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.
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[50] T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for farfield subdiffraction imaging at mid-infrared frequencies,” Optics express, vol. 21, no. 18,
pp. 20 888–20 899, 2013.
[51] D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems
using transformation optics,” Optics express, vol. 18, no. 20, pp. 21 238–21 251, 2010.
[52] T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New
Journal of Physics, vol. 10, no. 11, p. 115038, 2008.
38
CHAPTER 1. INTRODUCTION
Chapter 2
Transformation Optics and Their
Applications
2.1
Invisibility Cloaks
The mathematics of Transformation Optics are similar to the esoteric high level mathematics of general relativity. 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.
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[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.
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[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,
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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
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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, ??.
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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.
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[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.