n sin( n 0 n odd n x ) a (15.32) 15.2.5.5. The Periodic Function Constructed from x The function called repeating ramp is represented in Fig. 15.4 by the following graph. 601 Advances in Optics: Reviews. Book Series, Vol. 5 Fig. 15.4. Graph of the RR function x . The considered function is not continuous so that we can’t utilize (15.14) for calculating its Fourier series. Instead because its derivative function is identical to the square wave function so that its Fourier series is obtainable by integration of (15.32) then calculate the a constant of integration by putting x  2 f ( x)  x  C  4 a   n n cos( n 0 n odd n x a 1 n x )   4a  ( ) 2 cos( ) a 2 a n 0  n n odd From (15.33) we may obtain the value of pi squared and many sums over For examples by taking x  0 and x  2 8  (15.33) 1 . n2 a we get 3 1 1 1 1 2 1 1 1 1 .....,      ( 2  2  2  2  ...) 12 32 52 7 2 12 1 3 5 7 (15.34) 15.2.5.6. Fourier Series of a Function Corresponding to a Displaced Graph Let f ( x) be defined in the interval a  x  b then g ( x)  f ( x   ) is defined in the interval a    x  b  . The graph of g ( x)  f ( x   ) is that of f ( x) displaced left-hand side a distance equal to  . Similarly the graph of g ( x)  f ( x)   ) is that of f ( x) displaced vertically down a distance equal to  . By displacing up-down and left-right a graph we may centralize it and get its Fourier series. 602 Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers For example the Fourier series of the function represented by the graph in Fig. 15.5 is H a ( x   )      1  2 2   n sin( n 0 n odd n ( x   ) ) a (15.35) Fig. 15.5. Graph of H a ( x   )   . 15.2.5.7. Fourier Series of a Function Corresponding to a Symmetrised Graph Let f ( x) be defined in the interval a  x  b then g ( x)   f ( x) is defined in the same interval. The graph of g ( x)   f ( x) is symmetric with respect to the axis Ox of the graph of f ( x). Similarly the graph of the function g ( x)  2C  f ( x) is symmetric with respect to the line parallel to Ox at y  C. For example because the graph in the following Fig. 15.6 is obtainable from the graph of f ( x)  x described in (15.33) by a symmetrization with respect to the axis Ox following by a upward displacement of a distance to this graph is g ( x)   x  a  a, we see that the Fourier series corresponding a 1 n x )  4a  ( ) 2 cos( 2 a n0  n (15.36) n odd Fig. 15.6. Graph of the triangular signal. 603 Advances in Optics: Reviews. Book Series, Vol. 5 15.2.5.8. The Periodic Function Built from f ( x)  x2 , 0  x  a, P  a This function has f '( x)  2x and is described by the graph shown in Fig. 15.7. Fig. 15.7. Graph of f ( x)  x2 , 0  x  a, P  a. We have a a  x 2 dx   0 a3  , 3  2 i 2 n Px  2 x e  a ,  0 a  i 2 n Px   2 xe   2 a,  0 a   i 2 n Px   2e   0,  0 so that following (15.14) its Fourier series is x  i 2 n a2 a a 2 i 2 n ax 2a  a  ( x   a ( )e ) e , 0  x  a, a nZ 2i n 3 nZ 2i n 2 n0 x2  a2 a2   3 n0   1 x a2   sin(2 n )  2 a n 1 n 1 n n 1 2 cos(2n x ), 0  x  a a (15.37) The formula (15.35) is given by Spiegel in Ref. [4] with the inconvenient remark that at x  0 and x   the series converges to 2 2 . We think that the difficulty results in the fact that the second term in this formula is undetermined for x  0. To bypass this difficulty we utilize the formula (15.25) to transform the Fourier series (15.37) into the simpler one x2  a2 a a2  a( x  )  2  3 2  1 n n 1 2 cos(2n x ), 0  x  a a (15.38) From (15.36) we get the equation ( 604  1 x2 x 1 x 2    ) cos(2n ), 0  x  a   2 2 a a 6 a n 1 n (15.39) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers which gives by putting x  0 the values of pi squared and different sums over 1 such n2 as the Riemann zeta function of second order  1 n n 1 2  2 (15.40) 6 Another way of calculating the Fourier series of f ( x)  x 2 is possible by remarking that the derivative of f ( x)  x 2 is f '( x)  2x so that by an integration of (15.24) we get x 2  ax  a 2  1 i 2 n xa e C  2 nZ ( n) 2 n0 Integrating this function from x  0 to C xa and remarking that  a 0 e i 2 n x a dx  0 we get a2 so that 6 a2 a 2  1 i 2 n xa x   ax   e ,0 xa 6 2 nZ ( n) 2 2 (15.41) n0 This result furnishes in turn the formula for calculating pi squared and sums related to  2 1 x x 1 2 x cos(2 n  )   (   ), 0  x  a  2 2 a a a 6 n 1 n For examples we get corresponding to 1 n2 (15.42) 1 1 x  0, , a 2 8 1 2 1 1 1 1 1 1 1 1 1   2  2  2  2  ...,  2  2  2  2  2  ..., 6 1 2 3 4 12 1 2 3 4 11 2 1 1 1 1 (  2     ...) 12.16 2 12 32 52 7 2 Similarly for (15.43) x 1 1 1  , , . a 3 4 6 605 Advances in Optics: Reviews. Book Series, Vol. 5 15.2.5.9. The Ramp Function Built from f ( x)  0,  a  x  0, f ( x)  x,0  x  a, 2a The ramp function is represented by the graph in Fig. 15.8. Fig. 15.8. Graph of f ( x)  0,  a  x  0, f ( x)  x,0  x  a. This function f ( x) is not continuous so that we can’t apply the formula (15.14) for obtaining its Fourier series. Instead by remarking that by derivation it becomes a Heaviside-like function given in (15.29) we integrate (15.29) and get 1 f ( x)   (  2 x x 2 n x  1  sin( ))   2a     cos( n )  C a a 2 n0  n n odd  n  2 n 0 n odd Putting x  a then x  0 and summing the results we get the constant of integration a C  and the Fourier series of the ramp function 4 2 x a x  1  f ( x)    2a    cos( n ) , a  x  a a 2 4 n odd  n  n 0 15.2.5.10. The Parabolic Function f ( x)  x2 , a  x  a, P  2a This function is described by the graph shown in Fig. 15.9. 606 (15.44) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers Fig. 15.9. Graph of f ( x)  x2 , a  x  a. Because a a  x 2 dx      a x  i 2 2a  x 2 e 2 a   a 2 e  i  a 2 ei  0,  ,  3  a 3 a  i 2 2xa  i  i  2 xe   2ae  2ae  4a,  a we obtain from (15.14) the Fourier series of f ( x)  x2 , a  x  a x2    a2 xa 1 i 2 n x2aa a 2 1 2  2a 2    e a 4 cos(2 n )  2 2 a 3 3 nZ ( n) n 1 ( n) (15.45) n0 Another way to get the Fourier series of f ( x)  x2 , a  x  a is possible by a suitable integration of f '( x)  2x given by (15.28) x   ab P i 2 n x-aP P x-a   ( )e   ( )sin(2 n ) a  x  b, P  b  a, 2 P nZ 2i n n 1  n n0  x  2a  ( n 1 1 x-a )sin( n ),  a  x  a, P  2a n a Indeed by integration from 0 to  x 2  4a 2  ( n 1 x we get 1 2 x a   ) (cos  n   cos(n )),  a  x  a, P  2a n a   (15.46) 607 Advances in Optics: Reviews. Book Series, Vol. 5 Performing now once more integration from  0 a cos(n a to 0 and remarking that x-a )dx  0, a (15.47) we get the following results  1 a3  4a 3  ( ) 2 cos n , 3 n  n 1 which leads to 2 12    ( ) n n 1 1 , n2 (15.48) and the final form of (15.44) x2   a2 1 x-a    4a 2  ( ) 2 (cos  n ,  a  x  a 3 n a    n 1 15.2.5.11. The Periodic Function Built from f ( x)  x3 , 0  x  a, P  a This function is described by the graph shown in Fig. 15.10. Fig. 15.10. Graph of x3 , 0  x  a, P  a. We have a  3 i 2 n Px  a 3  x3d x   a ,  x e  a ,  0 4  0  608 4 a  2 i 2 n Px  2 3 x e   3a ,  0 (15.49) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers a  i 2 n Px  6 xe   6a ,  0 a  i 2 n Px  6e  0  0 so that its Fourier series following (15.14) is x i 2 n a3 a a 2 i 2 n ax a 3 i 2 n ax 2 a )e  3a  ( ) e  6 ( )e x   a ( 4 nZ 2i n nZ 2i n nZ 2i n 3 n0 n0 (15.50) n0 Thank to (15.25) and (15.39) we get a useful form of the Fourier series of x3 , 0  x  a, P  a x3   a2 3ax 2 a 3 i 2 n ax  6 ( x )e , 2 2 nZ 2i n (15.51) n0 and a beautiful formula for calculating pi tripled and sums containing ( 1 x sin(2n ) 3 n a  x3 3x 2 x 2 3 1 x  2    3 sin(2n ) ) 3 a 2a 2a 3 a n 1 n (15.52) 15.2.6. Polynomial Equivalent to an Individual Series in a Fourier Series From the above experiences in getting the Fourier series of x, x2 , x3 we realize that Lemma 1: By successive integrations from 0 to a A1 ( x)  x  2 x of the Fourier series of the binomial x  i 2 n a a )e a , 0  x  a , P  a , A1 ( x)  x    ( 2 nZ 2i n (15.53) n0 one obtains a set of polynomials Am ( x) Am ( x)   Am 1 ( x) dx  Am (0), x 0 (15.54) each equal to an individual series as shown in the following formulae 609 Advances in Optics: Reviews. Book Series, Vol. 5  Am ( x)   ( nZ n0 a m i 2 n xa ) e , 0  x  a, P  a 2i n (15.55) Moreover by remarking that  a i 2 n  a 0 x a dx  0, (15.56) Am ( x)dx 0, (15.57) e we get 0 and finally by (15.54) the useful formula Am (0)   x 1 a dx  Am 1 ( x)dx  0 0 a (15.58) According to (15.58) we get the interesting formula for calculating Am ( x) from Am1 ( x) Am ( x)   Am 1 ( x)dx  x 0 x 1 a dx  Am 1 ( x)dx  0 a 0 (15.59) For examples x a 1 a x 2 ax A2 ( x)   ( x  )dx   (  ) dx  0 2 a 0 2 2 x 2 ax 1 a x 2 ax x 2 ax a 2  (  )   (  ) dx  (  )  2 2 a 0 2 2 2 2 12 A3 ( x)  ( x3 ax 2 a 2 x 1 a x 3 ax 2 a 2 x x 3 ax 2 a 2 x )  (  )dx       3! 