The Imaging Properties of A Low Aberrations Transformation Optics Lens by Saul J. Wiggin A thesis submitted to the University in partial fulfilment of the requirements for the degree of Doctor of Philosophy October 1, 2014 Abstract This thesis proposes to design a low aberration lens using transformation optics based on the Cooke triplet. The Cooke triplet minimizes Seidel aberrations using only three glass elements. A reduced map triplet would possess no metamaterials (MTM)s and not be subject to losses. The MTM triplet is designed using ray tracing from commercial optical design software ZEMAX and a mesh representing the desired virtual space is generated from this. Discrete coordinate transformation optics is then used to find the permittivity values of the lens required to satisfy of original triplet. The Seidel aberrations are characterised as longitudinal and transverse spherical aberration given by the energy distribution in the focal region. The coma is characterised by an off-axis source incident on the original Cooke triplet, the MTM lens and the reduced dielectric map, which does not contain MTM. The field curvature and distortion are found according to the shift in focal length and the astigmatism included in the field curvature calculation. We find that the field of view in the proposed triplet is not limited as it is in the conventional case. The proposed triplet can and is matched to free space and it suffers less reflections from the incident waves and hence reduces the power loss of lenses. The wavefront is determined by the optical path difference though the lens and this is equivalent to the wavefront aberration function. This wavefront aberration function can then be further devolved into separate Zernike polynomials which are orthogonal unlike Seidel aberrations. The results are shown to have good agreement with the results for the Zernike aberrations in the same lens calculated using the ZEMAX software. These results have shown that the reduced map performs similarly to the conventional Cooke triplet but with slightly better off axis aberration and a larger depth of field for its on axis aberration. The lens with MTMs performs worse over the whole frequency range due to their limitation in bandwidth and losses. 3 This thesis is a presentation of my original research work. Wherever contributions of others are involved, every effort is made to indicate this clearly, with due references to the literature, and acknowledgement of collaborative research and discussions. 4 Acknowledgements I’d like to thank the metamaterials community and the antennas and electromagnetics group: Dr. Rui Yang, Dr. Oscar Quevedo-Teruel, Dr. Khalid Rajab, Dr Di Bao, Dr Rhiannon MitchellThomas, Dr Anestis Katsounaros, Dr. Quammer Abbasi, Dr Lianhong Zhang, Dr Yifeng Fan, Dr Tuda Yilmaz, Dr. Alice Pellegrini, Mrs Zhijiao Chen, Mr Alexander Sushko, Dr Rostslav Dubrovka and Dr Rob Donnan. Professors’s such as Professor Raj Mittra, Professor Douglas Werner, Professor John Pendry. My experience of them during my PhD were extremely positive and a group of people who work tirelessly on a subject which they are passionate about and which promises real benefits to our society and economy. I would like to thank Dr Ian Youngs and the EPRSC/DSTL for funding the research. 5 Journal Publications • Saul J. Wiggin, Wenxuan Tang, Yang Hao, Rob Foster and Ian Youngs ”MultiElement Optical Systems using Transformation Optics and Metamaterials, Optics Express (under submission). • Saul J. Wiggin, Yang Hao Optical Systems designed with Discrete Transformation Optics, Electronics Letters. 6 Conference Proceedings • S. J. Wiggin, W. Tang, Y. Hao and I. Youngs ”A Flat Profile Microwave Lens without Metamaterials using Transformation Optics” META 2012, Paris • Saul J. Wiggin, Wenxuan Tang, Yang Hao and Ian Youngs, ”A Low Aberration Microwave 2D GRIN Lens Antenna Realisable with Dieletrics”, DSTL internal conference. Contents 1 Introduction 23 1.1 Negative Index Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 The Motivation of Research and Original Contribution . . . . . . . . . . . . 29 1.3 The Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Transformation Optics and Their Applications 2.1 2.2 2.3 Invisibility Cloaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.1 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.2 The Invariance of Maxwell’s Equations . . . . . . . . . . . . . . . . . 40 2.1.3 A Design of the Cloaking Device . . . . . . . . . . . . . . . . . . . . 41 2.1.4 The Design of a Carpet Cloak . . . . . . . . . . . . . . . . . . . . . . 44 2.1.5 Thermal Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Transformation Optics Imaging Devices . . . . . . . . . . . . . . . . . . . . . 47 2.2.1 Single Flat Lens Design . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.2 Extraordinary Transmission Devices . . . . . . . . . . . . . . . . . . . 51 2.2.3 Other Optical TO Devices . . . . . . . . . . . . . . . . . . . . . . . . 53 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 The Design of a MTM Cooke Triplet 3.1 39 59 Ray Tracing of the Optical System . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.1 62 Verification with ZEMAX Model . . . . . . . . . . . . . . . . . . . . 7 8 CONTENTS 3.1.2 TO Lens in ZEMAX . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Structuring the Grid with Rays as Boundaries . . . . . . . . . . . . . . . . . 63 3.3 Discrete Transformation Optics Calculation . . . . . . . . . . . . . . . . . . 64 3.4 The Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . 66 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 FDTD Simulation of the MTM Lens 4.1 75 Ez Field Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1.1 MTM and Reduced Cooke Triplet . . . . . . . . . . . . . . . . . . . . 76 4.1.2 Conventional Cooke Triplet and Single Lens . . . . . . . . . . . . . . 77 4.2 Focal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Lens Power or Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Off Axis Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Focal Plot: On and Off axis for the Lenses . . . . . . . . . . . . . . . . . . . 83 4.6 Wide Angle Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7 The Effect of Dispersion on The Focal Length . . . . . . . . . . . . . . . . . 87 4.8 Transmission and Reflection Spectra . . . . . . . . . . . . . . . . . . . . . . 89 4.9 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Image Evaluation 5.1 The Wavefront Aberration Function . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 93 93 Sampling The Wavefront Aberration Function From The FDTD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1.2 Fitting a Polynomial to the Wavefront . . . . . . . . . . . . . . . . . 96 5.1.3 The Root Mean Squared (RMS) Error of The Wavefront Aberration . 97 5.1.4 Zernike Coefficients Calculation from Numerical Integration . . . . . 98 5.1.5 Spherical Aberration and Secondary Spherical aberration . . . . . . . 99 5.1.6 Defocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 CONTENTS 9 5.1.7 Verification with ZEMAX . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Modulation Transfer Function (MTF) . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6 Other Simulations of Transformation Optics Imaging Devices 107 6.1 Sub-Wavelength Imaging using a Drain and Time Reversal Techniques 6.2 A Square Fish Eye or Magnifier . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 Field Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7 Conclusion and Further Work 7.1 . . . 107 115 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.1.1 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 118 10 CONTENTS List of Tables 3.1 This table shows the values of the Seidel aberration coefficients in waves for a single lens and for a Cooke triplet, on which the transformation was based compared with the Seidel aberrations in the MTM Cooke triplet. . . . . . . 62 4.1 Fundamental performance metrics for the lenses. . . . . . . . . . . . . . . . . 79 5.1 This table shows the RMS of the aberration function sampled for the four lenses as a function of wavelength. The equivalent Strehl ratio is also given. The MTM lens and Cooke triplet are within typical optical design tolerances. 5.2 98 This table contains the full wave aberrations for all primary aberrations for the MTM lens, reduced map and the Cooke triplet. The zeros are aberrations which are present in the third dimension. These aberrations could not be calculated from a 2D simulation alone. The aberrations before astigmatism (tilt and defocus) are corrected for by finding the correct focal length. The remaining primary aberrations (field curvature and distortion) are zero because they are aberrations of the position of the focus rather than the image quality. 100 5.3 This table shows the focal length for the MTM lens and the reduced triplet. The defocus is the amount of Z0 in the Zernike aberration result. . . . . . . 100 11 12 LIST OF TABLES List of Figures 1.1 This figure shows (top left) the real and (top right)imaginary part of the refractive index and the real and imaginary part of the permeability (bottom left) and permittivity (bottom right) centred on 10 GHz. The metamaterial unit cell with the dimensions shown for a split ring resonator metamaterial simulated in CST studio where the unit cell is 5 mm by 3.33 mm. the metallic structure (grey) is silver and is placed on a dielectric FR4 and backed with a gold rod (0.25 mm x 3.33 mm) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 24 This is a SEM image 21-layer three dimensional optical metamaterial with a fishnet structure with a unit cell of 860 nm (30). The structure consists of alternating layers of 30 nm silver Ag and 50 nm magnesium fluoride (MgF2 ). 1.3 27 This figure shows in (a) the ray diagram for paraxial rays and meridonal rays passing through the lens (blue) and then corrected using the transformation media (yellow). The focal surface is curved when field curvature is present and results in the focus being different for different angles of incidence on the lens. zf shows the focal length for the paraxial ray and zp shows the focal length for the meridional ray and zc shows where the transformation media will be placed to flatten the surface. Panel (b) shows the inverse transformation grid (yellow) which possess the same electromagnetic properties as the original lens (white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 30 14 LIST OF FIGURES 2.1 This figure shows the spherically symmetric cloak designed by Sir J. Pendry. The inner sphere is the cloaked region radius, R1, and the outer sphere the radius of the cloak, R2. The lines show the ray propagation in the invisibility cloak being diverted around the cloaked region like water around a stone. . . 2.2 42 This figure shows an image of a nano fabricated carpet cloak. It has been placed in a silicon nitride waveguide on a nano porous silicon oxide substrate with very low refractive index for testing. The spatial index variation is realised by etching holes of various sizes in the nitride layer at deep wavelength scale creating a local effective medium index (10) cloaking the bump region. 2.3 44 This figure shows a cloaking device in the shape of an eye designed using an optimisation algorithm. This shows what is possible without using the transformation optics method to design the cloak and inside use optimisation, the initial requisite for a cloaked field and a genetic algorithm with single permittivity grid cells to create a cloaked effect. The panel (a) shows the uncloaked cyclinder for the Ez field and (b) for the field amplitude and (c) and (d) for the electric field distribution and amplitude when the eye cloak is used. 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 This figure shows in the top section the outline of the concave lens placed in the grid generation suite POINTWISE. A mesh is drawn inside the lens and then smoothed using control functions to make the grid as orthogonal as possible. The dimensions of the lens are 160 cm by 10 cm. The same as those used in a previous flat lens design (21). In the bottom section the permittivity map is shown. This is generated using the method previously outlined and known as discrete coordinate transformation. This lens share the same permittivity distribution as the paper (21). . . . . . . . . . . . . . 48 LIST OF FIGURES 2.5 15 These figures shows FDTD simulations of, on the left the original single lens on which the grid was designed. On the right is the lens fig. 2.4. The agreement in focal length is precise. The focus of the original lens on the left is 11.69 cm and the focus for the lens on the right is 10.19 cm. . . . . . . . . . . . . . . . 2.6 This figure shows the comparison between the compressed lens designed using the discrete transformation optics approach and the original concave lens. 2.7 50 . 51 This figure shows the curvature, thickness, d1 and d2 , n1 and n2 of a cemented doublet lens highly common in optical engineering systems. Panel (b) shows the virtual space and physical space in grid form for the doublet lens and the resulting refractive index profile generated using transformation optics where the refractive index is outside the metamaterial range. The panel (c) shows the electromagnetic simulation results for the original lens and the TO doublet lens where the focal regions match extremely successfully and the plot of the electric field is the image plane for both lenses showing extremely close agreement between the two devices (26). This shows a typical design of a transformation optics device and what is typically achievable using the method for a double optical system. . . . . . . . . . . . . . . . . . . . . . . . 2.8 52 This figure shows the virtual grid for the propagation characteristics of the Extraordinary Transmission (ET) device where an area is shrunk from 1.6 λ to 0.2λ to reduce the aperture to smaller than the wavelength where the field of the left in (a) is without any device, in (b) the field is with an ET setup on one side and (c) where one is on either side of the gap. The right hand side shows the design of the ET device. The rays are shown in the grid in figure (a) being squeezed to fit the small hole and the resultant virtual space surrounding. Panels (b) and (c) the permittivity map is shown at a high resolution and then at a lower resolution which possesses the same effective qualities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 16 LIST OF FIGURES 2.9 This figure shows the refractive index distribution from the Photonic Black Hole (32) with a negative index of distribution satisfied with metamaterials.The Continuous Index Photon Trap with a high permittivity which can be achieved with ferrites. 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 The Ray tracing of the Cooke triplet is optical software package OSLO. The key design parameters are given in the table below. The Cooke triplet does not have a unique solution therefore this is just one of many possible combinations. The key optics data is taken from the triplet lens library in OSLO. The focal length of this device is 50 mm, NA is 0.125 and the scale is given. This is the specification of the lens designed using the transformation optics method. Glass type SK16 and F2 are crown and flint types. 3.2 . . . . . . . . . . . . . . 61 Panel(a) shows the ray tracing diagram imported into numerical grid generation software POINTWISE for the triplet. The lateral lines are the rays and the structure of the lens is visible. In panel (b) only the top half is modelled to space space and the the design will be copied later in Matlab using symmetry consideration for the triplet. The grid size between the grid boundary nodes is 0.14. The grid is solved by Elliptic grid generation. This involves the commercial program solving a set of partial differential equations. Laplace smoothing functions create quasi-orthogonal grids by optimising the distribution of cells to the optimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 LIST OF FIGURES 3.3 17 This figure shows the original triplet in fig (a) with the key optical parameters and the table of key optical parameters beneath. Panel (b) shows the virtual space where the grid has been meshed around the rays which make up the boundaries from the triplet ray tracing. This grid was then transformed using transformation optics to panel (c) which gives the values of permittivity for a flat implementation in Matlab of the triplet. Panel (d) shows the permittivity map in the FDTD simulation domain before a signal is launched. Notice its symmetrical design which has been added at this stage to fully implement the triplet in panel (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 65 The triplet is centered at zero and extends 0.2 m by 0.2 m. A sinusoidal plane wave is launched and controlled using total field scattered field formulation. The simulation domain is terminated using a Perfectly Matched Layer. The dispersive MTM values in the triplet are modelled using the ADE approach. 3.5 67 A schematic of the time stepping, leap frog method for the calculation of the FDTD simulations. n-1, n and n+1 are the time steps, k is the grid cell number where the magnetic field, Hy, is at every half integer and the electric field Ex is at every whole integer. The two adjacent field components from the dual field vector are used to calculate the next field vector. This gives the possibility of time domain movies animations in FDTD for algorithm execution. 69 3.6 The figure shows the Yee grid unit cell in 3D where magnetic and electric fields are placed in a staggered arrangement to allow the solution of Maxwell’s equations for each grid cell (i,j,k). . . . . . . . . . . . . . . . . . . . . . . . . 70 18 LIST OF FIGURES 4.1 This figure shows the results of the field amplitude for the FDTD simulation results for the MTM (top panel) and reduced map (bottom panel). The red lines show the location of the PML and the permittivity boundaries on the top, left, right and bottom and the simulation domain. The red line signifies the location of the lens in the FDTD simulation too. The field amplitude is normalised to the maximum value. The source is a sinusoidal wave launched from the left of the domain. The permittivity map is superimposed onto the bounded structure for MTM and reduced lenses. . . . . . . . . . . . . . . . . 4.2 76 This figure shows the results of the field amplitude for the FDTD simulation results for the Cooke triplet (top panel) and single lens (bottom panel) at 8 Ghz. The field amplitude is normalised to the maximum value. The source is a sinusoidal wave launched from the left of the domain. The permittivity map for the Cooke triplet and single lens is placed on the simulation results where they were located in the MATLAB simulation. The focus is signified by an arrow and the Perfectly Matched Layer (PML) which acts as the boundary conditions for the numerical technique are shown as a red line. 4.3 . . . . . . . 78 The transverse intensity distribution for the single, Cooke, MTM and reduced map centred at the focus. Sidelobes peak for the Cooke is further out than for the MTM, with both offering similar peak levels. Conclusion MTM offers similar performance to Cooke, with the reduced map showing results similar to a single lens. The beamwidths for the lenses are 12.3, 11.1, 5.8 and 20.5. 80 LIST OF FIGURES 4.4 19 This figure shows the off axis source incident on the (1) single lens (2) the reduced lens, (3) the Cooke triplet with visible scattering and reflections and (4) the MTM Cooke triplet lens where the outline is shown is red. A focus can be seen at the maximum amplitude where the focus is highlighted with an arrow and the lens is highlighted in a box. We see significant electromagnetic scattering in the MTM lens. The coma is defined as the the peaks where coma is a set of smaller peaks after the main peak when the wave is incident at an angle to the lenses when a slice is placed at the focus for an off-axis beam. The Cooke triplet has a small coma. The coma for the MTM map is large and spread and the reduced map is comparable to the MTM triplet but less pronounced due to less focusing of light rays away from the lens at the edges of the device as in the Cooke and MTM lenses. 4.5 . . . . . . . . . . . . . . . . 82 This figure shows the focal points of the single lens (brown), Cooke triplet (cyan), MTM triplet (red) and the focal point of the reduced map is navy. The results are the focal points from the on and off axis FDTD simulations. The Field curvature is the deviation between focal points in the y axis and distortion in the x axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 83 This figure shows the intensity in dB along the transverse axis located at the focus where the coma is located. The side lobes to the right of the MTM lens being slightly more pronounced than the reduced map. The Cooke triplet is placed at 18 degrees where no focus is formed resulting in no large beam being formed at the centre and small coma to the right. . . . . . . . . . . . . . . . 