nature-inspired optimization of quasicrystalline arrays and all-dielectric optical filters and ...

The Pennsylvania State University

The Graduate School

Department of Electrical Engineering

NATURE-INSPIRED OPTIMIZATION OF QUASICRYSTALLINE

ARRAYS AND ALL-DIELECTRIC OPTICAL FILTERS AND

METAMATERIALS

A Dissertation in

Electrical Engineering

by

Frank (Farhad) A. Namin

c© 2012 Frank (Farhad) A. Namin

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2012

The dissertation of Frank (Farhad) A. Namin was reviewed and approved∗ by the

following:

Douglas H. Werner

Professor of Electrical Engineering

Dissertation Advisor, Chair of Committee

Pingjuan L. Werner


Julio Urbina


Brian Weiner

Professor of Physics

Kultegin Aydin


Head of the Department of Electrical Engineering

∗Signatures are on file in the Graduate School.

Abstract

Quasicrystalline solids were first observed in nature in 1980s. Their lattice geom-etry is devoid of translational symmetry; however it possesses long-range order aswell as certain orders of rotational symmetry forbidden by translational symmetry.Mathematically, such lattices are related to aperiodic tilings. Since their discoverythere has been great interest in utilizing aperiodic geometries for a wide varietyof electromagnetic (EM) and optical applications. The first thrust of this disser-tation addresses applications of quasicrystalline geometries for wideband antennaarrays and plasmonic nano-spherical arrays. The first application considered isthe design of suitable antenna arrays for micro-UAV (unmanned aerial vehicle)swarms based on perturbation of certain types of aperiodic tilings. Due to safetyreasons and to avoid possible collision between micro-UAVs it is desirable to keepthe minimum separation distance between the elements several wavelengths. Asa result typical periodic planar arrays are not suitable, since for periodic arraysincreasing the minimum element spacing beyond one wavelength will lead to theappearance of grating lobes in the radiation pattern. It will be shown that usingthis method antenna arrays with very wide bandwidths and low sidelobe levels canbe designed. It will also be shown that in conjunction with a phase compensationmethod these arrays show a large degree of versatility to positional noise. Nextaperiodic aggregates of gold nano-spheres are studied. Since traditional unit cellapproaches cannot be used for aperiodic geometries, we start be developing newanalytical tools for aperiodic arrays. A modified version of generalized Mie theory(GMT) is developed which defines scattering coefficients for aperiodic sphericalarrays. Next two specific properties of quasicrystalline gold nano-spherical arraysare considered. The optical response of these arrays can be explained in terms ofthe grating response of the array (photonic resonance) and the plasmonic responseof the spheres (plasmonic resonance). In particular the couplings between the pho-tonic and plasmonic modes are studied. In periodic arrays this coupling leads to

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the formation of a so called photonic-plasmonic hybrid mode. The formation ofhybrid modes is studied in quasicrystalline arrays. Quasicrystalline structures inessence possess several periodicities which in some cases can lead to the forma-tion of multiple hybrid modes with wider bandwidths. It is also demonstratedthat the performance of these arrays can be further enhanced by employing a per-turbation method. The second property considered is local field enhancementsin quasicrystalline arrays of gold nanospheres. It will be shown that despite aconsiderably smaller filling factor quasicrystalline arrays generate larger local fieldenhancements which can be even further enhanced by optimally placing perturbingspheres within the prototiles that comprise the aperiodic arrays.

The second thrust of research in this dissertation focuses on designing all-dielectric filters and metamaterial coatings for the optical range. In higher fre-quencies metals tend to have a high loss and thus they are not suitable for manyapplications. Hence dielectrics are used for applications in optical frequencies. Inparticular we focus on designing two types of structures. First a near-perfect opti-cal mirror is designed. The design is based on optimizing a subwavelength periodicdielectric grating to obtain appropriate effective parameters that will satisfy thedesired perfect mirror condition. Second, a broadband anti-reflective all-dielectricgrating with wide field of view is designed. The second design is based on anew computationally efficient genetic algorithm (GA) optimization method whichshapes the sidewalls of the grating based on optimizing the roots of polynomialfunctions.

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Table of Contents

List of Figures viii

List of Tables xiii

Acknowledgments xiv

Chapter 1Introduction 11.1 Ultra-Wideband Aperiodic Antenna Arrays . . . . . . . . . . . . . . 21.2 Quasicrystalline Plasmonic Nanoparticle Arrays . . . . . . . . . . . 31.3 All-Dielectric Optical Filters and Metamaterials . . . . . . . . . . . 51.4 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Ultra-Wideband Aperiodic Antenna Arrays . . . . . . . . . . 61.4.2 Quasicrystalline Plasmonic Nanoparticle Arrays . . . . . . . 61.4.3 All-Dielectric Optical Filters and Metamaterials . . . . . . . 7

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2Tilings, Crystals, and Quasicrystals 92.1 Periodic Tilings and Point Lattices and Symmetries . . . . . . . . . 92.2 Aperiodic Tilings and Quasicrystals . . . . . . . . . . . . . . . . . . 13

2.2.1 Penrose Aperiodic Tiling . . . . . . . . . . . . . . . . . . . . 162.2.2 Danzer Aperiodic Tiling . . . . . . . . . . . . . . . . . . . . 162.2.3 Ammann-Beenker Aperiodic Tiling . . . . . . . . . . . . . . 18

2.3 Higher-Dimensional Approach . . . . . . . . . . . . . . . . . . . . . 19

Chapter 3Ultra-Wideband Aperiodic Antenna Arrays 22

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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Planar Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Periodic Antenna Arrays . . . . . . . . . . . . . . . . . . . . 253.2.2 Aperiodic Antenna Arrays . . . . . . . . . . . . . . . . . . . 25

3.3 Optimization of Aperiodic Tiling Arrays . . . . . . . . . . . . . . . 273.4 Positional Noise Analysis and Error Correction . . . . . . . . . . . . 363.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5.1 Periodic Array . . . . . . . . . . . . . . . . . . . . . . . . . 423.5.2 Optimized Aperiodic Array: Broadside . . . . . . . . . . . . 43

Chapter 4Generalized Scattering Coefficients for Finite-Sized Spherical

Arrays 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 GMT with Incident Plane Wave . . . . . . . . . . . . . . . . . . . . 514.3 GMT Based on Finite Beamwidth Incident Wave . . . . . . . . . . 554.4 Generalized Scattering Coefficients . . . . . . . . . . . . . . . . . . 644.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 5Optimizations of Quasicrystalline Nanoparticle Arrays 675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Resonance Response in Periodic Gold Arrays . . . . . . . . . . . . . 68

5.2.1 Plasmonic Resonance of Gold Spheres . . . . . . . . . . . . . 685.2.2 Photonic Resonance . . . . . . . . . . . . . . . . . . . . . . 705.2.3 Hybrid Resonance . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Resonance Response in Quasicrystalline Gold Arrays . . . . . . . . 735.3.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Local Field Enhancements in Quasicrystalline Gold Arrays . . . . . 82

Chapter 6Optical Mirrors Based on Metamaterial Coatings 886.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 Effective Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.4 Fabrication and Measurements . . . . . . . . . . . . . . . . . . . . . 986.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Chapter 7Transmission Gratings Based on Efficient Optimization of Poly-

nomials 1027.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Appendix AGenetic Algorithms 110A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A.2 Binary Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . 111

A.2.1 Initial Population . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.2 Mating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.3 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.4 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . 113

Appendix BDerivation for Ωn(ka) and Ξn(ka) 114

Appendix CDerivation for Ψn(ka) 118

Appendix DDerivation for sν+1,ν(z) 120

Bibliography 122

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List of Figures

2.1 An example of a periodic tilings of 2D plane with two prototiles Aand B. Two sets of basis vectors are shown. . . . . . . . . . . . . . 10

2.2 A periodic point lattice (left) and its Fourier diffraction pattern(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 A random set of points (left) and its Fourier diffraction pattern(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 A set of points generated from the Penrose tiling (left) and thecorresponding Fourier diffraction pattern (right). . . . . . . . . . . . 15

2.5 Penrose aperiodic tiling. . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Prototiles of the Penrose tiling (θ = π/5 and τ = 1+

√5

2). . . . . . . . 17

2.7 Iterative inflation and substitution of Penrose prototiles. . . . . . . 172.8 Prototiles of the Danzer aperiodic tiling (θ = π/7). . . . . . . . . . 182.9 Two iterations of inflation and substitution applied to Danzer pro-

totile type III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Two iterations of inflation and substitution applied to square pro-

totile of A-B tiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11 Prototiles of the A-B tiling and their relative dimensions. . . . . . . 192.12 Higher-dimensional approach demonstrated to obtain a Fibonacci

sequence as a projection of I2p onto an irrational subspace. . . . . . 21

3.1 Radiation pattern of a periodic array of 441 isotropic radiators withmain beam steered toward broadside and operating wavelength λ =2dmin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Radiation pattern of a periodic array of 441 isotropic radiators withmain beam steered toward broadside and operating wavelength λ =dmin

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Prototiles of the Danzer aperiodic tiling (θ = π/7). . . . . . . . . . 283.4 Geometry of an aperiodic array of 659 elements generated from a

Danzer tiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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3.5 Radiation pattern of an aperiodic Danzer array of 659 isotropicradiators with main beam steered toward broadside and operatingwavelength λ = 2dmin. . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Radiation pattern of an aperiodic Danzer array of 659 isotropicradiators with main beam steered toward broadside and operatingwavelength λ = dmin

2. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7 Geometry of optimized Danzer array (267 elements) for maximumsidelobe suppression at f = 4f0. . . . . . . . . . . . . . . . . . . . . 32

3.8 Three optimized Danzer prototiles with additional perturbation el-ements used to generate the antenna array in Figure 3.7. . . . . . . 32

3.9 A segment of the Danzer tiling generated with the perturbed pro-totiles shown in Figure 3.8. . . . . . . . . . . . . . . . . . . . . . . . 33

3.10 Sidelobe level performance of the Danzer array optimized for side-lobe suppression at f = 4f0. The performances of a uniform peri-odic array as well as a base Danzer array are also shown for com-parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.11 The ideal radiation pattern for the 267 element aperiodic tilingarray shown in in Figure 3.7 at operating wavelength λ = dmin

2. . . . 34

3.12 Three optimized Danzer prototiles with additional perturbation el-ements used to generate the antenna array in Figure 3.13. . . . . . 35

3.13 Geometry of optimized Danzer array (505 elements) for maximumside-lobe suppression at f = 30f0. . . . . . . . . . . . . . . . . . . . 35

3.14 Side-lobe level performance of the Danzer array optimized for side-lobe suppression at f = 30f0. The performances of a uniformperiodic array as well as a base Danzer array are also shown forcomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.15 Normalized array factor for the optimized Danzer array with 505elements at f = 30f0. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.16 Geometry of a micro-UAV formation. In this illustration, rn repre-sents the nominal position of aircraft n, rn represents the error dueto turbulence, and n is the unit vector pointing in the direction ofthe far-field observation point at (θ, φ). . . . . . . . . . . . . . . . . 38

3.17 Effect of the phase correction factor in the direction of the main-beam. In this example, the mainbeam of the radiation pattern ispointed in the direction n0 = −z. The phase correction term βcnis used to ensure the phase of the signal from aircraft n is equal tozero at z = 0. This process is repeated for each aircraft, ensuring abeam can be resolved in the n0 direction. . . . . . . . . . . . . . . . 39

3.18 Illustration of angle ψ, the angle between the observation direction,n and the boresight of the array, n0. . . . . . . . . . . . . . . . . . 40

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3.19 The ideal radiation pattern for a micro-UAV swarm based on a 1793element periodic array. . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.20 The corrupted radiation pattern of a 1793 element periodic micro-UAV swarm. The swarm is corrupted by a Gaussian positionalnoise with σ = 0.1λ. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.21 The phase corrected radiation pattern of a 1793 element periodicmicro-UAV swarm. The swarm is corrupted by a Gaussian posi-tional noise with σ = 0.1λ. . . . . . . . . . . . . . . . . . . . . . . . 45

3.22 The effects of Gaussian noise with σ = 0.1λ on the optimized array.The intensity of gray in circles around the elements corresponds tothe probability of the element being in that region. . . . . . . . . . 46

3.23 The corrupted radiation pattern for a micro-UAV swarm based onthe 267 element aperiodic tiling array shown in Figure 3.7. Theswarm is corrupted by a Gaussian positional noise with σ = 0.1λ. . 47

3.24 The phase corrected radiation pattern for a micro-UAV swarm basedon the 267 element aperiodic tiling array shown in Figure 3.7. Theswarm is corrupted by a Gaussian positional noise with σ = 0.1λ. . 47

3.25 The phase corrected radiation pattern for a micro-UAV swarm basedon the 267 element aperiodic tiling array shown in Figure 3.7. Theradiation pattern is steered to an angle φ = 45, θ = 45 and theswarm is corrupted by a Gaussian positional noise with σ = 0.1λ. . 48

4.1 |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using the analytical expressions from Eq. (4.19). 60

4.2 |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using a four term VSWF expansion. . . . . . . . 60

4.3 |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using a six term VSWF expansion. . . . . . . . . 61

4.4 |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using an eight term VSWF expansion. . . . . . . 61

4.5 Geometry of an aperiodic Ammann-Beenker tiling. . . . . . . . . . 624.6 Normalized scattered field magnitude (dB) in the plane of the AB

aperiodic array illuminated by a plane wave. . . . . . . . . . . . . . 634.7 Normalized scattered field magnitude (dB) in the plane of the AB

aperiodic array illuminated by circular aperture diffracted waves. . . 644.8 Scattering response of infinite (solid lines) and finite (dashed lines)

periodic gold arrays obtained by CST MICROWAVE STUDIO andGMT with a finite beamwidth calculated using Eq. (4.41) andEq. (4.44). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

x

5.1 Real and imaginary parts of the gold dielectric function accordingto Eq. (5.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Extinction efficiency of a gold sphere with a radius of 20 nm in adielectric medium with an index of refraction of nM = 1.5. . . . . . 71

5.3 Extinction efficiency of a periodic array of 121 dielectric spheres(np = 3) of radius 50 nm and periodicity of Λ = 550 nm with anormally incident plane wave. . . . . . . . . . . . . . . . . . . . . . 72

5.4 Extinction efficiency of a periodic array of 100 gold spheres of radius80 nm and periodicity of Λ = 600 nm. . . . . . . . . . . . . . . . . . 73

5.5 Extinction efficiency for four periodic arrays of 100 gold spheres ofradius 80 nm with periodicities of 400 nm, 500 nm, 600 nm. . . . . 74

5.6 Narrow (vertex angles π/5 and 4π/5) and wide (vertex angles 2π/5and 3π/5) rhombi prototiles of Penrose tiling. The three markeddistances which are the large diagonal of the narrow rhombus (d1),the large diagonal of the wide rhombus (d2) and d3 which denotesthe side of both rhombi. . . . . . . . . . . . . . . . . . . . . . . . . 75

5.7 A segment of the Penrose quasicrystals composed of the narrow andwide rhombi and the corresponding distances d1, d2, and d3 fromFigure 5.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8 Diffraction pattern of Penrose quasicrystal with the first three re-ciprocal vectors displayed. . . . . . . . . . . . . . . . . . . . . . . . 77

5.9 Scattering response of two finite aperiodic Penrose gold arrays withdifferent tile sides (540nm, 630nm) obtained using GMT with afinite incident beamwidth produced by a circular aperture of radiusa = 2λ placed at a distance of 17µm from the array. Values of Tand R were calculated from Eq. (4.41) and Eq. (4.44) respectively. . 78

5.10 Prototiles of the Danzer aperiodic tiling (θ = π/7). . . . . . . . . . 805.11 Danzer tiling generated after two iteration applied to prototile type

III with elements placed at vertices. . . . . . . . . . . . . . . . . . . 805.12 Extinction efficiency for three Danzer of 349 gold spheres of radius

80 nm with c/2 values of 420 nm, 480 nm, 560 nm. The plasmonicregion of gold is highlighted. . . . . . . . . . . . . . . . . . . . . . . 81

5.13 A segment of an optimized Danzer tiling with additional elementsplaced at the circumcenter of type III prototile. . . . . . . . . . . . 82

5.14 Extinction efficiency for native Danzer array (349 spheres) and theoptimized Danzer array (521 spheres). Both arrays have been scaledsuch that c

2= 560 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.15 Extinction coefficient (E = 1 − T ) for native Danzer and the op-timized Danzer arrays evaluated using GMT method with a finitebeamwidth. Both arrays have been scaled such that c

2= 560 nm. . . 83

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5.16 Local fields (V/m) for a periodic array of 225 gold nanospheres withminimum surface to surface distance of 20 nm. . . . . . . . . . . . . 85

5.17 Local fields (V/m) for a Penrose array of 251 gold nanospheres withminimum surface to surface distance of 20 nm. . . . . . . . . . . . . 86

5.18 Local fields (V/m) for a Danzer array of 248 gold nanospheres withminimum surface to surface distance of 20 nm. . . . . . . . . . . . . 86

5.19 Local fields (V/m) for an A-B array of 225 gold nanospheres withminimum surface to surface distance of 20 nm. . . . . . . . . . . . . 87

6.1 The unit cell structure of the optimized metamaterial grating withp = 2.05µm and t = 468 nm. . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Simulated reflection and transmission coefficients of the optimizedmetamaterial structure in free space. . . . . . . . . . . . . . . . . . 95

6.3 Effective index of refraction for the optimized metamaterial mirror. 956.4 Effective normalized impedance for the optimized metamaterial mir-

ror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.5 Effective permittivity for the optimized metamaterial mirror. . . . . 966.6 Effective permeability for the optimized metamaterial mirror. . . . . 976.7 Electric field distributions for the optimized structure at resonance

(λ0 = 3.3µm) in the xy-plane. . . . . . . . . . . . . . . . . . . . . . 976.8 Electric field distributions for the optimized structure at resonance

(λ0 = 3.3µm) in the yz-plane. . . . . . . . . . . . . . . . . . . . . . 986.9 FESEM image of the fabricated metamaterial mirror with an inset

of a magnified unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . 996.10 Simulated and measured transmission and reflection spectra of 4×

4 mm2 sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.1 A fourth degree polynomial with roots in [0, 1]. . . . . . . . . . . . 1047.2 Resulting side-wall profile from the polynomial in Figure 7.1. . . . . 1057.3 Relative permittivity of ZnS over the visible range. . . . . . . . . . 1077.4 Unit cell of the optimized grating. . . . . . . . . . . . . . . . . . . . 1077.5 Four periods of the optimized grating structure. . . . . . . . . . . . 1087.6 Transmittance values for the optimized grating corresponding to

different values of incident TM radiation. . . . . . . . . . . . . . . . 108

A.1 Flowchart of a typical GA. . . . . . . . . . . . . . . . . . . . . . . . 111A.2 The mating process for two parents to produce two offspring. . . . . 112

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List of Tables

5.1 Geometrical properties of nano-spherical arrays. . . . . . . . . . . . 84

7.1 Mean transmittance values for different incidence angles over thevisible band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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Acknowledgments

First and foremost I would like to give my deepest and most sincere gratitude tomy parents and my dear brother, to whom this thesis is dedicated. Without theirlove, support, and encouragement this would not have been possible.

I would like to express my gratitude to my advisor Dr. Douglas Werner, for hissupport and commitment. His guidance and technical advice were instrumental inthe development of my research and this dissertation. I would also like to expressmy appreciation to Dr. Pingjuan Werner both as a member of my committee andfor her helpful assistance with many lab related issues.