4 12 a 0 3! 4 12 3! 4 12 (15.60) and by putting Am (0)  Am Am ( x) : xm x m 1 xm2 x mm  A1  A2  ...  Am (m  1)! (m  2)! (m  m)! m! By construction Am ( x) has the properties Am ( x)   ( nZ n0 610 P m i 2 n Px ) e , 2i n (15.61) Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers (15.62) A 'm1 ( x)  Am ( x),  P 2m ) , 2i n (15.63) P 2 m 1 P 2 m 1 )  ( ) )  0, m  0, 2i n 2i n (15.64) A2 m (0)   2( n 1  A 2 m 1 (0)   (( n 1 A1 (0)  0  a a  , 2 2 (15.65) (15.66) Am (1)  Am (0)   m1 15.2.7. Obtaining Bernoulli Polynomials as Fourier Series Defining the Bernoulli polynomial Bm ( x) by an operator applying on the monomial xm as followed [7] Bm ( x)  we get the properties Bm ( x  1)  x xm , e 1 (15.67) x x ( x  1) m e 1 x (15.68) B 'm ( x)  mBm1 ( x), and the generating function   Bm ( x) m0 tm t  e xt , m ! et  1 (15.69) which by identification gives B0 ( x)  1, B1 ( x)  x  1 2 (15.70) Now, from the Fourier series of B1 ( x) obtained from (15.14), (15.25) B1 ( x)  x   1 1 )ei 2 nx , 0  x  1   (  2 2 i n nZ (15.71) n0 we get by integration from 0 to x 611 Advances in Optics: Reviews. Book Series, Vol. 5  x 0  B1 ( x)dx  1! ( nZ n0 1 2 i 2 nx  1) ) (e 2i n (15.72) Taking another integration from 0 to 1 and utilizing the formula (15.57) we get  1 0  dx  B1 ( x)dx  1! ( x 0 nZ n0 1 2 ) , 2i n (15.73) so that  x 0 Baptizing  B1 ( x)dx  1! ( nZ n0  1 2 1 2 i 2 nx )  1! ( ) e 2i n nZ 2i n (15.74) n0 1 B2 ( x) the left hand side we get 2  B2 ( x)  2! ( nZ n0 1 2 i 2 nx ) e , 2i n (15.75) and see that B2 ( x) has the properties of a Bernoulli polynomial of order 2 B '2 ( x)  2B1 ( x), B2 (0)  B2 (1). By recurrence we get  Bm 1 ( x)  (m  1)! ( nZ n0 1 m 1 i 2 nx ) e , 2i n (15.76) 1 Bm 1 ( x)  (m  1)! Bm ( x)dx  (m  1)!  dx  Bm ( x)dx, x 0  Bm 1  (m  1)! ( nZ n0 0 x 0 1 m 1 ) 2i n (15.77) (15.78) From (15.78) we obtain immediately the well-known properties B2m1  0 if m  0 and B2 m B2 m 2  0. The formula (15.76) giving the equivalence between a Bernoulli polynomial with a Fourier series was announced by Hurwitz without proof in 1890 as private communication and may be found in Ref. [8].Formula (15.77) obtained from a property of ei 2 nx (15.57) allows the calculation of Bm1 ( x) from Bm ( x). With the formula (15.77) we get the set of Bernoulli polynomials  x B0 ( x)  1, B1 ( x)  1! 1dx  1! 0 612  1 0 xdx  x  1  B0 x  B1 , 2 Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers  x3  x2 B B x B x2  x B2 ( x)  2! B0  B1   2! 2 , B3 ( x)  3! B0  B1   3! 2  3! 3 , 2!  2! 1! 3! 2 1 2!  3!  (…) Bm ( x) B B x m2 x m B x m 1  B0  1  2  ...  m , m! m! 1! (m  1)! 2! (m  2)! m! m  m Bm ( x)    Bk x m  k : ( B  x)m k 0  k  (15.79) The formula (15.79) was established by Lucas in 1891 [9]. It is a symbolic formula where undefined notations Bm are to be replaced with well defined Bm . Combining (15.76) with (15.79) we finally get m m 1 Bm ( x)    Bk x m  k : ( B  x) m : m! e 2i  n x m k (2 )  i n k 0  nZ  (15.80) n 0 The above formula permits to obtain the Bernoulli polynomials Bm ( x) simultaneously with the Bernoulli number Bm . The inconvenience is that it is fastidious for calculate Bm with high values of m. Fortunately we have by operator calculus [7] a convenient formula due to Euler in 1738 [9] for calculating Bm m  m (1  m) Bm    ()k Bk Bm  k : ( B  B) m , m  1, k 0  k  (15.81) which leads because ()k  ()2m1k  0 immediately to the property (2m) B2 m1  ( B  B)2 m1  0 (15.82) From (15.81) we get for examples 3B2  2B1 B1 , 7 B6  30B2 B4  5 / 30, (…) 613 Advances in Optics: Reviews. Book Series, Vol. 5  20   20   20   20   20  21B20  2   B2 B18  2   B4 B16  2   B6 B14  2   B8 B12  2   B10 B10 2 4 6 8  10  (15.83) In Ref. [7] there is another method which is more convenient for calculating Bm . It consists in utilizing the Lucas formula (15.79) corresponded to x  1  m  m  m Bm (1 )  B m    Bm 1    Bm  2  ...    B0 , 1 2  m (15.84) B1 (1 )  B1 (0)  1 (15.85) m  m  m  m  Bm (1)  Bm   m1    B0    B1  ...    Bk  ...    Bm 1 , 0 1 k  m  1 (15.86) and the formula to get the recursion relation [7] Theorem: which may be written under the matrix form where the first line corresponds to m  0 as shown in the following formula 1 1 1 1 ... ... ... 3 ... 6 4 ... ... ... ... B0 1 B1 0 B2 0 = B3 0 ... ...  m  1  m  1  m  1 0     ... ... ...   Bm  0   1   m  2 3 4 ... (15.87) This matrix equation may be resolved by doing linear combinations over lines from the second one in order to replace them with lines each containing only some non-zero rational numbers. For instance for calculating successively following matrix equation 614 B0 , B1 , B2 , B4 , B6 ,..., B18 we may utilize the Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers 1 1 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 3 1 5 0 7 1 9/5 0 0 0 0 5 61 3 0 0  105 35 0 0 0 240 0 17 0 2052 0 0 775 9 0 11 0 0 13 0 0 0 0 0 0 0 15 0 0 17 0 0 0 19 B0 1 B1 0 B2 0 B4 0 B6 0 B8 = 0 B10 0 B12 0 0 B14 0 B16 B18 0. (15.88) We remark that the last line of this matrix has replaced the line 19     , i  0,1, 2, 4,6,8,10,12,14,16,18  i   For example from (15.88) we have 2052 B1  775B6  19 B18  0  B18  (1026  775 1 ) 42 19 This simply obtained value is to be compared with that given by Coen by another method [10] B18  43867 798 15.2.8. Obtaining Powered of Pi and Even Positive Zeta Functions From (15.71) we get the formulae for calculating the powers of pi Bm ( x)  m! ( nZ n0 (2 ) 2 m  () m 1 (2m)! 1 m i 2 nx ) e , m  1, 2i n  2 1 cos(2 nx),  B2 m ( x) n 1 n 2 m (15.89) 615 Advances in Optics: Reviews. Book Series, Vol. 5 (2 ) 2 m 1  () m 1 (2m  1)! 1 B2 m 1  1 n ( x) n 1 2m (15.90) sin(2 nx), 1 1 1 1 1 where we may conveniently take x  0, , , , , . 8 6 4 3 2 For example (2 ) 2  2!  2  1 4 1 1 1 1 cos( n)  () n 2  48( 2  2  2  ...)   2 1 n 1 n 1 1 1 n 1 n 1 2 3 B2 ( ) (   ) 2 4 2 6 As for positive even Riemann zeta functions we find again from (15.77) the formulae first proven by Euler as reported by Ayoub [6] 1 () m 1  (2m)   2 m  (2 ) 2 m B2 m (2m)!2 n 1 n  (15.91) For examples  (2)   2 B2\ ,  (4)   1 (2 ) 4 B4 48 (15.92) 15.2.9. Fourier Series of Polynomials From (15.14) we may calculate the Fourier series of monomials from the matrix equation   f ( x )   f ( x)   f '( x)   f "( x)  ... 1 f ( x) 0 1 1 1 1 0 0 0 ... f ( m 1 ) ... 1/2 1 0  1 x  1 ( x) 2  i 2  nx / (2i n ) n0 1!  x e e i 2  nx / (2i n ) 2 i 2  nx / (2i n ) 3 (15.93) n0 1/3 1 2! ...  e n0 x 3 ... x m 1/4 1 3 3! ... ... ... ... ... ... ... 1 / ( m  1) 1 m m(m-1) ... m! ...  e i 2  nx / (2i n ) m n0 For example for a Hermite polynomial of fourth order in the interval 0  x  1 we have 616 Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers H 4 ( x)  16 x 4  48 x 2  12    4 1 1 i i  32 (  6  12 ) ei 2 nx  2 3 5 (2 n) (2 n) (2 n) 4 n  0 (2 n)   4 1 16 48 96 ) cos(2 )) ( sin(2 nx)  32( x  )   (   nx   2 4 3 5 2 ( n) n 1 ( n) n 1 ( n) (15.94) and may write H 4 ( x)     76 16 48 96 sin(2 nx) (15.95) ) cos(2 nx )) 2 x ( ,   32 x   (    2 4 2 5 ( n) 2 nx n 1 ( n) n 1 ( n) so that for x  0 76 16 48    12 5 6 90 H 4 (0)  (15.96) 15.3. Remarks and Conclusion The main remark of this work is that it clarifies for us the question why Fourier transform and Fourier series are so important for the studies of Optics and Acoustics. That is because x both utilize the exponential functions exp(2n ) that represent waves of frequencies P n. They also are crucial in quantum mechanics where r and p represent the position and momentum of a wave that we‘ve resumed in the formula  k  FT  r [2]. Apart from recognizing this remark, in this work is given firstly a new formula for calculating the Fourier coefficients of continuous periodic functions defined on any interval a  x  b as so as of functions having primitive or derivative