4.7 85 This figure shows the focus for the Cooke triplet and the MTM lens at 5.53, 6.63, and 8.00 GHz which are the same harmonic as the blue green and red lines used in OSLO to determine the chromatic aberration of an optical system. These are calculated from a Fourier transformation in the FDTD simulation of a Gaussian pulse source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 20 LIST OF FIGURES 4.8 This figure shows the transmission and reflection spectra for the metamaterial triplet and the reduced triplet calculated from a Gaussian pulse incident in the Finite Difference Time Domain model centred at 8 GHz. The first panel shows the reflection coefficient where there is a peak for the reduced and metamaterial lenses and a smooth unvariation in the Cooke. The left hand side panel shows the transmission coefficient with a peak for the MTM and reduced maps and a smooth variation for the Cooke triplet The bottom panel shows than the sum of the reflection and transmission coefficients add to the Gaussian pulse accounting for all the energy in the simulation. . . . . . . . . 5.1 89 The wavefront aberration is characterised by the deviation of the aberrated beam / irregular wavefront from an ideal Gaussian for an incoming parallel beam with a plane wave. This Gaussian sphere represents the ideal wavefront from a point source from which any deviation is measured in terms of a wave aberration This is the dual of the ray aberration which is the failure of the individual rays to meet at a stigmatic point. All real lenses and optical systems are subject to some degree of wave or ray aberrations. . . . . . . . . . . . . . 5.2 This figure shows the wavefront for a Cooke triplet in Zemax and a comparison with the wavefront calculated from the FDTD method. . . . . . . . . . . . . 5.3 94 95 This figure shows a) the scatter plot containing all the Yee grid cells for the final wavefront with Ez greater than 0. In panel b) the figure shows this wavefront plotted and a polynomial fitted to give an analytical solution to the wavefront aberration. In panel c) the wavefront aberration is shown for the MTM lens, the Cooke triplet and the reduced map and in panel d) the original wavefront in FDTD is given. . . . . . . . . . . . . . . . . . . . . . . 96 LIST OF FIGURES 5.4 21 This figure shows wavefront on a unit disk, the set of Zernike polynomials, the amount of each Zernike polynomial which matches that function. The residual is the left over after all Zernike aberrations have been optimised to best match the input polynomial F. The original function in only 2D therefore results which are not 2D must are not valid. Panel a) shows the wavefront function for the Cooke triplet from the FDTD simulation rotated about a central axis. Panel b) shows the remaining wavefront which has not been fitted to the Zernike coefficient. Panel c) shows the Zernike coefficients first 9 modes and panel d shows the amount of each Zernike coefficient in the original wavefront. Notive that the majority of the Zernike wavefront is Z0 and Z4. . 5.5 99 The figure on the left shows the Zernike coefficients for the Cooke triplet in ZEMAX. In panel b) shows the figure for the Cooke triplet in the FDTD simulation calculation of the Zernike coefficients. 5.6 . . . . . . . . . . . . . . . 101 This figure shows the modulated transfer function MTF calculated from the results of the Zernike aberration coefficients for the Cooke triplet, the MTM triplet and the reduced map and a diffraction-limited aberration free case. 5.7 . 102 This figure shows the wavefront aberration function in panel a, the polynomial fitting in panel b to the sampled wavefront. In panel c) you can see the wavefront aberration function derived from the polynomial fitting and in panel d) is the bar chart of Zernike coefficients. . . . . . . . . . . . . . . . . . . . . 104 6.1 In the left hand figure a converged FDTD simulation for propagation inside a Maxwell’s fish eye lens surrounded by a mirror and with a source placed at one focus and a ’drain’ placed at the other. One the right hand side you will see the electric field distribution across the same device with two drains placed at a distance closer than the wavelength. . . . . . . . . . . . . . . . . 109 22 LIST OF FIGURES 6.2 This figure shows the coordinate system for the device described in this section. The grid boundaries were defined using a parametric equation. The centre was excavated and made into a small alcove in which the receiver may be placed.a) shows the grid in MATLAB. This is the initial step to implementing a coordinate transformation.b) shows the permittivity values calculated from the transformation placed in the real grid values of the original map.c) shows the permittivity map is placed in a FDTD grid. The relative permittivity is displayed here.d) shows the source placed at the region we modelled previously. The propagation following the expected ray path is observed in the device. . 110 6.3 This figure shows a flat lens. It is discrete as to be fabricated. It is designed on a standard convex lens. It is based on value calculated using the field transformation method (16) . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 1 Introduction 1.1 Negative Index Materials The negative index materials were theoretically predicted by Veselago long before it was possible to fabricate such materials (2). The extraordinary property of negative index materials was predicted that the classic Snell’s Law may be modified however his work was ignored at the time because it was thought that this material did not exist in nature (1). They predicted some interesting effects: Cherenkov radiation would travel backwards with the reversal of the Doppler shift and negative refraction (1). 23 24 CHAPTER 1. INTRODUCTION Figure 1.1: This figure shows (top left) the real and (top right)imaginary part of the refractive index and the real and imaginary part of the permeability (bottom left) and permittivity (bottom right) centred on 10 GHz. The metamaterial unit cell with the dimensions shown for a split ring resonator metamaterial simulated in CST studio where the unit cell is 5 mm by 3.33 mm. the metallic structure (grey) is silver and is placed on a dielectric FR4 and backed with a gold rod (0.25 mm x 3.33 mm) The first metamaterial was demonstrated, in 2000, by combining a negative permittivity medium, the wire medium, and a negative permeability medium, the split ring resonator (2). An example of a split ring resonator is visible in section 1.1. Pendry’s work re-assessed the merits of the wire medium (3) in addition to artificial magnetism from split ring resonators (4). This initiated a long list of novel structures for the construction of Negative impedance materials (NIM) and their applications such as perfect lens section 1.1 of which the omegastructure is the preferred for gain enhancement in low profile antennas (5) compared with traditional split ring resonators. Anisotropic metamaterials made from metallic nano-wire structures also enables negative refraction and focusing (6). Opposing chiral effects in bi- 1.1. NEGATIVE INDEX MATERIALS 25 anisotropic metamaterials such as the omega particles can achieve negative refractive index (7). The major limitation with bulk NIM metamaterials, which are typically split ring resonators combined with a wire medium, is that it has a very narrow bandwidth i.e. it can only work at a limited band of frequencies. This is due to the fact that the negative permittivity and permeability can only be achieved while the bodies are resonating at a specific frequency. At the resonant frequencies, the material possesses negative permeability and also has a large loss factor. This is sometime known as anomalous resonance, section 1.1. This resonance means that all metamaterials are lossy and operate in a very limited bandwidth around the resonance and this is a key limitation in the technology. Metamaterials have there genealogy in the 1960s, when a lot of work was done on artificial dielectrics. A wire medium was demonstrated at RF frequencies leading to a negative refraction below the cut off frequency (8). Lens antennas such as the parallel plate antenna or path length antenna used artificial dielectrics to focus electromagnetic rays and were used as an alternative for parabolic reflector antennas (9). This was the origins of the wire medium which was then added to by the split ring resonator in the design of a NIM metamaterial. Modern metamaterials research pushes towards the optical regime and relies on the addition of artificial magnetism to provide the combined negative permittivity and permeability to give a negative refractive index. NIMs can be also designed based on composite right/left-handed transmission line structures (10) which can be achieved with a combination of capacitor and inductor which has been used to experimentally verify perfect imaging in TL-NIM (11). Metamaterials created in transmission lines are not prone to being narrow-band and lossy. Negative Refractive index metamaterials therefore allow demonstration of negative index behaviour without the losses present in bulk materials (12). Evanescent wave growth (13), NIM lens focusing (14) and negative refraction in 2D waves (15) has been experimentally verified in TL-MTM. A negative index slab of MTM has been simulated and quantified in terms of the Seidel aberrations which define the image quality of an optical system (16; 17). These produce perfect images in negative index optical MTM lenses. This is not perfect in terms of beating the diffraction limit which is a physical law but in the sense that aberrations are min- 26 CHAPTER 1. INTRODUCTION imised. Specifically: a negative index slab lens is able to create an aplanatic image, one with minimised Spherical aberration and Coma (18). A perfect lens can magnify using curved boundaries rather than the planar slab and tailoring these curved boundaries (19) to tailor the subwavelength imaging effect. A lens made from a slab of negative refractive index MTM also makes a perfect lens (20). Other methods of achieving subwavelength imaging using metamaterials and transformation optics design techniques include using canalisation (21) and using transformation optics only (22). The first sub-wavelength imaging device (20) used negative index material as opposed to using transformation optics and is made from a combination of split ring resonator arrays and metallic wire arrays to generate the combined negative permittivity and negative permeability required for negative refractive index (23). The negative refractive index lens also known as a perfect lens. A superlens is a similar lens made from transformation optics and metamaterials but it is able to create a focal point far from the surface of the lens which is similar to how a typical glass lens focuses (24). MTMs allows the creation of a perfect imager, without the losses and narrow bandwidth of metamaterials, by controlling the flow of electromagnetic radiation at will. It would improve on existing microscopes and optics and allow the user to see small objects such as real DNA molecules or viruses with the naked eye through a microscope. Another limitation of the NIM superlens is that it operates in the near field. A transformation has allowed focusing in the far-field for a lens known as an optical hyper-lens (25). A perfect imager has been fabricated using only transformation optics and a drain placed at the source (26). Transformation optics can be used for designing superlenses (22) without the need for the metamaterial slab. This super lens which used transformation optics to focus in the far field was developed by the US navy (24) and would overcome the limitations of the NIM superlens which is intrinsically near field. Metamaterials and transformation optics can be used to create illusion devices which replace an image with an illusion (27). This process can be used to cloak a cup and replace it with an image of a spoon (28). This method of cloaking has been created using panels on the side of a tank to cloak in infrared replacing the signature of the tank with the signature of another object and is reported by BAE systems (29) This concept of illusion optics has also been applied to the projection of an image onto a physical cloak (27). 1.1. NEGATIVE INDEX MATERIALS 27 Plasmonics is the study of light interactions with metal which excite surface particles Figure 1.2: This is a SEM image 21-layer three dimensional optical metamaterial with a fishnet structure with a unit cell of 860 nm (30). The structure consists of alternating layers of 30 nm silver Ag and 50 nm magnesium fluoride (MgF2 ). called plasmons and these offer a way to create metamaterials at optical frequencies. Plasmonics can also offer a route to perfect imaging (31) and time reversal methods have been used to achieve perfect imaging in plasmonic structures (32). A Sievenpiper surface is a artificial surface plasmon (33). A plasma can be classified as a metamaterial and therefore a wire medium can be called an artificial plasma. A silver device was shown to be able to resolve sub-wavelength images (34) in the optical part of the spectrum and the device resolution is wideband. A metal, silver, has a negative permittivity below a certain plasma frequency as does gold section 1.1. For optical frequencies this occurs in silver at wavelengths around 800 nm. Plasmonics offer a route to using resonant metamaterial type structures for optical perfect imaging and energy harvesting devices. (35). Transformation optics techniques have been used to transform plasmonic structures to allow for light harvesting applications (36). One group of optical negative index metamaterials are called fish-net structures. This fish-net is made up of dielectric-metal layers with negative refractive index through layers achieved with low loss. The structures overcome the lossiness of bulk metamaterials and are 28 CHAPTER 1. INTRODUCTION valid at light frequencies for achieving negative refraction using silver. Frequency selective surfaces are artificially constructed surfaces to engineer interaction properties with electromagnetic radiation. This may seem very similar to metamaterials and it is. The key text book being by the authoritative Munk (37) who has also written a book critiquing metamaterials (38). This well established field of research is reviewed well in (39). Transformation optics and metamaterials can be applied to the study of wave propagation on surfaces. Here the wave is controlled on the surface by varying the impedance or resistivity rather than the permittivity and permittivity in bulk metamaterials. Variation in these surfaces takes place by changing the inclusions and shapes of the mushrooms. This is a standard metamaterial technique of engineering the sub-wavelength inclusions structurally by using different shapes in order to create macroscopic effects. Using structural variations rather than chemical ones typically found in nature. Colleagues in the group here at Queen Mary have published work on a surface wave cloak (40). Directivity or the ability to control the beam direction of an antenna is important for antenna engineers. A material with a refractive index of zero (ENZ) refracts incoming radiation into a plane wave perpendicular to the surface. An ENZ material could be used to direct a beam of radiation very effectively. Gradient index lenses are a hot topic in metamaterials and transformation optics. These are lenses with a refractive index which varies radially. They are very hard to fabricate with current techniques which involve diffusion with chemicals to create the gradient or by drilling holes or varying spaces or by using concentric shells. Metamaterials have been shown to improve the directivity of antennas using matching techniques (5). In optoelectronics there is the silicon photonic Luneburg lens which is a silicon based lens which focuses infra-red radiation for optical integrated circuitry. The lens is a theoretically perfect lens free from aberrations and in terms of the transformation optics theory, a simple device. The lens is a difficult lens to design due it its refractive index gradient however a lens has been fabricated as a silicon photonic device using an air spacing (41), greatly simplifying the device and allowing the possibility of use for optical communications. This is an alternative method for controlling the flow of light which can be achieved using photonic crystals where the propagation is controlled by periodic structures creating a band-gap channelling the light. This has implications for the burgeoning telecommunications industry investing in fibre optics for super-fast broadband because PC fibres (? ) allow greater control of the propagation reducing losses, achieving higher confinement and improving connectivity. 1.2. THE MOTIVATION OF RESEARCH AND ORIGINAL CONTRIBUTION 29 The commercial applications in THz communications for metamaterials and various applications are possible such as for MTM THz imaging systems for full body scanners and airports (42). Radiation at these frequencies is able to penetrate clothing and could be used in airport security. Raytheon have manufactured millions of metamaterial antennas for a range of mobile phones (? ). Metamaterials are widely used in antenna and microwave circuits, in order to increase the miniturization factor of the antenna, such as structuring the substrate (43). Metamaterial structures are actively loaded for miniaturization in mobile phones (44). The Chu limit is a fundamental limit for antennas which limits the bandwidth and antenna efficiency for small antennas significant for mobile phone antennas and this problem has been addressed for a canonical dipole surrounded by shells of metamaterial which approach or reportedly slightly improve on this limit (45). THz metamaterials have been reported to allow a 400 fold reduction of the radar cross section at 0.87 THz (46). 1.2 The Motivation of Research and Original Contribution The Cooke triplet was first designed and patented in 1893 by Dennis Taylor who sold the patent to TT&H Cooke and sons of York. The Cooke Triplet minimises all the primary Seidel aberrations: spherical, coma, astigmatism, distortion and field curvature as well as the chromatic aberration. The Cooke triplet uses a combination of two converging lenses of crown glass surrounding a diverging lens of flint glass. The TT&H Cooke lens catalog of 1897 states: Lack of sharp definition at the margins, and blackness and lack of detail in the shadows, are among the commonest defects of photographs. The introduction of lenses which, without the use of stops, yield definition uniformly fine throughout their plates, marks quite a new era in photography. referring to the distortion and vignetting which is reduced in the Cooke triplet. The Cooke triplet minimises all five Seidel aberrations from having more than one component for using a single lens Spherical aberration can be reduced using a combination of concave and convex lenses, off axis aberrations: coma and astigmatism can be reduced by tailoring the lens spacing and field curvature can be reduced by adjusting the lens spacing and use of a stop. Distortion can be controlled by use of a symmetrical system but all can not be minimised without adding more components and this is true generally for optical lens designers. The Cooke triplet is still used to capture images and reduce optical aberrations in 30 CHAPTER 1. INTRODUCTION a variety of applications today from mass-produced cameras to expensive ones used by the film industry. Aberration-free imaging is achieved theoretically using gradient index lenses such as the Luneburg Lens or Maxwell’s fisheye. The range of varying indexes in fish eyes is △n = 0.22 compared to △n = 0.03 in the human eye. Gradient index manufacturability has lagged behind theoretical developments, the largest gradient index which can be achieved using common ion exchange diffusion technique is △n = 0.05, meaning that while in theory gradient index optics (GRIN) might allow aberration free imaging the real GRIN lenses are often not as good in terms of aberrations and standard refractive optical systems such as the Cooke triplet. Aberrations in a lens image using NIM in optical design might allow mitigation of aberrations easier than with conventionally available materials (47). Gradient index with negative index materials have also been explored (48). A group used meta-surfaces to create a lens with no aberrations at telecom frequencies (49). These researchers at Havard have developed a distortion free flat lens using engineered metasurfaces (49) and have demonstrated the nano sized lens which eliminate distortion by tailoring the meta-elements on the surface. Another group used transformation optics to extend the flat negative index perfect lens and magnify the image into the far-field. (25). This hyper-lens has been demonstrated using graphene and dielectric layers (50). Transformation Optics can be used to improve existing optical systems such as a flattened Fisheye lens and a lens with reduced field curvature by flattening the focal surface (51),fig. 1.3. Many transformation optical devices possess large variations in permittivity. The invisibility cloak for example possesses infinity regions which can be smoothed away to provide working examples such as an Eaton lens or invisible sphere (52). The major novel contributions presented of this work are summarised as follows: - A ray tracing method is used to characterise an optical system for transformation into a transformation optics engineering optical system. This uses ray tracing of a Cooke triplet in ZEMAX and imports the rays to use as boundaries in a grid generation software package called Pointwise. - A triplet lens is engineered using a reduced transformation optics map which requires no metamaterials. The discrete coordinate transformation method is used to calculate the permittivity of individual grid cells in a triplet device. The metamaterial values 1.3. THE STRUCTURE OF THE THESIS 31 Figure 1.3: This figure shows in (a) the ray diagram for paraxial rays and meridonal rays passing through the lens (blue) and then corrected using the transformation media (yellow). The focal surface is curved when field curvature is present and results in the focus being different for different angles of incidence on the lens. zf shows the focal length for the paraxial ray and zp shows the focal length for the meridional ray and zc shows where the transformation media will be placed to flatten the surface. Panel (b) shows the inverse transformation grid (yellow) which possess the same electromagnetic properties as the original lens (white). are then removed around the edges and simulated. - A system for calculating the Zernike Aberrations in terms of a FDTD simulation. The source is placed at the centre of the focus and the distorted wavefront exiting the lens is sampled, interpolated using a polynomial and fitted to the Zernike polynomial set to give coefficients for the Zernike aberrations of the lens and these are compared to the results of a Cooke triplet in Zemax. - A Ray Tracing program for calculating a 2D distribution of refractive index is created. This is based on Bucahrel’s quasi-invariant method and applied to a 2D permittivity GRIN lens. This combines an axial gradient with a radial gradient. - A Flat Luneberg Lens is designed which transforms a plane wave to a point. This device is verified using an FDTD simulation. 