I would like to thank the members of my committee, Dr. Pingjuan Werner,Dr. Julio Urbina, and Dr. Brian Weiner, for their time and effort.

I would like to thank the members of my lab that have come and gone through-out the duration of my time at Penn State. Their friendship and assistance havemade my experience in graduate school a pleasurable one.

I would like to thank my dear friends Abteen Vaziri, Dario Ferdows and SanjeevIyengar for their great friendship and support.

Finally, I would like to acknowledge the Applied Research Lab for their generousfinancial support throughout the duration of my time at Penn State.

xiv

Dedication

To Behnam, Farzaneh, and Amir

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Chapter 1Introduction

In 1982 while studying rapidly solidified aluminum (Al) alloys at Johns Hopkins

University, Dr. Dan Shechtman noticed that the electron diffraction pattern of cer-

tain Al and manganese (Mn) alloys displayed 10-fold rotational symmetry which

corresponded to 5-fold rotational symmetry of the alloy. He published his results

in 1984 [1] which at the time was highly controversial and was met with great

resistance in the academic community. The reason was that it was mathematically

impossible for a periodic lattice to possess 5-fold rotational symmetry and at the

time it was assumed that all crystals possessed translational symmetry (periodic-

ity). In the following years his results were reconfirmed by other groups and new

observations of such solids were made. In time such materials which possessed

order (defined in terms of diffraction pattern) but lacked translational symme-

try (aperiodic) came to be known as quasicrystals. In 2011 Dr. Shechtman was

awarded the Nobel Prize in Chemistry for the discovery of quasicrystals.

Since their discovery there has been great interest in utilizing quasicrystalline

geometries for a wide variety of electromagnetic (EM) and optical applications

such as ultra-wideband antenna arrays [2], surface-enhanced Raman scattering

(SERS) substrates [3], and electronic band gap materials [4]. The first thrust of

research contained in this dissertation presents applications of aperiodic geometries

in wideband antenna arrays and plasmonic nanoparticle arrays. It will be shown

that novel properties of quasicrystalline arrays gives them unique properties that

cannot be obtained using periodic geometries. Additionally we will utilize nature-

inspired optimization algorithms in order to further enhance the desired properties

2

of these arrays.

The second thrust of research in this dissertation focuses on designing all-

dielectric filters and metamaterial coatings for the optical range. In higher fre-

quencies metals tend to have a high loss and thus they are not suitable for many

applications. Hence dielectrics are used for applications in optical frequencies. In

particular we focus on designing two types of structures. First a near-perfect opti-

cal mirror is designed. The design is based on optimizing a subwavelength periodic

dielectric grating to obtain appropriate effective parameters that will satisfy the

desired perfect mirror condition. Second, a broadband anti-reflective all-dielectric

grating with wide field of view is designed. The second design is based on a

new computationally efficient genetic algorithm (GA) optimization method which

shapes the sidewalls of the grating based on optimizing the roots of polynomial

functions.

1.1 Ultra-Wideband Aperiodic Antenna Arrays

Recent development and advancements of micro-UAV (unmanned aerial vehicle)

swarms have brought a lot of attention to their potential uses for remote sensing

and intelligent gathering applications. They can be used in military applications in

hazardous circumstances such as monitoring for nuclear or chemical contamination.

There are also various civilian applications for these swarms such as environmental

monitoring and pollution control. One of the major challenges is designing suitable

arrays for micro-UAV swarms. Due to safety reasons and to avoid possible collision

between micro-UAVs it is desirable to keep the minimum separation distance be-

tween the elements several wavelengths. As a result typical periodic planar arrays

are not suitable, since for periodic arrays increasing the minimum element spacing

beyond one wavelength will lead to the appearance of grating lobes in the radia-

tion pattern. The research in this dissertation focuses on design optimization of

suitable antenna arrays for these swarms based on perturbation of certain types of

aperiodic tilings. It will be shown that using this method antenna arrays with very

wide bandwidths and low sidelobe levels can be designed. It will also be shown

that in conjunction with a phase compensation method these arrays show a large

degree of versatility to positional noise.

3

1.2 Quasicrystalline Plasmonic Nanoparticle Ar-

rays

Gold and silver nanoparticle arrays have attracted immense attention due to their

optical properties in the visible and infrared (IR) regions. It has been demonstrated

that such metallic nanoparticles with sub-wavelength dimensions display strong

optical extinction in the visible or IR spectrum. This resonant behavior is known

as the particle plasmon [5]. This plasmonic resonance makes metal nanoparticle

arrays great candidates for wide variety of applications such as SERS substrates

and biosensors. The optical properties of these arrays can be explained in terms of

the grating response of the array (photonic resonance) and the plasmonic response

of the spheres (plasmonic resonance). The photonic resonances can be analyzed as

the Bragg grating modes, which are due to the coherent scattering experienced as

the wavelength of the incident radiation approaches the periodicity of the array. Of

particular interest are the phenomena that occur when the photonic and plasmonic

resonances are in close proximity. This allows for the plasmonic fields to radiate

in the plane of the array, which leads to stronger coupling and further resonance

enhancement. Subsequently, a so-called photonic-plasmonic hybrid mode is excited

[6].

In periodic structures these hybrid modes usually have a narrow bandwidth due

to the inherently narrowband nature of the photonic resonance. On the contrary,

hybrid modes in aperiodic structures can have more desirable properties because

they possess multiple periodicities that if designed properly can create hybrid mode

coupling over a relatively wide bandwidth.

A key challenge in evaluating the EM properties of aperiodic geometries is the

lack of analytical tools. Traditionally the EM properties of metamaterials and

photonic crystals have been evaluated by exploiting their translational symmetry

(periodicity). This approach significantly simplifies the analysis by applying ap-

propriate boundary conditions and only requiring Maxwell’s equations to be solved

for one unit cell, rather than the entire structure. However, quasicrystals gener-

ally lack translational symmetry. One particular approach that can be applied to

large aggregates of randomly sized spheres in a dielectric medium is the General-

ized Multiparticle Mie Theory (GMT) developed by Xu [7].The solution obtained

4

by GMT is a complete solution, based on rigorous analytical expressions which

account for all couplings and multi-particle interactions. This is particularly im-

portant when studying metal particles in their plasmonic resonance regions where

many approximate methods (e.g. discrete dipole approximation method) tend to

produce relatively inaccurate calculations. Furthermore, the GMT solution also

includes near-field results which make it suitable for studying phenomena such

as surface-enhanced Raman scattering (SERS). Traditionally the GMT method is

applied based on an incident plane wave excitation. However since a plane wave

has an infinite beamwidth, then it is not possible to define reflection and trans-

mission coefficients in the usual sense when considering the analysis of finite-size

(truncated) arrays via the GMT method. Obtaining reflection and transmission

coefficients are also necessary if one is interested in extracting effective medium

parameters for subwavelength arrays. To overcome this issue, we implemented

GMT based an in incident beam with finite beamwidth. This was done by emu-

lating conditions similar to experimental setup where a narrow incident beam is

created by placing a circular aperture in front of an illuminating plane wave. The

first step in this process was the expansion of diffracted waves in terms of spher-

ical harmonics. We were able to derive analytical expressions for the diffracted

fields. Expansion of diffracted fields in terms of spherical harmonics also led to

new analytical expressions for two important integrals involving Bessel, associated

Legendre and trigonometric functions which arise in electromagnetic diffraction

problems. This approach allowed for defining generalized scattering coefficients in

terms of far-field energy fluxes.

We also study local field enhancements in aperiodic arrays of gold nano-spheres.

It will be shown that despite the fact that aperiodic arrays are much sparser than

periodic arrays, they display larger local field enhancements. Local fields can be

further enhanced by optimally placing perturbing spheres within the prototiles

that comprise the aperiodic arrays.

5

1.3 All-Dielectric Optical Filters and Metamate-

rials

In recent years there has been considerable interest in the filtering properties of

subwavelength periodic dielectric gratings [8, 9, 10]. At infrared (IR) wavelengths

dielectric gratings have been shown to be superior to traditional metallic frequency

selective surfaces due primarily to their low absorption loss. If the periodicity is

extended to two dimensions and rotational symmetry is applied to the structure the

response will be identical for normal incident plane waves for both polarizations

(TE and TM). Traditionally these structures have been studied using rigorous

coupled-wave analysis (RCWA) [11] method which treats the periodic layers as

waveguides and the appropriate boundary conditions are imposed. The resulting

eigenvalue problem is then solved to find all the modes that can be supported by

the periodic structure. If the incident wave can excite one of the modes supported

by the grating, it can lead to resonant reflection or transmission [12]. For our

first design our goal is to design a filter with a near-perfect reflection band (i.e. a

near-perfect optical mirror) centered at the mid-IR band of 3.3 µm . We treat the

doubly periodic gratings as metamaterial structures and apply effective medium

theory to guide the design process. The appropriate optical properties that will

satisfy the desired ’perfect mirror’ condition are first derived. The structure of the

metamaterial grating is optimized using a single point cross-over binary GA. The

scattering parameters of the structures are evaluated using a periodic finite-element

boundary-integral (PFEBI) code. The scattering parameters are then employed to

derive the effective parameters using a suitable inversion algorithm. The goal for

the second all-dielectric filter design is a broadband anti-reflective grating with wide

field of view. We propose a computationally efficient GA optimization strategy

based on optimizing the roots of a polynomial. This method has the advantage

of having a relatively small number of variables to be optimized and at the same

time allows for more complex side-wall profiles.

6

1.4 Original Contributions

1.4.1 Ultra-Wideband Aperiodic Antenna Arrays

The research conducted in implementation and optimization of quasicrystalline

geometries for ultra-wideband antenna application has led to original contributions

in several areas including the

• Development of a new modified dynamic method for perturbing aperiodic

tiling arrays which can substantially increase their bandwidth.

• Development of volumetric aperiodic antenna arrays based on aperiodic tiling

of the 3D space. The same perturbation methods that were utilized for

planar aperiodic arrays can be used to enhance the radiation properties of

these volumetric arrays.

• Design and optimization of ultra-wideband antenna array for micro-UAV

swarms based on perturbation of aperiodic tilings. It was shown that along

with a phase compensation method, these arrays show a large degree of

robustness to positional noise.

• Some of the developments of this research have been published in the IEEE

Transactions on Antennas and Propagation [13].

1.4.2 Quasicrystalline Plasmonic Nanoparticle Arrays

The research conducted in the analysis and optimization of quasicrystalline plas-

monic nanoparticle arrays required a new implantation of GMT which led to orig-

inal contributions in several areas including the

• New closed form expressions for fields diffracted by a circular aperture in

terms of vector spherical wave functions (VSWF).

• New closed form expression for two definite integrals which can be of great

interest on vector diffraction problems.

• Generalized scattering coefficients defined for finite-sized spherical arrays.

7

• New optimization methods for aperiodic tilings for plasmonic nanoparticle

applications.

1.4.3 All-Dielectric Optical Filters and Metamaterials

The research conducted in design and optimization of all-dielectric optical filters

and metamaterials led to original contributions in several areas including the

• Derivation of a new set of effective parameters for a perfect mirror dielectric

slab with zero index of refraction.

• Design and optimization of an all-dielectric zero-index metamaterial layer.

The design was subsequently fabricated and the measured results were in

great agreement with the simulated results.

• Introduction of a new computationally efficient method to create sidewall

profiles for dielectric gratings based on optimizing the roots of polynomial

functions.

• Designed a broadband transmission grating with wide field of view using the

polynomial sidewall profile method mentioned above.

1.5 Overview

This section provides a brief overview of the material discussed in the remainder

of this dissertation. Since a large portion of this dissertation deals with quasicrys-

talline arrays, we dedicated Chapter 2 to provide a brief introduction and overview

of some basic concepts relating to crystals, tilings, and quasicrystals. Chapter 3

examines the applications of aperiodic tilings in wideband antenna designs. Chap-

ter 4 introduces the GMT method and includes the derivation required for imple-

mentation of the GMT method with a finite beamwidth incident field. Chapter 5

includes several examples of applications of metallic nanoparticle arrays based on

aperiodic geometries and methods to further enhance their properties. Chapter 6

considers the optimization of metamaterial coatings using nature-inspired algo-

rithms which includes one optimized design which was subsequently fabricated

8

and characterized. Chapter 7 introduces a new efficient methodology to design

transmission gratings based on the optimization of polynomial functions.

Chapter 2Tilings, Crystals, and Quasicrystals

In this chapter we provide some basic background regarding quasicrystals and how

they relate to aperiodic tilings. A rigorous and fundamental study of quasicrys-

tals is quite complicated and requires a deep understanding and application of

group theory [14]. However quasicrystals can also be arrived at using aperiodic

tilings which is less mathematically involved and still provides a sufficient intuitive

understanding of the subject.

2.1 Periodic Tilings and Point Lattices and Sym-

metries

A tiling T in an n-dimensional Euclidean space En is a partition of that space into

a countable, non-intersecting number of tiles. Using mathematical notation, T is

a set of tiles T1, T2, . . . such that [14]:

i 6= j ⇒ intTi ∩ intTj = ∅∞⋃i=1

= En (2.1)

where intT denotes the interior of T . In general there are no limitations on the

shapes of tiles and a tiling can be completely random. However, we limit our

attention to those tilings whose tiles are copies of a finite set of shapes. The

elements of this finite set are known as prototiles. A most basic example would be

10

covering the 2D plane by identical squares. The prototile set of such tiling consists

of a single square. Another example would be to cover the 2D plane by black and

while squares in a chessboard pattern. In this case the tiling is defined by two

prototiles: one black square and one white square with identical sides.

The periodicity of a tiling T of En can be established by determining n linearly

independent vectors ~b1, . . . ,~bn such that the translation M = m1~b1 + . . . + mn

~bn

where m1, . . . ,mn are integers will map T onto itself. Since the translation Mmaps the tiling onto itself, a periodic tiling is said to have translational symmetry.

A set of vectors ~b1, . . . ,~bn will form a basis set for T . It is also important to note

that the set of basis vectors are not unique. Figure 2.1 shows a periodic tiling

with two prototiles and two sets of basis vectors. Lattices are closely related to

tilings. Once a tiling is known a corresponding lattice can be generated, simply by

placing points at the vertices of each tile. Lattices resulting from periodic tilings

are known as point lattices and denoted by Lp.

Figure 2.1: An example of a periodic tilings of 2D plane with two prototiles A andB. Two sets of basis vectors are shown.

Order and symmetry in solids can be deduced from their diffraction pat-

terns. Mathematically there is a close relationship between diffraction patterns

11

and Fourier transform. Dirac delta function [15] introduces a convenient way to

represent a set of scatterers as a summation of infinitesimal points. We start by

considering Λ =~d1, ~d2, . . .

as a discrete set of points in En where ~dk is an

n-dimensional vector representing the location of the k-th point in Λ. We can

represent our set as the following summation of Dirac delta functions[14]

ρΛ(~x) =∑~dk∈Λ

δ(~x− ~dk) (2.2)

where δ(~x− ~a) is the Dirac delta function with the following properties [15]:

δ(~x− ~a) =

0 if ~x 6= ~a

∞ if ~x = ~a(2.3)

∫~x∈En

δ(~x− ~a)d~x = 1 (2.4)

The Fourier transform of a function f(~x) , ~x ∈ Rn is denoted by f(~s) and defined

by the following integral:

f(~s) =

∫Rn

f(~x) exp (−2πi~x · ~s) d~x (2.5)

It is simple to show that the Fourier integral represents a linear operation. Hence

we can denote it by F and write an equivalent form for Eq. (2.5) as

F(f) = f ⇔ f(~s) =

∫Rn

f(~x) exp (−2πi~x · ~s) d~x (2.6)

Using the integral in Eq. (2.5) it can be shown that F(δ(x−a)) = exp(−2πias) and

from the linearity of F it can be shown that the set of points defined in Eq. (2.2)

has the following Fourier transform

ρΛ(~s) = F

∑~dk∈Λ

δ(~x− ~dk)

=∑~dk∈Λ

exp(−2πi~dk · ~s

)(2.7)

The Fourier diffraction pattern is defined as the real-valued function J(~s) = |ρΛ(~s)|.Fourier diffraction pattern is a very useful tool in studying order and symmetry

12

of point sets. At the most fundamental level, a crystal is defined as a solid with a

discrete diffraction diagram. Figure 2.2 shows a periodic set of points (left) and the

corresponding Fourier diffraction pattern (right) calculated according to Eq. (2.7).

As it can be seen, the diffraction pattern displays discrete peaks which as defined,

correspond to a crystalline structure. Figure 2.3 shows a random set of points (left)

and the corresponding Fourier diffraction pattern (right). As it can be seen the

diffraction pattern is relatively flat and does not contain any discrete peaks (the

peak at the origin can be ignored since it would exist for any planar formation).

The diffraction pattern contains all the symmetry information of a given point set.

In fact when speaking of symmetry of a crystal, we are referring to the symmetry

implied by the diffraction pattern.

So far we have discussed the translation symmetry of the periodic point lattices.

A translation of a point lattice by any integer linear combination of its basis vectors

will map the lattice onto itself. Rotational symmetry is another symmetry which

is studied in crystals. It is trivial to verify that rotating a planar lattice by 2π/4

will map the lattice onto itself. Hence a planar point lattice is said to possess a 4-

fold rotational symmetry. Rotational symmetry is also displayed in the diffraction

pattern. For example the diffraction pattern shown in Figure 2.2 displays a 4-

fold rotational symmetry which corresponds to a point lattice. In fact rotational

symmetries are described in terms of the diffraction patterns rather than actual

point sets. Thus if the diffraction pattern of a point set is unchanged by a 2π/n

rotation, the point set is said to possess n-fold rotational symmetry. Here we note

an important theorem regarding periodic lattices in two-dimensional and three

dimensional spaces (a proof can be found in [14]):

The crystallographic restriction theorem: Rotational symmetries of order five

and those greater than six are impossible in periodic lattices in two-dimensional

and three dimensional spaces [14].

Figure 2.4 shows a set of points (left) and its corresponding diffraction pat-

tern. As it can be seen the diffraction pattern displays a discrete pattern, thus

it can be classified as a crystal. Also upon closer inspection it can be seen that

the diffraction pattern displays a 10-fold rotational symmetry hence it cannot be

periodic (crystallographic restriction theorem). The geometry shown in Figure 2.4

is generated from a Penrose tiling of the plane which is a member of aperiodic

13

Figure 2.2: A periodic point lattice (left) and its Fourier diffraction pattern (right).

tilings which we introduce in the next section.

2.2 Aperiodic Tilings and Quasicrystals

In the previous section, a set of necessary and sufficient conditions were stated

to establish periodicity in a given tiling. Basically to demonstrate periodicity in

n-dimensional space, we had to determine a set of n linearly independent basis

vectors, such that a translation of the tiling by an integer linear combination of

the basis vectors did not change the tiling. If such a set of basis vectors does not

exist, a tiling is said to be aperiodic.

There are several well-known examples of linear aperiodic tilings. Fibonacci

sequence is an example of a linear aperiodic tiling. The prototile set of the tiling

consists of two elements which we denote as L, S. The sequence can be generated

starting with one of the elements and iteratively applying the substitution rule

14

Figure 2.3: A random set of points (left) and its Fourier diffraction pattern (right).

L→ LS , S → L [16].