continuous currently known, including the Heaviside-like function and those derived from it by integration, displacement or/and symmetrization with respect to an horizontal axis. Secondly, it utilizes the fact that equivalent to the series  ( nZ n0  P 0 e i 2 n x P dx  0, n  Z for obtaining the polynomial P m i 2 n Px ) e permitting to divide the Fourier series of a 2i n function into sums of polynomials and series; to calculate positive even Riemann zeta functions and many serial forms for each power of pi. Thirdly concerning Bernoulli polynomials [8] this work proposes a concise method for obtaining Bm ( x) from Bm1 ( x) which leads to the symbolic formula Bm ( x) : ( B  m)m that was done in another way by Lucas in 1891 [9]. From this formula and the fact that Bm (1)  B m (0)   m1 we get an elegant matrix formula for calculating Bernoulli numbers 617 Advances in Optics: Reviews. Book Series, Vol. 5 which are omnipresent in many problems such as in heat equations, in powers sums on arithmetic and algebraic progressions [7]. We wish that the new formula for Fourier coefficients, the relation between Bm ( x) and Bm1 ( x), the matrix formula for Bm should be judged as worth to be added to lessons on Fourier series and Bernoulli numbers taught at universities. Acknowledgements The author thanks very much his adorable spouse and his responsible daughters Do Thi’s for the healthcare they devote to him during the hard period of corona virus isolation. He acknowledges Prof. Sergey Yurish for appreciations about his works now and in the past. References [1]. J. B.-J. Fourier, Oeuvres de Fourier (D. Darboux, Ed.), Gauthier-Villars et Fils, Paris, Gallica, 1888, pp. 218-219. [2]. D. T. Si, Principles of quantum mechanics and laws of wave optics from one mathematical formula, Applied Mathematics, Vol. 10, 2019, pp. 892-906. [3]. D. T. Si, The Fourier transform relation between Dirac Bras  k  FT  r and wave optics, Advances in Optics: Reviews Book Series, Vol. 4, 2019, pp. 117-146. [4]. Wikipedia, https://en.wikipedia.org/wiki/Fourier_series & M. R. Spiegel, Theory and Problems of Fourier Analysis, Schaum’s Outline Series, Mc Graw-Hill, 1974. [5]. G. B. Folland, Table of Fourier Series, Fourier Analysis and Its Applications, Wadsworth & Brooks/Cole, 2014, pp. 1-4. [6]. R. Ayoub, Euler and the Zeta function, The American Mathematical Monthly, Vol. 81, Issue 10, 1974, pp. 1067-1086. [7]. D. T. Si, Obtaining easily powers sums on arithmetic progressions and properties of Bernoulli polynomials by operator calculus, in New Insights into Physical Science, Vol 3, Book Publisher International, 2020, pp. 6-83. [8]. F. Costabile, F. Dell’Accio, M. I. Gualtieri, A New Approach to Bernoulli Polynomials, Rendiconti di Mathematica, Serie VII, Vol. 26, http://www1.mat.uniroma1.it/ricerca/ rendiconti/ARCHIVIO/2006(1)/1-12.pdf [9]. E. Lucas, Théorie des Nombres, Chapter 14, 1891 Topics: Number theory, Gauthier-Villars, https://archive.org/details/thoriedesnombr01lucauoft/page/n7/mode/2up [10] L. E. S. Coen, Sums of powers and the Bernoulli numbers, MD Thesis, Eastern Illinois University, 1996. [11]. Group of authors, Bernoulli Numbers, https://vdocuments.site/bernoulli-number.html 618 Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus Chapter 16 Obtaining Fourier Transforms of Functions by Differential Calculus Do Tan Si1 16.1. Introduction The notion of Fourier transforms was introduced 200 years ago, in 1822, by Joseph Fourier in his famous work “Analytical Theory of Heat” [1]. Afterward tables of Fourier transforms of functions, distributions are also available for examples in Refs. [2, 3]. Everything seems sufficient for researchers when dealing with the problems in Heat theory. However we now know that the Fourier transform is considered as a natural relation between a spatial 3D object described by a function f (r ) and a sum of infinity of plane waves each with wave vector k , wavelength   2 / k , amplitude being the Fourier transform f (k ) of f (r ). From this consideration we have enlarged the fields of applications of Fourier transform to the domain of principles of quantum mechanics [4] and in optics in confirming that in a diffraction of a plane wave k 0 by the object f (r ) the component wave has amplitude proportional to f (k  k0 ) [5]. k of the diffracted It has then for us a signification to re-actualize the study of the Fourier transform of functions. To realize this aim, we firstly will introduce the notions of operator calculus [6] which leads to the differential representation of the Fourier transform. Afterward a method Do Tan Si Ho Chi Minh-city Physical Association, Vietnam, Université libre de Bruxelles, UEM, Belgium 619 Advances in Optics: Reviews. Book Series, Vol. 5 free of integrations for calculating simply the Fourier transforms of functions, a list of current properties and a table of Fourier transforms are given. 16.2. Differential Operators Realizing Fourier Transform 16.2.1. Introduction to Operator Calculus Let Dx be the derivative operator, X̂ the Eckaert operator “multiply by the variable x ” and Iˆ the unity operator [7] Dx f ( x)  f '( x), (16.1) Xˆ f ( x)  x f ( x), (16.2) Iˆ f ( x)  f ( x), (16.3) we get the identity between operators ˆ  Iˆ, Dx Xˆ  XD x (16.4) as we can verify by applying both members of it on any entire function f ( x) Dx xf ( x)  xf '( x)  f ( x) From the identity (16.4) we may deduce by recurrence the following ˆ m  mD m1 , m  N Dx m Xˆ  XD x x (16.5) In fact suppose that (16.5) is correct we have ˆ m  mD m  XD ˆ m1  (m  1)D m1 QED Dx m1 Xˆ  Dx Dx m Xˆ  Dx XD x x x x Now let f ( x) be an entire function and f ´( x) its derivative function we get from (16.5) the identity f ( Dx ) Xˆ  Xˆ f ( Dx )  f '( Dx ) , i.e. f (Dx ) Xˆ f 1 (Dx )  Xˆ  f '(Dx ) f 1 (Dx ) (16.6) From (16.6) we get for example eaDx Xˆ eaDx  Xˆ  aIˆ, 620 (16.7) Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus 2 2 e aD Xˆ e  aD  Xˆ  2aDx (16.8) As first application we get from the identity (16.7) eaDx Xˆ meaDx  ( Xˆ  aIˆ)m , (16.9) then by applying the latter on a constant the formulae eaDx xm  ( x  a)m , (16.10) eaDx f ( x)  f ( x  a), (16.11) which means that eaD realizes the translation of entire functions. x A extremely precious remark is that because (16.7), (16.8) are obtained from and only from (16.4), by replacing the couple ( D , Xˆ ) with any other couples of operator ( Aˆ , Bˆ ) x verifying the same commutation relation with the couple ( Dx , Xˆ ), says ˆ ˆ  BA ˆ ˆ  Iˆ,  Aˆ , Bˆ   AB   (16.12) we will get from any identity between ( Dx , Xˆ ) exactly the same between ( Aˆ , Bˆ ). aD 2 ˆ  aDx2  Xˆ  2aDx , For example from (16.7), (16.8) we get e x Xe (16.13) ˆ2 ˆ  aAˆ 2  B ˆ  2aB ˆ eaA Be (16.14) ˆ ˆ ˆ  ˆ  ˆ ˆ ˆ ˆ ˆ ˆ  XD  x ,ln X   X  Dx ,ln X    X ,ln X  Dx  X ( Dx ln X  ln XDx )  I , (16.15) More concretely because we have the relations eaXDx ln Xˆ e  aXDx  ln Xˆ  aIˆ, ˆ ˆ (16.16) ˆ ˆ eaXDx (ln Xˆ ) m e  aXDx  (ln Xˆ  aIˆ) m , ˆ ˆ ˆ ˆ ˆ eaXDx Xˆ e  aXDx  eln X  a I  e a eln X  e a Xˆ , ˆ which give us the dilatation operator e aXD ˆ (16.17) x ˆ eaXDx x  e a eln x , e aXDx f ( x)  f (e a x), (16.18) 621 Advances in Optics: Reviews. Book Series, Vol. 5 that we will use hereinafter to calculate the Fourier transform of f (ea x). For curiosity we here give some other examples that may be found in Ref. [6]. 