1.3 The Structure of the Thesis This thesis introduces the novel concept of transformation optics (TO) in the design of gradient index lenses. The aim is to present evidence of the improvement achieved using a TO 32 CHAPTER 1. INTRODUCTION designed Lens. It reduces the aberrations, size, weight, cost and improving the manufacturability and performance compared with existing optical systems. Research in this field has shown that the a flat lens of negative refractive index of one will produce theoretically a resolution unlimited by the wavelength. For designers of optical systems the availability of refractive indices lower than one increases the design parameter space and allows aplanatic single lenses (18) and sub-wavelength resolution has been extended to a far-field lens using a transformation optics magnification (25) in theory. This thesis’s novel goal is to examine the aberrations in a single, flat lens designed using transformation optics which can be manufactured with no metamaterial values of permittivity and optimise the lens for aberrations to give an improvement in imaging for defence and civilian applications. In this thesis, we propose to apply transformation optics to design a single lens which preserves the optical property of the original Cooke triplet. Exotic material properties are usually generated from a transformation therefore metamaterials are required in the design of the proposed lens. The model is verified based on an in-house dispersive Finite-Difference Time-Domain software as having the same DNA as the original Cooke triplet by analysing the Seidel aberrations for the MTM lens compared with a conventional triplet. The lens is in the quasi-optical regime. Additionally, we compare the MTM lens with a simplified one, where MTMs are replaced with an all-dielectric reduced map. This reduced map will be valid for the TM mode. The lens designed will remain the same size as the Cooke triplet and will in fact be slightly larger due to extending the lines of light propagation before and behind to account for the curved surface. The lens will be flat and therefore allow increased compatibility with optical and optoelectronic system components. This thesis breaks down into the following chapters: - Chapter 2 explains the theory of transformation optics derived from the theory of invisibility. The key concept of carpet cloaking and how this allows more practical devices to be designed using dielectric materials. This chapter then examines some of the uses of transformation optics in antennas. electromagnetic and optical engineering applications. - Chapter 3 This chapter looks at the engineering of a low aberration triplet engineered using discrete transformation optics. It explains a virtual space and associated distorted grid was generated based on a Cooke triplet design, the theory behind the coordinate transformation based on this grid in order to compute a permittivity map BIBLIOGRAPHY 33 and finally the layout of the FDTD simulation which computed the electromagnetic response of the lens. The novel concept of ray tracing to specify the virtual space and the verification in terms of focal point. - Chapter 4 the FDTD simulation of the MTM triplet is presented in terms of the aberrations of the lens. The method for calculating each aberration is described and a comparison between the MTM lens, a simplified lens and the conventional Cooke triplet based on FDTD simulations. The MTF a key optical parameter presents the performance of the lenses. The RMS wavefront error is also calulcated showing good results for the all-dieletric triplet design. The novel concept of calculating aberrations for lenses in FDTD is also described based on sampling a wavefront aberration function. - Chapter 5 describes how the MTM lens, reduced lens and Cooke triplet and single lens results are verified using commercial electromagnetic simulation software on CST studio on a large high performance computer cluster. - Chapter 6 contains the novel flat Luneberg lens design. The virtual space, the FDTD simulation verification. It contains a test of the drain in perfect imaging of the Luneberg lens. It contains the comparison of the field transformation lens with a similar flat lens using transformation optics. Bibliography [1] V. G. Veselago, “”The Electrodynamics of substances with simultaneously negative values of epsilon and mu”,” Soviet Physics Uspekhi, vol. 10, no. 4, p. 509, 1968. [Online]. Available: http://stacks.iop.org/0038-5670/10/i=4/a=R04 [2] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. 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This discussion will therefore involve using mathematical concepts such as multiple dimensions, Non-euclidean geometry, covariance and unusual astrophysical phenomenon such as worm holes. The understanding of these mathematics is vital for the correct implementation of the equivalent transformation optics device. This section examines a classic example of the invisibility cloak and the latest development in the field of transformation optics including quasi-conformal mapping method which allows extremely simple implementation of the devices. 2.1.1 Conformal Mapping Transformation optics has been derived using a mathematical framework based on complex numbers known as conformal mapping which was originally used in one of the designs of the invisibility cloak (1). It is based on conformal mapping where different coordinate systems, w and z typically can be transformed maintaining their orthogonality between angles of the grid cells. It is possible to design a single lens which transforms a plane wave to a point source 39 40 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS using the conformal mapping Mobius transformation. z=i i+w i−w (2.1) for example and from the theory of optical conformal mapping (2) we find the refractive index is nz = nw dw dz (2.2) therefore the refractive index in this case is n(x, y) = 2n0 x2 + (1 + y)2 (2.3) This concept has been extended to include non-Euclidean geometries and multiple dimensions which allow one to minimise the singularities in the original cloak (3). This technique was later generalised to mapping to Riemann sheets (4). Schwarzchild-Christol mapping allows conformal mapping to regular polygons which may allow some interesting transformation optics antennas to be developed. I wrote a program to calculate the conformal mapping inside some interesting polygons including spheres and steps. 2.1.2 The Invariance of Maxwell’s Equations Maxwell’s equations are form invariant. Covariance states that a physical law should be the same independent of the frame of reference or coordinate system which the observer happens to be observing from. This is commonly stated as the special theory of relativity. The same is true for Maxwells equations, which should remain the same independent of the frame of reference to be truly universal. This principle of covariance in Maxwell’s equations means that the electromagnetic field must be independent of the co-ordinate system. Typical examples are Cartesian (x,y,z), cylindrical (z,r,θ) and spherical (r, θ, φ) co-ordinate systems. The field transformation means that the results to Maxwell’s equations must be the same for all possible co-ordinate systems used in the mathematical machinery of the calculation. The metric tensor is a measure of distance which is coordinate invariant and therefore 2.1. INVISIBILITY CLOAKS 41 very useful in the mathematics of general relativity which possess curved space-time and also in coordinate transformation theory. g ij is a tensor which is a matrix of differentials between the basis coordinate vectors (x, y and z). G is a coordinate system independent length scale. The permittivity and permeability are also covariant matrices which denotes an inhomogeneous and anisotropic material. Covariance and contravariance describes how a geometry responds to a change in basis vectors. A covariant properties such as the the length scale results in a transformation proportional to the length scale. A contravariant variable such as gradient results in an inverse transformation under a change in basis vector. The materials interpretation states that this process can be used as a design approach to yield a permittivity vector which gives the same propagation characteristics for electromagnetism of a given coordinate system in terms of permittivity and permeability (5). These equations are independent of the coordinate system. Maxwell’s equations are there independent of the coordinate system and covariant. Which are the most useful set of equations for any electromagnetic engineer as they provide much of the explanation of electromagnetism in them. Instead of interpreting this coordinate transformation as a change in topology we can interpret it as a change in material. 2.1.3 A Design of the Cloaking Device In order to create a cloaking device it is necessary to divert the field lines away from a region. This is achieved by stretching the coordinate system from a point in space to a sphere. The distorted coordinate system (u,v,w) is recorded in terms of the original coordinates system (x,y,z). u(x, y, x), v(x, y, z), w(x, y, z). (2.4) The change in coordinate system results is a change in permittivity or permeability of the medium to maintain covariance of Maxwell’s equations. µǫ′u = ǫu Qu Qv Qw ′ Qu Qv Qw ,µ = µ , 2 Qu Qu 2 (2.5) where Q is the length of the differential of the basis vector. We choose to cloak a spherical region, r < R1 and the cloak be an annulus R1 < r < R2, fig. 2.1. 42 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Figure 2.1: This figure shows the spherically symmetric cloak designed by Sir J. Pendry. The inner sphere is the cloaked region radius, R1, and the outer sphere the radius of the cloak, R2. The lines show the ray propagation in the invisibility cloak being diverted around the cloaked region like water around a stone. The coordinate transformation to compress the region r < R2 into the annulus is in spherical coordinates is r ′ = R1 + r(R2 − R1)/R2 (2.6) φ′ = φ (2.7) θ′ = θ. (2.8) Applying the transformation, eq. (2.4), for the annulus, R1 < r < R2, the permittivity and permeability are ǫ′r = µ′r = R2 (r ′ − R1) 2 R2 − R1 r2 ǫ′θ = µ′θ = ǫ′φ = µ′φ = R2 R2 − R1 R2 . R2 − R1 (2.9) (2.10) (2.11) 2.1. INVISIBILITY CLOAKS 43 The region r < R1 is now cloaked and no electromagnetic field can enter or exit the region. It is worth noting that the outer layer of the cloak is reflection-less and have the conditions which allow it to be compared to a perfectly matched layer ǫ′θ = ǫ′φ = 1 ′ 1 µθ = µ′φ = ′ ǫr µr (2.12) The electric permittivity and magnetic permeability are in terms of the geometry denoted by the metric tensor. Electromagnetic interaction with a medium is analogous to interaction with a geometry. This is the fundamental concept to transformation optics known as the materials interpretation (6). √ ǫij = µij = ± gg ij . (2.13) , or in another more common, tensor form, (5) in terms of the Jacobian Transformation matrix ǫ= ΛǫΛT det(Λ) (2.14) which is in an equivalent form for permeability, µ denoted here by the permittivity and permeability tensors. In a coordinate transformation this would be followed by a change in basis vectors and variables to account for the change in coordinate system and maintain the same electromagnetic behaviour as before maintaining the invariance of Maxwell’s equations (7). This coordinate transformation does also give us a permeability tensor and permittivity tensor describing the material in which say a Cartesian plane wave flowed like a cylindrical wave, or any other geometry imaginable for that matter. The transformation equation can be further simplified as long as we are always transforming to a flat Cartesian space. The all-dielectric approach was developed by (8) as a response to the cloak design fabricated by (9) which suffered the possession of a limiting factor such as highly anisotropic materials, in the form of highly anisotropic permittivity distributions with prohibitive ranges reaching infinity at the inner cloak boundary. If we choose our transformation so that it is almost orthogonal the design can be simplified (8). 44 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Figure 2.2: This figure shows an image of a nano fabricated carpet cloak. It has been placed in a silicon nitride waveguide on a nano porous silicon oxide substrate with very low refractive index for testing. The spatial index variation is realised by etching holes of various sizes in the nitride layer at deep wavelength scale creating a local effective medium index (10) cloaking the bump region. 2.1.4 The Design of a Carpet Cloak For a 2D E-polarised light where the material is invariant in the z direction and only ǫzz , µxx , µxy , µyy , µyx contribute and we use the 2D Jacobian matrix Λ, ǫ`z ≡ ǫ`zz = p ǫ0 det(g) µ0 ΛΛT . µ̀ = p det(g) , (2.15) (2.16) If the grid is quasi-orthogonal then this transformed medium can now be fabricated using dielectric ”lego” blocks. This description in terms of refractive index and anisotropy of the transformation makes sense if transforming a square cell δ by δ to a distorted parallelogram p ~x · δ by ~y · δ. A larger compression leads to a smaller area det(g)δ 2 and a larger refractive index. An expansion leads to a refractive index less than the original background medium 2.1. INVISIBILITY CLOAKS 45 and hence a ’metamaterial’ requirement. n2 ≈ p ǫz = det( (g))−1. ǫ0 (2.17) The permittivity and permeability can be normalised in terms of the refractive index to remove the problem of variation of permeability. The permeability equals unity and the refractive index is a function of permittivity. This eliminates the complicated permeability function, eq. (2.16) which requires materials with a permeability less than one, which does not occur in nature. If we treat the effective refractive index as a function of the permeability in the transverse and longitudinal direction nt = √ µl ǫ, (2.18) the permittivity and permeability of the distorted space can be totally described by the refractive index. nl = √ µt ǫ (2.19) From this definition if the relative permeability of the material is equal to free space µl µt = 1 (2.20) then the effective refractive index of the material is √ nl nt 1 = , n = √ ǫ0 det(g) 2 (2.21) where the anisotropy (α) is approximately one. This device had had material parameters, that all varied as a function of radial distance. The actual implementation of the cloak only the permittivity in the z direction varied while the other two parameters were held constant. This was possible so long as the effective medium condition was satisfied and the material parameters changed slowly as a function of wavelength. It was achieved by transforming the parameters into a single function of refractive index and then calculated back into a variation in only the ǫzz variables. An all dielectric cloak has been designed using only dielectric and an optimisation method 46 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS creating an eye type cloak, see fig. 2.3 as as using multilayer cladding (11). One group has engineered a cloak which works in a gradually changing medium (12) including arbitrary boundaries (13). This rather basic cloak might be further moved towards fabrication by using a layered approach (14). The cloak could be replaced by a waveguiding structure diverting the rays around an object using a grooved metallic cylinder(15). Figure 2.3: This figure shows a cloaking device in the shape of an eye designed using an optimisation algorithm. This shows what is possible without using the transformation optics method to design the cloak and inside use optimisation, the initial requisite for a cloaked field and a genetic algorithm with single permittivity grid cells to create a cloaked effect. The panel (a) shows the uncloaked cyclinder for the Ez field and (b) for the field amplitude and (c) and (d) for the electric field distribution and amplitude when the eye cloak is used. 2.1.5 Thermal Cloaking This technology is currently being heavily investigated and funded by defence industries along with thermal cloaking perhaps with mind to submarines which interestingly use magnetic cloaking to hide from detection. The Transformation for acoustic cloaking is derived by (16) as shown in the following substitution [p, vr , vφ , ρr , ρφ , λ−1 ] ↔ [−Ez , Hφ , −Hr , µφ , µr , ǫz ] (2.22) The conductivity equations in d.c. can also be mapped to the acoustic equations and has allowed the acoustic cloak to be generalised to a third dimension (17). It is theoretically 2.2. TRANSFORMATION OPTICS IMAGING DEVICES 47 possible to build a cloak for other waves as well as electromagnetic and this has been applied to acoustics. A cloak which guides an earthquake around the future city has been imagined and predicted (18). The cloak varies with the parameters density instead of permittivity as the wave in question, sound, is affected by density. This seismic cloak is able to divert earthquake around a region in space. Metafluids offer an implementation of the metamaterial concept to fluids. Fluids which are tailored to possess properties not typically found in nature. Wave fundamentally work the same and Maxwell’s equations and the invariance in them used to derive the transformation optics equations can equally be applied to the equation which govern sound wave propagation in matter (19). 2.2 Transformation Optics Imaging Devices Electromagnetic modelling is available in all sorts of commerical packages for metamaterials research. CST and HFSS are common packages which have been used to model split ring resonator arrays to give negative permeability. COMSOL is a commercial solver which can solve PDE’s and therefore has been used to model the invisibility cloak and plasmonic antennas. Outside of this, custom modelling has been done using the finite difference time domain method extensively as well as with finite element methods. Photonic crystals can be simulated using FDTD or with plane wave expansion yielding scattering parameters. Much work on FDTD has been carried out by yang (20). There are several ways commonly used to model Electromagnetic BandGap Materials (EBG) using Bloch waves and method of moments (MoM) and the Finite Difference Time Domain (FDTD) method. FDTD has been demonstrated to be the best application for modelling metamaterials. 