L→ LS → LSL→ LSLLS → LSLLSLSL→ . . . (2.8)

Historically it was always assumed that all crystals possess translational sym-

metry. When in 1984 Dan Shechtman published his results of observing a diffrac-

tion pattern with ten-fold rotational symmetry [1], he was met with great resistance

and skepticism in the academic community. According to the crystallographic

restriction theorem ten-fold rotational symmetry was impossible for a periodic

structure, and at the time the prevailing assumption was that crystals always have

translational symmetry. Eventually his findings were confirmed and this new class

of solids which lacked translational symmetry but possessed an order of rotational

symmetry forbidden by the crystallographic restriction theorem came to be known

as quasicrystals. In 2011 Dr. Shechtman was awarded the Nobel Prize in Chem-

istry for the discovery of quasicrystals.

Mathematicians established the theoretical foundations for quasicrystals before

15

Figure 2.4: A set of points generated from the Penrose tiling (left) and the corre-sponding Fourier diffraction pattern (right).

their actual discovery in nature as aperiodic tilings of the plane starting in 1960s.

Probably the best known aperiodic tiling is the Penrose tiling discovered by Sir

Roger Penrose in 1974 [17]. Figure 2.5 shows the pattern of the Penrose tiling

which is built from two triangle prototiles and has a 5-fold rotational symmetry.

From the set of prototiles, there are several ways to generate the tiling. The

most intuitive procedure is to place tiles next to each other according to specific

matching rules, which are meant to preserve the aperiodicity of the tiling. A more

systematic approach which lends itself better to programming is the use of an

iterative inflation and substitution process. Using this approach we can start with

a certain prototile and at each step it is inflated and filled with other prototiles

according to specific substitution rules. In order to obtain a point set from an

aperiodic tiling, points are placed at the vertices of each tile. Here we include a

more detailed description for three types of planar aperiodic tilings which will be

utilized in upcoming chapters.

16

Figure 2.5: Penrose aperiodic tiling.

2.2.1 Penrose Aperiodic Tiling

Discovered in 1974 by Sir Roger Penrose, the Penrose aperiodic tiling is built from

two triangular prototiles shown in Figure 2.6 where θ = π/5 and τ = 1+√

52

is the

golden ratio.

Figure 2.7 displays four iterations of the inflation and substitution process

applied to the red prototile from Figure 2.6. It is interesting to note that the

number of tiles grows according to the Fibonacci sequence (1→ 2→ 3→ 5→ 8→. . .). As mentioned before Penrose tiling possesses 5-fold rotational symmetry. The

rotational symmetry is clear in Figure 2.5. The diffraction pattern for a Penrose

set point is shown in Figure Figure 2.4.

2.2.2 Danzer Aperiodic Tiling

Danzer tiling of plane was discovered by Ludwig Danzer [18]. The prototile set for

this tiling consists of three triangles shown in Figure 2.8 where θ = π/7 and a, b,

17

Figure 2.6: Prototiles of the Penrose tiling (θ = π/5 and τ = 1+√

52

).

Figure 2.7: Iterative inflation and substitution of Penrose prototiles.

and c are related by the law of sines

a

sin θ=

b

sin 2θ=

c

sin 4θ(2.9)

We refer to the three prototiles in Figure 2.8 simply as type I, type II, and type

III (displayed from left to right). Danzer tiling has 7-fold rotational symmetry and

from the set of prototiles, the tiling can be generated either based on matching rules

or using the inflation and substitution method. Figure 2.9 displays two iterations

18

Figure 2.8: Prototiles of the Danzer aperiodic tiling (θ = π/7).

Figure 2.9: Two iterations of inflation and substitution applied to Danzer prototiletype III.

of the inflation and substitution applied to type III prototiles of the Danzer tiling.

2.2.3 Ammann-Beenker Aperiodic Tiling

Ammann-Beenker (A-B) aperiodic tilings were discovered in 1970s by Robert Am-

mann [19]. A-B tiling possesses 8-fold rotational symmetry and it consists of three

prototiles: a square, a rhombus (vertex angles π/4 and 3π/4), and an isosceles right

triangle. Similar to the previous two tilings, the simplest way to generate large

tilings is utilizing inflation and substitution method. Figure 2.10 shows two iter-

19

Figure 2.10: Two iterations of inflation and substitution applied to square prototileof A-B tiling.

Figure 2.11: Prototiles of the A-B tiling and their relative dimensions.

ations of the inflation and substitution applied to a square prototile. Figure 2.11

shows the relative dimensions of the prototiles where as it is shown, a denotes the

side of the square prototile.

2.3 Higher-Dimensional Approach

Using aperiodic tilings is one way to generate quasicrystalline arrays. The main

advantage of this method is that it is rather simple and does not require a large

amount of mathematical rigor. The main disadvantage of aperiodic tilings is that

only a very limited number of them have been discovered. The three aperiodic

20

tilings we have covered thus far possess 5, 7, and 8-fold rotational symmetries

respectively. There is another method which can generate quasicrystalline geome-

tries of any required order of rotational symmetry and does not require aperiodic

tilings. This method known as the higher dimensional approach [16], is based

on the fact that a quasicrystalline lattice in En with k-fold rotational symmetry

(k > n) can be obtained as an irrational projection of a point lattice in Ek onto

En.

As it will be seen in the upcoming chapters the lack of translational symme-

try makes it very challenging to analyze the electromagnetic (EM) properties of

systems based on quasicrystalline geometries. In recent years there has been some

efforts to use the higher-dimensional approach to study the EM properties of qua-

sicrystals [20]. Here as a simple example we demonstrate how a Fibonacci sequence

can be obtained as a projection of a 2D point lattice onto a line.

Figure 2.12 shows a standard 2D integer point lattice I2p . Thus using the

standard basis each lattice point can be represented as an integer pair (m1,m2).

Let l be a line through origin with a slope of 1τ. This would ensure that the line

l does not cross any other lattice points. Hence l forms an irrational subspace of

E2. We also define l′ as the orthogonal subspace to l. Furthermore, we define

two mappings Π and Π⊥ as orthogonal projections onto l and l′ respectively. The

Voronoi cell at the origin is denoted by V (0).

Let X be a subset of I2p whose cells are cut by l. It can be shown that Π⊥(X)

belongs to Π⊥(V (0)). Thus mathematically the X can be defined as [14]:

X =x ∈ I2

p |Π⊥(x) ∈ Π⊥(V (0))

(2.10)

It can be shown that Π(X) forms a Fibonacci sequence [14].

21

Figure 2.12: Higher-dimensional approach demonstrated to obtain a Fibonaccisequence as a projection of I2

p onto an irrational subspace.

Chapter 3Ultra-Wideband Aperiodic Antenna

Arrays

3.1 Introduction

The recent development of micro-UAV (Unmanned Aerial Vehicle) swarm technol-

ogy has generated a great amount of interest within the intelligence gathering and

remote sensing communities. Micro-UAV swarms generally consist of a variety of

simple, inexpensive sensors distributed across multiple airborne platforms. These

platforms are capable of sharing data and resources, leading to an aggregate sys-

tem that gathers and processes the information in order to perform higher order

operations and tasks. Because the sensors in a micro-UAV swarm are distributed

across multiple platforms, the system lacks a single point of failure. This makes

micro-UAV swarms a favorable solution for hazardous and hostile situations. For

example, a potential application for this technology is in the monitoring of the

environment for nuclear and chemical contamination [21]. In such a situation, a

swarm of micro-UAVs containing appropriate sensors could be flown over a large

area, relaying information back to the base if contaminates are discovered. This

methodology is advantageous to the single sensor approach because the micro-

UAVs could be distributed over a wide area with simple flight plans, reducing the

complexity and size requirements of the airborne platforms and allowing for the

collection of real time information across a wide region of interest.

23

In addition to environmental monitoring, there is interest in applying micro-

UAV swarms to radar imaging applications. In a conventional radar imaging sys-

tem, a single aircraft platform carries the instrumentation. In order to reduce the

risk to the system in hostile situations, the aircraft must operate from a safe dis-

tance with respect to the area of interest. In this scenario, the system must utilize

a high gain antenna that operates over a wide bandwidth in order to obtain the

required resolution. On the other hand, the micro-UAV approach is designed to

fly into hostile areas and gather data directly. The swarm is designed to fly over a

wide area, creating a large aperture that can provide high resolution imagery with-

out the use of wide-band receivers or other expensive components and integration.

In addition, the impact of aircraft attrition can be minimized by distributing re-

sources and allowing the flight formation to adapt in real time. For these reasons,

micro-UAV swarms can provide a cost effective solution for a variety of intelligence

gathering, surveillance, and reconnaissance applications.

However, two major challenges face micro-UAV swarm-based radar technology.

The first challenge is maintaining coherent beamforming despite the ever chang-

ing nature of the aperture. The effects of turbulence and other positional errors

can have a dramatic impact on the radiation pattern. In order to address this

issue, a robust phase compensation algorithm that allows for the resolution of the

mainbeam even in turbulent environments is introduced. The second challenge is

designing formations that can reduce the risk of midair collisions between micro-

UAVs. In order to resolve targets of interest, these systems must be functional at

frequencies that place the operating wavelength on the order of the aircraft size or

smaller. Therefore in order to avoid possible collisions, it is necessary to keep the

minimum separation distance between the aircraft on the order of several wave-

lengths. However, conventional array theory dictates that interelement spacings

beyond one wavelength in a periodic array will lead to the appearance of radiation

pattern grating lobes. These grating lobes essentially alias the radiation pattern,

causing errors that can seriously limit the accuracy and effectiveness of the system.

In this chapter, we discuss antenna arrays based on aperiodic tilings that can

operate effectively over wide bandwidths and inter-element spacings. Such arrays

were initially investigated in [22] and improved upon in [2]. Furthermore, we intro-

duce a modified and more dynamic optimization method for aperiodic arrays which

24

vastly improves their radiation properties. This method utilizes global optimiza-

tion techniques with robust nature-inspired geometries to form the sparse planar

array layouts. Due to their optimized aperiodicity, the radiation patterns of these

arrays possess low peak side-lobe levels and are devoid of grating lobes even over

multi-wavelength interelement spacings. In addition, these arrays offer efficient

solutions to the micro-UAV swarm problem by covering the same aperture with

a considerably smaller number of elements than a comparable size periodic array.

Several examples of an aperiodic micro-UAV swarm are discussed. Through these

examples, we illustrate how the combination of the phase compensation algorithm

with aperiodic array apertures can achieve high radiation pattern resolution in a

micro-UAV swarm-based radar imaging application.

3.2 Planar Antenna Arrays

Phased antenna arrays are employed in a wide variety of modern radar and high

performance communication systems. They offer a degree of agile pattern control

and low sidelobe levels well beyond single aperture antennas. We start by con-

sidering the far-field radiation pattern of an arbitrary planar antenna array in the

xy-plane. For simplicity isotropic radiating elements and ejωt time-dependence are

assumed where j =√−1 and ω = 2πf denotes the angular frequency. A general

expression for the normalized far-field radiation pattern of an array of N isotropic

radiators (unit excitation) is given by:

AF (θ, φ) =1

N

N∑n=1

exp j [kn · rn + βn] (3.1)

where rn is the position vector for element n and k = 2πλ

is the free-space wavenum-

ber. Also, n = [sin θ cosφ, sin θ sinφ, cos θ] is the unit vector pointing in the di-

rection of the far-field observation point at (θ, φ) and βn represents the phase of

the current excitation for element n. As mentioned we are considering arrays in

the xy-plane, hence rn = [rn cosφn, rn sinφn, 0] , where (rn, φn) denote the polar

coordinates of element n. Thus the array factor expression from Eq. (3.1) can be

25

simplified as:

AF (θ, φ) =1

N

N∑n=1

exp j [krn sin θ cos(φ− φn) + βn] (3.2)

The phase values βn can be adjusted to steer beam in a given direction. By

inspecting Eq. (3.2) it is evident that to steer the main beam toward (θ0, φ0), we

must have:

βn = −krn sin θ0 cos(φ0 − φn) (3.3)

3.2.1 Periodic Antenna Arrays

The most common lattices used to construct antenna arrays are periodic lattices

where all elements are spaced an equal distance apart. While providing a simple

architecture and a predictable performance, the periodic nature of these arrays

provides a limited bandwidth. The limited bandwidth is due to the appearance of

grating lobes, undesired beams of radiation equal to the intensity of the main beam.

Considering the standard periodic array with element spacing dmin, the grating

lobes appear as the the element spacing dmin exceeds the operating wavelength λ.

As an example Figure 3.1 shows the radiation pattern of a periodic array of

441 elements with main beam steered toward broadside and operating wavelength

λ = 2dmin. In all the radiation pattern plots throughout the paper the elevation

angle (θ) is measured radially along the x and y axes whereas the azimuthal angle

(φ) is measured azimuthally in the xy-plane. As it can be seen from the plot, since

the operating wavelength is larger than the element spacing the radiation is devoid

of any grating lobes. Figure 3.2 shows the radiation pattern of the same periodic

array of with main beam steered toward broadside and λ = dmin

2. In this case, the

radiation pattern displays several grating lobes which are highly undesirable.

Thus it can be seen that periodic antenna arrays cannot provide large operating

bandwidths due to the presence of grating lobes at higher frequencies.

3.2.2 Aperiodic Antenna Arrays

Antenna arrays based on aperiodic tilings were first proposed in [22]. Converting

an aperiodic tiling to an antenna array is a straightforward process and simply

26

Figure 3.1: Radiation pattern of a periodic array of 441 isotropic radiators withmain beam steered toward broadside and operating wavelength λ = 2dmin.

requires placing antenna elements at the vertices of the tiling. As a direct con-

sequence of the lack of translational symmetry, the radiation patterns of these

aperiodic tiling antenna arrays are void of grating lobes regardless of the oper-

ating frequency. However in their native form, these arrays are not suitable for

wideband applications due to their relatively high side-lobes. Danzer aperiodic

tiling was introduced in Chapter 2. The tiling consists of three prototiles shown

in Figure 3.3 and can be generated using the inflation and substitution process.

Figure 3.4 shows the geometry of an array of 659 elements that was generated

from a Danzer tiling. We denote the minimum, maximum, and average spacing

of the array as dmin, dmax, and davg. Of course such distinction does not apply to

standard periodic arrays. In a Danzer array these three values are roughly related

bydmin

1≈ davg

1.05≈ dmax

2.25(3.4)

Figures 3.5 and 3.6 show the normalized radiation patterns of the Danzer ar-

ray displayed in Figure 3.4 at operational wavelengths of λ = 2dmin and λ = dmin

2

27

Figure 3.2: Radiation pattern of a periodic array of 441 isotropic radiators withmain beam steered toward broadside and operating wavelength λ = dmin

2.

respectively. As it can be seen from Figure 3.6 even as the minimum spacing ex-

ceeds the operating wavelength the radiation pattern does not display any grating

lobes. However using the standard value of −10dB as the sidelobe criteria for our

definition of bandwidth, we see that sidelobe values are too high.

3.3 Optimization of Aperiodic Tiling Arrays

As it was shown in the previous section, antenna arrays based on aperiodic tilings in

their native form are not suitable for wideband applications due to their relatively

high sidelobes. To overcome this issue, a perturbation method was developed in

[2]. This method starts by placing a fixed and equal number of additional points

within the boundary of each prototile of an aperiodic set. The inflation process

is then carried out to generate the tiling. Converting this perturbed tiling to

an array yields elements at the vertices of the tiling as well as elements at the

perturbation points. The resulting array also needs to be scaled to obtain the

desired minimum spacing. As a result of the inflation process, the relative position

28


of each perturbation element within a prototile is conserved. For each perturbation

element, the location is specified by two variables and these variables are optimized

to obtain the best radiation properties. The optimal location of the perturbation

elements can be chosen by a genetic algorithm (GA). GAs are very well established

methods for performing global optimization of electromagnetic problems. They are

based on evolutionary principles and natural selection. A brief introduction of GA

is included in Appendix A. For a more complete description of GAs and their

applications in electromagnetics, the reader is referred to reference [23].

In this chapter a modified perturbation method is used to enhance the radiation

properties of the arrays. In the method developed in [2] the number of perturbation

elements for each prototile is predetermined and all the prototiles have the same

number of perturbation elements. The process can be made more dynamic by

letting the optimizer choose the number of perturbation points for each prototile

and then find their optimal locations. Using this method, in the final design,

different prototiles can (and usually do) have a different number of perturbation

points. To make the optimization less complex usually the range for the number

of perturbation points for each prototile is fixed and it is left to the GA to pick the

29

Figure 3.4: Geometry of an aperiodic array of 659 elements generated from aDanzer tiling.

optimal number of points within this range. Intuitively, this modified approach

makes more sense, since different prototiles have different areas and one can place

more perturbation elements in larger prototiles.

At this point, we discuss two examples of optimized arrays based on Danzer

aperiodic tilings. The radiation properties of the arrays are further improved

by optimally placing the appropriate number of radiating elements inside each

prototile. Assuming there are m, n, and p points for the type I, type II, and

type III prototiles respectively, a total of 2(m + n + p) variables are required

to determine the location of these perturbation points. In addition, the optimal

number of points in each prototile (m,n, p) has to be determined. Hence a total of

2(m+n+p)+3 variables must be optimized for each array. Each tiling is scaled so

that the elements will have a minimum spacing of 0.5λ at the principal frequency

f0 (with corresponding wavelength λ) and we consider a circular aperture with a

radius of 12λ for the array. Also, the range for the number of perturbation points

30

Figure 3.5: Radiation pattern of an aperiodic Danzer array of 659 isotropic radia-tors with main beam steered toward broadside and operating wavelength λ = 2dmin.

for each prototile is 0 − 5. Hence for a Danzer tiling which has three prototiles

there are a total of 33 optimization variables. As mentioned earlier these variables

are optimized using a GA. For our purpose we use a binary coded GA with single

point cross-overs. Moreover, even though the optimal number of perturbation

points for each prototile can be less than the maximum number allowed, within the

GA optimization process we have incorporated the maximum number of variables

in each chromosome. When the number of perturbation points is less than the

maximum, the additional variables are simply disregarded.

For the first design our goal is to have an array which minimizes the side-lobe

level at f = 4f0 which corresponds to a minimum spacing of 2λ. Figure 3.7 shows

the geometry of the optimized array. It has a total of 267 elements. The number

of perturbation points for the type I, type II, and type III prototiles are 3, 4,

and 4 respectively. Figure 3.8 shows the perturbed prototiles with the additional

optimized elements and Figure 3.9 shows a segment of the tiling generated using

these perturbed prototiles.

A closer inspection of the geometry in Figure 3.7 also reveals the rotational

31

Figure 3.6: Radiation pattern of an aperiodic Danzer array of 659 isotropic radia-tors with main beam steered toward broadside and operating wavelength λ = dmin

2.

symmetry which is characteristic of aperiodic tilings. The array has a PSLL of

−16dB at f = f0 and a PSLL of −13.62dB at f = 4f0 (the array performance was

optimized for the latter frequency). Figure 3.10 shows the sidelobe level perfor-

mance of the array and for comparison the sidelobe level performance of a periodic

array and a standard Danzer array are also shown. Using the standard value of

−10dB as the sidelobe criteria for our definition of bandwidth, it can be seen that

the optimized array far outperforms the periodic and the standard Danzer array.

Also from the plot it can be seen that the bandwidth of the optimized array ex-

tends well beyond the targeted design frequency. From the plot it can be seen

that this array has a bandwidth that extends up to 14.4f0. Figure 3.11 shows

the normalized radiation pattern for this array at f = 4f0 which corresponds to a

minimum spacing of 2λ.