2 x   Xˆ 2 Dx , Xˆ 1   I  e  aXˆ Dx f ( x)  f ( ),   1  ax (16.19)  Xˆ ln Xˆ Dx ,ln ln Xˆ   I  eln Xˆ ln Xˆ Dx f ( x)  f ( x a )   (16.20) 16.2.2. Operators Realizing the Fourier Transform Consider the Fourier transform of a function f ( x) defined by the formula 1 2 f (k )  FTf ( x)     (16.21) e  ikx f ( x)dx, we have by direct calculations the properties FTxf ( x)  1 2    e  ikx xf ( x)dx  1 2    iDk e  ikx f ( x )dx, i.e. FTxf ( x)  (iDk ) f (k ) (16.22) On the other side FTf '( x)   ik 1 e  ikx f ( x)    2 2    e  ikx f ( x)dx, i.e. FTf '( x)  (ik ) f (k ) (16.23) Thirdly FT  ( x)  1 2    FT  ( x)  e  ikx ( x)dx, i.e. 1 2 (16.24) Searching for a differential transform having the properties of (16.22), (16.23) we see from the differential identities (16.8), (16.9) that e 622  i ˆ2 X 2 i (e 2 Dx 2 (e  i ˆ2 X 2 i ˆ2 i i ˆ2 i ˆ2 i i i ˆ2  Dx  X  Dx X X Dx X Xˆ e 2 )e 2 )e 2  e 2 (e 2 Xˆ e 2 )e 2  2 2 2 Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus e  i ˆ2 X 2 i ˆ2 X ( Xˆ  iDx )e 2  Xˆ  Xˆ  iDx  iDx , (16.25) i ˆ2 X (16.26) and similarly e  i ˆ2 X 2 i (e 2 Dx 2 (e  i ˆ2 X 2 )e Dx e 2  i Dx 2 2 i ˆ2 X  iXˆ )e 2 From (16.25), (16.26) we arrive to define the operator FT FT   e  i ˆ2 X 2 i e2 i ˆ2 X FT 1   1e 2 e Dx 2 e  i  Dx 2 2 i ˆ2 X 2 i ˆ2 X e2 (16.27) , (16.28) , having the property that when applying on a function, ( Dx , Xˆ ) are changing into ( Dk , Kˆ ) FTf ( x)   e i  Kˆ 2 2 i e2 Dk 2 e i  Kˆ 2 2 (16.29) f (k ) The operator FT has the properties FTDx FT 1  iXˆ , (16.30) FT Xˆ FT 1  iDx (16.31) At this state the crucial problem is the calculation of the factor  . For this let us remark that x2 i ( Dx  iXˆ )e 2  0, FT ( Dx  iXˆ ) FT 1 FTe i x2 2  (iKˆ  Dk ) FTe (16.32) i x2 2 0 (16.33) x2 i But ( Dx  iXˆ )e 2  0, so that FTe i x2 2  C e i k2 2 (16.34) Similarly we have FTe i x2 2  C e i x2 2 (16.35) 623 Advances in Optics: Reviews. Book Series, Vol. 5 Combination of (16.34) and (16.35) gives e CC  1 i x2 2  FTFTe i x2 2  C FTe k2 2 i  C C e i x2 2 , i.e. Taken i  (16.36) C  e 4 , we get finally the differential form of the Fourier transform i  FT  e 4 e i  Kˆ 2 2 i e2 Dk 2 e i  Kˆ 2 2 (16.37) As first application of (16.37) we have x x  k  k i i i i i i x2 k2  2 FT cos( )  FT (e 2  e 2 )  (e 4 e 2  e 4 e 2 )  2cos(  ), 2 2 4 2 2 2 2 FT cos( 2 x2 k2 k2 ) (cos( )  sin( )) 2 2 2 2 (16.38) FT sin( 2 x2 k2 k2 ) (cos( )  sin( )) 2 2 2 2 (16.39) Similarly Eqs. (16.38), (16.39) were proven by Easton [8]. Terminating beautifully this example let us remark from (16.21) that for k  0 1 2 f (0)  1 2  x2 2    1 2 f ( x)dx, so that (16.34), (16.35) give    e i x2 2 dx  C  e i  4 ,   e dx  C  e 4 , and (16.38), (16.39) give i i 2 1  2 2 x2 cos(  2 )dx, (16.40) 2 1  2 2  x2 )dx 2 (16.41)    sin( Another operator representing the Fourier transform may be obtained by taking the i  i 1 Fourier transform of itself FT  FT FT FT  e 4 e 2 624 Dk 2 e  i ˆ2 K 2 i e2 Dk 2 . Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus For completeness we add that according to Wolf [9] i  FT  e 4 e i  4 ( Dk 2  Kˆ 2 ) (16.42) As conclusion we may state that the Fourier transform may be represented by any one of the three differential operators (16.40), (16.41) or (16.42). 16.2.3. The Fourier Transform of f ( x)  1 Applying (16.37) on the Dirac delta distribution  ( x) we get by (16.24) i  i  Kˆ 2 4 2 FT  ( x)  e e e i 2 i Dk  Kˆ 2 2 2 e 1 , 2  (k )  (16.43) i.e. because x ( x)  0 i  i 2 e 4 e 2 Dx 2 i x2 (16.44) i  x2 2 (16.45)  ( x)  e 2 Replacing x with ix we get formally i  2 e 4 e i  Dx 2 2  (ix)  e From the property of Dirac delta  (ax)  1  ( x), a (16.46) 1 i (16.47) we recommend that  (ix)   ( x) With this recommendation we get i e2 Dx 2 e i  x2 2  2 e i  4  i 1  ( x)  2 e 4  ( x), i (16.48) and finally i  FT 1  e 4 e i  Kˆ 2 2 i e2 Dk 2 e i  Kˆ 2 2 i  1,  e 4 e i  Kˆ 2 2 i  ( 2 e 4 e i  4 i ˆ2 K e 2  ( x))  2 (k ) (16.49) 625 Advances in Optics: Reviews. Book Series, Vol. 5 Classically this result must come from (16.21) FT 1  1 2    e  ikx dx, but for us it is not easy to prove it. 16.2.4. Geometric Signification of a Convolution Product Consider 3D objects represented by functions f (r ) equal to zero if r is outside the object and to unity if inside. The integral  r R 3 f (r ) g (r )dr represents then the volume of the intersection between two corresponding objects. If we displace the object f (r ) with respect to the origin by a vector r0 then the volume of the intersection between the objects described by f (r  r0 ) and g (r ) becomes  r R3 f (r  r0 ) g (r )dr (16.50) i.e. the value at r0 of the convolution product between f (r ) and g (r ), the all denoted by ( f  g )(r0 )   r R3 f (r0  r ) g (r )dr (16.51) As example let f (r ) represents the following square and g (r )  f (r ) Fig. 16.1. Graph of Sq( x) . When we right-displaced one of the squares an amount x with 0  x  a / 2 the new area of the intersection is equal to (a  x)a1 so that the convolution product Sq( x)  Sq( x) corresponding to a  1 is the triangular function tri( x). 626 Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus Fig. 16.2. Area of Sq( x)Sq( x)  (a  x)a 2 . Fig. 16.3. Graph of tri( x)  a  x . As extension of the above method we think that it is feasible for calculating the convolution products tri( x)  tri( x), Sq( x)  tri( x) etc. and the Fourier transforms of them as we can see hereafter. 16.2.5. Fourier Transform of a Convolution Product  Consider the convolution product ( f  g )( x)   f (x  x0 ) g ( x0 )dx0 .  We may write from the formula for translated functions (16.11) FTf ( x  a)  FTeaDx f ( x)  eiak FTf ( x), that     FT ( f  g )( x)   FTf (x  x0 ) g ( x0 )dx0   e  ix0 k FTf ( x ) g ( x0 )dx0    FTf ( x)  e  ix0 k g ( x0 )dx0  2 FTf ( x) FTg ( x),  (16.52) i.e. ”The Fourier transform defined by (16.21) of a convolution product of two functions is 2 time the product of their Fourier transforms”. 627 Advances in Optics: Reviews. Book Series, Vol. 5 The above formula is well-known for example in Ref. [10]. It may also be proven by differential calculus as we will see in the hereafter paragraph. 16.3. General Properties of the Fourier Transform From the differential realization of the Fourier transform (16.30) FTf ( x)  1 2   i  e  ikx f ( x)dx  e 4 e i  Kˆ 2 2 i i Dk 2  Kˆ 2 2 f (k ), (16.53) f (k )  FTf ( x)  FTf ( Xˆ )1  f (iDk ) FT1  2 f (iDk ) (k ), (16.54)  e2 e we get simply the following general properties. 16.3.1. The Primordial Property Thank to (16.49) we may write down the formula which seemingly is a newly highlighted property of the Fourier transform. It may be utilized for obtaining in a simple manner the following formulae. 16.3.2. The Convolution Product  FTf ( x) g ( x)  f (iDk ) g (k )  f (iDk )   (k  k0 ) g (k0 )dk0   1  2    f (k  k0 ) g (k0 )dk0  1 f (k )  g (k ), 2 (16.55) i.e. FT ( f ( x)  g ( x))  2 f (k ) g (k ) (16.56) 16.3.3. The double Fourier Transform FT FTf ( x)  2 FTf (iDk ) (k )  2 f (i 2 Xˆ ) FT  (k )   f (i 2 x)  f ( x) (16.57) 16.3.4. The Inverse Transform Eq. (16.57) gives FT f (k )  f ( x) 628 (16.58) Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus 16.3.5. Transform of Complex Conjugate FTf * ( x)  f * (iDk ) (k )  ( f (iDk ) (k ))*  f * (k ) (16.59) 16.3.6. Transform of Translated Functions Following (16.11) FTf ( x  a)  FTeaDx f ( x)  eiak f (k ) (16.60) 16.3.7. Transform of a Dilated Function ˆ (16.18) we get From the operator realizing a dilatation e a XD x (i) For f ( x) even, f ( x)  f ( x) ˆ ˆ ˆ FTf (e a x)  FTe aXDx f ( x)  e  a Dk K f (k )  e  a ( KDk 1) f (k )  e  a f (e  a k ), (16.61) FTf (ea x)  ea f (ea k ) (16.62) (ii) For f ( x) odd, f ( x)   f ( x) FTf (ea x)  ea f (ea k ) (16.63) Joint these two cases we get the well-known formula FTf ( x)  1 k f( )   (16.64) 16.3.8. Transform of a Sum of Two Functions FT ( f ( x)  g ( x)  f (k )  g (k ) (16.65) It is evident that (16.65) is necessary for calculating the transforms of sinusoidal functions. 