2.2.1 Single Flat Lens Design The Cooke triplet design was compressed and the transformation optics method verified, Figure 2.6. Compression is a possible transformation which may have huge improving effects for antenna applications by reducing size, weight and power. The permittivity map was 48 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS transformed in the custom MATLAB code to a real space 1/16 of the size of the original map. This figure shows that the relativity permittivity have increased across the board in the horizontal direction due to the horizontal compression. The condition of matching at the corner means that this does not result in increases reflections at the edges but does feature a wider range of permittivity. There is no theoretical reason I can see that would limit the amount of compression possible beyond the wavelength of light which determines how much matter can see the wave and how much the wave can see the different variations in gradient index. Figure 2.4: This figure shows in the top section the outline of the concave lens placed in the grid generation suite POINTWISE. A mesh is drawn inside the lens and then smoothed using control functions to make the grid as orthogonal as possible. The dimensions of the lens are 160 cm by 10 cm. The same as those used in a previous flat lens design (21). In the bottom section the permittivity map is shown. This is generated using the method previously outlined and known as discrete coordinate transformation. This lens share the same permittivity distribution as the paper (21). The flat lens, which possesses advantageous qualities to the parabolic reflector for mm wave communications (22) is used as an example of a device which might be optimised by transformation optics (21; 23) and designed hence-forth. The ability to create conformal flat surfaces allows simplification of optical systems while maintaining the original optical 2.2. TRANSFORMATION OPTICS IMAGING DEVICES 49 performance. This involves using a discrete coordinate transformation for each individual cell which is defined by the matrix g = [g11 , g21 ; g12 , g22 ] (2.23) and the anisotropy is the angle between the two sides of the cell θ = cos−1 g21 g12 g11 g22 (2.24) A simplification is allowed for a 2D design where the Jacobian matrix becomes J = [dx′ /dx, dy/dx, 0; dx/dy, dy/dy, 0; 0, 0, 1] and the refractive index is given by n= 1 det(J) (2.25) (2.26) A compressed single lens was modelled and simulated in FDTD then compared with the original refractive lens in order to validate the theory. This is a common example in which to validate the modelling suite as it provides a simple and well studied standard. The dimensions used for the grid were matched to the parameters in the paper (21). The inverse transformation was then used going in reverse. Typically one would transform from the virtual space containing the desired ray propagation to the flat real space in which the geometry desired. This would require the ray characteristics. Instead a reverse transformation from the virtual flat space to the real curved space was carried out. In this model, fig. 2.4, a variation on this analytical approach is used called the discrete coordinate transformation (21). The difference being that in the discrete method the material is split up into discrete unit cells. The equation given here is then applied to each individual cell. In an analytical approach the g tensor is typically analytical and describes a mathematically defined geometry. In the discrete coordinate transformation it allows a more general calculation of a geometry and resultant material parameters and requires only a mathematically calculated grid as a source input. 50 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Figure 2.5: These figures shows FDTD simulations of, on the left the original single lens on which the grid was designed. On the right is the lens fig. 2.4. The agreement in focal length is precise. The focus of the original lens on the left is 11.69 cm and the focus for the lens on the right is 10.19 cm. The simulation in FDTD is shrunk an order of magnitude to for it to be computationally viable. This means that the dimensions of the cloak are divided by ten. So is the wavelength so 8 GHz becomes 0.8 GHz. This simulation is however still valid and can be interpreted better as a function of wavelength. The focus of the original lens is 11.69 cm and the focus of the compressed lens is 9.405 cm, fig. 2.5. The back surface of the lens is at 0 cm and the original at 1.5 cm therefore the true focal length of the original lens is 10.19 cm. 2.2. TRANSFORMATION OPTICS IMAGING DEVICES 51 Figure 2.6: This figure shows the comparison between the compressed lens designed using the discrete transformation optics approach and the original concave lens. Planar antennaes use transformation optics to make curved surfaces such as the luneburg lens flat (23; 21; 24) and curved reflectors such as parabolic antennas flat (21) which have been used to reduce the size of satellite communications systems. A flattened achromatic doublet may be formed using transformation optics for, fig. 2.7. The term can also be applied to flat lenses which remove the distortion present in curved refractive lenses to create an aberration free flat camera lens. One might be a long way off from replacing conventional camera lenses however one has been demonstrated at Havard where Capasso has pioneered using metamaterials in optical reflect and transmit arrays (25). 2.2.2 Extraordinary Transmission Devices Transformation Optics allows the compression of an EM wave to allow it to propagate through a hole smaller than its wavelength without diffracting. An example of transformation media is the extraordinary transmission device which allows electromagnetic waves 52 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Figure 2.7: This figure shows the curvature, thickness, d1 and d2 , n1 and n2 of a cemented doublet lens highly common in optical engineering systems. Panel (b) shows the virtual space and physical space in grid form for the doublet lens and the resulting refractive index profile generated using transformation optics where the refractive index is outside the metamaterial range. The panel (c) shows the electromagnetic simulation results for the original lens and the TO doublet lens where the focal regions match extremely successfully and the plot of the electric field is the image plane for both lenses showing extremely close agreement between the two devices (26). This shows a typical design of a transformation optics device and what is typically achievable using the method for a double optical system. to propagate through a region smaller than its wavelength without suffering diffraction. This device has been verified experimentally. It has been fabricated without metamaterials. It is therefore broadband. This device design, fig. 2.9. (27) have proposed an idea for a subwavelength cavity made from metamaterials due to compression. Another use with acoustics is to make an anti-extraordinary acoustic device. An extraordinary transmission is when a beam is compressed and allowed to pass through a hole smaller than the size of its wavelength without diffracting. A neat trick allowed by transformation optics. An anti-device has been 2.2. TRANSFORMATION OPTICS IMAGING DEVICES 53 Figure 2.8: This figure shows the virtual grid for the propagation characteristics of the Extraordinary Transmission (ET) device where an area is shrunk from 1.6 λ to 0.2λ to reduce the aperture to smaller than the wavelength where the field of the left in (a) is without any device, in (b) the field is with an ET setup on one side and (c) where one is on either side of the gap. The right hand side shows the design of the ET device. The rays are shown in the grid in figure (a) being squeezed to fit the small hole and the resultant virtual space surrounding. Panels (b) and (c) the permittivity map is shown at a high resolution and then at a lower resolution which possesses the same effective qualities. demonstrated which device which behaves as a soundproof window which is transparent to the airflow (28). 2.2.3 Other Optical TO Devices Novel optical devices which control the propagation of electromagnetic waves have been suggested based on the TO method using metamaterials. The manipulation of electromagnetic wave propagation is a key feature in many lenses and optical systems. For example the single lens focuses a plane wave to a single point. Many classic optical devices such as beamsplitters and lenses (29) have been redesigned using the methods of transformation optics. New types of absorber have been introduced such as the optical black hole. These transformation designed devices are simple optical instruments and can therefore be understood in terms of 54 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Figure 2.9: This figure shows the refractive index distribution from the Photonic Black Hole (32) with a negative index of distribution satisfied with metamaterials.The Continuous Index Photon Trap with a high permittivity which can be achieved with ferrites. quantum optics (30). Integrated optical systems into a single metamaterial device such as in optical modulators for communications modulators integrates three lenses for low aberrations into one metamaterial lens. A high gain lens can be designed using transformation optics (31). The technology has allowed a transformation optics engineered flat doublet lens fig. 2.7 Metamaterials share an analogy with general relativity which can be used to generate table top copies of astrophysical phenomenon. A continuous index photon trap, ?? is a model of a planet orbiting a star and transformation optics would allow a hardware study of this phenomenon. This has been used to generate black holes at optics (32). FDTD simulation of these black hole structures have been carried out by this group (33). Contiguous Index Photonic Index traps allow light confinement for optical computing and telecommunications, for example, providing an on-demand time delay or optical memories (34). 2.3 Summary Transformation Optics has spawned many interesting and practical devices. The original designs for the invisibility cloaks were fascinating and deliverable. The innovation of carpet cloaking provided a way to realise these devices more effectively using more achievable values for permittivity and permeability. The materials interpretation of invariance in Maxwell’s equation provided an interesting insight which had not previously been examined, or only considered but not fully explored (35). BIBLIOGRAPHY 55 Useful devices which this change in thinking could bring about where able to surpass fundamental limited such as using negative index metamaterials to surpass the resolution limit in imaging or surpassing the diffraction limit for transmission through small holes in the extraordinary transmission device. The transformation optics method could be applied to many fields of science such as heat and thermal and pressure and sound for cloaking in heat, quantum and seismic wave scenarios. Bibliography [1] Ulf Leonhardt, “”Optical Conformal Mapping”,” Science, vol. 312, p. 1777, 2006. [2] U. Leonhardt, “Optical conformal mapping,” Science, vol. 312, no. 5781, pp. 1777–1780, 2006. [3] U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidean cloaking,” Science, vol. 323, no. 5910, pp. 110–112, 2009. [4] L. Xu and H. Chen, “Transformation optics with artificial riemann sheets,” New Journal of Physics, vol. 15, no. 11, p. 113013, 2013. [Online]. Available: http://stacks.iop.org/1367-2630/15/i=11/a=113013 [5] D. Schurig, J. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” arXiv preprint physics/0607205, 2006. [6] A. J. Ward, “Refraction and geometry in Maxwell’s equations,” Journal of Modern Optics, vol. 43, pp. 773–793, Apr. 1996. [7] J. B. P. D. S. Schurig and D. R. Smith, “”calculation of material properties in transformation media”,” Optics Express, vol. 14, 2006. [8] J. Li and J. B. Pendry, “”Hiding under the carpet: a new strategy for cloaking”,” ArXiv, 2008. [9] D. Smith, D. Vier, T. Koschny, and C. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Physical Review E, vol. 71, no. 3, p. 036617, 2005. [10] J. V. Majid Gharghi and X. Zhang, “A carpet cloak for visible light,” Nano lett, 2011. 56 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS [11] C. A. Valagiannopoulos and P. Alitalo, “Electromagnetic cloaking of cylindrical objects by multilayer or uniform dielectric claddings,” Physical Review B, vol. 85, no. 11, p. 115402, 2012. [12] J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B.-I. Wu, “Cloak for multilayered and gradually changing media,” Physical Review B, vol. 77, no. 3, p. 035116, 2008. [13] C. Li, K. Yao, and F. Li, “Invisibility cloaks with arbitrary geometries for layered and gradually changing backgrounds,” Journal of Physics D: Applied Physics, vol. 42, no. 18, p. 185504, 2009. [14] C.-W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Physical Review E, vol. 79, no. 4, p. 047602, 2009. [15] S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Physical review letters, vol. 103, no. 10, p. 103905, 2009. [16] S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New Journal of Physics, vol. 9, no. 3, p. 45, 2007. [17] H. Chen and C. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Applied physics letters, vol. 91, no. 18, pp. 183 518–183 518, 2007. [18] P. Sheng, “A step towards a seismic cloak,” Physics Online Journal, vol. 7, p. 34, 2014. [19] G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New Journal of Physics, vol. 8, no. 10, p. 248, 2006. [20] ”FDTD Modeling of Metamaterials: Theory and Applications”. Artech house, 2009. [21] Wenxuan Tang, Christos Argyropoulos, Efthymios Kallos, W. Song, and Y. Hao, “”Discrete Coordinate Transformation for Designing All-dielectric Flat Antennas”,” IEEE Transactions on Antenna and Propagation, 2010. [22] R. Mittra and Y. Zhang, “A low-reflection flat-lens design for microwave imaging system,” in Antennas and Propagation (EUCAP), Proceedings of the 5th European Conference on. IEEE, 2011, pp. 851–852. BIBLIOGRAPHY 57 [23] F. Kong, B.-I. Wu, J. au Kong, J. Huangfu, S. Xi, and H. Chen, “”Planar focusing antenna design by using coordinate transformation technology”,” Applied physics letters, 2007. [24] N. K. D. A. Roberts and D. R. Smith, “”Optical lens compression via transformation optic”,” Optics express, 2009. [25] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature materials, vol. 13, no. 2, pp. 139–150, 2014. [26] Q. Wu, J. P. Turpin, X. Wang, D. H. Werner, A. Pogrebnyakov, A. Swisher, and T. S. Mayer, “Flat transformation optics graded-index (to-grin) lenses,” in Antennas and Propagation (EUCAP), 2012 6th European Conference on. IEEE, 2012, pp. 1701–1705. [27] N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” Antennas and Wireless Propagation Letters, IEEE, vol. 1, no. 1, pp. 10–13, 2002. [28] S.-H. L. Sang-Hoon Kim, “Air transparent soundproof window,” pre-print, 2013. [29] J. B. P. D. Schurig and D. R. Smith, “”Transformation-designed optical elements”,” Optics Express, 2007. [30] U. Leonhardt, “”Quantum physics of simple optical instruments”,” Reports on progress in physics, 2003. [31] I. Aghanejad, H. Abiri, and A. Yahaghi, “Design of high-gain lens antenna by gradientindex metamaterials using transformation optics,” 2012. [32] X. Z. D. A. Genov, S. Zhang, “”Mimicking Celestial Mechanics in Metamaterials,” Nature, vol. 5, p. 687, 2009. [33] Christos Argyropoulos, Efthymios Kallos, and Yang Hao, “”FDTD analysis of the optical black hole”,” J. Opt. Soc. Am. B, 2010. [34] D. P. San-Román-Alerigi, A. B. Slimane, T. K. Ng, M. Alsunaidi, and B. S. Ooi, “A possible approach on optical analogues of gravitational attractors,” Optics express, vol. 21, no. 7, pp. 8298–8310, 2013. [35] E. J. Post, Formal Structure of Electromangetics: General Covariance and Electromangetics. Holland Publishing Company, 1962. 58 CHAPTER 2. TRANSFORMATION OPTICS AND THEIR APPLICATIONS Chapter 3 The Design of a MTM Cooke Triplet The Cooke triplet is the most simple system of lenses which is able to minimise all five Seidel aberrations. It remains in use today in cameras. Ray tracing was used for the implementation of the MTM triplet. This is the first time this has been reported in the literature. For the design of the Cooke triplet we first need to specify the electromagnetic propagation characteristics for the device. The motivation for this is to use transformation optics to design the lens as outlined in the motivation section of the thesis. Ray tracing allows this to be done in the optical part of the spectrum. Light is within the geometrical limit and can be approximated as a plane wave moving in a straight line. This method should be able to be generalised to implement any optical systems for example sight finders on a rifle or periscopes as well as the multitude of other applications found for optical systems outside the military. The design of the triplet lens was implemented originally with the help from Wenxuan Tang. The project was inherited from her were she did the design of the triplet which I optimised and re-implemented at the same resolution and then a multitude of higher resolutions. The FDTD code to calculate the field pattern was written by her as was the TO code. 3.1 Ray Tracing of the Optical System The Cooke triplet was first designed and patented in 1893 by Dennis Taylor who sold the patent to TT&H Cooke and sons of York which still makes the lenses under the name Cooke Optics in Thurmaston near Leicester. The Cooke Triplet eliminates five third-order 59 60 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET Seidel aberrations (spherical, coma, astigmatism, distortion and field curvature) and chromatic aberration (1) by using an elegant combination of two converging lenses surrounding a diverging lens. Spherical aberration can be reduced using a combination of concave and convex lenses, off axis aberrations: coma and astigmatism can be reduced by tailoring the lens spacing and field curvature can be reduced by adjusting the lens spacing and use of a stop and distortion can be controlled by use of a symmetrical system. The TT&H Cooke lens catalogue of 1897 states: “Lack of sharp definition at the margins, and blackness and lack of detail in the shadows, are among the commonest defects of photographs. The introduction of lenses which, without the use of stops, yield definition uniformly fine throughout their plates, marks quite a new era in photography.” The Cooke triplet is still used to capture images and reduce optical aberrations in a variety of applications today cheap from mass-produced cameras to expensive ones used by the film industry. The Cooke triplet can be designed by with the following tailing parameters to find an optical combination of a parameters for three lenses K = K1 + P = h2K2 h3K3 + h2 h1 K1 K2 K2 + + n1 n2 n3 h22 K1 h22 K2 h32 K3 + + V1 V2 V3 h1h1K1 h2h2K h3h3K3 C2 = + + v1 V2 V1 C1 = (3.1) (3.2) (3.3) (3.4) Where K is the power, P is the Petzval sum and C1 and C2 are the the longitudinal and transverse color. The Cooke triplet is optimised using these equations containing the independant variables: distances between the lenses d1 and d2, six curvatures of the lenses which can be considered as three powers K1, K2 and K3 and three shape factors. A glass type is chosen before. There is no unique solution and typically an estimate is made and then optimised for a given merit function. It is an achromatic lens in that it minimises chromatic aberration by balancing the powers of two converging lenses with a central diverging lens. A Cooke Triplet comprises a negative flint glass element in the centre with a crown element on each side. 3.1. RAY TRACING OF THE OPTICAL SYSTEM 61 Figure 3.1: The Ray tracing of the Cooke triplet is optical software package OSLO. The key design parameters are given in the table below. The Cooke triplet does not have a unique solution therefore this is just one of many possible combinations. The key optics data is taken from the triplet lens library in OSLO. The focal length of this device is 50 mm, NA is 0.125 and the scale is given. This is the specification of the lens designed using the transformation optics method. Glass type SK16 and F2 are crown and flint types. Ray tracing is an elegant and analytical method which describes the ray and the interaction with materials via Snell’s law of refraction n1 sinθ = n2 sinθ (3.5) This equation is applied at each surface for a multiple surface system such as a combination of lenses. Snell’s law describes the angle deviated by light when it enters a medium. This is due to light having a slower effective speed in a medium with a higher refractive index. Ray tracing has long been used to design optical systems and is an effective way of determining primary aberrations and imaging transformations such as inversion and magnification. Ray tracing of the Cooke Triplet is carried out by optical system design software such as ZEMAX 62 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET and OSLO. For a Cooke Triplet existing in the lens library in OSLO the paraxial rays were traced as they traversed the Cooke Triplet, see Fig. 3.1. The rays were then exported to MATLAB with the rays shortened so they finished somewhere between the final surface and the focal point so the device focused outside the transformation media. The rays used in the grid generation and device specification included the rays which exited the Cooke triplet. 