For the second design our goal was to design an ultra-wideband array. We

attempted to maximize the sidelobe suppression at f = 30f0 which corresponds to

a minimum spacing of 15λ. Figure 3.12 shows the perturbed prototiles with the

additional perturbation elements and Figure 3.13 shows the resulting array. The

32

Figure 3.7: Geometry of optimized Danzer array (267 elements) for maximumsidelobe suppression at f = 4f0.

Figure 3.8: Three optimized Danzer prototiles with additional perturbation ele-ments used to generate the antenna array in Figure 3.7.

optimized array has a total of 505 elements. As it can be seen from Figure 3.12,

33

Figure 3.9: A segment of the Danzer tiling generated with the perturbed prototilesshown in Figure 3.8.

the number of perturbation points for the type I, type II, and type III prototiles

are 3, 3, and 4 respectively. The array has a PSLL of −17.8dB at f = f0 and

a PSLL of −11.66dB at f = 30f0 (the array performance was optimized for the

latter frequency). A significant sidelobe suppression exits far beyond the targeted

optimization frequency resulting in a very large operating bandwidth of 63 : 1.

Figure 3.14 shows the sidelobe level performance of the array and for comparison

the sidelobe level performance of a periodic array and a standard Danzer array are

also shown. Figure 3.15 illustrates the normalized radiation pattern for this array

at f = 30f0 which corresponds to a minimum spacing of 15λ.

34

Figure 3.10: Sidelobe level performance of the Danzer array optimized for side-lobesuppression at f = 4f0. The performances of a uniform periodic array as well as abase Danzer array are also shown for comparison.

Figure 3.11: The ideal radiation pattern for the 267 element aperiodic tiling arrayshown in in Figure 3.7 at operating wavelength λ = dmin

2.

35

Figure 3.12: Three optimized Danzer prototiles with additional perturbation ele-ments used to generate the antenna array in Figure 3.13.

Figure 3.13: Geometry of optimized Danzer array (505 elements) for maximumside-lobe suppression at f = 30f0.

36

Figure 3.14: Side-lobe level performance of the Danzer array optimized for side-lobe suppression at f = 30f0. The performances of a uniform periodic array aswell as a base Danzer array are also shown for comparison.

3.4 Positional Noise Analysis and Error Correc-

tion

Due to their distributed nature, micro-UAV swarm-based arrays are subject to the

effects of turbulence and other positional errors. In order to resolve a coherent

radiation pattern, it is essential to understand the nature of these errors and pro-

vide appropriate phase compensation for each of the individual antenna elements.

We start by considering the effects of positional noise on the radiation pattern of

an array. The expression in Eq. (3.1) describes the ideal radiation pattern of an

antenna array. By ideal we mean that the position of each element n is exactly

rn. However in actuality due to their distributed nature, the elements are subject

to some type of positional noise. A general expression for the normalized far-field

37

Figure 3.15: Normalized array factor for the optimized Danzer array with 505elements at f = 30f0.

radiation pattern of an array in the presence of turbulence is given by:

AF (θ, φ) =1

N

N∑n=1

exp j [kn · (rn + rn) + βn] (3.5)

where rn defines the positional noise associated with element n. This noise

is represented by a three dimensional Gaussian random variable [rnxrnyrnz] ∼N (0,Σ). It can be reasonably assumed that the covariance matrix is diagonal

Σ = diag(σ2x, σ

2y , σ

2z

)which implies that the error terms in each of the three di-

mensions are uncorrelated. Also for simplicity and without loss of generality it is

assumed that σ2x = σ2

y = σ2z = σ2. Now the positional error vector can be defined

as rn = [rnxrnyrnz] · [xyz]T . Figure 3.16 illustrates how these vectors are used

to describe the overall micro-UAV swarm geometry. At this point in the discus-

sion, it is desirable to separate the phase term of each antenna element into two

components.

βn = βin + βcn (3.6)

38

Figure 3.16: Geometry of a micro-UAV formation. In this illustration, rn representsthe nominal position of aircraft n, rn represents the error due to turbulence, andn is the unit vector pointing in the direction of the far-field observation point at(θ, φ).

where βin represents the ideal phase of element n and βcn represents the correction

phase of element n. The goal of the ideal phase term is to point the mainbeam of

the array in the direction of (θ0, φ0), creating a uniform phase front in the planes

normal to a vector n0. The correction phase term is used to minimize the effects

of positional noise on the mainbeam of the radiation pattern. By separating the

phase term into these two variables, it becomes possible to define the radiation

pattern in terms of element patterns based on ideal positions, AFn, and terms

governing the positional error and subsequent phase correction of each antenna

element, AF n:

AF (θ, φ) =1

N

N∑n=1

[1 + AF n(θ, φ)

]AFn(θ, φ) (3.7)

where,

[1 + AF n(θ, φ)

]= exp j [kn · rn + βcn] (3.8)

AFn(θ, φ) = exp j [kn · rn + βin] (3.9)

39

At the center of the mainbeam, the expression simplifies to be:

AF (θ0, φ0) =1

N

N∑n=1

exp j [kn0 · rn + βcn] (3.10)

Therefore the phase required to eliminate the effects of positional noise on the

mainbeam is defined as:

βcn = −kn0 · rn (3.11)

where rn is determined by onboard inertial navigation systems (INS) or another

swarm based guidance network. Figure 3.17 illustrates how this phase correction

factor is used to create a coherent mainbeam. In this figure, several aircrafts are

jostled from their ideal positions in the xy plane. The swarm is still capable of

pointing its mainbeam in a negative z direction by ensuring the transmitted signals

from each element are in phase as they cross the xy plane. At this point it is

Figure 3.17: Effect of the phase correction factor in the direction of the mainbeam.In this example, the mainbeam of the radiation pattern is pointed in the directionn0 = −z. The phase correction term βcn is used to ensure the phase of the signalfrom aircraft n is equal to zero at z = 0. This process is repeated for each aircraft,ensuring a beam can be resolved in the n0 direction.

possible to define a linear expression for the positional error and phase correction

term. If the argument of the expression is small, we can use the Taylor series

approximation expx ≈ 1 + x to define an approximation for AF n given by

AF n(θ, φ) ≈ j [kn · rn + βcn] (3.12)

40

Since the orientation of rn is uniformly random in all directions, we can replace

the vector by the magnitude of the random variable in any given direction, rn ∼N (0, σ2). If we include the phase correction term used to minimize errors in the

mainbeam, expression in Eq. (3.12) reduces to:

AF n(θ, φ) ≈ j [k(n− n0) · rn] (3.13)

Here in order to analyze AF n, it is helpful to define AF n(θ, φ) = [k(n− n0) · rn]

such that AF n(θ, φ) = jAF n(θ, φ). At this point, the angle ψ, illustrated in

Figure 3.18, is defined as the angle between n0 and n (i.e. the angle the observation

point deviates from the boresight of the array). Applying the definition of ψ, an

Figure 3.18: Illustration of angle ψ, the angle between the observation direction,n and the boresight of the array, n0.

expression for the magnitude of the error is derived:

AF n(θ, φ) = krn

√sin2 ψ + (1− cos2 ψ)

AF n(θ, φ) = krn√

2(1− cosψ) (3.14)

This can be represented as a ψ dependent Gaussian random variable:

AF n(θ, φ) ∼ N(

0,(kσ√

2(1− cosψ))2)

(3.15)

Finally the overall error of the phase corrected system can be represented by:

AF (θ, φ) =1

N

N∑n=1

AF n(θ, φ)AFn(θ, φ) (3.16)

41

which is a weighted sum of N independent random variables, AF n. It is also

important to note that in Eq. (3.16) we have disregarded j since |j| = 1 and it will

have no effect on the overall value of the error. In general if a random variable y is

defined as the average of a linear combination of N uncorrelated Gaussian random

variables xi for i = 1, 2, ..., N such that

y =1

N

N∑n=1

anxn , xn ∼ N (µn, σ2n) (3.17)

it can easily be shown that y will also be Gaussian with mean µy and variance σ2y

of

µy =1

N

N∑n=1

anµn (3.18)

σ2y =

1

N2

N∑n=1

a2nσ

2n (3.19)

Hence the variance of the overall system can be expressed as

σ2system ≈

1

N2

N∑n=1

[AFn(θ, φ)]2 σ2n (3.20)

where σ2n is the variance of AF n. It is important to note that since the observation

point and mainbeam focus are in the far-field, the variance, σ2, is equivalent for

every element of the array. Therefore, we can use the expansion

σ2system ≈

1

N2

N∑n=1

[AFn(θ, φ)]2 σ2 (3.21)

Finally, since excitation amplitudes are equal and normalized, max (AF 2n) = 1, we

can establish a bound on the variance of the system:

σ2system ≤

2k2σ2(1− cosψ)

N(3.22)

This expression indicates how our swarm-based array will operate in a turbulent

environment and what factors drive overall system level performance. First it is im-

42

portant to note that the error of the system increases as the operational frequency

increases; however, as long as the tolerance of the electrical spacing remains the

same, the error remains independent of aperture size. This indicates that sparse

arrays are no more susceptible to turbulence than standard arrays. Next, the error

of the system is inversely proportional to the number of elements in the array,

indicating that large-N arrays are more robust to positional tolerances than ar-

rays with a smaller number of elements. Finally, the error of this system varies

with respect to distance from boresight. The error correction algorithm reduces

the error for observation points within 60 from broadside. It remains possible for

larger errors at angles greater than 60; however, because most element patterns

have beamwidths much less than 120, the algorithm is effective in reducing the

performance degradation seen by the swarm through turbulence.

3.5 Examples

In this section we present some examples of error correction algorithm that was

developed in the previous section applied to periodic and aperiodic planar arrays

with simulated positional noise. For the first example we consider a standard

periodic planar array. For our second example we consider the optimized Danzer

array from Section 3.3 with 267 elements that was optimized for minimum side-

lobe level at f = 4f0 which corresponds to a minimum spacing of 2λ. As it

will be demonstrated while the algorithm does restore the main beam for both

types of arrays, it did not eliminate the grating lobes associated with the periodic

arrays. However the aperiodic arrays still displayed no grating lobes and a large

suppression of the side-lobes over a wide operating bandwidth after the application

of the error correction algorithm. These desirable properties of aperiodic arrays

in conjunction with the error correction algorithm can be exploited for creating

robust swarm formations.

3.5.1 Periodic Array

At this point we consider the performance of a micro-UAV swarm based on a

periodic array. In order to reduce the risk the micro-UAVs will collide in midair,

43

the minimum spacing between elements is set to be 2λ. The lattice is truncated

by a circular aperture with a radius of 48λ, for a total number of 1793 elements.

Figure 3.19 illustrates the ideal radiation pattern for this periodic micro-UAV

formation. As it can be seen from the plot, the radiation pattern exhibits several

grating lobes. When the formation is corrupted by a random zero-mean Gaussian

Figure 3.19: The ideal radiation pattern for a micro-UAV swarm based on a 1793element periodic array.

positional noise (σ = 0.1λ), the mainbeam of the radiation pattern disappears

into the incoherent sidelobes of the array (see Figure 3.20). However, Figure3.21

illustrates that when the phase compensation algorithm is applied to the array,

the mainbeam of the pattern once again is able to be resolved. Nevertheless, it is

apparent that the grating lobes nearest to the mainbeam are still present in the

corrected periodic micro-UAV swarm pattern. It is for this reason that the periodic

lattice is insufficient for micro-UAV swarm applications.

3.5.2 Optimized Aperiodic Array: Broadside

We first consider the example of a micro-UAV swarm that is based on the 267

element aperiodic tiling array shown in Figure 3.7 with its main beam steered at

44

Figure 3.20: The corrupted radiation pattern of a 1793 element periodic micro-UAV swarm. The swarm is corrupted by a Gaussian positional noise with σ = 0.1λ.

broadside. While the nominal positions of the array elements are restricted to the

xy plane, the positional noise of each aircraft system is estimated by a spherical

Gaussian distribution. Cross-sections of this distribution are defined to have a

standard deviation σ = 0.1λ. Figure 3.22 illustrates the positional noise effect.

The circles around each element show the 3σ likely locations of aircraft. The dis-

tribution is Gaussian which has a drop off ∝ e−r2, where r denotes the distance

of the element from its ideal location (magnitude of positional noise). The ideal

normalized radiation pattern for the aperiodic array was shown in Figure 3.11 at

f = 4f0 which corresponds to a minimum spacing of 2λ. As it can be seen from the

plot, the radiation pattern does not exhibit any grating lobes. Figure 3.23 illus-

trates the normalized radiation pattern of the same array, where element locations

are corrupted by a random zero-mean Gaussian positional noise with σ = 0.1λ.

Figure 3.23 demonstrates that even small amounts of positional noise will cause

significant changes in the radiation pattern effectively eliminating the main beam.

To compensate for the errors around the main beam, the phase correction algo-

rithm introduced in Section 3.4 is applied to the array. Since we are dealing with

45

Figure 3.21: The phase corrected radiation pattern of a 1793 element periodicmicro-UAV swarm. The swarm is corrupted by a Gaussian positional noise withσ = 0.1λ.

broadside arrays, according to Eq. (3.11) the correction phase for each element

is simply −krnz, where k is the free space wave number and rnz is the positional

error of the n-th aircraft from turbulence in the z direction. Applying this method

will restore the main beam for any planar array geometry. Figure 3.24 shows the

radiation pattern of the aperiodic array with positional noise after the correction

algorithm has been applied. The phase corrected radiation pattern has a PSLL of

−12.95dB. Because of the optimized aperiodic nature of the array, grating lobes

do not exist in the phase corrected radiation pattern. In addition, grating lobes do

not appear even if the mainbeam of the radiation pattern is steered significantly

from broadside. Figure 3.25 illustrates the phase corrected radiation pattern of the

micro-UAV swarm when the mainbeam is steered to the angle φ = 45, θ = 45.

In this example, the progressive phase shift is applied to steer the nominal pattern

and the phase correction adjusts for positional noise normal to the desired pointing

direction of the mainbeam.

46

Figure 3.22: The effects of Gaussian noise with σ = 0.1λ on the optimized array.The intensity of gray in circles around the elements corresponds to the probabilityof the element being in that region.

47

Figure 3.23: The corrupted radiation pattern for a micro-UAV swarm based on the267 element aperiodic tiling array shown in Figure 3.7. The swarm is corruptedby a Gaussian positional noise with σ = 0.1λ.

Figure 3.24: The phase corrected radiation pattern for a micro-UAV swarm basedon the 267 element aperiodic tiling array shown in Figure 3.7. The swarm iscorrupted by a Gaussian positional noise with σ = 0.1λ.

48

Figure 3.25: The phase corrected radiation pattern for a micro-UAV swarm basedon the 267 element aperiodic tiling array shown in Figure 3.7. The radiationpattern is steered to an angle φ = 45, θ = 45 and the swarm is corrupted by aGaussian positional noise with σ = 0.1λ.

Chapter 4Generalized Scattering Coefficients

for Finite-Sized Spherical Arrays

4.1 Introduction

The optical properties of quasicrystals have been a topic of immense interest in

recent years. In the previous chapter we studied the application of aperiodic ge-

ometries in designing ultra-wideband antenna arrays. Another area of immense

interest has been aperiodic nanoparticle arrays of metallic particles. It has been

demonstrated that such metallic nanoparticles with sub-wavelength dimensions

display strong optical extinction in the visible or IR spectrum. This resonant be-

havior is known as the particle plasmon [5]. This plasmonic resonance makes metal

nanoparticle arrays great candidates for wide variety of applications such as SERS

substrates and bio-sensors [24, 3].

Traditionally the EM properties of metamaterials and photonic crystals have

been evaluated by exploiting their translational symmetry (periodicity). This ap-

proach significantly simplifies the analysis by applying appropriate boundary con-

ditions and only requiring Maxwell’s equations to be solved for one unit cell, rather

than the entire structure. However quasicrystals generally lack translational sym-

metry. This characteristic requires the development of new analytical tools that

can efficiently and accurately evaluate the EM properties of aperiodic metamate-

rials.

50

One particular approach that can be applied to large aggregates of randomly

sized spheres in a dielectric medium is the Generalized Multiparticle Mie The-

ory (GMT) developed by Xu [7]. The solution obtained by GMT is a complete

solution, based on rigorous analytical expressions which account for all couplings

and multiparticle interactions. This is particularly important when studying metal

particles in their plasmonic resonance regions where many approximate methods

(e.g. discrete dipole approximation method) tend to produce relatively inaccu-

rate calculations. Furthermore, the GMT solution also includes near-field results

which make it suitable for studying phenomena such as surface-enhanced Raman

scattering (SERS).

Traditionally the GMT method is applied based on an incident plane wave

excitation. However since a plane wave has an infinite beamwidth, then it is not

possible to define reflection and transmission coefficients in the usual sense when

considering the analysis of finite-size (truncated) arrays via the GMT method. Ob-

taining reflection and transmission coefficients are also necessary if one is interested

in extracting effective medium parameters for subwavelength arrays. To overcome

this problem, we implement an approach which is, in principle, similar to what was

introduced in [25] to define generalized scattering parameters for two-dimensional

finite-sized arrays of cylindrical elements.

In this chapter, we extend the approach in [25] to three dimensions by placing

a circular aperture in front of a plane wave to obtain a beamwidth smaller than

the array dimensions to avoid diffraction. Such a setup is very similar to realistic

experimental conditions. Fields diffracted by the circular aperture must first be

expanded in terms of vector spherical wave functions [26] (VSWF). We were able to

obtain complete analytical expansions for the diffracted fields. Once the incident

field was expanded in terms of VSWFs, the expansion coefficients also had to be

calculated at the displaced coordinate systems defined by the sphere centers. In

the case of an incident plane wave, it can be shown that the expansion coefficients

at the displaced systems differ from the primary expansion coefficients only by a

constant phase term. However in the case of aperture diffracted waves, the results

are not trivial and addition theorems for VSWF [27] have to be applied to obtain

the expansion coefficients for displaced systems. Once the expansion coefficients

have been determined at all the displaced coordinate systems, the GMT method

51

can be utilized and a solution for the scattered fields is obtained. Using the far-field

expressions for scattered fields, we define generalized transmission and reflection

coefficients based on total far-field energy fluxes. To test the validity of our method,

it was applied to a truncated periodic array of gold nano-spheres and compared

with the results obtained for an infinite array when periodic boundary conditions

are imposed, with excellent agreement between the two approaches.

4.2 GMT with Incident Plane Wave

We consider an array of L homogeneous, nonintersecting spheres of diameter d,

and a complex permittivity ε embedded in a non-absorbing dielectric medium

characterized by permittivity εrε0 and permeability µrµ0. Let (Xj, Y j, Zj) denote

the center of the jth sphere. Without loss of generality we consider a z-propagating

plane wave where the electric field is x-polarized with unit amplitude. Assuming

e−iωt time-dependence, where ω is the angular frequency, the incident electric field

vector is

Einc = xeikz (4.1)

where k = 2πλ

is the wave number and λ is the incident wavelength in the dielectric

medium and i =√−1. Also x, y, z denote the unit vectors in the Cartesian

coordinate system. In order to apply the GMT method, the incident field has to

be expanded in terms of VSWFs in all L displaced coordinate systems defined

by the sphere centers. We start by considering the primary coordinate system

centered at the origin. The incident electric field can be expanded in terms of

vector spherical basis functions N(1)mn, and M

(1)mn as

Einc = −∞∑n=1

n∑m=−n

iEmn[pmnN(1)mn(ρ, θ, φ) + qmnM

(1)mn(ρ, θ, φ)] (4.2)

where (r, θ, φ) are the spherical coordinates and ρ = kr. The terms Emn, N(1)mn,

and M(1)mn found in Eq. (4.2) are defined as [7]

N(1)mn(ρ, θ, φ) =

rn(n+ 1)Pm

n (cos θ)jn(ρ)

ρ

52

+[θτmn(cos θ) + φiπmn(cos θ)]ψ′n(ρ)

ρ

eimφ

M(1)mn(ρ, θ, φ) =

[θiπmn(cos θ)− φτmn(cos θ)

]jn(ρ)eimφ

Emn = in[

(2n+ 1)(n−m)!

n(n+ 1)(n+m)!