16.4. Table of Fourier Transform of Differential Operators ˆ Operator A Dx ˆ : FT A ˆ FT 1 Fourier transform of A i Kˆ X̂ iDk Dx Xˆ  Kˆ Dk 629 Advances in Optics: Reviews. Book Series, Vol. 5 Xˆ Dx  Dk Kˆ Dx  Xˆ D  Xˆ i( Dk  Kˆ )  i( D  Kˆ ) Dx 2  Xˆ 2 f ( D , Xˆ ) x ˆ 2)  ( Dk 2  K ˆ , iD ) f (i K f ( Dx , Xˆ ) f ( iKˆ ,iDk ) x e k k e 16.5. Table of Fourier Transform of Functions f ( x) f (k )  FTf ( x)  e  ikx f ( x)dx  2  (k ) 1 (2 )1/ 2 f (  x) f ( k ) f ( x) f ( k ) Dx f ( x) ik f (k ) f ( x)  f ( x)1 2 f (iDk ) (k ) e aDk f (k )  f (k  a) eiax 2 (k  a) cos(ax) 2 ( (k  a )   (k  a )) 2 sin(ax) 2 ( (k  a)   (k  a)) 2 H ( x)  Heaviside function, H '( x)   ( x), Dx ( H ( x)  C )   ( x), Dx ( H ( x)  C )   ( x), H ( x)  H (  x)  1 H ( x) H ( x  a) H ( x  a)  H ( x  a ) 630 2  ( x) eiax f ( x)  1 ik ( H (k )  C 2 (k ))  ( 2 )1 , ik ( H (k )  C 2 (k ))  ( 2 )1 , H (k )  H (k )  2C 2 (k )  2 (k )   2C  1 1 1 2 (k )  ( 2 ) 1 2 ik 1 1 1 1 1 2 (k ))  (   ( k ))  ik 2 2 2 ik 1 1 (   (k ))eiak 2 ik 1 2 sin(ak ) 2 k  H (k )  Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus Rectan gular function Sq ( x, a )  0, a Sq( x, a)  x a/2 1 1 sin(ak / 2) (ak / 2) 2 x  a / 2, a a 1 H (( x  )  H ( x  )) a 2 2 1 2 2 ik sign( x)  H ( x)  H ( x) 1 ix  Dx sign( x)  2 ( x) 2 sign(k ) Note : FT f (k )  f ( x) 2 1 2 2 ikFTsign( x)  ik  2 ik 2 tri( x)  Sq(1, x)  Sq(1, x) 1 sin 2 ( k / 2) 2 (k / 2) 2 ( Dx 2  a 2 )e a x  2a ( x) (k 2  a 2 ) FTe a x  2aFT  ( x) e 2a 1 2 2 (k  a ) a x 2 2 exp( a k ) 2a 1 x2  a2 1 Xˆ  1 x 1 x iDk FT  2 2 Dk ( H (k )  H (k ))  Dk sign(k ) 2 2 2 sign(k )  C 2i 1 Notification : C  0 because odd x 1 () n 1 1  Dx n 1 n (n  1)! x x 1 Dx (ln x  C )  , x  0 ln x, x  0 x 1 1  e iaDx , a  0 x  ia x 1 1  eiaDx , a  0 x  ia x 1  2  (k )  x () n 1 1 (ik ) n 1 FT (n  1)! x ikFT (ln x  C )   2 sign(k ), k  0 2i 2 sign(k )  C 2 (k ) 2k  2 ak e H (k ) i 2  ak e H (k ) i 631 Advances in Optics: Reviews. Book Series, Vol. 5 ( Dk 2  a 2 ) FT 1 1 2 ( x  a2 ) ( x2  a2 ) 1  2 (k ), x  a2 2 Note : ( Dk 2  a 2 )e a k  2a (k ) 1 ( x2  a2 ) 1  a k e a 2 2 ( Dx  Xˆ )e  x / 2  0, 2 i( Dk  Kˆ ) FTe k /2  0, e x 2 e k /2 1 e (2a ) 2 e  ax2 , a  0 or a  i , Notification : eax is dilatation of e x ( Dx  iXˆ )eix 2 /2 i C e 2 C e i 2 x 2  i 2 x 2 i Note : e 2 x2 k2 i  FTFTe 2 i x2  C FTe   C C  1  C  e 4 , C  e 2 x i x2 1 ix )  (e 2  e 2 ) 2 2 sin( x2 ) 2 FT cos i i  x2 2   4 i k  i x 2 1 i4 i k2  (e e e 4e 2 )  2 2 2 k2 k2 (cos( )  sin( )) 2 2 2 2 cos( /2 2 ( Dk  iKˆ ) FTeix /2  0 0 2 2 2 ( Dk  iKˆ ) FTeix /2  0 i  x2 2 e /2 2 2 ( Dx  iXˆ )e ix / 2  0 e 1 2  k 4a 2 2 2 k2 k2 (cos( )  sin( )) 2 2 2 ((eiDk  eiDk )ik  (eiDk  eiDk )) f (k )  0  (ch( x) Dx  sh( x)) f ( x)  0  (ik  ii  1) f (k  i)  (ik  ii  1) f (k  i)  0   f ( k  i )  f (k  i )  0 1 f ( x)  ch( x)   f (k )  ch( k ) or ch 1 ( k ), 2 2   Note : ch( (k  i)  ch( (k  i)  0 2 2 1 1 ch(  2 x)  2 k) eax  eaX 1, FTeax  eiaDk 2  (k ), e ax 2  (k  ia) ˆ 632 ch( Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus ˆ  1 ) g ( x)  0 ( XD x 2 1 ˆ  1 ) g (k )  0 ( Dk Kˆ  ) g (k )  ( KD k 2 2 1 g ( x)  1 g (k )  k x e  x2 2   (2n  1)e 1  x2 2  x2 2  x2 2 i 1 i 2   e 4e H n ( x) i  4 (2 n 1) e 1  i 2 e H n (k )  H n (k )  e  in  2 e 1  k2 2 H n (k ), Note : FT  e e H n ( x) H n ( x)   x3 x2  e 1  k2 2 1  k2 2 ˆ2 i (( Dk  K )) 4 2   i i ( Dk 2  Kˆ 2 ) 4 4  2 4 1 1 ,  4 3 1 x k 1 4 xk ( x  xk ) x1   e 4e H n ( x), Note : ( Dx 2 Xˆ 2 )e e i   in  2 e 1  k2 2 H n (k ) sign(k )(ch(ak )sin(ak )  sh(ak ) cos( ak )    ch(ak ) cos(ak ), 2 1 2  ak Note : FT e H (k ), a  0  x  ia i   x4 16.6. Remarks and Conclusion Firstly this work introduces some knowledge about differential transforms in mathematical analysis based on the Eckaert operators Dx , Xˆ . From which we may factorize ˆ ˆ  aD 2 aD 2 sums of operators such as Dx  aIˆ  e  aX Dx e aX , 2aDx  Xˆ  e x Xˆ e x , then obtain the differential forms of the Fourier transform say i  FT  e 4 e i  Kˆ 2 2 i e2 Dk 2 e i  Kˆ 2 2 i   e 4e i  Dk 2 2 i ˆ2 K e2 e i  Dk 2 2 i   e 4e i  4 ( Dk 2  Kˆ 2 ) Secondly, from these differential forms we may deduce almost all properties of the Fourier transform of functions without needs of integrations. Thank to this facility, we establish a table of Fourier transforms of operators and of many functions rather easily with concise details, some may be added to the existing Tables of Fourier transforms we may find in Refs [2], [3]. By the way we obtain the formula FT1  2 ( x), the geometric interpretation of convolution product which happens to be very useful for calculations of Fourier transforms. As for Fourier transforms of functions in plane and space such as points, straight lines, planes, pyramids, ellipses, cylinders, spheres, the readers may find in Refs. [5], where applications of Fourier transform and Dirac delta functions in optics are shown. 633 Advances in Optics: Reviews. Book Series, Vol. 5 Acknowledgments The author acknowledges Prof. Sergey Y. Yurish at Barcelona for the appreciation and encouragement about his contributions in mathematics and optics. He thanks sincerely the referee who justly demands him many precisions. He think that this work gives many joys to his spouse and daughters who carefully take care of him in this hard period of worldwide Covid disaster. References [1]. J. Fourier, Analytical Theory of Heat, The Cambridge University Press, 1878. [2]. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Tables of Integral Transforms, Vols. 1-2, McGraw-Hill, New York, 1954. [3]. D. Kammler, A First Course in Fourier Analysis, Prentice Hall, 2000. [4]. D. T. Si, Principles of quantum mechanics and laws of wave optics from one mathematical formula, Applied Mathematics, Vol. 10, 2019, pp. 892-906. [5]. D. T. Si, The Fourier transform relation between Dirac bras  k  FT  r and wave optics, Chapter 5, in Advances in Optics, Vol. 4, International Frequency Sensor Association Publishing, Barcelona, 2019, pp. 117-146. [6]. D. T. Si, Operator Calculus, Edification and Utilization, LAP Lambert Academic Publishing, Saarbrücken, Deutschland, 2016. [7]. C. Eckart, Operator calculus and the solutions of the equations of quantum dynamics, Phys. Rev., Vol. 28, 1926, pp. 711-726. [8]. R. L. Easton Jr., Fourier Methods in Imaging, John Wiley & Sons, 2010. [9]. K. B. Wolf, Hyperdifferential operators and integral transforms, Rev. Mexicana Fis. Math., Vol. 25, Issue 1, 1976, pp. 55-60. [10]. E. W.Weisstein, Convolution Theorem, https://mathworld.wolfram.com/ ConvolutionTheorem.html, 2010 634 Index Index 2 2.5D interposer chip, 159 package, 135 3 3D package, 135 torus network, 136 A a square-law detector, 27 ablation, 474, 477-479, 482 process, 477 threshold, 474, 477, 479, 482 absorption, 477, 478, 482, 519, 528, 534, 564 measurement, 301, 305–311, 319, 321, 327 mechanism, 477 spectral dependence, 301, 309–311 spectrum, 422, 429, 436 two-photon absorption, 474, 477-479 accelerating charge, 29, 30 accelerator processors, 115 Acceptance angle, 182, 185, 186, 192-194, 213, 214, 216, 220, 222, 226, 231 acoustic waves, 535, 560 acousto-optic superlattice modulator, 560, 561 adhesive technology, 151 AGN. See noise, Gaussian AI processor, 115 Airy function, 338 Allan deviation, 519, 522 Altazimuth, 433, 436, 443, 446 Ampere’s law, 23, 29 amplitude difference comparison of coherent envelope (ADCCE), 541 amplitude division, 76, 79, 81 Amplitude Point Spread Function (APSF), 66 Transfer Function (ATF), 66 Analog-to-digital converter, ADC, 318 Analyte deposition, 491 meniscus-guided coating, 491 spin coating, 491 thermal inkjet printing, 491 thin film, 491 analytical symbolic computation, 276 analyzer, 88 Angular momentum flow tensor, 354 spectrum, 50, 54, 58, 333 decomposition method (ASDM), 334 of Airy light-sheet, 339 of Bessel pincer light-sheet, 339 of Gaussian light-sheet, 340 of initial electric field, 335 anti-reflection coating (AR), 154 Aperiodic Sequence, 239 Zone Plates, 239 Aplanatic lens, 426, 429 achromatic, 432 Apparent Sun semidiameter, 424 Arbitrary size, of particles, 333 Arbitration, 146 architecture license, 119 ARM processor, 119 assembly language, 110 Astronomical unit, 424 ATAC architecture, 143 Atoms, 512 autocorrelation, 45, 46, 68 avalanche emission, 512 axial image chromatic aberration, 275 B back focal length, 272 Bandgap, 461 BBAR. See Broadband antireflection coating Beam 635 Advances in Optics: Reviews. Book Series, Vol. 5 order, 335, 339 quality factor, 476 shape coefficients (BSCs), 334, 341 General expressions, 348 of Bessel pincer-sheet, 349 of Gaussian light-sheet, 350 of TE-polarized Airy, 349 of TM-polarized Airy light-sheet, 349 waist, 475, 476 bending-dependent loss, 548 Bernoulli numbers, 594 polynomial, 593, 594, 610 big cores, 119 bilinear function, 269, 270, 272, 285, 287, 288 birefringence, 89-91, 93 birefringent, 549 bit-level, 115 Blackbody, 424 BLIM Bidirectional Lambertian Irradiation Method, 188, 198-202 Bolosilicate glass, 433 SBNACZ, 433 boundary conditions, 35, 36, 38, 104, 107, 108, 343 Bow-tie aperture, 438, 443, 444 branch instruction, 116 prediction, 116 predictor unit (BPU), 129 Brewster’s angle, 87, 88 Brillouin scattering, 517 optical time-domain analysis (BOTDA), 516 reflectometry (BOTDR), 