3.1.1 Verification with ZEMAX Model In order to verify the results for the FDTD simulation of the lenses I will use ZEMAX to compare the results with. ZEMAX is a popular optical design software package which provides a large range of results for optical systems. There is a Cooke triplet in the ZEMAX library. It is of comparable size to the one in the FDTD simulations. The wave-front error for this Cooke triplet is shown in the following figure with the wave-front aberration taken from the results of the FDTD simulation. The RMS error in terms of wave are in good agreement. The Cooke triplet is at 0.18 waves and the Cooke triplet in the custom simulation is 0.15 waves. In addition the Cooke triplet is ZEMAX has a spherical aberration of 0.000125 which is of the order of -0.0059 waves in the model. The lenses in the model were corrected for defocus and it is possible to do this to the conventional triplet in ZEMAX resulting in a larger thickness in the final lens. It is also possible to compare the values for our wave-front aberrations calculated in the ray tracing simulations with those in ZEMAX. Singlet Cooke Triplet Eyepiece GRIN SA 4.3583 1.3912 0.6942 Coma 14.527 -1.309 - Astig 10.9279 -3.3721 - Fcur 15.2966 6.57330 - Distort 156.633 109.045 - CLA -3.14942 0.258051 - CTR -15.7121 -2.861521 - Table 3.1: This table shows the values of the Seidel aberration coefficients in waves for a single lens and for a Cooke triplet, on which the transformation was based compared with the Seidel aberrations in the MTM Cooke triplet. 3.2. STRUCTURING THE GRID WITH RAYS AS BOUNDARIES 3.1.2 63 TO Lens in ZEMAX It is possible to import gradient index materials to model them. ZEMAX contains a GRIN lens in sequential and non-sequential modes. The gradient index dispersion is modelled using the Sellmeier equation n2 − 1 = K1 λ 2 K2 λ 2 K3 λ 2 + + λ2 − L1 λ2 − L2 λ2 − L3 (3.6) where the L’s represent the wavelength where the resonances exist. It is possible to model metamaterials in ZEMAX because metamaterials are a resonant phenomenon with anomalous dispersion yielding negative values of refractive index near the resonance. The metamaterial lens has an x and a y distribution which can be fitted analytically. It can be then imported into ZEMAX gradient surfaces. The radial numbers are n0 = 0.83 and n2 is 0.025 where 0.83 comes from the metamaterial outer edges. The reduced map is n0 = 1 and n2 is 0.025. The gradient index can further be optimised in ZEMAX in order to get the smallest spherical aberration by changing the gradient index. 3.2 Structuring the Grid with Rays as Boundaries A non-euclidean coordinate system in which the light will propagate is generated using grid generation software as an approximation to curved space-time. The purpose here is to engineer a geometry in which the electromagnetic propagation will behave as defined by the ray tracing calculation. A 2D, structured, quadrilateral grid is generated, fig. 3.2. Mesh generation is achieved using iterations of an elliptic Partial Differential Equation (PDE) of the Laplace equation △φ = 0 (3.7) The grid, fig. 3.2 is made orthogonal using a smoothing function to make a better match the boundary points and to improve cell orthogonality. The modified Liao functional (2) is used with the slipping condition at the boundaries which allows a quasi orthogonal grid to be generated with the condition that the boundary grid points are allowed to slip. Grid generation was undertaken using the program GridGen95. In this program grid generation is split into three stages. The first stage involves defining the geometry. For the Cooke triplet eight equi-distant rays from the top hemisphere where used as well as a rectangular box which enclosed the rays to define nine different grid boundaries. In the second stage the boundary is defined. This is done by selecting each line in the geometry and setting it 64 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET as having a north, south, east or west boundary for a specified grid number. Therefore for the Cooke triplet design the left side of the rectangular box was defined at west for all grids and the right hand side was defined as east for all grids. The rays which defined the Cooke triplet along the centre were were defined as north boundary for the grid below and south boundary for the grid above. In the third part part the block was defined. In this section the grid point spacing is defined, here as 14 cm. The method of grid generation selected is algebraic. The points on the boundary are declared. Then in a Cartesian coordinate system the grid lines are interpolated inside the grid using a solution to a set of algebraic equations. The grid is output to a MATLAB code for transformation and calculation of permittivity map. Figure 3.2: Panel(a) shows the ray tracing diagram imported into numerical grid generation software POINTWISE for the triplet. The lateral lines are the rays and the structure of the lens is visible. In panel (b) only the top half is modelled to space space and the the design will be copied later in Matlab using symmetry consideration for the triplet. The grid size between the grid boundary nodes is 0.14. The grid is solved by Elliptic grid generation. This involves the commercial program solving a set of partial differential equations. Laplace smoothing functions create quasi-orthogonal grids by optimising the distribution of cells to the optimum. 3.3. DISCRETE TRANSFORMATION OPTICS CALCULATION 3.3 65 Discrete Transformation Optics Calculation The permittivity map and anisotropy map, fig. 3.3, were generated by the transformation optics design algorithm. A simplified map is also used where all metamaterials are removed and replaced with low index material. The relative permittivity of the transformation optics lens is between 1.5 and 0.8. Figure 3.3: This figure shows the original triplet in fig (a) with the key optical parameters and the table of key optical parameters beneath. Panel (b) shows the virtual space where the grid has been meshed around the rays which make up the boundaries from the triplet ray tracing. This grid was then transformed using transformation optics to panel (c) which gives the values of permittivity for a flat implementation in Matlab of the triplet. Panel (d) shows the permittivity map in the FDTD simulation domain before a signal is launched. Notice its symmetrical design which has been added at this stage to fully implement the triplet in panel (a). The principle of covariance central to Einstein’s theory of relativity states that these physical laws are independent of there coordinate system u(x, y, z), v(x, y, z), w(x, y, z) (3.8) 66 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET or space time using the terminology of the theory of general relativity. It is possible to satisfy the condition of covariance of Maxwell’s equations so long as there is also a change in the material parameters and a new set of tensor values for permittivity and permeability are generated (3) renormalising the permittivity and permeability ′ ǫu = ǫ QuQvQw ′ QuQvQw µu = µu , etc 2 Qu Q2u (3.9) where the constituent relation then becomes ′ Eu′ = Qu Eu , Hu = Qu Hu , etc (3.10) δx 2 δy 2 δz 2 δx 2 δy 2 δz 2 2 δx 2 δy 2 δz 2 2 + + , Qv = + + , Qw = + + δu δu δu δv δv δv δw δw δw (3.11) where Q2u = This equation can be shown to be equivalent to this equation (4) ǫìj̀ = µìj̀ = Λìi ǫij Λj̀j det(Λìi ) , Λìi µij Λj̀j det(Λìi ) , (3.12) (3.13) where Λ is the Jacobian matrix. Reduced map In order to create a cloak without singular values of permittivity (5) choose a transformation which wouldn’t need to possess such extreme characteristics. A problem is that the calculation using this full transformation optics approach often yield less-than-unity values of permittivity such as in the invisibility cloak (4) which require metamaterials, which are materials not found in nature therefore a subset Quasi-Conformal Method is utilized known hereafter as the reduced triplet. This is also an important consideration in designing optics because metamaterials introduce losses and narrowband operation limiting the performance. 3.4. THE FINITE-DIFFERENCE TIME-DOMAIN METHOD 3.4 67 The Finite-Difference Time-Domain Method FDTD is better than other methods such as plane wave expansion or finite element methods in modelling metamaterials (16) because it allows the easy implementation of dispersive and anisotropic materials. Finite element time domain methods also allows a time domain simulation which is more intuitive to grasp compared to frequency domain methods. The Finite Difference Time Domain method allows chopping up of complex permittivity matrices and simulation. The Finite Difference Time domain method can also be extended to parallel architectures to increase simulation domain size and accuracy of modelling results which allows paralisation and the computation of very large electrical structures. Finite Difference Time domain codes can also to written in C rather than MATLAB as in this thesis to increase computation time to get closer to the metal and improve run times even more. Figure 3.4: The triplet is centered at zero and extends 0.2 m by 0.2 m. A sinusoidal plane wave is launched and controlled using total field scattered field formulation. The simulation domain is terminated using a Perfectly Matched Layer. The dispersive MTM values in the triplet are modelled using the ADE approach. There follows a brief introduction to the FDTD method which is later used to simulate electromagnetic devices (6; 7) and our Cooke triplet fig. 3.4. FDTD has been used to model antennas as well as other standard antenna modelling methods such as The Method of Moments (MoM) (8). For brevity, only the 2D case is examined. The Finite Difference Time Domain (FDTD) Method is a numerical method which solves Maxwell’s equations and can be used to simulate the electromagnetic field in the presence of a dielectric material (9). This 68 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET method has been used successfully many times in the study of the cloaking devices (10; 11) and in the MTMs (12) and subwavelength imaging in silver. The FDTD Method a solution to Maxwell’s equations which describe electromagnetic fields: ∇Ė = 4πρ (3.14) ∇=0 ˙ (3.15) ∇×E =− 1 δB c δt δB 1 ∇ × B = (4πJ + c δt (3.16) (3.17) These can be split into TE and TM mode which do not interact and simplify the equations: ǫ0 δHz δEx = δt δy δEy δHz =− δt δx δEx δEy δHz = − µ0 δt δy δx ǫ0 (3.18) (3.19) (3.20) where the component Ex , Ey and Hz are non-zero. A finite-difference approximation which allows the solution of the differential equations to be solved for the time differentials. Maxwell’s equations in TE mode with a finite difference are therefore: Hz (i + 21 , j + 21 − Hz (i + 21 .j − 12 ) 1 ǫ0 Ėx (i + , j) = 2 △y Hz (i + 21 , j + 12 − Hz (i − 12 .j + 12 ) 1 ǫ0 Ėy (i, j + ) = 2 △x (3.21) (3.22) Ex (i + 21 , j + 21 − Ex (i + 21 .j) Ey (i + 1, j + 12 − Ey (i, j + 12 ) 1 1 µ0 Ḣz (i + , j + ) = − (3.23) 2 2 △y △x The time-stepping equations, fig. 3.5, are where n is the time step. The separation of the time field into discrete units is known as the leap frog method. The space and time stepping grid is arranged in a leapfrog arrangement where one component is calculated and then used sort of recursively in the next calculation. The electromagnetic field components calculated on the previous iteration are the only input in calculating the field components for the next 3.4. THE FINITE-DIFFERENCE TIME-DOMAIN METHOD successive field component E n = E n−1 + 69 △t △xH n−1/2 ǫ0 H n+1/2 = H n−1/2 − (3.24) △t △xE n µ0 (3.25) This is a very convenient set of equations for computational calculations using iterations. This provides a differential solution to Maxwell’s equations which is time dependent. Giving the name Time Difference in Finite Difference Time Domain. Figure 3.5: A schematic of the time stepping, leap frog method for the calculation of the FDTD simulations. n-1, n and n+1 are the time steps, k is the grid cell number where the magnetic field, Hy, is at every half integer and the electric field Ex is at every whole integer. The two adjacent field components from the dual field vector are used to calculate the next field vector. This gives the possibility of time domain movies animations in FDTD for algorithm execution. To model the interaction with conventional dielectrics in the simulation using FDTD the finite difference equations, here in 2D, material properties such as conductivity and permittivity and permeability are combined into variables n− 1 n+ 1 n n + Hxni− 1 ,j − Hxni− 1 ,j+1 ) Ezi− 12,j+ 1 = Ca Ezi− 12,j+ 1 + Cb (Hyi,j+ 1 − Hy i−1,j+ 1 2 2 2 2 2 2 2 n+ 1 n+ 1 Hxn+1 = Da Hxni− 1 ,j+1 + Db (Ezi− 12,j+ 1 − Ezi− 12,j+ 3 ) i− 1 ,j+1 2 2 2 2 2 n+1 n 2 Hyi,j+ − Ezi− 12,j+ 1 ) 1 = Da Hyi,j+ 1 + Db (Ez i+ 1 ,j+ 1 2 2 2 2 2 (3.27) 2 n+ 1 n+ 1 (3.26) 2 2 (3.28) 70 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET according to the geometry of the Yee grid where Ca and Cb describe conventional dielectrics. A Yee grid, named after the author of the original paper (13), fig. 3.6 is used to place the field components, see Fig.fig. 3.6. The Magnetic field components are staggered compared to the electric so the code can increment from magnetic to electric. The Yee grid gives a structure to the simulation domain which allows discrimination in the space and a solution to Maxwell’s equations. Figure 3.6: The figure shows the Yee grid unit cell in 3D where magnetic and electric fields are placed in a staggered arrangement to allow the solution of Maxwell’s equations for each grid cell (i,j,k). In FDTD, for the simulation to remain stable the Courant stability condition c△t ≤ △x (3.29) must be satisfied. The physical interpretation of this equation is that information can travel 3.4. THE FINITE-DIFFERENCE TIME-DOMAIN METHOD 71 in the grid no faster than the speed of light. Therefore when the Courant condition is satisfied, the grid is causally connected and the results are stable solutions. This is generally allowed so long as the grid cells are smaller than 10 times the wavelength. Boundary conditions avoid reflections at the edges of the problem obscuring the electromagnetic solution are available using a Perfectly Matched Layer PML (14). The boundary conditions to the simulation domain are a rectangular PML µ2 δHx −δEz + σyHx = δt δy (3.30) δEz δHy + σxHy = δt δx δEzx δHy ǫ2 + σx Ezx = δt δx −δHx δEzy + σyEzy = ǫ2 δt δy (3.31) µ2 (3.32) (3.33) which acts as an absorbing layer to the incident radiation from all angle of incidence and reduces the intensity to zero. The lens is modelled by a conventional, lossless medium for areas which permittivity is greater than one and by a dispersive medium based on the Drude model for the remains where the permittivity is less than one. The lenses when a pulse is incident are modelled using the Auxillery Differential Equation method 1 B n+1 = B n − δt· ≈ △ × E n+ 2 1 B n+1 = B n + δt· ≈ △ × H n+ 2 (3.34) (3.35) as this has been shown to give more accurate results in the modelling of left-handed media which are inherently dispersive (15). Other models of lossy dielectrics include Debye and Lorentz media with there own formalisms for various combinations of poles. The frequency of the incidence field at 8 GHz which is in the microwave regime and therefore the geometric approximation used for ray tracing remains valid at optical frequencies. The resolution is λ/20 satisfying the stability condition. A high quality sinusoidal source is launched using the Total-Field Scattered Field formulation Hx,total = Hx,scat + Hx,inc (3.36) 72 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET for TM mode(Hx, Hy and Ez). 3.5 Summary In this section the Ray tracing of the Cooke triplet has been presented. This involved commerical software package OSLO where the Seidel aberrations were minimised using the merit function. In addition to this the virtual space for the MTM triple was generated using commercial grid generation software Pointwise and GenGrid where Laplace smoothing functions were used to increase cell orthogonality. The Discrete coordinate transformation approach was used to design the MTM triplet using a calculation on a cell by cell basis. The Finite Difference Time Domain is one of the most popular computational electromagnetic solvers which are used. It is a differential equation solver which solves in the time domain which allows time series movies to be produced. The lens is placed in the FDTD domain and a plane wave source at 8 GHz launched at the lens. The simulation area is terminated by PML and the metamaterial regions are modeled using the Drude model. The Finite Difference Method used to model the Transformation Optics lens is formulated in this chapter. Metamaterials are dispersive and therefore require use of the auxiliary differential equation formulation. The boundary areas are closed using PML and the transformation optics lens is placed in the centre of the simulation domain. The simulation is stable so long as the Courant condition is satisfied. Bibliography [1] R. Kingslake, Lens design fundamentals. academic press, 1978. [2] J. Li and J. B. Pendry, “”Hiding under the carpet: a new strategy for cloaking”,” ArXiv, 2008. [3] E. J. Post, Formal Structure of Electromangetics: General Covariance and Electromangetics. Holland Publishing Company, 1962. [4] D. Schurig, “”Metamaterial Electromagnetic Cloak at Microwave Frequencies”,” Science, vol. 314, p. 977, 2006. [5] D. J. B. Pendry, D. Schurig, “”Controlling Electromagnetic Fields”,” Science, vol. 321, June 2006. BIBLIOGRAPHY 73 [6] A. F. Peterson, S. L. Ray, and R. Mittra, ”Computational methods for electromagnetics”. IEEE Press, 1998. [7] A. Taflove and S. C. Hagness, ”Computational Electrodynamics: the finite difference time-domain method”. Artech House, 2005. [8] H. A. El-Raouf, V. Prakash, J. Yeo, and R. Mittra, “Fdtd simulation of a microstrip phased array with a coaxial feed,” IEE Proceedings-Microwaves, Antennas and Propagation, vol. 151, no. 3, pp. 193–198, 2004. [9] D. M. Sullivan, Electromagnetic simulation using the FDTD method. Wiley-IEEE Press, 2013. [10] Christos Argyropoulos, Efthymios Kallos, and Yang Hao, “”FDTD analysis of the optical black hole”,” J. Opt. Soc. Am. B, 2010. [11] Y. H. Christos Argyropoulos, Yan Zhao, “”A radially-dependant dispersive finitedifference time-domain method for the evaluation of electromagnetic cloaks”,” IEEE Transactions of Antennas and Propagation, 2009. [12] ”FDTD Modeling of Metamaterials: Theory and Applications”. Artech house, 2009. [13] K. S. Yee, “”Numerical solution of inital boundary value problems involving maxwells equations in isotropic media”,” IEEE Transactions on antennas and propagation, 1966. [14] J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” Journal of computational physics, vol. 127, no. 2, pp. 363–379, 1996. [15] Y. Zhao, C. Argyropoulos, time-domain simulation press, 16, vol. no. of 9, and Y. electromagnetic pp. 6717–6730, Hao, “”Full-wave cloaking Apr finite-difference structures”,” 2008. [Online]. Opt. Ex- Available: http://www.opticsexpress.org/abstract.cfm?URI=oe-16-9-6717 [16] Y. Hao and R. Mittra, FDTD modeling of metamaterials: theory and applications. Artech house, 2008. 74 CHAPTER 3. THE DESIGN OF A MTM COOKE TRIPLET Chapter 4 FDTD Simulation of the MTM Lens This section presents the analysis of the conventional Cooke triplet and the MTM and reduced maps based on numerical simulations. The simulation results are in the form of electric field components. Typical optical designers will use ray’s as a first basic approach to understanding the properties of an optical system. A ray tracing model will show a similar distribution of rays along a plane and the distribution for an electric field. A typical ray tracing analysis will look at the the five Seidel aberrations: Spherical, coma, astigmatism, field curvature and distortion. Remaining properties typically used in optical system design will be compartmentalised into wave behaviour which will be presented in the next section and this includes: wavefront error, Zernike aberrations and Modulation Transfer Function. This is the basis for the quantitative results for the ray based aberrations in the following section for the analysis of the optical properties of the MTM Lens. 75 76 4.1 4.1.1 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS Ez Field Results MTM and Reduced Cooke Triplet Figure 4.1: This figure shows the results of the field amplitude for the FDTD simulation results for the MTM (top panel) and reduced map (bottom panel). The red lines show the location of the PML and the permittivity boundaries on the top, left, right and bottom and the simulation domain. The red line signifies the location of the lens in the FDTD simulation too. The field amplitude is normalised to the maximum value. The source is a sinusoidal wave launched from the left of the domain. The permittivity map is superimposed onto the bounded structure for MTM and reduced lenses. 4.1. EZ FIELD RESULTS 77 The results for the simulation at 8 GHz for the MTM lens as described in the methodology is given in this section. The edges of the TO-designed Cooke Triplet have values of epsilon less than free space because of there inverted and more extreme deviation from orthogonal cells. A Cooke triplet without these extreme material parameters would be made up of an almost constant permittivity. A device with an almost constant value of permittivity would be much easier to fabricate. The device is tested using an FDTD simulation. The simulation domain was 8 m by 4 m for the lenses. The frequency used was 8 GHz which is a wavelength of 3.75 cm. A simulation for the reduced map was executed using the same simulation set up. This lens was generated by setting the values of permittivity in the MTM triplet with permittivity less than one to be air to the permittivity of a standard vacuum. The permittivity being a smooth function the reduced map was quantised into bands of discrete permittivity where λ/10. The bands were smaller than the wavelength so the incident wave effectively seeing a smooth distribution. To allow a comparison a simulation of a conventional Cooke triplet was carried out using the specifications of the Cooke triplet used in the ray tracing software. The permittivity map is placed in a domain surrounded by PML fig. 4.1. 