]1/2

(4.3)

where Pmn (cos θ) is the associated Legendre function of the first kind of degree n

and order m, jn is the spherical Bessel function of the first kind, and ψn(ρ) = ρjn(ρ)

is the Riccati-Bessel function. In addition, the functions πmn(cos θ) and τmn(cos θ)

are defiend as

πmn(cos θ) =m

sin θPmn (cos θ)

τmn(cos θ) =d

dθPmn (cos θ) (4.4)

Here we adopt the convention for the associated Legendre function which omits

the (−1)m term [28].

Pmn (x) = (1− x2)m/2

dm

dxmPn(x) (4.5)

where Pn is the Legendre polynomial of order n. In the remainder of this chapter,

for the sake of brevity, we forgo writing the argument for functions πmn, τmn, Pmn ,

and Pn. Unless otherwise stated, it is always assumed that these functions have

cos θ as their argument. In practice the expansion is truncated to Nmax terms

(n = 1, 2, . . . , Nmax). The expansion coefficients pmn and qmn are obtained using

the orthogonality of VSWFs and integrating Eq. (4.2) which leads to

qmn =i∫ 2π

0

∫ π0

Einc ·M(1)∗mn sin θdθdφ

Emn∫ 2π

0

∫ π0

∣∣∣M(1)mn

∣∣∣2 sin θdθdφ(4.6)

pmn =i∫ 2π

0

∫ π0

Einc ·N(1)∗mn sin θdθdφ

Emn∫ 2π

0

∫ π0

∣∣∣N(1)mn

∣∣∣2 sin θdθdφ(4.7)

53

In the case of an incident plane wave of the form in Eq. (4.1), the expansion

coefficients reduce to following simple expressions [7]

pmn = qmn = 0, |m| 6= 1

p1n = q1n =

√2n+ 1

2

p−1n = −q−1n = −√

2n+ 1

2(4.8)

As mentioned, it is required that expansion coefficients be evaluated in all displaced

coordinate systems defined by the sphere centers. In the case of an incident plane

wave, this is a trivial matter. It can easily be shown that if qjmn and pjmn denote

the expansion coefficients in the jth system with its origin at (Xj, Y j, Zj), they

only differ from primary expansion coefficients by a constant phase term [7]. For

a z-propagating plane wave, the displaced expansion coefficients are given by

pjmn = exp(ikZj)pmn , qjmn = exp(ikZj)qmn (4.9)

Hence in the jth system the expansion of the electric field has the form

Ejinc = −

∞∑n=1

n∑m=−n

iEmn[pjmnN(1)mn(ρj, θj, φj) + qjmnM

(1)mn(ρj, θj, φj)] (4.10)

Similarly, for each sphere the scattered and internal fields can be expanded in

terms of spherical harmonics. Denoting the internal and scattered electric fields of

the jth sphere by EI(j) and ES(j) respectively, they can be expanded as

EI(j) = −∞∑n=1

n∑m=−n

iEmn[djmnN(1)mn(ρj, θj, φj) + cjmnM

(1)mn(ρj, θj, φj)] (4.11)

ES(j) =∞∑n=1

n∑m=−n

iEmn[ajmnN(3)mn(ρj, θj, φj) + bjmnM

(3)mn(ρj, θj, φj)] (4.12)

where N(3)mn and M

(3)mn are defined in a similar manner as N

(1)mn and M

(1)mn in Eq. (4.3),

except that the spherical Hankel function of the first kind (h(1)n ) is substituted for

the spherical Bessel function of the first kind (jn). The expansion coefficients

54

ajmn, bjmn, cjmn, and djmn are known as the interactive scattering coefficients and

they can be obtained by solving the linear systems that are obtained by applying

boundary conditions to each sphere [7, 29, 30]. Here it is important to note that for

each sphere, the total incident field consists of the original incident wave Einc plus

the scattered fields of all other spheres. Thus in order to obtain the appropriate

boundary conditions it is essential to be able to express the expansion coefficient

in one coordinate system in terms of the basis set of another coordinate system.

The connection between expansion coefficients in translated coordinate systems

(ρl, θl, φl) and (ρj, θj, φj) were derived by Stein [31]and Cruzan [32]:

M(3)mn(ρl, θl, φl) =

∞∑ν=0

ν∑µ=ν

[Amnµν (l, j)M(1)

µν (ρj, θj, φj) +Bmnµν (l, j)N(1)

µν (ρj, θj, φj)]

N(3)mn(ρl, θl, φl) =

∞∑ν=0

ν∑µ=ν

[Bmnµν (l, j)M(1)

µν (ρj, θj, φj) + Amnµν (l, j)N(1)µν (ρj, θj, φj)

](4.13)

where Amnµν (l, j) and Bmnµν (l, j) are the so-called translation coefficients. If

(dlj, θlj, φlj) denotes the spherical coordinates of the origin of the lth coordinate

system in the jth coordinate system, Amnµν (l, j) and Bmnµν (l, j) are given by [33]

Amnµν (l, j) = (−1)µiν−n2ν + 1

2ν(ν + 1)

n+ν∑p=|n−ν|

(−i)p [n(n+ 1) + ν(ν + 1)− p(p+ 1)]

a(m,n,−µ, ν, p)jp(kdlj)P (m−µ)p (cos θlj) exp (i(m− µ)φlj)

Bmnµν (l, j) = (−1)µiν−n

2ν + 1

2ν(ν + 1)

n+ν∑p=|n−ν|

(−i)pb(m,n,−µ, ν, p, p− 1)

jp(kdlj)P(m−µ)p (cos θlj) exp (i(m− µ)φlj)

(4.14)

where a(m,n, µ, ν, p) is the Gaunt coefficient defined by [34]

a(m,n, µ, ν, p) =2p+ 1

2

(p−m− µ)!

(p+m+ µ)!

∫ 1

−1

Pmn (x)P µ

ν (x)Pm+µp (x) dx (4.15)

55

and b(m,n,−µ, ν, p, p− 1) is defined as

b(m,n,−µ, ν, p, p− 1) =2p+ 1

2p− 1[(ν − µ)(ν + µ+ 1)a(m,n,−µ− 1, ν, p− 1)−

(p−m+ µ)(p−m+ µ− 1)a(m,n,−µ+ 1, ν, p− 1)

+2µ(p−m+ µ)a(m,n,−µ, ν, p− 1)] (4.16)

No closed form solution is known to exist for the integral in Eq. (4.15) and direct

numerical evaluation can be problematic, especially due to the presence of the

factorial terms. There is a vast amount of literature regarding the efficient and

fast evaluation of vector translation coefficients based on recursive methods to

calculate Gaunt coefficients [27, 35, 36]. Once all the interactive coefficients have

been determined, the total scattered electric field can be expressed in the primary

coordinate system (j0) as

ES =∞∑n=1

n∑m=−n

iEmn[amnN(3)mn(ρ, θ, φ) + bmnM

(3)mn(ρ, θ, φ)] (4.17)

where

amn =L∑l=1

∞∑ν=1

ν∑µ=−ν

[alµνA

µνmn(l, j0) + blµνB

µνmn(l, j0)

]bmn =

L∑l=1

∞∑ν=1

ν∑µ=−ν

[alµνB

µνmn(l, j0) + blµνA

µνmn(l, j0)

](4.18)

4.3 GMT Based on Finite Beamwidth Incident

Wave

Since a plane wave has an infinite beamwidth, it is not possible to define reflection

and transmission coefficients in the usual sense when considering the analysis of

finite-size (truncated) arrays via the GMT method. To define reflection and trans-

mission coefficients for the array, an incident beamwidth smaller than the sample

dimensions is required to avoid diffraction. A simple way to obtain such an inci-

dent beam is to place a circular aperture in front of the array. For simplicity we

assume that all spheres are located in the upper half-space (z > 0) and we place

56

a circular aperture of radius a in the xy-plane (z = 0). Since the fields diffracted

by the aperture in the z > 0 region will acts as the incident fields on the array,

we denote them by Einc. Using the Kirchhoff integral [37], we can show that for

an incident electric field xeikz in the z < 0 region, the scattered field in the z > 0

region is Einc = Eθθ + Eφφ with Eθ and Eφ given by

Eθ =−iakeiρJ1(ka sin θ) cosφ

ρ sin θ

Eφ =iakeiρJ1(ka sin θ) sinφ cos θ

ρ sin θ(4.19)

where J1 is a first order Bessel function of the first kind.

We start by expanding the incident field in Eq. (4.19) in terms of VSWFs

as shown in Eq. (4.2) in the primary coordinate system. Comparing the basis

functions in Eq. (4.3) and the incident field in Eq. (4.19) it can be seen that the

expansion is problematic since the basis functions N(1)mn and M

(1)mn are well-defined

and finite everywhere whereas the expression in Eq. (4.19) contains a singularity at

the origin. This issue can be resolved since the expression for the diffracted fields

are derived assuming far-field conditions, thus we can replace the expressions in

Eq. (4.19) with the following far-field equivalents which are well-defined and finite

everywhere

Eθ =iakJ1(ka sin θ) cosφ[j1(ρ) + ij2(ρ)]

sin θ

Eφ =−iakJ1(ka sin θ) sinφ cos θ[j1(ρ) + ij2(ρ)]

sin θ(4.20)

It can easily be shown that pmn = qmn = 0 for all values of m 6= ±1. Furthermore

it can be shown that q−1n = q1n and p−1n = −p1n. Thus the only quantities that

need to be evaluated are q1n and p1n. We start by determining q1n, using Eq. (4.6).

Evaluating the integral in the denominator, we arrive at∫ 2π

0

∫ π

0

∣∣∣M(1)1n (ρ, θ, φ)

∣∣∣2 sin θdθdφ =4πn2(n+ 1)2

2n+ 1j2n(ρ) (4.21)

57

The numerator in Eq. (4.6) contains the integral∫ π/2

0

(J1(ka sin θ)[π1n + cos θτ1n]

)dθ (4.22)

A closed from expression for Eq. (4.22) is not available in the literature, including

the mathematical handbooks, and cannot be found using commercial software

packages such as Mathematica. However, in this paper a derivation of an exact

solution is presented, which is expressed in terms of two auxiliary functions Ωn(ka)

and Ξn(ka): ∫ π/2

0


)dθ

= n(n+ 1)Ωn(ka) + Ξn(ka) (4.23)

Key steps of the derivation and the resulting expressions for functions Ωn(ka) and

Ξn(ka) are given in Appendix B. Upon substituting the results from Eq. (4.23)

and Eq. (4.21) into Eq. (4.6) we arrive at

q1nin2n(n+ 1)√

2n+ 1j2n(ρ) = akijn(ρ) [j1(ρ) + ij2(ρ)] [n(n+ 1)Ωn(ka) + Ξn(ka)] (4.24)

It is still necessary to eliminate the dependence on the radial component ρ. Since

our derivations for the diffracted fields assumed far zone conditions, we can use

the following large argument properties Bessel functions [38]

limρ→∞

j1(ρ) = in−1jn(ρ) if n is odd

limρ→∞

j2(ρ) = in−2jn(ρ) if n is even (4.25)

Next using the orthogonality property of spherical Bessel functions, we can inte-

grate with respect to the radial component

∫ ∞−∞

ijn(ρ) [j1(ρ) + ij2(ρ)] dρ ≈∫ ∞−∞

ijn(ρ)in−1jn(ρ) dρ =inπ

2 + 4n(4.26)

58

Finally we arrive at the following expression for q1n

q1n =ak√

2n+ 1n(n+ 1)Ωn(ka) + Ξn(ka)

2n(n+ 1)

(4.27)

The next step in the derivation is to evaluate p1n. We start by evaluating the

integral in the denominator in Eq. (4.7), which has the following solution∫ 2π

0

∫ π

0

∣∣∣N(1)1n (ρ, θ, φ)

∣∣∣2 sin θdθdφ =

4πn2(n+ 1)2

2n+ 1

[n(n+ 1)

j2n(ρ)

ρ2+

(ψ′n(ρ)

ρ

)2]

(4.28)

The numerator in Eq. (4.7) contains the integral∫ π/2

0

(J1(ka sin θ)[τ1n + cos θπ1n]

)dθ (4.29)

A closed form solution to Eq. (4.29) is apparently not available in the literature.

However, an exact solution is derived here for this integral in terms of an auxiliary

function Ψn(ka):∫ π/2

0


)dθ = n(n+ 1)Ψn(ka) (4.30)

Key steps of the derivation and the resulting closed form representation for the

function Ψn(ka) is provided in Appendix C. Substituting the results from Eq. (4.30)

and Eq. (4.28) into Eq. (4.7) will result in an expression with radial component.

A similar procedure that was applied to Eq. (4.24) can be used which allows for

integrating out the radial component using asymptotic and orthogonal properties

of the spherical Bessel functions resulting in:

p1n =ak√

2n+ 1Ψn(ka)

4(4.31)

Here it is important to note that analytical expressions for both Eq. (4.22) and

Eq. (4.29) can be of great interest for vector diffraction problems that arise in elec-

tromagnetic theory, especially when partial wave decomposition is required. These

59

analytical expressions will render computationally intensive numerical quadrature

methods unnecessary. There have been recent papers published which provide

analytical results for other types of diffraction related integrals [39, 40]. The

closed form representations derived here for auxiliary functions Ωn(ka), Ξn(ka),

and Ψn(ka) are all expressed in terms of finite summations involving Lommel

functions [41]. In general, evaluation of Lommel functions involves time consum-

ing numerical integrations or infinite summations. The specific Lommel functions

encountered in our derivations are of the form sν+1,ν(z). We were able to derive a

concise analytical expression for this particular order of Lommel functions as

sν+1,ν(z) = zν − 2νΓ(ν + 1)Jν(z) (4.32)

where Γ is the gamma function. The general expression for sµ,ν(z) as well as the

derivation of Eq. (4.32) is provided in Appendix D. Our finite summation expres-

sions for the auxiliary functions Ωn(ka), Ξn(ka), and Ψn(ka) in addition to the

concise analytical expression obtained for the Lommel functions in Eq. (4.32) pro-

vide exact evaluations and render computationally expensive numerical techniques

unnecessary.

To verify our results, we consider an example where a plane wave with electric

field xeikz is incident upon a circular aperture of radius a, where we set k = 2π

and a = 1. Figure 4.1 shows the values for |Eθ| and |Eφ| in the xy-plane at a

distance of D = 15 from the aperture evaluated using the analytical expressions

from Eq. (4.19). Figures 4.2, 4.3 , and 4.4 shows the corresponding values ob-

tained using 4, 6, and 8 term VSWF expansions (respectively) using the expansion

coefficients from Eq. (4.27) and Eq. (4.31). As it can be seen from the plots, great

convergence is obtained using a fairly small number of terms.

The expansion coefficients for the incident fields also have to be calculated in all

the L displaced coordinate systems defined by the sphere centers. As it was shown

in the case of incident plane wave, the expansion coefficients in displaced coordinate

systems only vary by a phase term. However in our case, the relationship is not as

simple and the displaced expansion coefficients have be evaluated by application

of vector translational addition theorems. It can be shown that in general qjmn and

60

Figure 4.1: |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using the analytical expressions from Eq. (4.19).

Figure 4.2: |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using a four term VSWF expansion.

pjmn may be calculated from

pjmn =∞∑ν=1

ν∑µ=−ν

EµνEmn

[pµνA

µνmn(j0, j) + qµνB

µνmn(j0, j)

]qjmn =

∞∑ν=1

ν∑µ=−ν

EµνEmn

[pµνB

µνmn(j0, j) + qµνA

µνmn(j0, j)

](4.33)

where pµν and qµν are primary expansion coefficients and Eµν and Emn are defined

according to Eq. (4.3). Aµνmn(j0, j) and Bµνmn(j0, j) are vector translation coefficients

and are evaluated according to Eq. (4.14) except that the spherical Bessel function

61

Figure 4.3: |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using a six term VSWF expansion.

Figure 4.4: |Eθ| (left) and |Eφ| (right) values for fields diffracted by a circularaperture evaluated using an eight term VSWF expansion.

of the first kind (jp(kdlj)) is replaced for the spherical Hankel function of the

first kind(h

(1)p (kdlj)

). As previously mentioned direct evaluation of translation

coefficients is not feasible. Here we utilized a recursive approach described in [36].

As an example, here we consider the scattered field in the plane of an array

of gold nano-spheres. The array is based on an Ammann-Beenker (AB) aperiodic

tiling, which is shown in Figure 4.5. The prototile set consists of a rhombus ( vertex

angles π/4 and 3π/4 ) and an isosceles right triangle[16]. Using specific matching

rules, these prototiles can cover the two-dimensional plane without translational

symmetry. The tiling can be converted to an array by placing elements at the

62

Figure 4.5: Geometry of an aperiodic Ammann-Beenker tiling.

vertices of each tile followed by the proper scaling and truncation. The array

is composed of 353 gold spheres of radius 80nm and scaled such that there is a

minimum center to center spacing of 450nm (290nm surface to surface) between

spheres. A modified Drude model[42] is used to represent the dielectric function

of gold

ε(ω) = ε∞ −ω2p

ω2 + iΓω(4.34)

with ε∞ = 9, ωp = 13.8× 1015s−1, and Γ = 0.11× 1015s−1. It is assumed that the

array is contained in the xy-plane. Figure 4.6 shows the normalized local scattered

fields in the plane of the array when illuminated by a linearly polarized plane wave

Einc = xeikz with a wavelength of λ = 500nm. To obtain the scattered fields in

this case, primary expansion coefficients qmn and pmn were obtained from Eq. (4.8)

and displaced expansion coefficients qjmn and pjmn from equations Eq. (4.9). As it

can be seen from the field plot, all of the elements are illuminated and there is

considerable diffraction of fields around the periphery of the structure. Next we

place a circular aperture with radius of a = 1µm at a distance D in front of the

array with the plane wave incident upon it. The distance of the aperture from the

array (D) must be such that diffraction of the fields at the periphery of the array

63

Figure 4.6: Normalized scattered field magnitude (dB) in the plane of the ABaperiodic array illuminated by a plane wave.

is avoided. Considering the expression of the incident fields in Eq. (4.20), a simple

way to determine the appropriate distance is using the first zero of the J1(ka sin θ)sin θ

term which approximately occurs at ka sin θ0 ≈ 3.832 . Denoting the set of spheres

on the periphery of the array by S and their spherical coordinates by (rs, θs, φs)

where s ∈ S, we define α ≡ argmins∈S

θs. The condition

θα ≤ θ0 , D = rα cos (θα) (4.35)

ensures that the incident fields at the peripheral elements of the array are zero

or much diminished compared to the incident fields on the interior elements of

the array. Using Eq. (4.35) we set D = 15µm. In this case the primary expan-

sion coefficients in Eq. (4.27) and Eq. (4.31) are used and displaced expansion

coefficients are obtained from Eq. (4.33). Figure 4.7 shows the normalized local

scattered fields evaluated in the plane of the array when illuminated with aperture

diffracted waves. As it can be seen from the field plot, there are almost no fields

at the periphery of the array (‖ES‖ < −30 dB), whereas and fields are highly

localized to the interior of the array which confirms the validity of our derivations.