516 Broadband antireflection coating, 450 Built on ARM Cortex, 119 butt coupling, 150 C cache coherence, 116 part, 111 caching/home agent (CHA), 131 calcium fluoride (CaF2), 519 Carriers and Photons Density Dynamics 636 Rate Equations, 407 Cavity loss. See Round-trip resonator loss Cavity mirror, 420, 422 photon lifetime, 420 CCD, 180, 182, 187 camera, 164, 166-169, 172, 176 Center Point Strategy, 172, 173, 175, 176 Chirp, 585, 586, 587, 588 Chirped pulse amplification (CPA), 575 Chromatic aberration, 260, 429, 432, 434 circular polarization, 27, 97 Cladding, 436, 438, 455, 464 Clear Imaging, 252 coherence, 44-46, 48, 50, 68, 69, 82-85, 103 Spatial coherence, 48, 85 Temporal coherence, 46, 50 time, 46, 82 coherent Coherence area, 48 detection, 517, 540 Illumination, 68 light source, 511, 512 volume, 48 Colourful Imaging, 257 Coma aberration, 426 complex instruction set computer (CISC), 110 Composite rod, 454, 455 2D, 456, 457 3D, 456, 457 compute node, 137 Concentration quenching, 452 Continuous wave, 449 contrast difference between the second peak and the second trough (CDSPST), 541 converged mesh stop (CMS), 130 Conversion efficiency, 418, 423, 444, 462, 463 Convex lens, 425, 427 Convolution Product, 626 Core, 436, 439, 455, 464 Corona Architecture, 143 correlation function, 45, 46 matrix, 102 cortex license, 119 Cortex-A78, 118 CPC, 180, 181, 184, 187, 192, 193, 202208, 210-215, 220, 222, 225, 231, 232 Index Cr-codoped Nd:YAG, 418, 427, 440, 450, 461 Cross sections, extinction and scattering, 352 cross-correlation, 45 cursor effect, 522 CW. See Continuous wave cycles per instructions (CPI), 115 D damping rate, 518, 530 data cache L1, 128 Data preprocessing fast Fourier transform, 507 Multiplicative Scatter Correction, 504 Normalization, 501, 503 Savitzky-Golay Algorithm, 504 Soft independent modeling by class, 506 Standard Normal Variate, 504 data-level, 115 DCF. See Double-clad fiber De-excitation, 512 Deformation theory, 312 delayed self-heterodyne interferometer (DSHI), 523, 525 dense wavelength division multiplexing (DWDM), 514 Detailed valance theory, 462 dextrorotatory, 94 diattenuation, 88 differential phase-shift keying (DPSK), 514 quadrature phase-shift keying (DQPSK), 514 diffracted wave, 598 diffraction, 26, 37-44, 48, 50, 54, 56-66, 85, 105-107, 312, 320 limit, 474, 483 limited, 267, 294, 296 Diffuser, 179, 180, 200, 223, 224 digital coherent receivers, 514 technology, 514, 515 cross-correlation, 544 signal processor (DSP), 115 thermal sensors (DTS), 134 Direct Lambertian absorptance, 189 Conductance, 190 Irradiation Method, 188 reflectance, 189 Disk-type LM, 423, 432, 433, 461 Displaced Graph, 602 displacement current, 23 Dissipative soliton resonance (DSR), 575, 576, 578, 579, 588 distributed acoustic sensing (DAS), 516 Feedback (DFB) laser, 512 Feed-Back (DFB) laser, 581 strain sensing (DSS), 516 temperature sensing (DTS), 516 vibration sensing (DVS), 516 DLIM, 187-189, 195, 196, 199, 202-204, 207, 211 double precision, 116 Double-clad fiber, 436 Down-conversion, 419 DSR-like square pulses, 576, 577 dual cavity feedback structure (DCFC), 537 dual-wavelength, 547-550, 557-559 DynamIQ technology, 119 E EDF (erbium-doped fiber), 512 Effective aspect ratio, 422 focal length, 273, 274 Einstein A coefficient, 420 Electric field amplitude, 338 electrical feedback, 527-529 electromagnetic waves, 23, 29 electrostatic, 526 electrostriction, 534 EMesh, 143 Energy efficiency, 334 absorption, 352 extinction and scattering, 352 of Airy light-sheet, 357 of Bessel pincer light-sheet, 369 of Gaussian Light-sheet, 385 scattering, absorption, and extinction, 333 transfer, 452, 453, 458, 464 entrance pupil, 64 637 Advances in Optics: Reviews. Book Series, Vol. 5 equalization, 551 Equatorial mounting, 431, 446, 447 Erbium, 513, 547-549, 566 Erbium-doped fiber amplifier (EDFA), 515 étendue, 190 Ethernet switch, 138 evanescent coupling effect, 526 wave, 31-33 Excitation, 512 spectra, 429 exit pupil, 64-68 external parallel feedback, 522, 524 Extinguished and scattered energies, 352 extraordinary- or e-ray, 90 F Faraday’s Law, 23 far-field approximation, 476 patterning, 474 processing, 474 regions, 478 resolution, 474 scattering intensity, 333 dimensionless far-field scattering intensity, 352 dimensionless normalized far-field scattering intensity, 352 of Airy light-sheet, 355 of Bessel pincer light-sheet, 366 of Gaussian Light-sheet, 383 fat-tree network, 136 architecture, 141 femtosecond, 474, 477-479, 482 few-mode fiber (FMF), 515 fiber Fabry-Perot tunable filter (FFP-TF), 555 laser, 512-515, 527, 534, 548-550, 552555, 557-561, 563-566 Fiber-type LM, 423, 432, 433, 464 Fibonacci Zone Plate, 249 finite measurement time, 538 Flash-lamp pumped laser, 417 floating point operations per second (FLOPS), 115 Fluorescence, 433 lifetime, 433 Fluoride glass, 418, 433, 437, 438 638 ZBLAN, 433, 436, 437, 441, 464 Focal point, 335 Force and torque, negative, 333 Fourier series, 50, 51, 593, 594 transform, 25, 41-43, 47, 50, 52, 54-59, 61, 63, 66- 69, 82, 597, 619 Dispersive Fourier transform (DFT), 576-578, 583-586, 588 double Fourier transform, 628 infrared spectroscopy, FTIR spectroscopy, 301 fast Fourier transform, 41 frequency transform (FFT), 547 Spectroscopy, 68 Four-level scheme, 420 four-wave mixing, 552, 557 FPB. See full Poincaré beam FPB Polarimetry, 164 Fractal Zone Plate, 243 Fraunhofer conditions, 598 diffraction, 38, 42 free-electron lasers, 512 frequency discriminator, 527, 528 jitter, 538 Fresnel biprism, 78 diffraction, 38, 41, 42, 65 double-sided mirror, 78 integral formula, 241 lens, 418, 427, 450, 461 fringes, 37, 70, 71, 76-81, 83-85 front focal length, 272 FSO, 395, 397, 404 Fujitsu’s Fugaku (442 PFlops), 136 full Poincaré beam, 164, 166, 168, 172 width at half maximum (FWHM), 518 fused multiply-add (FMA), 116 G gain medium, 512, 513, 526, 530, 531, 544, 547, 549, 551, 555, 557, 558, 563 gallium arsenide, 513 GaP prism, 152 gas lasers, 512 Gaussian, 474-476 Index beam, 315, 475, 476 broadening, 543 beam spreading, 475 Law, 23 mode, 475 Generalized Lorenz-Mie theory (GLMT), 333 Glan-Foucault prism, 92 GLMT, 334, 341 Global solar irradiance, 447, 448 Goos-Hänchen Shift, 32 Gordon, 136 Graphene, 563-565 graphics processing unit (GPU), 115 grating coupling, 150 Green’s theorem, 38, 104, 105 group velocity, 74, 75 dispersion, 585 H half-wave plate, 94, 99 Hanbury-Brown Twiss effect, 547 Heaviside Function, 600 Heliostat, 427 Helmholtz equation, 26, 38, 39, 52, 104, 105, 107 heterodyne detection, 538-540 high finesse FP resonator, 528 High-reflectivity coating, 450 HR coating, 450 hit quality index, 500 homogeneous line broadening, 551 host channel adapter (HCA), 137 Huygens-Fresnel’s principle, 37, 39 hydrofluoric acid, 563, 564 hyper threads, 114 I I/O node, 137 IBM's Summit (148.6 PFlops), 136 Ideal linear polarizers, 98 ILIM, 186-189, 195, 200, 202, 213-215, 230 Illuminometer, 447 image processing unit (IPU), 115 signal processor (ISP), 115 imaging system, 64-68 impedance of the light field, 27 impulse response, 39, 64 Impurities concentration measurement, 303–305, 309, 310, 322, 327–328 Cu, 306, 320, 327, 328 Fe, 305–308, 311, 327, 328 OH group, 301, 305, 308–311 Inclusions. See. Impurities incoherent illumination, 50, 68 ligh, 46, 64, 67, 103 light, 67 light source, 511, 512 index of refraction, 24, 25, 27, 30, 32, 33, 37, 87, 90 Indium, 513 Inductively Coupled Plasma Mass Spectrometry, ICP-MS, 304 detection limit, 304, 306 Infiniband switch, 138 infrared lasers, 512 light, 513, 564 Initial electric field, 335 of Gaussian light-sheet, 340 of the Airy laser-sheet, 338 of the Bessel pincer light-sheet, 339 in-line amplifier, 513 Instruction cache L1, 128 instruction-level, 115 instructions per cycle (IPC), 115 Integrating sphere, 180, 199, 200, 225, 226, 433, 440, 441 intensity distribution, 475, 476, 478, 480 modulation (IM), 514 Interference Phenomena of Partial Coherent Light, 81 Interferometric fringes, 581, 582, 588 Internal and scattered field, 335, 342 Internal field coefficients, 342 Inverse Lambertian absorptance, 189 Conductance, 190 Irradiation Method, 186, 188, 230 reflectance, 189 639 Advances in