4.1.2 Conventional Cooke Triplet and Single Lens The simulation was also carried out for a conventional Cooke triplet and a conventional single lens. The conventional Cooke triplet possessed the same surface curvature, power, glass type and spacings as the one modelled in ZEMAX for the MTM triplet design, fig. 4.2. The single lens was the first converging lens in the Cooke triplet. The refractive index was then adjusted to give the same focal length, fig. 4.2. The Cooke triplet has a tighter focal region as you would expect for a higher performing optical system when compared directly to the electromagnetic field pattern of the single lens, fig. 4.2. The reflection parameter is greater for the single lens when compared to the Cooke Triplet. The far field is also more scattered according to our FDTD simulations when compared to the FDTD simulation of the Cooke triplet lens. This analysis would suggest that the Cooke triplet is an improvement on the Single convex lens in terms of the electromagnetic field pattern shown here. 78 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS Figure 4.2: This figure shows the results of the field amplitude for the FDTD simulation results for the Cooke triplet (top panel) and single lens (bottom panel) at 8 Ghz. The field amplitude is normalised to the maximum value. The source is a sinusoidal wave launched from the left of the domain. The permittivity map for the Cooke triplet and single lens is placed on the simulation results where they were located in the MATLAB simulation. The focus is signified by an arrow and the Perfectly Matched Layer (PML) which acts as the boundary conditions for the numerical technique are shown as a red line. 4.2 Focal Length In microwave lens design aberrations have been studied by (2). To calculate the spherical, coma, and astigmatism. They then take measurements for the intensity distribution on axis 4.2. FOCAL LENGTH 79 and then off axis at the focal region. The aberration is quantified as the fitted function with a given Seidel coefficient. An aberration free or diffraction limited intensity distribution is derived to be a solution to the diffraction integral which is solved using Bessel functions. Seidel aberration are the main means which the performance of the MTM triplet is verified. Variation in the source were the main way that quantification of Seidel aberrations were made. The longitudinal spherical aberration, transverse spherical aberration, coma and field curvature were calculated. The S1 1 parameter gave a measure of the amount of reflections from each lens. The dispersion from each lens was calculated from a Fourier transform of a Gaussian pulse source which gives a measure of the chromatic aberration. The focal length is a key indication of the performance of the lens. The focal length of the system is found by taking the maximum intensity along the optical axis, ?? for the MTM triplet, conventional triplet and reduced map, table 5.3. Table 4.1: Fundamental performance metrics for the lenses. Back Focal Length (cm) Gain (dB) Beamwidth Singlet 29.1750 3.4115 12.3000 Triplet 27.0375 10.8488 11.1000 MTM lens 28.7985 13.1876 5.8000 Reduced map 27.2625 8.4221 20.5000 80 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS 4.3 Lens Power or Gain Figure 4.3: The transverse intensity distribution for the single, Cooke, MTM and reduced map centred at the focus. Sidelobes peak for the Cooke is further out than for the MTM, with both offering similar peak levels. Conclusion MTM offers similar performance to Cooke, with the reduced map showing results similar to a single lens. The beamwidths for the lenses are 12.3, 11.1, 5.8 and 20.5. The resolution is one of the most important and widely used measure for a lens. The transverse spherical aberration tSA is generated the same as the longitudinal spherical aberration but measured in the transverse axis. tSA and resolution and beamwidth are used interchangeably. tSA was measured by plotting the field amplitude (Ez) for all 4 lenses along the transverse axis at the focal point fig. 4.3. The focal point being taken as the point of maximum intensity along the optical or x axis. This plot is equivalent to the resolution of the image given by the Rayleigh criterion sinθ ≈ 1.22 λd . The resolution for all four lenses is much greater than the resolution limit. The resolution of the MTM is smaller than the Cooke triplet. Th resolution of the 4.4. OFF AXIS INCIDENCE 81 reduced map and single lens are much larger, almost double. This would suggest that the resolution and imaging performance of the MTM lens is slightly better than the conventional Cooke triplet with the reduced amp and single lens being much worse. This should be cavetated by saying that this would certainly be a benefit in some specific applications for example where you wanted to focus power in a small area such as high gain applications and this property has been reported in the literature (3). In more complicated optical systems however the imaging system as a whole is considered and the depth of field in this imaging system would be less for the MTM lens. Regarding the specific focusing ability on it’s own then the MTM lens performs better than the Cooke triplet and the reduced map and single maps worse. 4.4 Off Axis Incidence Seidel aberrations are the geometric aberrations in an image which occur when the Gaussian paraxial limit ceases to apply. This includes the real case where light rays strike a lens with an angle not close to perpendicular. There are 5 major third order Seidel aberrations: spherical, coma, astigmatism, field curvature and distortion. Seidel aberrations can be understood in terms of ray diagrams in which straight parallel rays interact with the lens and the rays do not intersect at a sharp image point but rather a spread. The off-axis aberrations such as coma are analysed by placing the incident source at π/2, π/3 and π/4 radians, ??. 82 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS Figure 4.4: This figure shows the off axis source incident on the (1) single lens (2) the reduced lens, (3) the Cooke triplet with visible scattering and reflections and (4) the MTM Cooke triplet lens where the outline is shown is red. A focus can be seen at the maximum amplitude where the focus is highlighted with an arrow and the lens is highlighted in a box. We see significant electromagnetic scattering in the MTM lens. The coma is defined as the the peaks where coma is a set of smaller peaks after the main peak when the wave is incident at an angle to the lenses when a slice is placed at the focus for an off-axis beam. The Cooke triplet has a small coma. The coma for the MTM map is large and spread and the reduced map is comparable to the MTM triplet but less pronounced due to less focusing of light rays away from the lens at the edges of the device as in the Cooke and MTM lenses. The four lenses were then tested with a source at a range of angles from 14 to 27 degrees to the lens. The resulting field distribution in the focal plane shows the coma in the lens. The coma is a result of rays far from the paraxial axis being refracted more or less than those at the centre. This results in a focal plane with multiple peaks, a distinguishing feature of coma aberration (4). A wide-angle antenna with low coma is the Rotman lens (5). The intensity in the transverse axis at the focus for a source at 18 degrees is plotted for the single lens, Cooke triplet, MTM lens and reduced lens, fig. 4.4. The coma is the smallest peaking at 0.18 for the original Cooke triplet. The MTM triplet peaks at just above one and the 4.5. FOCAL PLOT: ON AND OFF AXIS FOR THE LENSES 83 reduced map about 0.7 with additional scatter and coma spread over a larger area. 4.5 Focal Plot: On and Off axis for the Lenses Figure 4.5: This figure shows the focal points of the single lens (brown), Cooke triplet (cyan), MTM triplet (red) and the focal point of the reduced map is navy. The results are the focal points from the on and off axis FDTD simulations. The Field curvature is the deviation between focal points in the y axis and distortion in the x axis. Aplanatic means without spherical aberration. Aplanatic lens antennas have been studied by (1). The spherical aberration is the first aberration which we examine. This is the result of paraxial rays incident on a lens being refracted greater at the edges than the centre resulting in a smearing of the stigmatic focus across a region. The transverse spherical aberration is a Seidel aberration which is due to rays far from the optical axis, outside the paraxial approximation limit being refracted more. The intensity at the focus is typically defined by the Airy or Rayleigh Disk and the resolution is proportional to the width of the main beam. This can be quantified by the 3 dB width of the main beam which is approximately the distance from the peak of the main beam to the first null. It is clear from the results that the Cooke triplet has a wide main beam and several smaller beams at the start due to reflections. The plot for the spherical aberration shows that the MTM lens is better the the Cooke triplet as expected. The reduced map and single lens are more difficult to explain. The Cooke lens has lobes to the left hand side of the figure because these are due to increases in amplitude 84 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS due to reflected power from the dielectric surfaces which are not like the reduced and MTM lens matched to free space. The reduced map appears to have a large focus however this is an image aberration due to poor focusing of the light by the reduced map. The single lens has reflections at the start and a wide beam width. The focus and the 3 dB beamwidth are shown in Tab. table 5.3. The cells size is 0.375 cm or one tenth of the wavelength. The field curvature is a Seidel aberration which is results in a curvature in the focal plane (6). The variation in position of the focus was calculated from measuring the focus for each of the four lenses for a plane wave at 0 degrees, 14 degrees, 18 degrees and 27 degrees. This curvature is also due to astigmatism which can not easily be separated from field curvature (7). The field curvature and distortion are quantified in the variation in the focal point for the source at an angle of incidence. A full analysis of astigmatism would require both sagital and tangential planes modelled simultaneously which could be achieved using the Body of Revolution Finite Difference Time Domain Method. The field curvature for the Cooke triplet is the smallest and the reduced map is slightly larger and the metamaterial lens is the worst as shown in fig. 4.5. The change is focal position is an aberration related to the field curvature, astigmatism and distortion of the lens. 4.6. WIDE ANGLE INCIDENCE 4.6 85 Wide Angle Incidence Figure 4.6: This figure shows the intensity in dB along the transverse axis located at the focus where the coma is located. The side lobes to the right of the MTM lens being slightly more pronounced than the reduced map. The Cooke triplet is placed at 18 degrees where no focus is formed resulting in no large beam being formed at the centre and small coma to the right. The MTM lens design presented in the results section has a larger field of view than the standard Cooke triplet. The Gradient index ray tracing calculations show that coma increases sharply with angle of incidence. This behaviour is not present in the FDTD simulations of the MTM triplet at large angles of incidence suggesting that the MTM triplet has a larger field of view not unlike other wide angle lens antennas such as the Luneburg lens. Another example of a lens which is used because of its large field of view is the Rotman lens (8) and this facility is a valuable ability to have in enhancing communications systems. For standard optical systems the Cooke triplet has a field of view at 20 degrees yet the MTM triplet is still able to form a focus at 27 degrees, fig. 4.6. Wide angle beam scanning is a desirable feature for dielectric lenses (9; 10) to compare with the performance of alternative reflector antennas. Dielectric lens antennas might have applications in millimetre communications instead of reflector antennas (11) fig. 4.6 which shows that the Cooke triplet can not focus at 27 degrees but the MTM and reduced maps are able to. This confirms that transformation 86 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS optics designed dielectric lenses provide good performance in wide-angle scanning roles such as intelligent car management systems or microwave dielectric lenses for communications and is something which might be investigated as a further application. 4.7. THE EFFECT OF DISPERSION ON THE FOCAL LENGTH 4.7 87 The Effect of Dispersion on The Focal Length Figure 4.7: This figure shows the focus for the Cooke triplet and the MTM lens at 5.53, 6.63, and 8.00 GHz which are the same harmonic as the blue green and red lines used in OSLO to determine the chromatic aberration of an optical system. These are calculated from a Fourier transformation in the FDTD simulation of a Gaussian pulse source. 88 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS The chromatic aberration is due to dispersion in a medium therefore this is related to the band-width. The permittivity is a function of frequency and for a resonant metamaterial goes negative at a narrowband width. The chromatic aberration for the four lenses is calculated using a Gaussian pulse which was launched at 0 degrees incidence fig. 4.7. The Fourier amplitude for various frequencies was calculated at three frequencies contained in the pulse. Optical engineers typically use the difference in focus between green, red and blue light to measure the chromatic aberration. Here the centre frequency is 8 GHz which we treat as green and using the linearity of the frequency of the electromagnetic spectrum use a scale factor to find the focus for an equivalent ’red’ and ’blue’ light in this microwave range. The Fourier Transform was calculated and used to find the chromatic aberration at three frequencies. The inverse Fourier transform is used to transform this from a frequency domain equation to a time domain equation. This possesses second order time differentials which a finite difference approximation is made to around a central time step n. An equation for the electric field can then be calculated for each time step in cell regions which possesses lossy Drude materials such as metamaterials. In addition, it is possible to use Fourier transformation to transform the field into a frequency dependant domain from time domain in order to determine the different behaviour at a range of frequencies.S parameters are those used in microwave engineering in modelling microwave systems. s11 , s21 , s12 , s22 are the four components of the S matrix which describes all possible behaviour between 2 ports. Reflection and transmission coefficients are calculated for the computation using the same method as outlined in a paper which calculated the reflection and transmission coefficients in Pendrys invisibility cloak (12). 4.8. TRANSMISSION AND REFLECTION SPECTRA 4.8 89 Transmission and Reflection Spectra Figure 4.8: This figure shows the transmission and reflection spectra for the metamaterial triplet and the reduced triplet calculated from a Gaussian pulse incident in the Finite Difference Time Domain model centred at 8 GHz. The first panel shows the reflection coefficient where there is a peak for the reduced and metamaterial lenses and a smooth unvariation in the Cooke. The left hand side panel shows the transmission coefficient with a peak for the MTM and reduced maps and a smooth variation for the Cooke triplet The bottom panel shows than the sum of the reflection and transmission coefficients add to the Gaussian pulse accounting for all the energy in the simulation. The lens can be treated as a microwave system. Here typically S parameters are used to measure transmission and reflection of the system. The reflection coefficient, S11 is calculated to determine the amount of reflection that occurs for each lens system. The reflection and transmission parameters are calculated by capturing the Gaussian signal at the beginning of the simulation 1 m from the launch position. This is the input signal, a1 . There is no b1 , output signal at the other end so this is not collected. Once the wave packet hits the lens the return signal and transmitted signal begin being recorded, a2 and b2 . Once the simulation is finished and the signal has almost been totally reflected or transmitted then the reflection 90 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS and transmission coefficients are calculated as R = 4.9 a2 a1 and T = b2 . a1 Discussion and Conclusion The main results from the Finite Difference Time Domain Simulation were the field the lenses match focal length confirming the structures performance as a Cooke triplet as is the case in other designs of optical systems. In the next chapter we continue to confirm these results using a large simulation in commercial software CST Studio. In terms of directivity compared to the traditional Cooke triplet the MTM and reduced maps have a wider angle of incidence which is a benefit of using transformation optics to design wide angle imaging lenses or antennas this is in agreement with papers published on using transformation optics for wide angle beam scanning. The frequency results are calculated and transformed into s parameters to find the transmittance and reflectance of the lenses. The reduced map has been matched to free space and the reduction in reflections is a significant feature of this lens when compared to the conventional case. The Gain of the MTM lens is much greater than the conventional lens which also agrees with results published on high gain antennas using transformation optics. The gain is not supported when the reduced map is used. Bibliography [1] G. Cloutier and G. Bekefi, “Scanning characteristics of microwave aplanatic lenses,” Antennas and Propagation, IRE Transactions on, vol. 5, no. 4, pp. 391–396, 1957. [2] M. Bachynski and G. Bekefi, “Aberrations in circularly symmetric microwave lenses,” Antennas and Propagation, IRE Transactions on, vol. 4, no. 3, pp. 412–421, 1956. [3] W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Applied Physics Letters, vol. 93, no. 22, p. 221906, 2008. [4] R. Kingslake, Lens design fundamentals. academic press, 1978. [5] W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” Antennas and Propagation, IRE Transactions on, vol. 10, no. 1, pp. 82–95, 1962. BIBLIOGRAPHY 91 [6] T. A. Milligan, Modern antenna design. John Wiley & Sons, 2005. [7] J. Cogdell and J. Davis, “Astigmatism in reflector antennas,” Antennas and Propagation, IEEE Transactions on, vol. 21, no. 4, pp. 565–567, 1973. [8] W. Rotman and R. Turner, “Wide-angle microwave lens for line source applications,” Antennas and Propagation, IEEE Transactions on, vol. 11, no. 6, pp. 623–632, 1963. [9] F. Friedlander, “A dielectric-lens aerial for wide-angle beam scanning,” Electrical Engineers-Part IIIA: Radiolocation, Journal of the Institution of, vol. 93, no. 4, pp. 658–662, 1946. [10] W. Rotman, “Analysis of an ehf aplanatic zoned dielectric lens antenna,” Antennas and Propagation, IEEE Transactions on, vol. 32, no. 6, pp. 611–617, 1984. [11] R. Shavit, “Dielectric spherical lens antenna for wireless millimeter-wave communications,” Microwave and Optical Technology Letters, vol. 39, no. 1, pp. 28–33, 2003. [12] Christos Argyropoulos, Efthymios Kallos, and Yang Hao, “”FDTD analysis of the optical black hole”,” J. Opt. Soc. Am. B, 2010. 92 CHAPTER 4. FDTD SIMULATION OF THE MTM LENS Chapter 5 Image Evaluation 5.1 The Wavefront Aberration Function The wave aberrations is a way to characterise the optical system from the RMS wavefront error and modulated transfer function and Zernike aberrations and includes effects like diffraction, fig. 5.1. The mathematical functions were described by Frits Zernike in 1934. Frits Zernike went on to win the Nobel prize in 1953 in physics for the development of Phase Contrast Microscopy. This approach is commonly used by opticians for spectacles and contact lenses (1). Zernike aberration coefficients are the standard way to quantify the aberrations (2) as they provide a method which each aberration coefficient is independent of the others. It is an innovation on the traditional Seidel aberrations so much so that it is widespread in the optical industry for aberration measurements in the eye (3). Zernike aberrations are used in astronomy, optics, optometry, and ophthalmology. It has also yielded some interesting results in experiments on optical cloaking (4). In this section the aberration is classified as a deviation from an ideal Gaussian sphere and this is the subject of analysis from the FDTD simulation. This is the first work to the authors knowledge that the aberrations of a lens have been calculated from an FDTD simulation. The source was placed at the focus so that the output of the lenses was expected to be a plane wave, fig. 5.1. This was then sampled to give a measure of the distorted wavefront. The wavefront aberration are presented for the MTM, reduced map and triplet lens in the following section in a similiar way to an optical designer might approach his task. The wavefront aberrations, Zernike aberrations and MTF are compared to optical design software ZEMAX for the conventional triplet to validate the results. 