64

Figure 4.7: Normalized scattered field magnitude (dB) in the plane of the ABaperiodic array illuminated by circular aperture diffracted waves.

4.4 Generalized Scattering Coefficients

In the far-field much simpler asymptotic expressions for the total scattering coef-

ficients were derived by Xu [29] as

amn =L∑j=1

exp(−ik∆j)ajmn

bmn =L∑j=1

exp(−ik∆j)bjmn (4.36)

where ∆j = Xj sin θ cosφ + Y j sin θ sinφ + Zj cos θ and (Xj, Y j, Zj) denotes the

center of the jth sphere. It can be shown that for θ = 0, the scattered far-field has

the form

ES(ρ, 0, 0) =ieiρ

ρ

Nmax∑n=1

√2n+ 1[a1n + b1n]θ (4.37)

Also from Eq. (4.19) it follows that the incident field for θ = 0 is

Einc(ρ, 0, 0) = −i(ak)2eiρ

2ρθ (4.38)

65

Hence the total far-field for θ = 0 can be expressed as the sum of incident and

scattered fields according to

ETotal(ρ, 0, 0) =ieiρ

ρ

(−(ak)2

2+

Nmax∑n=1

√2n+ 1[a1n + b1n]

)(4.39)

Here we define a generalized transmission coefficient (T ) in terms of the total

far-field energy flux relative to that of the incident field energy flux for θ = 0

T =

∣∣∣∣ETotal(ρ, 0, 0)

Einc(ρ, 0, 0)

∣∣∣∣2 (4.40)

After some simplification, we arrive at

T =

∣∣∣∣∣1− 2

(ak)2

Nmax∑n=1

√2n+ 1[a1n + b1n]

∣∣∣∣∣2

(4.41)

Similarly a generalized reflection coefficient (R) can be defined in terms of the

scattered energy flux for θ = π relative to that of the incident energy flux for θ = 0

R =

∣∣∣∣ ES(ρ, π, 0)

Einc(ρ, 0, 0)

∣∣∣∣2 (4.42)

It can be shown that for θ = π, the scattered far-field is given by

ES(ρ, π, 0) =ieiρ

ρ

Nmax∑n=1

(−1)n√

2n+ 1[a1n − b1n]θ (4.43)

Substituting Eq. (4.43) and Eq. (4.38) into Eq. (4.42) and after simplification, we

arrive at the following expression for R

R =

∣∣∣∣∣ 2

(ak)2

Nmax∑n=1

(−1)n√

2n+ 1[a1n − b1n]

∣∣∣∣∣2

(4.44)

66

Figure 4.8: Scattering response of infinite (solid lines) and finite (dashed lines)periodic gold arrays obtained by CST MICROWAVE STUDIO and GMT with afinite beamwidth calculated using Eq. (4.41) and Eq. (4.44).

4.5 Example

To test the validity of the results, we consider a finite-sized periodic array of gold

nano-spheres. In this case, the results for the truncated structure can be com-

pared with results obtained for an infinite planar array applying periodic bound-

ary conditions. The array consists of 316 gold nano-spheres of radius 60nm and

periodicity of 350nm, and it is placed at a distance of 10µm away from the circu-

lar aperture. The radius of the aperture is a function of the incident wavelength

a = 2λ. We also employed the full wave commercial package CST MICROWAVE

STUDIO[43] to evaluate the scattering response of an infinitely periodic array of

gold nano-spheres with the same radius and periodicity based on a single unit cell

with periodic boundary conditions.

Figure 4.8 shows the results obtained for an infinitely periodic structure and

those obtained by applying GMT to a finite array and determining the generalized

scattering parameters from Eq. (4.41) and Eq. (4.44). As it can be seen from the

plot, there is excellent agreement between the two results.

Chapter 5Optimizations of Quasicrystalline

Nanoparticle Arrays

5.1 Introduction

In recent years gold and silver nanoparticle arrays have attracted immense atten-

tion due to their optical properties in the visible and infrared (IR) regions. It

has been shown that sub-wavelength gold and silver nanoparticles display a strong

resonant behavior known as the particle plasmon in the visible and IR spectrum

[5]. When placed in regular 2D periodic arrays, the particle plasmon response can

be further enhanced by coupling to the grating resonance of the structure [6] and

forming a so-called photonic-plasmonic hybrid mode. This plasmonic resonance

makes metal nanoparticle arrays great candidates for wide variety of applications

such as surface-enhanced Raman scattering (SERS) substrates and biosensors.

In this chapter we study the optical properties of quasicrystalline arrays of

gold nano-spheres. A key challenge in evaluating the EM properties of aperiodic

geometries is the lack of analytical tools. However in the case of spherical arrays

Generalized Multiparticle Mie Theory (GMT) which was introduced in Chapter 4

can be used as a great analytical tool for aperiodic geometries. Furthermore im-

plementing GMT with an incident beam with finite beamwidth which was derived

in Chapter 4 will allow us to define scattering coefficients for aperiodic spherical

arrays.

68

Section 5.2 starts by describing the formation of hybrid modes in periodic

arrays. In Section 5.3 it is shown that hybrid modes also exist in quasicrystalline

arrays and their location can be deduced from the Fourier diffraction pattern of

the array. Furthermore, it will be shown that scattering properties of arrays based

on aperiodic tilings can be further enhanced using a perturbation method.

Section 5.4 studies the formation of local hot spots in gold arrays. It was first

shown that aperiodic arrays of gold nanoparticles have regions with larger local

field enhancements than periodic arrays [3]. Here we study local field enhancements

in quasicrystalline arrays and show that local fields can be further enhanced by

optimizing the aperiodic tilings.

5.2 Resonance Response in Periodic Gold Ar-

rays

The optical response of arrays of gold nanospheres can be explained in terms of the

plasmonic response of the spheres (plasmonic resonance) and the grating response

of the array (photonic resonance). The plasmonic resonance is a function of particle

shape and their dielectric function. The photonic resonances can be analyzed as

the Bragg grating modes, which are due to the coherent scattering experienced as

the dielectric wavelength of the incident radiation approaches the periodicity of

the array. The photonic resonance is due to the morphology of the array rather

than constitutive particles and material properties. We start by giving a brief

description of each of these resonances.

5.2.1 Plasmonic Resonance of Gold Spheres

Consider a single sphere of radius R with an index of refraction np embedded

in a dielectric medium with an index of refraction nM and illuminated by a z-

propagating x-polarized plane wave such that the electric field is of the form

Eplane = x eikz (5.1)

69

where k = 2πλ

is the wave number and λ is the wavelength in the dielectric medium.

Using Mie theory [26]the scattered fields can be expanded in terms of spherical

harmonics:

Esca(ρ, θ, φ) =∞∑n=1

En[ianN(3)e1n(ρ, θ, φ)− bnM(3)

o1n(ρ, θ, φ)] (5.2)

where (r, θ, φ) denote the spherical coordinates and ρ = kr. The terms En, N(3)e1n,

and M(3)o1n found in Eq. (5.2) are defined as [26]

N(3)e1n(ρ, θ, φ) =

n(n+ 1)h(1)n (ρ)

ρcosφP 1

n(cos θ)r+

cosφdP 1

n(cos θ)

dθ

1

ρ

[ρh(1)

n (ρ)]′θ − sinφ

dP 1n(cos θ)

dθ

1

ρ

[ρh(1)

n (ρ)]′φ (5.3a)

M(3)o1n(ρ, θ, φ) =

cosφ

sin θP 1n(cos θ)h(1)

n (ρ)θ − sinφdP 1

n(cos θ)

dθh(1)n (ρ)φ (5.3b)

En = in[

(2n+ 1)

n(n+ 1)

](5.3c)

The expansion coefficients an and bn are obtained by applying the boundary

conditions at the surface of the sphere and are given by [26]

an =ψn(x)ψ′n(mx)−mψ′n(x)ψn(mx)

ξn(x)ψ′n(mx)−mξ′n(x)ψn(mx)(5.4)

bn =mψn(x)ψ′n(mx)− ψ′n(x)ψn(mx)

mξn(x)ψ′n(mx)− ξ′n(x)ψn(mx)(5.5)

where m = np

nM, x = kR and ψn(z) = zjn(z) and ξn(z) = zh

(1)n (z) are the Riccati-

Bessel functions. The denominators in Eq. (5.4) and Eq. (5.5) can become very

small and in essence form complex numbered poles. At these poles due to the large

value of the expansion coefficient the scattered fields exhibit a resonant behavior.

Considering the denominators in Eq. (5.4) and Eq. (5.5), the resonances are

mξ′n(x)

ξn(x)=ψ′n(mx)

ψn(mx)(5.6)

1

m

ξ′n(x)

ξn(x)=ψ′n(mx)

ψn(mx)(5.7)

70

For very small particles we can use the quasi-static approach which allows us to

approximate the Riccati-Bessel functions using their first order approximations.

In this case Eq. (5.6) and Eq. (5.7) will simplify to [44]

εp = −n+ 1

nεM (5.8)

εp = −2εM (5.9)

The resonances resulting from Eq. (5.9) will lead to a trivial solution and are of no

interest. Eq. (5.8) represents the particle plasmon mode of the sphere. The lowest

order (n = 1) of Eq. (5.8) requires a dielectric function with a real part of −2εM

and a vanishing imaginary part.

As an example here we consider a gold sphere with a radius of 20 nm in a

dielectric medium with an index of refraction of nM = 1.5 (εM = 2.25). A modified

Drude model [42] is used to represent the dielectric function of gold

ε(ω) = ε∞ −ω2p

ω2 + iΓω(5.10)

with ε∞ = 9, ωp = 13.8×1015s−1, and Γ = 0.11×1015s−1. Figure 5.1 shows the real

and imaginary parts of of the modified Drude model shown in Eq. (5.10) over the

visible range. Figure 5.2 shows the extinction efficiency of the sphere evaluated

using Mie theory. As it can be seen the location of resonance corresponds to

ε(ω) ≈ −2εM .

5.2.2 Photonic Resonance

The photonic resonances can be analyzed as the Bragg grating modes, which are

due to the coherent scattering. In order to achieve coherent scattering, the period-

icity of the structure must approach an integer multiple of the wavelength of the

incident radiation. If Λ denotes the periodicity of the structure, the wavelength

corresponding to the m-th photonic resonance λm is given by:

λm =Λ

mm = 1, 2, . . . (5.11)

71

Figure 5.1: Real and imaginary parts of the gold dielectric function according toEq. (5.10).

Generally the first order resonance (m = 1) is strongest one since there is consid-

erable diffraction at higher orders. As an example the the scattering response of a

periodic array of 121 dielectric spheres of radius 50 nm and periodicity of Λ = 550

Figure 5.2: Extinction efficiency of a gold sphere with a radius of 20 nm in adielectric medium with an index of refraction of nM = 1.5.

72

nm is considered. It is assumed that the spheres are loss-less and dispersion-less

with an index of refraction of np = 3. A linearly polarized plane wave is incident

on the array. The scattering properties of this array can be evaluated using GMT

method with an incident plane wave which was introduced in Section 4.2. The

extinction efficiency of the array is shown in Figure 5.3. As it can be seen from the

plot as the incident wavelength approaches the lattice constant around 550 nm a

resonance is observed.

Figure 5.3: Extinction efficiency of a periodic array of 121 dielectric spheres (np =3) of radius 50 nm and periodicity of Λ = 550 nm with a normally incident planewave.

5.2.3 Hybrid Resonance

The optical properties of gold nanoparticle arrays can be explained in terms of the

plasmonic resonance and the photonic resonance. Figure 5.4 shows the extinction

efficiency a periodic array of 100 gold spheres of radius 80 nm and periodicity of

Λ = 600 nm. As it can be seen from the plot the spectrum displays two distinct

resonances. The first resonance in the 450 nm region is due to the plasmonic

resonance of gold and the second resonance in the 600 nm region is the photonic

resonance of the array.

73

Figure 5.4: Extinction efficiency of a periodic array of 100 gold spheres of radius80 nm and periodicity of Λ = 600 nm.

Figure 5.5 shows the extinction efficiency for three periodic array of 100 gold

spheres of radius 80 nm and different periodicities as indicated in the plot. As

it can be seen for the larger lattice constant of 600 nm plasmonic and photonic

resonances are distinct. Of particular interest are the phenomena that occur when

the photonic and plasmonic resonances are in close proximity. This allows for

the plasmonic fields to radiate in the plane of the array, which leads to stronger

coupling and further resonance enhancement. Subsequently, a so-called photonic-

plasmonic hybrid mode is excited [6]. The formation of the hybrid mode can be

clearly identified in Figure 5.4 for lattice constants of 400 nm and 500 nm where

instead of two distinct resonances, one enhanced hybrid resonance is formed. In

periodic structures these hybrid modes usually have a narrow bandwidth due to

the inherently narrowband nature of the photonic resonance.

5.3 Resonance Response in Quasicrystalline Gold

Arrays

Optical properties of aperiodic gold nanoparticle arrays where first considered in

Ref.[24]. However they only considered aperiodic geometries and no quasicrys-

74

Figure 5.5: Extinction efficiency for four periodic arrays of 100 gold spheres ofradius 80 nm with periodicities of 400 nm, 500 nm, 600 nm.

talline formations were studied. Quasicrystals in essence possess multiple photonic

resonances due to specific real-space distances in the structure.

An intuitive approach to study the photonic resonances of quasicrystals is by

analyzing their Fourier diffraction pattern which was introduced in Chapter 2. The

discrete peaks in the diffraction pattern of quasicrystals can be associated with

reciprocal vectors F that satisfy exp (iF ·R) = 1 for apertures having coordinates

R [45]. The reciprocal vectors can be indexed according to their distance from the

origin. These distanced correspond directly and are inversely proportional to half

of specific distances in the structure |Fi| ∝ 2di

.

As an example here we consider a Penrose quasicrystal. The Penrose aperiodic

tiling was introduced in Chapter 2. It possess 5-fold rotational symmetry and

can be generated from two triangular prototiles. Alternatively, Penrose tiling can

be generated by placing narrow (vertex angles π/5 and 4π/5) and wide (vertex

angles 2π/5 and 3π/5) rhombi tiles with equal sides based on specific matching

rules [16]. Figure 5.6 shows the two prototiles of the Penrose tiling along with

three marked distances which are the large diagonal of the narrow rhombus (d1),

the large diagonal of the wide rhombus (d2) and d3 which denotes the side of both

rhombi. Figure 5.7 shows a segment of the Penrose quasicrystals composed of the

narrow and wide rhombi shown in Figure 5.6. As it was discussed in Chapter 2

an interesting property of the Penrose tiling is that different distances between

75

Figure 5.6: Narrow (vertex angles π/5 and 4π/5) and wide (vertex angles 2π/5and 3π/5) rhombi prototiles of Penrose tiling. The three marked distances whichare the large diagonal of the narrow rhombus (d1), the large diagonal of the widerhombus (d2) and d3 which denotes the side of both rhombi.

elements can be expressed in terms of τ , the golden ration (τ = (1 +√

5)/2). It

can be shown thatd1√

4− τ−2=d2

τ=d3

1(5.12)

Fig. 5.8 shows the Fourier diffraction pattern of the Penrose quasicrystal. The

first three reciprocal vectors have been indexed according to their distance from the

origin. Due to the 5-fold rotational symmetry of diffraction pattern, the following

basis for the reciprocal vector space can be defined [46]

ei =

(cos

(i− 1)π

5, sin

(i− 1)π

5

), i = 1, . . . , 5 (5.13)

All reciprocal vectors can be written as an integer linear combination of ei. For

F1, F2, and F3, we have

F1 = e2 , F2 = e2 + e5 , F3 = e4 + e5 (5.14)

76

Figure 5.7: A segment of the Penrose quasicrystals composed of the narrow andwide rhombi and the corresponding distances d1, d2, and d3 from Figure 5.6.

From Eq. (5.14) it can be easily verified that

|F1|d1

=|F2|d2

=|F3|d3

(5.15)

Thus the first three reciprocal vectors correspond to aperture spacings d12

, d22

, andd32

. These three values also correspond to the lowest order photonic resonances

of the quasicrystal. Thus if λ denotes the wavelength of the incident field in the

medium, the lowest order photonic resonance occurs as λ → d12

and the next

photonic resonance occurs as as λ → d22

and so on. In general the wavelength

corresponding to the m-th photonic resonance λm of a quasicrystal is given by:

λm =dm2

m = 1, 2, . . . (5.16)

77

In periodic structures all spacings are integer multiples of each other, and thus

photonic resonances are far apart. The same is not true for quasicrystals. For

example in the case of Penrose quasicrystal which was just analyzed, using the

results from Eq. (5.12) and Eq. (5.16), the first two resonances are roughly related

by λ2 ≈ 1.17λ1.

Figure 5.8: Diffraction pattern of Penrose quasicrystal with the first three recip-rocal vectors displayed.

This novel property of quasicrystals which allows them to have several photonic

resonances in close proximity can be particularly useful for gold nano-spherical

arrays. As it was mentioned, in periodic arrays the hybrid mode is a narrow band

phenomenon due to the inherently narrowband nature of the photonic resonance.

On the contrary, hybrid modes in aperiodic structures can have more desirable

properties because they possess multiple resonances that if designed properly can

create hybrid mode coupling over a relatively wide bandwidth.

To demonstrate this, we consider two aperiodic arrays of gold nano-spheres

based on Penrose tilings. Both arrays consist of 466 gold nano-spheres with a

78

Figure 5.9: Scattering response of two finite aperiodic Penrose gold arrays withdifferent tile sides (540nm, 630nm) obtained using GMT with a finite incidentbeamwidth produced by a circular aperture of radius a = 2λ placed at a distanceof 17µm from the array. Values of T and R were calculated from Eq. (4.41) andEq. (4.44) respectively.

diameter of 160nm. For the first array, we set the tile side (s) to 540nm. As a

result we would expect the two resonances to occur roughly around 440nm and

515nm region. In the case of the second array we set the tile side (s) to 630nm

for which we would expect resonances around 510nm and 600nm. Figure 5.9

shows the reflectance and transmittance values calculated for both arrays based

on Eq. (4.41) and Eq. (4.44) which were derived in Chapter 4. Considering the

first array (s = 540nm), the first resonance is clearly distinguishable around λ =

550nm. This resonance is further enhanced since it is in the plasmonic region of

gold and can form a photonic-plasmonic hybrid mode. The second resonance is

much weaker around λ = 450nm since it is not in the plasmonic region of the gold.

The second array (s = 630nm) displays much more interesting properties. Even

though it has a smaller filling factor than the first array, since both resonances are

in the plasmonic region of gold, they are both enhanced and in essence we observe

two hybrid modes.

79

5.3.1 Optimization

The performance of the aperiodic arrays can be further enhanced by a perturbation

method similar to what was done for antenna arrays based on aperiodic tilings in

Chapter 3. The main difference is that in the case of antenna arrays, array factor

calculations are fairly fast. Furthermore array factor calculations do not require

large amounts of memory which allows for the GA to be parallelized. As a result

using a GA algorithm is a suitable method to optimize the location of perturbation

points, since the cost function evaluations are not too computationally expensive.