Optics: Reviews. Book Series, Vol. 5 J Jones vector, 95, 96, 97, 99 Judd-Ofelt analysis, 433 K Kidney-shaped spool, 436, 437 KX-6000, 126 L L2 cache, 128 Lambertian absorbance, 183, 205-213 conductance, 183, 184, 190-192, 199, 204, 207, 208, 211, 212, 223, 225, 231 diffuser, 179 irradiation, 179, 188-190, 198, 199, 201, 202, 211, 230, 231 light, 180-182, 186-189, 193, 194, 196, 199, 202, 208, 213, 214, 225, 231 radiation, 179, 186, 187, 213-216, 221, 222, 231 reflectance, 183, 203-212 screen, 187 source, 181, 185, 187, 188, 195-197, 200 transmittance, 183, 184, 188, 190, 192, 195, 202-204, 206-208, 210, 211 large effective area fiber (LEAF), 532 LASCAD, 453, 455 Laser, 179, 180 Ablation Inductively Coupled Plasma Mass Spectrometry, LA-ICP-MS, 304, 305 detection limit, 304, 306–7, 327, 328 cavity, 450, 451, 453, 458 Characterizations, 395 medium, 417, 418, 437, 443, 454 oscillation mode, 422 rod, 422, 428, 441, 445, 449 LASER (Light Amplification by Stimulated Emission of Radiation), 511 Lasing threshold, 439 latency, 117 lateral magnification, 273 law 640 of reflection, 35, 36 of refraction, 35 levorotatory, 94 Light cones, 181, 187, 192, 193, 207-210, 216, 220, 221, 231 detection and ranging (LIDAR), 516 intensity, 26, 27, 64, 75-77, 83, 88, 89, 477, 478 Light-sheet, 333 Airy light-sheet, 333, 334, 338 auto-focusing Bessel pincer light-sheet, 334 Bessel pincer light-sheet, 333, 334 Bessel pincer-sheet, 340 circularly-polarized Airy light-sheet, 334 Gaussian light-sheet, 333, 334, 340 Nonparaxial Bessel pincer light-sheet, 334 linear polarizer, 164, 168, 174, 175 Linewidth, 513-522, 524-539, 541-546, 565-567 compression, 511, 519, 520, 522, 524530, 532, 535-537, 569 measurements, 511 liquid state lasers, 512 little cores, 119 Lloyd’s mirror, 79 LM. See Laser medium local oscillators (LO), 514 Lock-in amplifying, 302 logic part, 111 Longitudinal-mode hopping, 438 Loongson, 140 3B4000, 140 X-CPU, 140 loss-compensated recirculating delayed self-heterodyne interferometer (LCRDHSI), 524 low pressure chemical vapor deposition (LPCVD), 149 M machine language, 110 Mach-Zehnder interferometer (MZI), 576, 577, 581, 583, 588 macro-operations (MOPs), 128 Made-to-order stage, 428 Index Magnesium-based energy cycle, 417 magnetic field relativistic velocity, 512 main focus, 243 Malus’ law, 88, 89 Mass production, 418, 423, 428, 433, 462 matrix, 269-271 transformations, 268 Maxwell’s equations, 23, 27, 31, 33 stress tensor, 353 Mechanical effect, 333 mesh interconnect architecture, 130 MESI, 117 Method BLIM, 181, 187, 188, 189, 199-202, 204, 207, 211, 213-215 Parretta-Herrero, 187 PH-Method, 187 MgF2 (magnesium fluoride), 520 Michelson Interferometer (MI), 545 Stellar Interferometer, 85 micro-FBG (MFBG), 563 Microfiber, 526, 527 microlenses, 151 microlensing, 480, 482 micro-operations (μOP), 128 microspheres, 474, 479-483 MMF (multimode fiber), 522 Mode-matching efficiency, 422, 436, 439, 464 Mode-matching Efficiency, 453 Modulation parameter, 335, 338 transfer function, 68, 253 Molecules, 512 Monochromatic, 512, 540 coherent laser beam, 417, 418 wave, 25, 26 monochromaticity, 513 Mueller matrix, 163, 164, 166, 167, 172176 multi-conical surface, 292, 293 multi-core fiber (MCF), 515 multi-level modulation formats, 514, 515 multimode interferometer (MMI), 524 Multipole expansion, 335 Multivariate analysis classification table, 506 exploratory data analysis (EDA), 504 loadings plots, 504, 505 Partial Least Squares (PLS), 506 principal component, 504 analysis (PCA), 504 score plot, 504 Multivariate analysis, MVA, 488 multi-wavelength, 547, 550, 552, 553, 554, 557 N Natural air convection, 462, 464 Nd-doped yttrium aluminum garnet, 418 Nd YAG, 418, 430, 440, 449, 457 Nd-doped Y3Al5O12, 420 Nearest Neighbor Strategy, 173, 175, 176 near-field focusing, 474 zone, 476 regions, 478 negative uniaxial crystal, 91 neural processing unit (NPU), 115 noise, 165, 167, 168, 174-176, 305, 322–28, 324 accumulated, 168 characterization, 511, 544, 545 component of the measured signal, 305, 308, 317, 320 dark shot, 167 fixed pattern, 167, 168 Gaussian, 172, 174 photon shot, 167 read, 167 RMS deviation, 323–27 source, 323, 325 non-blocking network, 141 Non-diffraction, 333 nonlinear polarization rotation (NPR) , 552, 577 scattering, 537 semiconductor optical amplifier (NLSOA), 557 nonmonochromatic, 25, 43, 95 Non-paraxial Airy “acoustical-sheet”, 334 Non-periodic Functions, 51 non-uniform memory access (NUMA), 136 north bridge, 130 numerical aperture, 292, 474 641 Advances in Optics: Reviews. Book Series, Vol. 5 O Odeillo solar furnace, 427, 428 Off-axis parabolic mirror, 418, 434, 464 OAP, 418, 434, 443, 451, 462 Offset angle, 435 on-axis spherical aberration, 277, 288, 291, 295, 296, 298 ONet, 143 on-package interconnect (OPI), 130 Operator Calculus, 620 Optical absorption edge, 464 apparatus, 180, 181 axis, 182, 186, 187, 197, 202 Bidirectional Lambertian Conductance, 179, 187, 190, 191, 198, 231 circulator-based feedback circuit, 524, 525, 526 characterization, 179, 180, 186, 187 code, 195, 200, 201 comb filter (OCF), 558 concentration ratio, 182, 185, 186 conductance, 181, 191, 224, 231 conduction, 191 configuration, 179 Direct Lambertian Irradiation Method, 187 efficiency, 180, 181, 182, 185, 186, 187, 188, 213, 215, 216 element, 190, 191, 195, 196, 198, 199, 211, 213, 231 elements, 181, 182, 187, 191, 192, 198, 213, 222, 225, 231 force, 353 of Airy light-sheet, 360 of Bessel pincer light-sheet, 373 of Gaussian Light-sheet, 387 frequency domain reflectometry (OFDR), 516 Imaging, 252 Interferometry, 69, 75 Lambertian conductance(OLC) , 187, 190-192, 195, 231 Bidirectional Lambertian Conductance (OBLC), 187, 190, 198, 202 law, 181, 231 manipulation, 333 maps, 194 642 network chip (ONC), 159 path, 204 difference, 577, 582, 583 properties, 179, 185, 187, 231 quantities, 181, 186, 187, 188, 189, 192, 203, 206, 211, 230 receiver, 514 reversibility, 181, 231 self-injection feedback, 524, 532 simulation, 196, 197 simulations, 182, 192, 194, 195, 202, 231 solar concentration, 186 spectrum analyzer, 432 spin torque, 334, 354 of Airy light-sheet, 362 of Bessel pincer light-sheet, 380 system, 181 systems, 186, 231 terminal, 179, 184, 185, 191 terminals, 191, 231 Transfer Function (OTF), 68 transmission efficiency, 192 transmittance, 202, 231 transmitters, 514 Tweezers, 243 Wireless Communications, 395 wireless power transmission, 463 ordinary- or o-ray, 90 oscillating dipole, 29, 30, 88 Oscillation wavelength, 418, 420, 464 out-of-order, 116 Output coupler (OC), 443, 445, 446, 450, 451 OWC, 395-397, 403, 409 P Package on Package (PoP), 135 Parabolic Function, 606 mirror, 418, 427, 432, 439 parallel computing, 114 thin films, 79 paraxial approximation, 42, 55, 57 partially coherent light, 46 polarized light, 100, 103 Particles, spherical, 333 Index Passively mode-locked fiber laser, 576, 581, 588 PCH, 130 Periodic Functions, 50 Phase jump, 581 or frequency noise, 513-517 shift, 37, 39, 52, 68, 93 Transfer Function, 68 velocity, 73, 74, 75 phase-sensitive optical time-domain reflectometry (φ-OTDR), 516 phase-shifted FBG, 528 Phosphate, 418, 433 Phosphide (InGaAsP), 513 Photoelasticity tensor, 313 Photonic crystal fiber (PCF), 549 Photons, 512 Photothermal Common-path Interferometry, PCI, 302, 307–309, 311, 319 detection limit, 307–9, 311 photo-thermal effect, 564 pipelines, 114 plane of incidence, 33-35, 87 Poincaré sphere, 166, 167, 172, 173 polarimetric system, 164 polarimetry, 163, 164 polarization, 27, 163, 164, 166, 172, 173, 176 Elliptical polarization, 27, 96 input state, 166 linear polarization, 27, 96, 97 state, 163, 164, 166, 167, 172, 173, 176 polarization division multiplexing (PDM), 514 state analyzer, 164, 168, 172, 176 state generator, 163, 172 polarization-maintaining fiber (PMF), 532 polarized light, 27-29, 86-89, 92-97, 100103 Completely Polarized Light, 95, 102, 103 polarizers, 86, 91, 95, 98, 100 Polluting chemical elements. See. Impurities Polynomial Equivalent, 609 Population inversion, 459 post-amplifier, 513 Power feeding, 463, 464 meter sensor, 431, 446 pre-amplifier, 513 principal section, 90, 91 PROMES-CNRS laboratory, 427, 428 PSA. See