93 94 CHAPTER 5. IMAGE EVALUATION Figure 5.1: The wavefront aberration is characterised by the deviation of the aberrated beam / irregular wavefront from an ideal Gaussian for an incoming parallel beam with a plane wave. This Gaussian sphere represents the ideal wavefront from a point source from which any deviation is measured in terms of a wave aberration This is the dual of the ray aberration which is the failure of the individual rays to meet at a stigmatic point. All real lenses and optical systems are subject to some degree of wave or ray aberrations. The wavefront aberration were investigated and provided a better understanding due to the wave nature employed in terms of Seidel aberrations (5). A perfect lens should perfectly transform a point source placed at the focus into a plane wave. Any deviation from this plane wave can be expressed as aberrations of the lens as a sum of Seidel polynomials (6). a(Q) = C40 r 4 + C31 h′ r 3 cosθ + C22 h′2 r 2 cos2 θ + C20 h′2 r 2 + C11 h′3 rcosθ (5.1) where C40 is spherical, C31 is coma, C22 and C22 are the astigmatism and field curvature and C11 is the distortion Seidel coefficient. The aberration function can be decomposed into a sum of Zernike polynomials W (ρ, θ) = ∞ X n X n=0 m=0 znm p (n + 1)Vnm (ρ)cos(mθ) (5.2) 5.1. THE WAVEFRONT ABERRATION FUNCTION 95 The Zernike polynomial is another way to characterise the wave front aberration of an optical system. The radial coefficients for the polynomial are given by the formulae. The wavefront aberration can be written as a sum of weighted Zernike polynomials, jmax W (x, y) = X Wj Zj (x, y) (5.3) j=0 These have been place in a rectangular co-ordinate system for ease of use and indexed according to a single indexing scheme, j, where j= n(n + 2) + m 2 (5.4) Polynomials are an advantage because they are orthogonal. This property allows the Zernike coefficients to be distinct from one another unlike the Seidel aberration calculation. 5.1.1 Sampling The Wavefront Aberration Function From The FDTD Simulation The source was placed at the focus in the FDTD simulation described in the chapter on Seidel aberrations. The results from this when the wave has passed through the lens is seen in figure fig. 5.2. The lenses were then simulated again. The aim was to calculate all aberrations from the distorted wavefront function. A program was written in MATLAB in order to find the point where Ez was greater than zero for the region one wavelength beyond the MTM triplet. This program sampled at every grid cell in the transverse axis. The result of this program was a line which represented the actual distorted wavefront in the FDTD simulation. SCATTERPLOT was used to plot all the points where the E-field was above zero. This gave an outline of a single wave contour. 96 CHAPTER 5. IMAGE EVALUATION Figure 5.2: This figure shows the wavefront for a Cooke triplet in Zemax and a comparison with the wavefront calculated from the FDTD method. 5.1.2 Fitting a Polynomial to the Wavefront Figure 5.3: This figure shows a) the scatter plot containing all the Yee grid cells for the final wavefront with Ez greater than 0. In panel b) the figure shows this wavefront plotted and a polynomial fitted to give an analytical solution to the wavefront aberration. In panel c) the wavefront aberration is shown for the MTM lens, the Cooke triplet and the reduced map and in panel d) the original wavefront in FDTD is given. 5.1. THE WAVEFRONT ABERRATION FUNCTION 97 The problem with finding a handle to analyse the MTM triplet was answered with the use of optical path difference. The optical properties can be reduced to an understanding of the path of individual rays through-out the MTM triplet. Once we know the optical path difference beyond the MTM lens we can calculate other various properties of the optical system and characterise it’s performance compared with conventional optical systems. In order to derive the properties for the optical path difference the wave aberration function was used. The wave front aberration function is an equation derived by Seidel in the 18th century to characterise the aberrations in an optical system. The wave front aberration equation is a polynomial where the coefficients of each term give a quantity for each of the Seidel aberrations: Spherical Aberrations, coma, astigmatism, field curvature and distortion. A tenth order polynomial was the highest order required to give a better fit to the wavefront which was sampled from the FDTD simulation and then fitted with a polynomial to give an analytic wave front aberration function. The source was placed at the focus of the three lenses: MTM, conventional and reduced. The lenses were then simulated using the same FDTD code as presented for the previous section of Siedel aberrations but this time we are extracting the wavefront aberration function to give a measure of the aberrations of the lens which can be seen for the Cooke triplet in panel d fig. 5.3. The residual was lowest for this order. The fitted polynomial is shown in panel 3, fig. 5.3 for the MTM, reduced and Cooke triplet. Low order polynomial fitting is less successful at fitting the correct distribution. Fitting a cubic spline gives the best result for interpolation of the sampled data points for reconstructing the wavefront, fig. 5.3. This polynomial which is fitted to the wavefront to give the wavefront aberration function for the cooke triplet is shown in panel b, fig. 5.3. The wavefront was extracted from the FDTD simulation by sampling the field values for a single wave above the central zero. This gave a contour to the wavefront for the wave at the one from the simulation edge, see panel d fig. 5.3. It was more accurately modelled at higher FDTD grid resolution therefore originally the simulation was done at a cell size of 1/10th of the wavelength whereas the graphs shown here are for 1/35 of the wavelength to improve the sampled data which the polynomial was fitted to. The results of this wavefront extraction procedure gives a value for the wavefront aberra- 98 CHAPTER 5. IMAGE EVALUATION tion function which is commonly used in commercial optical design software and is archetypal to many other optical systems aberrations. The smaller the wavefront aberration the better the performance. The root mean squared wavefront deviation gives a rough performance of each lens. The MTM lens possesses the largest wavefront RMS error which is an average deviation from a flat plane wave in fig. 5.3, panel c. The MTM triplet suffers from losses from the MTM regions and scattering which deteriorate the performance of the lenses ability to form a good image. The reduced lens has the smallest wavefront deviation suggesting it is an improved design in terms of imaging than the other two devices. 5.1.3 The Root Mean Squared (RMS) Error of The Wavefront Aberration The average deviation of the wavefront yields the root mean squared optical path difference, see Tab. 5.1 which is related to the Stehl ratio using the following equation Strehl = (1 − 2π 2 ω 2 )2 (5.5) This is a common measure for optical systems as is the Peak to Valley amount which is the difference between the height of the peak of the wavefront at the maximum minus the depth of the valley. RMS Error (/λ) P-V Ratio MTM Lens 0.43 0.1333 Reduced Map 0.59 0.3429 Cooke Triplet 0.15 0.1840 Singlet 0.693 0.3506 Cooke Triplet (ZEMAX) 0.1438 0.5470 Table 5.1: This table shows the RMS of the aberration function sampled for the four lenses as a function of wavelength. The equivalent Strehl ratio is also given. The MTM lens and Cooke triplet are within typical optical design tolerances. 5.1.4 Zernike Coefficients Calculation from Numerical Integration From the curve fitting of the distorted wavefront a polynomial for each lens is derived, fig. 5.4 and this is the input to a MATLAB script called deco.m which deconvolves, as in de-convolution, the polynomial into its constituent Zernike coefficients for each aberration, fig. 5.4. The coefficients for each order of the polynomial are used to calculate the coefficients for the radial Zernike polynomials and because these form an orthogonal set of basis functions will combine and add up to the total RMS wavefront error. The ’closeness’ to the original polynomial and the example Zernike aberration is quantified in terms of an integration. The 5.1. THE WAVEFRONT ABERRATION FUNCTION 99 area under the polynomial is calculated in MATLAB using dblquad. It is integrated over the unit cell of a Zernike disk from 0 to 2π. The bar chart shows the Q value which is plotted in the bar chart is the difference between the Zernike polynomial and the wavefront aberration (7). Figure 5.4: This figure shows wavefront on a unit disk, the set of Zernike polynomials, the amount of each Zernike polynomial which matches that function. The residual is the left over after all Zernike aberrations have been optimised to best match the input polynomial F. The original function in only 2D therefore results which are not 2D must are not valid. Panel a) shows the wavefront function for the Cooke triplet from the FDTD simulation rotated about a central axis. Panel b) shows the remaining wavefront which has not been fitted to the Zernike coefficient. Panel c) shows the Zernike coefficients first 9 modes and panel d shows the amount of each Zernike coefficient in the original wavefront. Notive that the majority of the Zernike wavefront is Z0 and Z4. 5.1.5 Spherical Aberration and Secondary Spherical aberration Zernike aberrations give a value for aberrations which can be separated, table 5.2. This means that there are no overlaps such as occurs in Seidel aberrations. This is achieved by 100 CHAPTER 5. IMAGE EVALUATION using orthogonal basis functions. The relationship between the aberrations which have integer numbers n and m are given here with their classical aberration relatives. The third order spherical aberration can be calculated from decomposition of the wavefront and the full list of calculated aberrations. Name Back Focal Length (dm) RMS Error (waves) Z1 (Piston) Z4 (Defocus) Z10 (Spherical Aberration) Z22 (Secondary SA) MTM lens 28.7985 0.043 0.015 0.0054 -0.0248 -3.9E005 Reduced map 27.2625 0.59 -0.08 0.0049 0.1800 -0.0045 Cooke Triplet 27.0375 0.15 -0.0059 -0.0016 0.0290 0.01 Single Lens 29.1132 -0.0021 0.072 0.0090 -0.015 Table 5.2: This table contains the full wave aberrations for all primary aberrations for the MTM lens, reduced map and the Cooke triplet. The zeros are aberrations which are present in the third dimension. These aberrations could not be calculated from a 2D simulation alone. The aberrations before astigmatism (tilt and defocus) are corrected for by finding the correct focal length. The remaining primary aberrations (field curvature and distortion) are zero because they are aberrations of the position of the focus rather than the image quality. 5.1.6 Defocus The wavefront aberration polynomial calculated for each lens in the previous section was validated by adjusting for defocus. One of the major factor the wavefront polynomial depends on is the position of the source, which is placed at the focus. The focus was calculated at the maximum of intensity for the lenses. The defocus is the first order coefficient. The focus and the first order coefficient are shown in table 5.3. The defocus was used as a measure of how Focus defocus MTM lens 28.7985 0.172 Reduced 27.2625 0.16 Table 5.3: This table shows the focal length for the MTM lens and the reduced triplet. The defocus is the amount of Z0 in the Zernike aberration result. successful the Zernike aberration polynomial was. The polynomial fitted to the wavefront was determined using MATLAB. Then the first order coefficient for x, in the case of the MTM Cooke Triplet this being 0.172. The location of the point source, which was previously placed at the focal length, was placed at the true focal length for this lens in FDTD. This corrected the defocus and improved the calculation of the focal length in the MTM 5.1. THE WAVEFRONT ABERRATION FUNCTION 101 triplet. This method was also applied to the reduced map, conventional and singlet lenses. The method of calculating Zernike aberrations was validated by adjusting the lens according to correct defocus. Defocus or Z0 in the previous section was used to adjust the original position of the focal point and so recursively improve the aberration correction algorithm. 5.1.7 Verification with ZEMAX Figure 5.5: The figure on the left shows the Zernike coefficients for the Cooke triplet in ZEMAX. In panel b) shows the figure for the Cooke triplet in the FDTD simulation calculation of the Zernike coefficients. The model can be verified using ZEMAX. ZEMAX give data for the Zernike aberration coefficients in a Cooke triplet. They show excellent agreement within 10 % for the first aberration and also for the remaining aberrations, see fig. 5.5. This validates the method for calculating the aberration using the method presented here. This is the first adoption of calculation of the aberrations from an FDTD simulation to the author’s knowledge. 102 5.2 CHAPTER 5. IMAGE EVALUATION Modulation Transfer Function (MTF) Figure 5.6: This figure shows the modulated transfer function MTF calculated from the results of the Zernike aberration coefficients for the Cooke triplet, the MTM triplet and the reduced map and a diffraction-limited aberration free case. The spatial frequency along the bottom axis represents how many black and white lines or cycles can fit in one mm therefore the higher the spatial frequency the more difficult it getting to resolve between individual illuminations or slits until a cut of frequency is reached. The y axis is the modulation or contrast between these solid 100 black or white lines. It is one of the most widely used and scientific measure of lensing performance in the optics industry. The MTF is calculated from the Zernike aberration polynomials and is closely related to the pupil function and the point spread function via Fourier transforms and is used widely in optical systems software to give a value to the resolution of the optical system. An aberrated system, as we have already seen, can be described by its deviations. These can be included 5.2. MODULATION TRANSFER FUNCTION (MTF) 103 in the pupil function to give a general pupil function which gives a description of an aperture where the wavefront aberrations are described by a theoretical phase retarding plate P(x, y) = P (x, y)exp[jkW (x, y)] (5.6) The point spread function is the Fourier transform of the wavefront (8). The Modulation Transfer Function MTF is related to the pupil function and the Point Spread Function PSF via Fourier transform. It is used to analyse spatial resolving power in industrial application. Any real system, as we have already seen, can be described by its aberrations. P SF(x,y) = 1 (λ2 d2 Ap ) F T (p(x,y) )(e−i2π(W (x,y))λ )2 (5.7) where p(x, y) = P (x, y)exp[jkW (x, y)] (5.8) is the pupil function and is the aberration function eq. (5.2). The MTF can be calculated from the optical transfer function of which it is the absolute value. OT F (sx, sy) = F T (P SF ) F T P SF sx =0,sy =0 MT F (sx, sy) = ||OT F(sx,sy ) || (5.9) (5.10) The Metamaterial map appears to perform much worse in terms of image evaluation. This is not surprising due to the large metamaterial regions which are lossy and causes absorption and scattering and would interfere with the image. This is not the case as much for the reduced map which has roughly the same performance in terms of Strehl ratio as the Cooke triplet. There is significant aliasing in the metamaterial lens. None of the lenses approach the diffraction limited case which would be an image free of aberrations. The MTF can be calculated from the optical transfer function of which it is the absolute value. The optical transfer function is the Fourier transform of the point spread function. The point spread function is a Fourier transform of the pupil function and as we have already shown an aberrated lens can be described by a generalised pupil function fig. 5.6. The results show reasonable agreement with the MTF of a Cooke triplet simulated in OSLO. 104 5.3 CHAPTER 5. IMAGE EVALUATION Summary Figure 5.7: This figure shows the wavefront aberration function in panel a, the polynomial fitting in panel b to the sampled wavefront. In panel c) you can see the wavefront aberration function derived from the polynomial fitting and in panel d) is the bar chart of Zernike coefficients. The wavefront is extracted from the FDTD simulation according the procedure shown in fig. 5.7. The first figure shows the results of the FDTD simulation where the source was placed at the focus. Panel b) shows the sampled wavefront and fits a polynomial to this. Panel c shows this polynomial on a unit disk and panel d) shows the Zernike aberrations calculated from this wavefront aberration. This is the first recorded way of finding aberrations from an FDTD simulation in the literature. The wavefront for the three lenses was sampled and fitted using a tenth order polynomial to find the function governing the wavefront. The wavefront can be split up into Zernike polynomials which is a common method used to measure the aberrations in the eye when one visits the opticians. The MTM lens shows a better imaging performance in terms of RMS wavefront error, BIBLIOGRAPHY 105 aberrations and MTF. The reduced or map which does not require metamaterials shows a similar or worse performance overall when compared to the conventional device. The optical path difference is better for the MTM lens and worse for the reduced lens. The Zernike aberrations shows that the MTM lens has a larger Zernike aberration than the reduced lens and the method was validated using the measure of defocus. Bibliography [1] W. Charman, “Wavefront aberration of the eye: a review,” Optometry & Vision Science, vol. 68, no. 8, pp. 574–583, 1991. [2] D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” Biomedical Engineering, IEEE Transactions on, vol. 51, no. 11, pp. 1969–1980, 2004. [3] J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” JOSA A, vol. 14, no. 11, pp. 2873–2883, 1997. [4] T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett., vol. 107, p. 173901, Oct 2011. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.107.173901 [5] E. Hecht, “Optics 4th edition,” Optics, 4th Edition, Addison Wesley Longman Inc, 1998, vol. 1, 1998. [6] A. J. Janssen, “Extended nijboer–zernike approach for the computation of optical pointspread functions,” JOSA A, vol. 19, no. 5, pp. 849–857, 2002. [7] W. H. Press, Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007. [8] J. Goodman, “Introduction to fourier optics,” 2008. 