The same is not true for GMT calculations. The amount of memory required

for an array of Np spheres using VSWF expansions with Nmax terms is of the

order N2pN

3max + N4

max. Also the resulting linear systems are very large and have

to be solved via iterative methods. Using GA for such cost functions is not a

feasible option since cost function evaluations tend to be intensely computationally

expensive and time consuming. As a result a more analytical approach should be

utilized to optimize the nano-spherical arrays based on aperiodic tilings.

The Danzer tiling was introduced in Chapter 2. The prototile set the tiling is

composed of three triangles shown in Figure 5.10 where θ = π/7 and a, b, and c

are related by the law of sines

a

sin θ=

b

sin 2θ=

c

sin 4θ(5.17)

Figure 5.11 shows the resulting structure after two iteration of inflation and

substitution is applied to type III prototile with elements placed at vertices. As it

can be seen from Figure 5.11 the largest element spacing in the array correspond

to c and a corresponds to the minimum spacing between the spheres. Thus the

lowest order photonic resonance should occur at λ = c/2.

Figure 5.12 shows the extinction efficiency of three quasicrystalline nano-

spherical array based on Danzer tiling. All three arrays consist of 349 gold spheres

with a radius of 80 nm. However the prototiles have been scaled such that the

values for c/2 are 420 nm, 480 nm, and 560 nm as indicated in the plot. The plas-

monic region region of the gold has been highlighted in the plot. The resonance

response for the Danzer array with c2

= 560 nm is much stronger since photonic

resonance is in the plasmonic region and hence a hybrid mode is formed.

80


Figure 5.11: Danzer tiling generated after two iteration applied to prototile typeIII with elements placed at vertices.

81

Figure 5.12: Extinction efficiency for three Danzer of 349 gold spheres of radius 80nm with c/2 values of 420 nm, 480 nm, 560 nm. The plasmonic region of gold ishighlighted.

Thus just by proper scaling, the performance of the array can be considerably

improved. The performance of the the quasicrystalline array can be further en-

hanced by introducing additional elements inside prototiles. However one issue

that must be taken into account is that the perturbation should not significantly

alter the minimum spacing of the tiling since the minimum spacing often represents

a fabrication constraint.

A close inspection of Danzer tiling reveals that it is possible to place an ad-

ditional element in the circumcenter of type III prototile without disturbing the

minimum spacing of the tiling. As a result the density of the arrays goes up by

roughly 50%. Figure 5.13 shows a segment of the Danzer tiling with additional

elements placed at the circumcenter of type III prototile. Figure 5.14 shows the

extinction coefficients of the native and optimized Danzer arrays, incident with a

plane wave. Both arrays consists of gold spheres of radius 80 nm and have been

scaled such that c2

= 560 nm. The native array is composed of 349 spheres and the

optimized array is composed of 521 spheres. It is important to note even though

the number of spheres of the optimized array is roughly 50% higher than the native

array, the resonance is almost twice as strong. Hence the perturbation has actually

caused a lot more coupling in the array.

82

Figure 5.13: A segment of an optimized Danzer tiling with additional elementsplaced at the circumcenter of type III prototile.

Figure 5.15 shows the extinction coefficient (E) of the same two array defined

as E = 1− T . These results are obtained using the modified GMT method which

was developed Chapter 4 based on an incident beam with finite beamwidth. As it

can be seen, the results are consistent with those in Figure 5.14 and these set of

results can be qualitatively compared with experimental measurements.

5.4 Local Field Enhancements in Quasicrystalline

Gold Arrays

In recent years there has been a wide variety of applications for Surface-Enhanced

Raman Scattering (SERS) techniques in rapid and label-free chemical and bio-

logical sensing applications[47]. While the underlying physical phenomena is not

completely understood, it has been shown that SERS is mainly driven by the en-

hanced electromagnetic fields on metal surfaces. In particular it has shown that

83

Figure 5.14: Extinction efficiency for native Danzer array (349 spheres) and theoptimized Danzer array (521 spheres). Both arrays have been scaled such thatc2

= 560 nm.

Figure 5.15: Extinction coefficient (E = 1−T ) for native Danzer and the optimizedDanzer arrays evaluated using GMT method with a finite beamwidth. Both arrayshave been scaled such that c

2= 560 nm.

84

Raman enhancement FSERS is roughly proportional to the fourth power of local

field enhancement:

FSERS ∝(|Eloc||Einc|

)4

(5.18)

where Eloc and Einc denote the local and the incident fields respectively. Local field

enhancements in aperiodic arrays were considered in [3]; however these structures

were not truly planar arrays. Rather they were linear aperiodic arrays extended

into two dimensions. As a result they lacked any higher order rotational symme-

tries. We start by considering periodic, Penrose, Danzer and Ammann-Beenker

(A-B) arrays.

For all the arrays considered in this section gold nanospheres with a radius of 80

nm are assumed. The array is placed in the xy-plane and the incident electric field

is linearly polarized along the x axis with a magnitude of 1 V/m. The incident

radiation has a free space wavelength of 550 nm. Gold has a complex index of

refraction of n = 0.35 + 2.4j at the operating wavelength. The minimum surface

to surface separation between spheres is 20 nm which corresponds to a center to

center separation of 180 nm. The number of spheres for all the arrays are in the

225-251 range, however due to the fact that they have different filling factors, the

total area of the array will vary in each case. Table 5.1 displays the properties

of the four array. The terms dmin, dmax and davg correspond to the minimum,

maximum and the average surface to surface distance. The number of elements in

each array is denoted by N and D denotes relative density which assigns a value

of unity to the periodic array since it has the highest density.

Table 5.1: Geometrical properties of nano-spherical arrays.

Array N dmin (nm) dmax (nm) davg (nm) D

Periodic 225 20 20 20 1Penrose 251 20 135 24 0.78Danzer 248 20 244 46 0.46

A-B 225 20 75 34 0.72

Figure 5.16 shows the local fields for the periodic array. As mentioned before,

this formation has the highest filling factor and provides a maximum local field

intensity of 7.24 V/m. Figure 5.17 shows the local fields for an aperiodic Penrose

85

Figure 5.16: Local fields (V/m) for a periodic array of 225 gold nanospheres withminimum surface to surface distance of 20 nm.

array. From the array geometry it can be observed that this formation has a 5-fold

rotational symmetry and provides a maximum local field intensity of 9.5 V/m.

Figure 5.18 shows the local fields for an aperiodic Danzer array. This array has

a 7-fold rotational symmetry and provides a maximum local field intensity of 13.6

V/m, while Figure 5.19 shows the local fields for an aperiodic Ammann-Beenker

array with 8-fold rotational symmetry. The maximum local field intensity for this

array is 9.55 V/m.

86

Figure 5.17: Local fields (V/m) for a Penrose array of 251 gold nanospheres withminimum surface to surface distance of 20 nm.

Figure 5.18: Local fields (V/m) for a Danzer array of 248 gold nanospheres withminimum surface to surface distance of 20 nm.

87

Figure 5.19: Local fields (V/m) for an A-B array of 225 gold nanospheres withminimum surface to surface distance of 20 nm.

Chapter 6Optical Mirrors Based on

Metamaterial Coatings

6.1 Introduction

Resonant reflection in subwavelength gratings were first identified and studied in

early 1990s [48]. With appropriate patterning, such periodic gratings composed of a

single dielectric layer could perform as well as 25-40 layer dielectric Bragg reflectors

[49]. In addition to being considerably more compact than dielectric stacks, these

structures when properly designed provide new optical features such as polarization

and phase control [50, 51, 52, 53]. Resonance typically occurs when the first order

diffracted mode is a guided mode supported by the grating and hence trapped.

The guided mode is then reradiated into zeroth order mode where it will interfere

with the incident field to create a pronounced reflection [48]. By exploring 1D/2D

periodic nanostructures, coupling with external illumination can be controlled to

produce complex polarization sensitive or insensitive resonance line shapes that

can be used to create a variety of passive optical elements, including mirrors and

filters.

Traditionally, the optical response of planar subwavelength periodic dielectric

nanostructures has been analyzed by calculating the dispersion of the supported

guided mode resonances [54]. In these devices, the response is strongly dependent

on the optical properties of all of the dielectric materials as well as the nanostruc-

89

ture unit cell geometry and dimensions [55]. Despite this dependence, the com-

plexity of designing and fabricating the structures has constrained most earlier

guided mode devices to extremely simple pillar and hole geometries, which places

limits on their optical properties [12]. Recently, nature-inspired design methods

have been used to optimize more sophisticated periodic dielectric nanostructure

geometries that produce advanced user-specified scattering properties arising from

the guided modes [56]. By properly tailoring the scattering parameters with con-

trolled scattering phases and amplitudes, this design approach can be extended to

engineer the effective refractive index and impedance of the subwavelength dielec-

tric nanostructure, which greatly increases the optical functionalities that can be

achieved.

In this chapter we treat the periodic gratings as metamaterial structures and

apply effective medium theory to guide the design process. Our goal is to design

a filter with a near-perfect reflection band (i.e. a near-perfect optical mirror)

centered at the mid-IR band of 3.3µm. We start by deriving the appropriate optical

properties that will satisfy the desired perfect mirror condition. The structure of

the metamaterial grating is optimized using a single point cross-over binary genetic

algorithm (GA) [23]. The scattering parameters of the structures are evaluated

using a periodic finite-element boundary-integral (PFEBI) code. The scattering

parameters are then employed to derive the effective parameters using a suitable

inversion algorithm. The metamaterial filter designed here consists of a thin layer

of amorphous silicon (a-Si) with a doubly-periodic array of air holes inserted into

it. The grating is supported by a 500µm layer of fused silica. Since the thickness

of the supporting layer is much larger that the wavelength of interest, it can be

treated as a semi-infinite half-space. Once the structure was optimized, it was

fabricated and measurements were made of its reflection/transmission properties.

6.2 Effective Parameters

Our objective is to obtain the set of effective material properties for a dielectric

slab which will highly reflective. For a dielectric slab of thickness l with an index of

refraction n and an impedance η, the reflection coefficient R and the transmission

coefficient T for normal incidence radiation with free-space wavelength λ0 are given

90

by:

R =jZ sin(nk0l)− j sin(nk0l)

Z

2 cos(nk0l) + jZ sin(nk0l) + j sin(nk0l)Z

(6.1)

T =2

2 cos(nk0l) + jZ sin(nk0l) + j sin(nk0l)Z

(6.2)

where Z = ηη0

is the normalized impedance of the slab with respect to the free-

space impedance (η0) and k0 = 2πλ0

is the free-space wavenumber. To achieve unity

reflection, the numerator and the denominator in Eq. (6.1) should be of equal

magnitude. This requirement can be satisfied under two extreme cases:

• Perfect electric conductor (PEC): Z → 0 R→ −1

• Perfect magnetic conductor (PMC): Z →∞ R→ 1

Recently all dielectric metamaterials with PEC and PMC properties based on

cylindrical and spherical elements were proposed [57]. However the proposed ge-

ometries were multilayer structures and the high reflection was due to the photonic

bang-gap of the structure. As a result, the final structure was relatively thick. Here,

we derive and apply an alternative set of conditions that can be realized using a

thin metamaterial coating to fabricate a perfect dielectric mirror.

We begin by considering a complex index of refraction n = n′+ jn′′ with near-

zero real part and large imaginary part (n′ ≈ 0 , |n′′| 0). The trigonometric

functions for complex arguments are defined as [15]:

sin(x+ jy) = sinx cosh y + j cosx sinh y

cos(x+ jy) = cos x cosh y − j sinx sinh y (6.3)

where sinh and cosh are the hyperbolic sine and cosine functions. Using our as-

sumption n′ ≈ 0 , |n′′| 0, it can be shown that:

sin(nk0l) ≈ sin(jk0ln′′) = j sinh(k0ln

′′)

cos(nk0l) ≈ cos(jk0ln′′) = cosh(k0ln

′′) (6.4)

91

Furthermore for |n′′| 0, following approximation is valid:

cosh(k0ln′′) ≈ − sinh(k0ln

′′) (6.5)

Thus substituting the results from Eq. (6.4) and Eq. (6.5) into Eq. (6.1) we have

R =sinh(k0ln

′′)(Z − 1

Z

)2 cosh(k0ln′′)− sinh(k0ln′′)

(Z + 1

Z

) =sinh(k0ln

′′)(Z − 1

Z

)− sinh(k0ln′′)

(2 + Z + 1

Z

) (6.6)

In order to have a high reflection surface we must have |R| → 1. Applying this

condition to Eq. (6.6) we arrive at∣∣Z − 1Z

∣∣∣∣2 + Z + 1Z

∣∣ =|Z − 1||Z + 1|

= 1 (6.7)

At this point it is helpful to rewrite Eq. (6.7) with the normalized impedance Z

expressed in terms of its real and imaginary parts as

|Z − 1||Z + 1|

=|Zr + jZi − 1||Zr + jZi + 1|

= 1 (6.8)

where Zr and Zi denote the real and imaginary parts of the normalized impedance,

respectively. A close inspection of Eq. (6.8) reveals that for a purely imaginary

normalized impedance (Zr → 0), the numerator and the denominator have equal

magnitudes and |R| → 1. Thus, a thin near zero refractive index dielectric coating

with a large extinction coefficient (i.e., an evanescent wave condition) and a purely

imaginary impedance will result in high reflection. These three requirements can

also be expressed as:

<(n)→ 0 (6.9a)

=(n)→∞ (6.9b)

<(Z)→ 0 (6.9c)

92

6.3 Design Process

The metamaterial coating considered here is composed of a doubly periodic a-Si

nanostructure grating with a unit cell period(p) that is small enough to suppress

higher order diffraction modes. The targeted wavelength is 3.3µm and our objec-

tive is to obtain a reflectance of at least 98%. Denoting the indices of refraction for

half-spaces above and below the structure ninc and next respectively, the condition

for zero-th order reflection and transmission for a normally incident plane wave is:

p < min

(λ0

ninc

,λ0

next

)(6.10)

For design optimization, the unit cell is divided into a 16 × 16 grid of pixels

assigned with the value of either 1 for the high-index a-Si features or 0 the low-index

air regions. To mitigate polarization sensitivity, eight-fold rotational symmetry is

enforced. Along with the geometry of the unit cell, the unit cell period, and a-

Si layer thickness are also encoded into a binary string to completely describe

the metamaterial structure. The scattering parameters of each candidate design

are calculated using a full-wave periodic finite-element boundary-integral (PFEBI)

method [58], and the corresponding effective parameters (n, Z) are retrieved using

an established inversion algorithm [59]:

Z = ±

√(1 +R)2 − T 2

(1−R)2 − T 2(6.11a)

ejnk0l = X ± j√

1−X2 , X =1

2T (1−R2 + T 2)(6.11b)

where the sign ambiguities are resolved by enforcing the conditions <(Z) ≥ 0 and

=(n) ≤ 0.

The grating is optimized via a single point crossover binary GA. GAs are very

well established methods for performing global optimization of electromagnetic

problems. They are based on evolutionary principles and natural selection. A

brief introduction of GA is included in Appendix A. For a more complete descrip-

tion of GAs and their applications in electromagnetics, the reader is referred to

reference [23]. Defining a proper cost function is one of the most essential steps in

93

implementing a GA. The metamaterial conditions shown in Eq. (6.9) for perfect

reflection are embodied in the following cost function:

Cost = |<(n)|+ |<(Z)| − |=(n)| (6.12)

Once the cost function has been properly defined GA can proceed to minimize

it. As mentioned previously, eight-fold rotational symmetry is enforced on the

structure, thus the pixelated geometry can be described using 36 bits. We use

8 bit discretization for the unit cell period and the a-Si layer thickness. Hence

the complete geometry of the structure is encoded in 52 bits. Furthermore we

incorporated two fabrication constraints into our GA to eliminate geometries which

include single a-Si pixels or diagonally connected a-Si pixels by assigning them large

costs.

Figure 6.1 illustrates one unit cell of the GA optimized wavelength-selective di-

electric ZIM mirror design. When replicated in 2D, this pattern results in an array

of isolated a-Si blocks surrounded by an interconnected a-Si dielectric structure.

The a-Si layer thickness is t = 468 nm and the unit cell period is p = 2.05µm,

which gives a pixel dimension of 125 nm. At the target 3.3µm free-space op-

erating wavelength the zero-th order scattering condition from Eq. (6.10) is also

satisfied (ninc = next = 1). The simulated reflection and transmission coefficients

for the structure are shown plotted in Figure 6.2. Figure 6.3 and Figure 6.4 show

the extracted index of refraction (real and imaginary parts) and the normalized

impedance respectively. As it can be seen from the plots of the effective parame-

ters near the targeted region (∼ 90 THz) all the conditions specified in Eq. (6.9)

are satisfied. The slab has a purely imaginary index of refraction as well as a

purely imaginary impedance and the resonance region corresponds to the largest

value for the imaginary part of the index of refraction.The values for the complex

refractive index and normalized impedance at λ0 = 3.3µm are n = 0.072 − 5.41j

and Z = 0.001 + 0.794j, which satisfy the metamaterial conditions for a perfect

dielectric mirror.

Once the index of refraction and the normalized impedance have been cal-

culated, then the relative permittivity and permeability can be determined from

the following simple relations εr = n/Z and µr = nZ respectively. Figure 6.5

94

Figure 6.1: The unit cell structure of the optimized metamaterial grating withp = 2.05µm and t = 468 nm.

and Figure 6.6 show the effective permittivity and permeability of the lossy ZIM

respectively. Figure 6.7 shows the electric field distributions in the optimized

structure at the targeted wavelength of λ0 = 3.3µm at the xy-plane. It is assumed

that the structure lies in the xy-plane with a x-polarized plane wave normally

incident upon it. Figure 6.7a shows the field distributions in the xy-plane at the

top of the structure. Figure 6.7b shows the field structures in the xy-plane in the

95

Figure 6.2: Simulated reflection and transmission coefficients of the optimizedmetamaterial structure in free space.

Figure 6.3: Effective index of refraction for the optimized metamaterial mirror.

96

Figure 6.4: Effective normalized impedance for the optimized metamaterial mirror.

Figure 6.5: Effective permittivity for the optimized metamaterial mirror.

middle of the structure. Figure 6.8a and Figure 6.8b show the field distributions

looking sideways in the yz-plane at two different values of x. The relative values of

x are shown with a red line in the unit cell of the structure in the lower left hand

corner of these two figures. As it can be seen from all four figures fields are much

97

Figure 6.6: Effective permeability for the optimized metamaterial mirror.

(a) xy-plane on the top of the struc-ture.

(b) xy-plane in the middle of thestructure.

Figure 6.7: Electric field distributions for the optimized structure at resonance(λ0 = 3.3µm) in the xy-plane.

stronger in low dielectric (air) regions which confirms that the optimized structure

displays guided mode resonance.

98

(a) yz-plane with relative value of xin the unit cell shown with the redline in the lower left hand corner ofthe figure.

(b) yz-plane with relative value of xin the unit cell shown with the redline in the lower left hand corner ofthe figure.

Figure 6.8: Electric field distributions for the optimized structure at resonance(λ0 = 3.3µm) in the yz-plane.