Polarization state analyzer PSG. See Polarization state generator pulse duration, 473, 474, 477, 478 Pump, 512, 521, 526, 531, 532, 536-538, 548, 549, 554, 555, 563-566, 568 Pumping efficiency, 420, 421, 458 Pyrheliometer, 431, 446, 448 Q Q-switched mode locking (QML), 564 Quad Small Form-factor Pluggable (QSFP), 138 quadrature amplitude modulation (16-QAM), 515 phase-shift keying (QPSK), 514 Quantum cascade lasers, 487 conduction band, 488 Diffuse reflectance, 492 diode lasers, 488 discrete transition, 488 grazing angle, 487, 489, 490 incident angle, 489, 492 Population inversion, 488 tunable lasers, 490 Quantum defect, 419, 421, 422 efficiency, 421, 433, 459, 461 fluctuations, 518 limit, 518 quarter wave phase plate, 164, 172, 174 quarter-wave plate, 93, 99 Quartz glass, ultrapure, UQG, 301, 303– 311, 316, 318–320, 321, 326-328 quick path interconnect (QPI), 132 QWP. See quarter wave plate R Radiance, 179-184, 186-190, 195-201, 208, 213-225, 231 Radiant flux power, 424, 427-430 Raman amplification, 517 microspectroscopy, 496 optical time-domain reflectometry (ROTDR), 516 643 Advances in Optics: Reviews. Book Series, Vol. 5 scattering, 516, 517 Ramp Function, 605 rare-earth-doped fiber, 512 Rayleigh backscattering (RBS), 529 length, 479, 483 range, 475 scattering, 517, 520, 521, 530, 532, 533, 535, 538 Rayleigh-Sommerfeld formula, 38, 39, 40, 104, 108 reabsorbing effect, 548 Red shift, 429, 432, 448 reduced instruction set computer (RISC), 110 refractive index, 149, 543, 566 matching, 456 nonlinearity, 316, 317 register part, 111 relative intensity noise (RIN), 513, 528 permeability, 24 permittivity, 24 Renewable energy, 417 ReOrder Buffer, 128 residual chromatic aberration, 291 spherical aberration, 296 resonant cavity, 512, 566 Riemann zeta functions, 594 rigorous analytical dependences, 268, 288, 298 ring bus, 127 Ronchi’s gratings, 59 Round-trip resonator loss, 420 S sagittal radius, 278-280, 282, 292 radius-coefficient, 277-279, 285 Sagnac loop interferometers, 558 sample thermal decomposition, 490 saturable absorber (SA), 563 saturation power, 551 Scaling parameter, 335, 339 scanning electron microscope (SEM), 153 scattering, 30, 88-90, 333, 353 field coefficients, 342 Self-healing, 333 644 self-heterodyne, 519, 531, 540, 542, 566 self-homodyne detection, 540 self-injection locked, 520, 521 semiconductor laser, 512, 515, 519, 528 optical amplifier (SOA), 513 Shannon capacity, 515 side mode suppression ratio (SMSR), 526 Sierra (94.64 PFlops), 136 Signal-to-noise ratio (SNR), 305, 306, 308, 316, 319–321, 325, 326, 513 silicon, 303, 304, 473, 474, 477-480 dioxide (SiO2), 148 nitride (Si3N4), 148 oxynitride (SiOxNy), 148 single longitudinal mode (SLM), 513 precision, 116 Single-mode fiber (SMF) , 515, 577, 578, 584, 585 single-wall carbon nanotubes saturable absorber (SWCNT-SA), 564 single-wavelength, 547, 549, 550 Size parameter, 335 Slope efficiency, 439, 444, 453, 456, 458 slope efficiency, external, 444, 445, 448, 449 Small Form-factor Pluggable (SFP), 138 Snell’s law, 31, 87 Solar angular dispersion, 186 cell, 185, 417, 418, 463, 464 concentration ratio, 422, 424, 430, 436, 442 concentrator, 179-182, 185, 187, 188, 190-193, 196, 202, 205, 214, 230, 231, 424, 427, 434, 449, 462 compound parabolic, 179 constant, 424 furnace, 418, 427 lighting system, 428, 462 radiation, 179, 185, 186, 223, 224, 430, 463 AM0, 463 AM1.5, 463 diffuse, 429, 464 direct, 429, 436, 443, 446, 462 global, 429, 430, 431 spectrum, 419, 432, 436, 440, 463 sunlight spectrum, 422, 437, 438 Index model by Bird and Riordan, 429, 430 telescope, 446, 447 tracker, 181, 186 tunnel, 181, 187, 192, 194, 210-213, 221-223, 225, 231 Solarization, 449, 461 Solar-pumped laser, 417, 420, 432, 462 fiber laser, 433, 439, 464 SPFL, 432, 433, 436, 438, 464 SPL, 417, 423, 433, 449, 457 sun-pumped laser, 417 μSPL, 441, 445, 447, 450, 462 Solar-tracking system, 428, 436, 443, 462 coordinated, 462 solid-state lasers, 512 south bridge, 130 Space solar power station, 417, 463 spatial division multiplexing (SDM), 514, 515 light modulator, 243 Specific Optical Bidirectional Lambertian Conductance (SOBLC) , 184, 191, 192, 204, 207, 208, 211, 212 Lambertian Conductance (SOLC), 191 spectral correlation algorithms, 500, 501 matching, 439, 440, 461 quantum efficiency, 463 SPECTRAL2, 429, 430, 431, 432 specular reflectance, 492, 508 Spherical aberration, 426 Bessel function of first kind, 339 Spontaneous emission, 420, 440, 458, 459, 461, 518 spot size, 474-476 Square Wave Function, 600 SRAM, 112 Standing Waves, 72 Stiffness tensor, 314 stimulated Brillouin scattering (SBS), 531 emission, 513, 518 emission, 418, 420, 433, 458, 461 Raman scattering (SRS), 535 Rayleigh scattering (STRS), 532 stochastic light fields, 86, 100 Stokes parameters, 163, 164, 166, 168, 172 polarimeter, 164 vector, 163, 164, 166, 172 submicron, 473, 482, 483 subsidiary foci, 243 Sunway SW26010 1.45 GHz, 136 TaihuLight (93.01 PFlops), 136 Super Invar alloy, 462 Super-Gaussian pulse profile, 585 Superposition of Waves, 70, 72 Super-resolution imaging, 333 superscalar processor, 114 Surface albedo, 438 Sylvester's matrix, 277 symbolic computation, 287 Symmetrised Graph, 603 Synthetic crystalline quartz, SCQ, 303–304, 306, 310, 316, 319–321, 328 system on chip (SoC), 115 T tangential radius, 278 radius-coefficient, 278 tapered boron codoped germanosilicate fiber, 560 optical fiber, 535 task-level, 115 TE and TM polarized, 334 TE-polarized laser sheet, 335 TM-polarized laser sheet, 335 Tellurite glass, 418, 433 TEM00-mode, 450 temperature coefficient, 566 dependent refractive index, 423 Temporal coherence, 576-579, 581, 583, 584, 588 TE-polarized electromagnetic fields, 336 tetrahedron, 167, 172-175 thermal design power (TDP), 134 dissipation, 423, 428, 432, 462 effect, 566 expansion, 462 expansion coefficient, 566 lens effect, 423 Thermodynamical limit, 462 thermo-electric cooler (TEC), 565 645 Advances in Optics: Reviews. Book Series, Vol. 5 Thermo-optical parameter, 313, 314, 319 thin film filter, 155 threads, 114 three-cornered-hat method, 519, 520 Threshold power, 430, 436, 443, 449, 456 pumping efficiency, 421 through silicon via (TSV), 135 Thue-Morse Zone Plate, 250 Thulium, 513 Ti sapphire laser, 433 Time dependence, 336 Time-resolved Photothermal Common-path Interferometry, TPCI, 311–321, 327, 328 calibration, 313, 315, 320, 327, 328 detection limit, 319, 327, 328 TM-polarized electromagnetic fields, 337 Total efficiency, 439, 444, 449 internal reflection, 31, 33 Townes and Schawlow, 511, 512 Traditional Mie scattering coefficients, 343 transfer function, 527 transmission loss, 149 transverse electric, 552 magnetic, 552 scale, 335, 338 tunable bandpass filter (TBF), 554 tunable optical filter (TOF), 558 two equal-intensity foci, 258 two-photon absorption, 474, 477-479 U ultrafast lasers, 474, 482 ultra-path interconnect (UPI), 132 unpolarized light, 86, 89, 102 Upconversion, 418 UV (Ultra-Violet) lasers, 512 filter, 439, 443, 449 V van Cittert-Zernike theorem, 48, 49, 50 van der Waals, 526 VCSEL, 133 Chaos Dynamics, 405 Chaos Synchronization, 406 646 L-I Curve Characteristics, 396, 399, 400, 401, 402, 403, 404, 405, 406, 409 Nonlinearity, 404 Optical and Electrical Properties, 400 Polarization Switching, 402 vector instruction, 115 processor, 115 spherical wave functions (VSWFs), 334 verification mode, 500 Vertical cavity surface emitting semiconductor laser, 399, 461 VECSEL, 461, 464, 465 VIA, 126 visible lasers, 512 Voight fitting, 543 VSWFs, 341 W Waist radius, 335 wave aberration, 65 impedance, 352 speed, 24, 25 vector, 335 wavefront division, 76-78, 81 wavelength blocking, 547 converted lasers, 547 converter, 513 division multiplexing (WDM), 514 switching, 511, 547, 548, 550, 560, 561, 569 tuning, 511, 547, 553-555, 557, 563565, 567-569 wavelength-tunable or switchable, 513, 514, 515 Wavenumber, 335 WDM, 155 weiner effect, 522 whispering gallery mode, 519 white frequency noise, 543 Wollaston prism, 92 X x86 instruction set, 110 Xeon Platinum 9282, 130 Index X-ray lasers, 512 Z Y Zaoxin, 126 ZEMAX, 453, 454 Zeta Functions, 615 Young's experiment, 76, 77, 81, 84 Ytterbium, 513, 547 647