106 CHAPTER 5. IMAGE EVALUATION Chapter 6 Other Simulations of Transformation Optics Imaging Devices 6.1 Sub-Wavelength Imaging using a Drain and Time Reversal Techniques Spherical lenses are a closely related field to lenses designed by transformation optics. Maxwell’s initial derivation of the refractive index of the fish eye lens which carries his name was based on the orbit of a planet in a Newtonian gravitational field. This class of lenses posses a radial distribution of refractive index. This makes them hard to fabricate. Luneburg lenses which transform a point on the surface to a plane wave are a slight deviation in the design. These lenses are used in modern communications (1) and radio astronomy (2). The Luneburg lens is a lens with a variation of material parameters which can be realised without requiring metamaterials hence it can be assumed that it will not suffer the inherent limitations present in metamaterials. A Luneburg Lens can be realised with the several materials one important one for possible future integration with computers is silicon (3). One key application of negative index material is in a perfect imaging device (4). Pendry theoretically proved that negative index material amplifies evanescent waves. A slab of negative index material can therefore focus an object placed in the near field without losing information contained in the near field. A device like this was experiential verified using a slab of silver which possess negative index at various frequencies which was able to resolve the word NANO with a resolution greater than the illumination frequency. As the Science has developed and moved away from negative index material other methods of perfect imag107 108CHAPTER 6. OTHER SIMULATIONS OF TRANSFORMATION OPTICS IMAGING DEVICES ing have been suggested with transformation optics but without metamaterials (5; 6; 7). Some claim that this really is not perfect imaging and the image is an artefact of the drain placed at the image point rather than what you would typically call the conjugate image (8). The device is 2D but can be created for 3D if the demonstration for 2D proves successful (9). Electromagnetic simulations have shown that this is not an artefact but it does require a theoretically perfect drain (7) which could be replaced by a reciprocal antenna or a wire. Transformation optics allows lenses and other optical devices to be developed or improved that until recently were the realm of science fiction. One example of this is the perfect imaging device. Recent work has been on developing a perfect lens using transformation optics but without metamaterials. This involves using a Maxwells fish eye but surrounded by a mirror and with a drain placed at the image point. Other attempts have involved, for example, using time reversal (10; 11) which is arguably how the results in this Maxwell’s fisheye were obtained, however this is disputed by the original progenitor (12). This device has been demonstrated in the laboratory to possess sub-wavelength imaging properties (13) by resolving two probes at a distance apart smaller than the wavelength of operation. This is in disagreement with various papers who doubt this effect takes place. The drain in question is the subject of intense debate. Is it physical . It has been modelled as a coaxial cable port without success my this experimenter. A Maxwells fish eye has a permittivity distribution, n= r 2− r2 R (6.1) as a function of radius, much like the Minano lens which is another class of radially dependant spherical lenses. A sinusoidal source is placed at 0.6 times the radius on the left hand side along the central axis. The fish-eye is surrounded by an high permittivity layer which reflects incident electromagnetic waves and acts like a mirror. The drain is placed at 0.6R along the central axis. The drain is a modelled here as a point which records the field incident at that point at every time step and then cancels it. Effectively behaving as the perfect drain described in the paper. This drain has no known physical parallel. The drain is first modelled as a PEC point. The result is scattering and no field being present at the image. The drain was then modelled as a time reversed source. We found that the field cancelled at the centre and no image was found. We modelled the drain as a point which recorded and then cancelled the incoming field and go the peaks shown. We then placed two of these drains and two sources 0.3 wavelengths apart equidistant from the central axis. The field 6.1. SUB-WAVELENGTH IMAGING USING A DRAIN AND TIME REVERSAL TECHNIQUES109 pattern appears to show sub-wavelength imaging, fig. 6.1. This is most likely an artefact from the drains. The consensus amongst the scientific community is now generally that this is not a perfect imaging device. Figure 6.1: In the left hand figure a converged FDTD simulation for propagation inside a Maxwell’s fish eye lens surrounded by a mirror and with a source placed at one focus and a ’drain’ placed at the other. One the right hand side you will see the electric field distribution across the same device with two drains placed at a distance closer than the wavelength. I would not call this perfect imaging because it is a discrete effect and does not therefore require the same wavelength resolution. The drain appears to be an absorber much like a photo-detector. It is therefore not a free wavelength image. Further work would involve corroborating this without the Fisheye as this effect would still remain. It is unfortunate that this work was published (14) without recognizing my contribution. The key problem which was to be solved was how to achieve sub-wavelength imaging i.e. an image with a resolution greater than the wavelength. This had been proposed but not verified. Using a transformation optics method it was possible to describe flow of electromagnetic waves. The key introduction is the drain which allows the gathering of near field. Time reversal is then used to play back the field propagation recreating the near field. This has been verified here. 110CHAPTER 6. OTHER SIMULATIONS OF TRANSFORMATION OPTICS IMAGING DEVICES 6.2 A Square Fish Eye or Magnifier It should be considered that this device has been patented by BAE systems. It and it’s methodology is similar to start-up Kymeta (? ) in its ability to create conformal communication antennas. Following the work I realized that it is the same grid as a ’magnifier’ designed using transformation optics by (15). This device has been designed using grid generation software Pointwise to generate a coordinate system which will allow this behaviour, fig. 6.2. The grid lines were generated from a set of parametric equations. An improvement in the design might occur if an adaptive mesh were used to more accurately model the behaviour of EM wave propagation in space time. Pointwise does not possess the ability to model distorted coordinate systems in terms of Einstein field equations. This may provide source of further work using more advanced methods of mesh generation. Figure 6.2: This figure shows the coordinate system for the device described in this section. The grid boundaries were defined using a parametric equation. The centre was excavated and made into a small alcove in which the receiver may be placed.a) shows the grid in MATLAB. This is the initial step to implementing a coordinate transformation.b) shows the permittivity values calculated from the transformation placed in the real grid values of the original map.c) shows the permittivity map is placed in a FDTD grid. The relative permittivity is displayed here.d) shows the source placed at the region we modelled previously. The propagation following the expected ray path is observed in the device. This grid was imported into Matlab in order to use the transformation optics code pre- 6.2. A SQUARE FISH EYE OR MAGNIFIER 111 viously described in this thesis. This required some reformatting of the data as Pointwise is a slightly different grid generation software program than Gridgen, fig. 6.2. Only on half of the system is modelled as it will behave the same as the other due to symmetry. The device was modelled using traditional coordinate transformation approach used previously. This relies on 2D TM mode polarisation incident radiation to function correctly. It is transformed to a single block in real space. The permittivity map in the original grid can be seen in fig. 6.2b and in the real space orthogonal block fig. 6.2c. The permittivity map was placed in an FDTD grid. The source was placed just inside the top left hand corner of the device. The surrounding edges are encased in PEC reflecting layers. A point source was used and a plane wave expected. The final result we see an excellent plane wave emerging and propagating in the far field from the device, fig. 6.2d. The final plane wave is highly evident in this example. The earlier time-steps in the FDTD simulation are very interesting and show the expected ’flow’ of radiation inside the device which is obscured by reflections in later stages of the time evolution. 112CHAPTER 6. OTHER SIMULATIONS OF TRANSFORMATION OPTICS IMAGING DEVICES 6.3 Field Transformation Figure 6.3: This figure shows a flat lens. It is discrete as to be fabricated. It is designed on a standard convex lens. It is based on value calculated using the field transformation method (16) Field transformation is an alternative approach to transformation optics developed by (16), (17). It takes the opposite and complementary approach of transforming the field rather than the coordinate system, the amplitude and phase rather than the grid. It is effectively reverse engineering transformation optics and offers improved performance for TO devices (16) It should be possible to apply field transformation to the aberration problem. The field following the lens should be flat. We can reverse engineer a strip of permittivity values which can transform a cylindrical wave to a flat wave from the equation. The Field transformation BIBLIOGRAPHY 113 flat lens has been fabricated and incorporated in the FDTD mechanism, fig. 6.3. For the transverse, T, and longitudinal, z, permittivity ǫT T µT T = 2 = ǫT T 2 v u (6.2) ǫzz u2 = µzz (6.3) v 2 = ǫzz (6.4) and for the off diagonal, Tz and zT permittivity where the transformation in field polarisation from TE to TM comes about due to the cross product in the off diagonal terms. It is possible to design a polarisation converter using transformation optics which transforms TE to TM mode in an arbitrary shape such as an arrow, (18). A polarisation converter has been fabricated using gold nano-ribbons (19). A polariser using field transformation has been modelled in COMSOL by associate professor Yunming Zhao. I assisted with measurements with my QMUL colleague Lianhong to measure the polarisation of the device at microwave frequencies created using dielectric layers. Bibliography [1] C. Fernandes, V. Brankovic, S. Zimmermann, M. Filipe, and L. Anunciada, “Dielectric lens antennas for wireless broadband communications,” Wireless Personal Communications, vol. 10, no. 1, pp. 19–32, 1999. [2] G. James, A. Parfitt, J. Kot, and P. Hall, “A case for the luneburg lens as the antenna element for the square kilometre array radio telescope,” submitted to Radio Science, 1999. [3] A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” arXiv preprint arXiv:1101.1293, 2011. [4] R. Pendry, “”Negative refraction makes a perfect lens”,” New journal of physics, 2000. [5] U. Leonhardt, “”perfect imaging without negative refraction”,” New journal of physics, vol. 11, p. 093040, 2009. [6] R. Blaikie, “Comment on’perfect imaging without negative refraction’,” New Journal of Physics, vol. 12, no. 5, p. 058001, 2010. 114CHAPTER 6. OTHER SIMULATIONS OF TRANSFORMATION OPTICS IMAGING DEVICES [7] U. Leonhardt, “Reply to comment on’perfect imaging without negative refraction’,” New Journal of Physics, vol. 12, no. 5, p. 058002, 2010. [8] S. Guenneau, A. Diatta, and R. McPhedran, “Focusing: coming to the point in metamaterials,” Journal of Modern Optics, vol. 57, no. 7, pp. 511–527, 2010. [9] U. Leonhardt, “Perfect imaging without negative refraction,” New Journal of Physics, vol. 11, no. 9, p. 093040, 2009. [10] J. de Rosny and M. Fink, “Overcoming the diffraction limit in wave physics using a time-reversal mirror and a novel acoustic sink,” Physical review letters, vol. 89, no. 12, p. 124301, 2002. [11] M. Fink and C. Prada, “Acoustic time-reversal mirrors,” Inverse problems, vol. 17, no. 1, p. R1, 2001. [12] U. Leonhardt and S. Sahebdivan, “Perfect imaging: they do not do it with mirrors,” Journal of Optics, vol. 13, no. 2, p. 024016, 2011. [13] Y. G. Ma, C. Ong, S. Sahebdivan, T. Tyc, and U. Leonhardt, “Perfect imaging without negative refraction for microwaves,” arXiv preprint arXiv:1007.2530, 2010. [14] O. Quevedo-Teruel, R. Mitchell-Thomas, and Y. Hao, “Frequency dependence and passive drains in fish-eye lenses,” Physical Review A, vol. 86, no. 5, p. 053817, 2012. [15] A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Optics letters, vol. 33, no. 1, pp. 43–45, 2008. [16] F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Physical review letters, vol. 111, no. 3, p. 033901, 2013. [17] S. Jain, M. Abdel-Mageed, and R. Mittra, “Flat-lens design using field transformation and its comparison with those based on transformation optics and ray optics,” arXiv preprint arXiv:1305.0586, 2013. [18] Y. Zhao, M. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nature communications, vol. 3, p. 870, 2012. BIBLIOGRAPHY 115 [19] J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science, vol. 325, no. 5947, pp. 1513–1515, 2009. 116CHAPTER 6. OTHER SIMULATIONS OF TRANSFORMATION OPTICS IMAGING DEVICES Chapter 7 Conclusion and Further Work By following a similar mathematical technique as was followed for the invisibility cloak it is possible to design other transformation media with interesting abilities. It is an important point and one worth repeating here. The basic procedure followed is to define the desired property i.e. field rotation (1); specify the field transformation, and in this case it would be a variation in angle with radius; use the canonical transformation optics equations which calculates the permittivity and permeability vector in terms of the field transformation ǫ= Λi Λj ǫ det(Λ) (7.1) where this is the same form for permeability. This gives the permittivity of the medium which possess the desired field transformation. The Quasi-Conformal Transformation Technique is an interesting way to implement curved and arbitrary surfaces. This theory tells us that the field must be the same at the boundary so if the original virtual space is conformal ( a special case of this being a Cartesian mesh ) then a transformation which conserves the angles locally will possess the same electromagnetic propagation characteristics. This is especially useful as the mesh generation program features grid smoothing functions which allow a conformal or quasi-conformal mesh to be generated inside a boundary. The possibilities for the surface are only limited by ones imaginableness and could be used to create devices which fit exactly a given body as is the case with the spherical Lunburg lens which has been transformed to feature a flat surface which allows compatibility with any sensor (2). It is possible to extend the current transformation optics theory into multiple polari117 118 CHAPTER 7. CONCLUSION AND FURTHER WORK sations by including anisotropic permittivity matrices (3). There is an insurmountable trade off in transformation optics design. The more complicated the device design the better the simulated performance however the more difficult it is to realise. The realisation is being incorporated into the design methodology. Here I simulate a MTM triplet which has a lower resolution to allow easier fabrication with solid dielectric blocks. Taking an engineering approach to transformation optics allows the design and possible fabrication of many exotic devices such as invisibility cloaks and black hole devices. Some have been explored here such as flat lenses, Graphene based devices based on engineering conductivity, gravitational lenses and sub-wavelength images. I have applied this method to the design of a gradient index optics lenses. The principles are the same as for optics and microwaves and a device which can be manufactured has been proposed for prototype construction. 7.1 Discussion and Conclusion Metamaterials are artificially made materials or meta ’atoms’ which derive there properties from there structure rather than there chemical composition. These have been used to achieve values of negative permittivity and permeability resulting in a narrow-band and lossy interaction which has been modelled using the Drude model in FDTD. Transformation optics has been used in the development of cloaking devices to control the flow of electromagnetic radiation. This has been modelled using anisotropic and inhomogeneous permittivity and permeability parameters to create a broadband, lossless lens. This has been designed using the ray tracing approach from a lens in a commercial optical design package. The virtual space was then modelled using a grid generation program and discrete coordinate transformation gave the material properties and these were verified in FDTD simulation. We have designed a low aberration lens using transformation optics based on the Cooke triplet. Ray tracing has been used to model the flow of electromagnetic radiation. The rays from ZEMAX modelling software were used to characterise the behavior of the Cooke triplet. The virtual space has been modelled using commercial grid generation software package. These have given quasi-orthogonal grids which allow isotropic blocks of permittivity to create the device. A discrete transformation optics method has been used that operates at a single polarization but across a wide bandwidth. The proposed lens would match this 7.1. DISCUSSION AND CONCLUSION 119 performance but possess the benefit of being made of a single block of dielectrics reducing size, weight and price for military applications. The Cooke triplet minimizes Seidel aberrations in an image: spherical, coma, astigmatism, field curvature and distortion using only three glass elements. A reduced map triplet would possess no metamaterials and not be subject to losses and narrow-band operation. The MTM triplet is designed using ray tracing from commercial optical design software ZEMAX and a mesh representing the desired ’virtual space’ is generated from this. Transformation optics is then used to find the dielectric values of the lens required to satisfy this property. We find that the field of view in the proposed triplet is not limited as it is in the conventional case. We also find that as the proposed triplet can and is matched to free space, reflections internally are less, this improved the gain or power of the lens. The reduced lens posesses an extended field of view compared with the conventional lens which was tested in FDTD for comparison. FDTD Simulations were confirmed using commercial package CST. A method of calculating the aberrations in an FDTD calculation is presented using the wavefront aberration function. A source was placed at the source focus. The distorted wavefront has been sampled and then fitted to a polynomial to measure the wavefront aberration for the MTM, reduced, Cooke and single lenses.The Zernike aberrations have been calculated from the wavefront aberrations by deconvolving the wavefront into orthogonal Zernike polynomials. The reduced transformation optics lens possesses the smallest mean wavefront aberration error. The point spread function and modulation transfer function has been calculated from the wavefront using the pupil function and transmittance functions. The results for the Zernike aberrations calculated are shown to have good agreement with the results for the Zernike aberrations for the same lens in ZEMAX. The reduced transformation optics lens performs comparably with the conventional Cooke triplet in terms of optical imaging criteria with MTF being the key overall measure. These results have shown that the reduced map performs similarly to the conventional Cooke triplet but with slightly better off axis aberration performance and reduced on axis performance. The lens with Metamaterials performs worse over the whole range due to disruption resulting from the losses. Additional work from the thesis is presented in the further work section. Key parts of this are the rectangular Luneburg lens design which was patented by BAE systems. This shows that a flat Luneburg lens has been fabricated using the ray tracing technique and grid 120 CHAPTER 7. CONCLUSION AND FURTHER WORK generation software and the electromagnetic behaviour verified in electromagnetic experiment validating it’s ability. Perfect Imaging using drains to allow sub-wavelength imaging was tested as a hypothesis and we find that drains artificially create the impression of a sub-wavelength image where subwavelength imaging might be but this is an artefact of imposing the drain. I have implemented the field transformation lens and calculated the field distribution which demonstrates that a flat field transformation lens performs equally as well as one designed using transformation optics designed in this thesis validating the field transformation method for designing flat lenses. 7.1.1 Recommendations for Future Work Arising from the research carried out during this work, areas of potential further research could include: - Demonstration is a requirement. The device could be tested and fabricated using dielectrics in a antenna and or microwave laboratory. This would verify the results here and also provide a confirmation of the results in experiments. The experiment would be carried out on a 2D scanner fed by a waveguide with a probe measuring the field. - Research into GRIN optical materials is required. If the device is to be fabricated at optics then more research must be done on optical metamaterials and the limitations of current gradient index optical lenses. In order for the device to be implemented at optical frequencies more research would have to be done on the optical materials. The current design would only function for microwaves and antennas. - Implementing the design into a functioning device such as a camera or binoculars. The proof of the benefit of using discrete coordinate transformation optics in the design of optical systems would be to create a prototype device which is used commercially. The obvious suggestion would be a camera but also binoculars or contact lenses would suffice. Bibliography [1] H. Chen and C. Chan, “Transformation media that rotate electromagnetic fields,” Applied physics letters, vol. 90, no. 24, pp. 241 105–241 105, 2007. BIBLIOGRAPHY 121 [2] N. Kundtz and D. R. Smith, “”Extreme-angle broadband metamaterial lens”,” Nature letters, 2008. [3] F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Physical review letters, vol. 111, no. 3, p. 033901, 2013.