6.4 Fabrication and Measurements

The optimized mirror structure has isolated a-Si blocks which are surrounded by

air holes, making the fabrication of a free-standing structure infeasible. In practical

applications, however, the metamaterial mirror would be used as a coating. Hence,

we included a 500µm thick fused silica substrate as a supportive layer that is

transparent throughout the targeted range of wave-lengths. Replacing fused silica

for air will not largely affect the performance of the optimized structure since there

is a large contrast between the dielectric constants of fused silica and a-Si which

allows for the confinement of fields in the grating and formation of guided mode

resonance. We would however expect a slight shift in the location of the resonance.

Fabrication started with the deposition of a 468 nm thick a-Si layer on a cleaned

fused silica substrate using a plasma enhanced chemical vapor deposition (PECVD)

system (Applied Materials P-5000 cluster). Following E-beam exposure (Leica

EBPG5 HR & Vistec EBPG 5000+ HR) and development of the resist (Nippon

Zeon ZEP 520A), a 60 nm Cr layer was then evaporated and lifted off to form

an etching mask. The pattern was transferred into the a-Si layer to define air

holes by highly anisotropic RIE (Tegal 6540) using chlorine (Cl2) and argon (Ar).

Finally, the Cr mask layer was wet-etched and the samples were then ready for

characterization. The electron micrograph of the fabricated sample is shown in

Figure 6.9 with highly anisotropic sidewalls.

For validating the performance of the mirror, two different sized samples were

99

prepared. First, a 4 × 4 mm2 sample was fabricated using the above-mentioned

process steps for characterization of the scattering parameters, and then another

fabrication of a 1 × 1 in.2 sample was conducted for uniformity mapping mea-

surements. To account for the loading effect of the substrate, another numerical

simulation was performed with the measured dispersive optical properties of fused

silica, which induced a slight reflection peak shift from 3.33µm to 3.47µm as shown

in Figure 6.10. Despite this minor shift, the profile of the resonance was main-

tained, and the peak reflection was further enhanced from 0.997 to 0.999 due to

the fused silica substrate.

Figure 6.9: FESEM image of the fabricated metamaterial mirror with an inset ofa magnified unit cell.

The angular sensitivity of the fabricated mirror sample was also tested. In

many practical applications for filters and mirrors, angular tolerance as well as po-

larization insensitivity is preferred for operation when illuminated by an unpolar-

ized light beam with a non-negligible divergence. As expected from the eight-fold

symmetry of the unit cell geometry, the spectral response of this mirror is nearly in-

100

Figure 6.10: Simulated and measured transmission and reflection spectra of 4 ×4 mm2 sample.

dependent of the incident wave polarization. Further experiments showed that this

mirror also can be utilized up to 10 incidence without a noticeable degradation

in reflection efficiency (> 0.98%) and shift of resonance position (< 6%).

6.5 Conclusion

In conclusion, a thin high-efficiency metamaterial mirror coating was successfully

synthesized at the mid-IR wavelength of 3.3µm based on an effective medium

design approach. It was demonstrated that the derived metamaterial conditions

could be exploited to achieve high reflection originating from a guided mode reso-

nance. The optimized structure was fabricated in large scale and characterized to

show good agreement with the theoretical predictions, thereby demonstrating that

the metamaterial mirror could be used in practical coating applications. We have

shown that a metamaterial-based approach can be used to design optical coatings

which, when coupled to a suitable GA optimization scheme, provides a more gen-

eral and robust alternative to conventional methods based on direct manipulation

101

of guided mode resonances.

Chapter 7Transmission Gratings Based on

Efficient Optimization of

Polynomials

7.1 Introduction

Recently there has been considerable interest in utilizing subwavelength periodic

gratings to generate surfaces with wideband properties for solar cell applications

[60, 61, 62]. These gratings are usually placed on top of photovoltaic cells and thus

are required to be highly transmissive over a wide field of view and be broadband.

In [60] an optimized triangular grating was utilized to generate large transmission

values over a wide bandwidth; however only normal incident radiation was con-

sidered. In [63]aperiodic metal-dielectric stacks were considered. The resulting

structure did provide high transmission with a wide angular tolerance; however

the bandwidth was rather narrow.

One of the recent approaches that has been investigated is to employ metallic

gratings. Using this method absorption is enhanced as a result of surface plasmons

which are supported by the metallic gratings [64]. While this method does provide

a high transmission value, the losses in metals which tend to be large in the visible

range can reduce the efficiency of the solar cells. As a consequence it has been

proposed that dielectric gratings might be more suitable for this purpose [65].

103

A sawtooth shaped grating made of silicon carbide was considered in [66]. The

structure was shown to provide a good angular response, however the average

absorption was only about ∼ 66%.

Traditional transmission filters are usually comprised of triangular, sinusoidal,

or rectangular gratings [67]. The specific dimensions need to be optimized ac-

cording to the design objectives. However, these basic shapes are too limited and

often they are not sufficient to meet challenging design specifications. One way to

overcome these limitations is to use gratings with more complicated side-wall pro-

files. Basically the side-wall is discretized into a certain number of layers of equal

thickness and the width of each layer is then optimized to meet the design goals.

This will greatly increase the degrees of freedom in the design process. However

there are two problems associated with this approach.

First, depending on the number of discretized layers the associated number of

variables that require optimization can be rather large and difficult to manage.

Secondly since the width of each layer is optimized separately there might be

large discontinuities between neighboring layers that can lead to designs that are

impractical to fabricate.

In this chapter, we propose a computationally efficient genetic algorithm (GA)

strategy based on optimizing the roots of a polynomial. This method has the ad-

vantage of only requiring that a relatively small number of variables be optimized

while at the same time allowing for more complex side-wall profiles to be achieved.

It also overcomes the issue of discontinuity between neighboring layers since poly-

nomials are continuous functions. Using this method a filter is designed to provide

mean transmittance values larger than 95% over the visible band and with a field

of view of 120. The filter we designed consists of a layer of periodic zinc sulfide

(ZnS) and it is supported by a semi-infinite layer of the same material (ZnS). ZnS

was used due to its low loss in the visible range of the spectrum.

7.2 Design Process

As mentioned earlier, we attempt to shape the side-wall profiles of the grating

based on an optimized polynomial. An n-th degree polynomial with real roots can

104

be represented as:

Pn(x) = (x− x1) (x− x2) . . . (x− xn) , x1, . . . , xn ∈ R (7.1)

In order to simplify the process, it is further assumed that all the roots are

between 0 and 1. To create the profile, the polynomial is normalized and sampled

according to fabrication constraints imposed on the height and width of the ridges

in the structure. To create the profile, the polynomial is normalized and sampled

according to fabrication constraints imposed on the height and width of the ridges

in the structure. Figures 7.1 and 7.2 show an example of this process for a fourth

degree polynomial. Figure 7.1 shows the shape of the forth degree polynomial and

Figure 7.1 shows the structure of the resulting side-wall.

Figure 7.1: A fourth degree polynomial with roots in [0, 1].

Hence in order to find the best choice for the polynomial, all the roots must

be optimized. In addition, the periodicity of the structure and its height are also

included in the optimization. There is also the question of what the degree of

the polynomial should be. Two different approaches can be taken regarding the

degree of the polynomial. First, the degree of the polynomial can be fixed. In

this case there will be a total of n + 2 variables that require optimization. In the

second approach, a range (minimum and maximum value) can be specified for the

105

Figure 7.2: Resulting side-wall profile from the polynomial in Figure 7.1.

degree of the polynomial and then it is left to the optimizer to first determine the

optimal degree of the polynomial and then the optimal choice for the corresponding

roots. For the second approach there will be a total of n+ 3 variables that require

optimization. For our problem we used the second approach and allowed the degree

of the polynomial to be in the 3− 6 range.

As mentioned earlier the design variables are optimized by a GA. The GA is a

global optimization strategy that is based on the evolutionary principles of natural

selection and mutation. Their use in electromagnetic optimization problems is well

established. A brief introduction of GA is included in Appendix A. For a more

complete description of GAs and their applications in electromagnetics, the reader

is referred to reference [23].

106

7.3 Results

As mentioned earlier our targeted range is the visible band of the spectrum which

covers 400 − 770 THz (0.39µm< λ < 0.75µm). Also it was required that the

structure provide a high transmission for a FOV of 120(−60 < θ < 60). It

is important to mention that for the current design only one polarization was

considered. It is assumed that the electric field of the incident wave is polarized

along the periodicity of the grating (TM). Based on the above specifications, the

following cost function was defined:

F =∑θ,f

(1− |Tθ,f |)2

θ = 0, 15, 30, 45, 60 , f = 400, 450, 500, 550, 600, 750THz (7.2)

where Tθ,f is the transmission coefficient at frequency f and incident angle of θ.

The summation in Eq. (7.2) contains a total of 24 terms.

A single cross-over binary GA was used to minimize the cost function defined

in Eq. (7.2). For each design candidate the scattering parameters were calculated

using a periodic finite-element boundary-integral (PFEBI) code [58]. Once the

scattering parameters have been determined, the cost function can be evaluated

and the GA will proceed.

The filter we designed consists of a layer of periodic zinc sulfide (ZnS) and it is

supported by a semi-infinite layer of the same material (ZnS). ZnS was used due

to its low loss in the visible range of the spectrum. Figure 7.3 shows the relative

permittivity of ZnS over the visible range [68].

Figure 7.4 shows a unit cell of the optimized structure. The optimized structure

has a periodicity of 113 nm and a thickness of 77 nm. Figure 7.5 shows several

periods of the optimized grating. Figure 7.6 shows the transmittance of the grating

for a TM polarized plane wave over the visible band for different angles of incidence.

Table 7.1 shows the tabulated mean transmittance values for different incidence

angles over the visible band. The average transmittance for all angles of incidence

up to 60 is greater than 95%.

107

Figure 7.3: Relative permittivity of ZnS over the visible range.

Figure 7.4: Unit cell of the optimized grating.

108

Figure 7.5: Four periods of the optimized grating structure.

Figure 7.6: Transmittance values for the optimized grating corresponding to dif-ferent values of incident TM radiation.

7.4 Conclusion

A new method for designing all-dielectric gratings with high transmission, broad

bandwidth and wide FOV was introduced. The method was based on shaping

the side-wall profiles of the gratings according to polynomial functions. It was

demonstrated that by using this method it is possible to generate gratings with

complicated side-wall profiles while using a relatively small number of variables.

The roots and degree of the polynomial along with dimensions of the grating

109

Table 7.1: Mean transmittance values for different incidence angles over the visibleband.

Angle of Incidence (degrees) Mean Transmittance

0 98%10 98.1%20 98.4%30 98.7%40 98.9%50 98.4%60 96%

(periodicity and height) were optimized to meet a specific set of design goals. A

GA was employed along with a suitable cost function in order to optimize the

design parameters.

To illustrate the new design methodology, a design example was considered for

a grating (filter) that provides mean transmittance values greater than 95% with

a FOV of 120 over the entire visible band.

Appendix AGenetic Algorithms

A.1 Introduction

Traditional optimization methods such as Newtons method and conjugate gra-

dient method [69] generally require an analytical expression of the cost function

and its derivative (gradient). When dealing with well-behaved convex analytical

cost functions, the traditional calculus-based optimization methods are the most

efficient methods to find the optimal solution. However many realistic problems

arising in a variety of applications do not fall in this category. The cost functions

generally contain discontinuities and on additionally in many instances the cost

function is evaluated using numerical methods and cannot be defined explicitly.

Over the recent years a number of natural optimization algorithms including sim-

ulated annealing, particle swarm optimization, and ant genetic algorithms have

been developed [70]. These natural optimization algorithms do not have the limi-

tations of calculus-based optimization methods and have shown great results in a

variety of optimization problems. Genetic algorithms in particular have been used

in a variety of electromagnetic problems with great results [23]. Here we give a

brief introduction of components of genetic algorithm with discrete variables (bi-

nary genetic algorithm). It is important to note that genetic algorithms can also

be applied with continuous variables. For a complete description of genetic algo-

rithms and their applications in electromagnetic problems, the reader is referred

to reference [23, 70].

111

A.2 Binary Genetic Algorithms

Genetic algorithm (GA) is a heuristic optimization method based on the evo-

lutionary principles of natural selection and evolution. Like other optimization

problems, the problem is defined by optimization variables (x1, x2, . . . , xN) and

the cost function f(x1, x2, . . . , xN) where N denotes the number of variables. Each

solution can be regarded as a chromosome and since we are considering binary

genetic algorithms, each chromosome can be represented as a binary string of 0s

and 1s. Figure A.1 shows a general flowchart of a typical GA. Here we give a brief

description of some of the key steps in the process.

Figure A.1: Flowchart of a typical GA.

112

A.2.1 Initial Population

The initial population is generated randomly. Assuming our general population

consists of Npop chromosomes and each chromosome in encoded with M bits, a

random binary matrix of size Npop ×M can generate the initial population with

each row corresponding to a single chromosome.

A.2.2 Mating

Mating is the creation of offspring from the parents. The parents can be selected

using a variety of pairing methods such as random or tournament selection [70].

Once the parents have been determined a crossover point is randomly selected.

One parent passes its binary code to the left of the crossover point to one offspring

and its binary code to the right of the crossover point to the other offspring. The

other parent does the same process however the order is switched. As a result

each offspring contains portions from both parents. The process is demonstrated

in Figure A.2.

Figure A.2: The mating process for two parents to produce two offspring.

A.2.3 Mutation

During each iteration, a mutation process randomly alters a certain percentage of

the bits in each chromosome. The main purpose of introducing mutations in the

113

algorithm is to prevent convergence at a local minimum point. However mutations

are not applied to the best solutions.

A.2.4 Stopping Criteria

A variety of conditions can be used as stopping criteria for the algorithm. The

algorithm can stop once the cost function has reached a certain value or the number

of iterations has reached a preset limit.

Appendix BDerivation for Ωn(ka) and Ξn(ka)

Our goal is to evaluate the definite integral in Eq. (B.1)∫ π/2

0


)dθ (B.1)

We start by rewriting the integrand with τ1n and π1n defined in terms of the

associated Legendre function P 1n based based on Eq. (4.4)

J1(ka sin θ)[π1n + cos θτ1n] = J1(ka sin θ)

[P 1n

sin θ+ cos θ

dP 1n

dθ

](B.2)

As it was previously noted, unless otherwise stated, it is always assumed that

functions πmn, τmn, Pmn , and Pn have cos θ as their argument. As the next step,

we consider Legendre’s differential equation [38]

d

dθ

(sin θ

dPndθ

)= −n(n+ 1) sin θPn (B.3)

By performing the differentiation in Eq. (B.3) and using the relation dPn

dθ= −P 1

n ,

it can be shown that

P 1n

sin θ+ cos θ

dP 1n

dθ= n(n+ 1) cos θPn + sin θP 1

n (B.4)

115

Thus the integral in Eq. (B.1) can be decomposed as

n(n+ 1)

∫ π/2

0

J1(ka sin θ) cos θPndθ

+

∫ π/2

0

J1(ka sin θ) sin θP 1ndθ (B.5)

Here we define two auxiliary functions Ωn(ka) and Ξn(ka) based on the integrals

in Eq. (B.5)

Ωn(ka) ≡∫ π/2

0

J1(ka sin θ) cos θPndθ (B.6)

Ξn(ka) ≡∫ π/2

0

J1(ka sin θ) sin θP 1ndθ (B.7)

We start by considering Ωn(ka). From Gradshteyn and Ryzhik [41] we have

the following expansion for Pn(x)

Pn(x) =1

2n

bn2c∑

k=0

(−1)k(2n− 2k)!

k!(n− k)!(n− 2k)!xn−2k (B.8)

where b·c denotes the floor function. The cos θPn term in the integrand can be

written as the following finite summation [38]

cos θPn =1

2n

bn2c∑

t=0

(−1)t(2n− 2t)!(cos θ)(n−2t+1)

t!(n− t)!(n− 2t)!(B.9)

Substituting Eq. (B.9) into Eq. (B.6), the integrand can be represented as a finite

summation. Furthermore, due to the linearity of the integral operator, the order

for the integration and summation can be interchanged. As a result, the problem

is reduced to dealing with a summation of integrals of the form∫ π/2

0

(cos θ)n−2t+1J1(ka sin θ)dθ (B.10)

116

The integral in Eq. (B.10) can be evaluated using the following result [41]∫ π/2

0

Jµ(z sin θ)(sin θ)1−µ(cos θ)2ν+1dθ =s(µ+ν,ν−µ+1)(z)

2µ−1zν+1Γ(µ)(B.11)

where Γ is the gamma function and sµ,ν(z) is the Lommel function defined as

sµ,ν(z) =π

2

[Yν(z)

∫ z

0

zµJν(z)dz − Jν(z)

∫ z

0

zµYν(z)dz

](B.12)

and Yν is the Bessel function of the second kind of order ν. Letting µ = 1,

2ν = n− 2t, and z = ka in Eq. (B.11) we arrive at∫ π/2

0

cos θ(n−2t+1)J1(ka sin θ)dθ =s(1−t+n

2,n2−t)(ka)

(ka)(1−t+n2

)(B.13)

Numerical evaluation of Lommel functions can be in general a challenging un-

dertaking. However, at this point a closer inspection of Eq. (B.13) reveals that it

is only necessary to consider Lommel functions of the form sν+1,ν . We were able to

derive a concise analytical expression for this particular order of Lommel functions

as

sν+1,ν(z) = zν − 2νΓ(ν + 1)Jν(z) (B.14)

The derivation of Eq. (B.14) is provided in Appendix D. Combining all these

results and after some further simplifications we arrive at the following expression

for Ωn(ka)

Ωn(ka) =1

ka− 1

2n

bn2c∑

t=0

(−1)t(2n− 2t)!

t!(n− t)!(n− 2t)!

Γ(1− t+ n2)2(n

2−t)J(n

2−t)(ka)

(ka)(1−t+n2

)(B.15)

Next we consider Ξn(ka) defined in Eq. (B.7). By differentiating Eq. (B.8) with

respect to θ and using the fact that P 1n = −dPn

dθthe following expansion for P 1

n can

be obtained:

P 1n(cos θ) =

sin θ

2n

bn−12c∑

t=0

(−1)t(2n− 2t)!

t!(n− t)!(n− 2t− 1)!(cos θ)n−2t−1 (B.16)

117

Using this expansion for P 1n , the integrand in Eq. (B.7) can be cast in the form of

the summation given below:

J1(ka sin θ) sin2 θ

2n

bn−12c∑

t=0

(−1)t(2n− 2t)!(cos θ)(n−2t−1)

t!(n− t)!(n− 2t− 1)!(B.17)

Substituting Eq. (B.17) back into Eq. (B.7), the individual terms in the summation

can be evaluated using [41]∫ π/2

0

Jµ(α sin θ)(sin θ)µ+1(cos θ)2%+1dθ

= 2%Γ(%+ 1)α(−%−1)J(%+µ+1)(α) (B.18)

Setting µ = 1, α = ka, and % = n2− t− 1 leads to

∫ π/2

0

J1(ka sin θ)(sin2 θ)(cos θ)(n−2t−1)dθ

= 2(n2−t−1)Γ

(n2− t)

(ka)(t−n2

)J(n2−t+1)(ka) (B.19)

Finally, by using this result we arrive at the following expression for Ξn(ka)

Ξn(ka) =1

2n

bn−12c∑

t=0

(−1)t(2n− 2t)!2(n2−t−1)Γ

(n2− t)

(ka)(t−n2

)J(n2−t+1)(ka)

t!(n− t)!(n− 2t− 1)!(B.20)

Appendix CDerivation for Ψn(ka)

Our goal is to evaluate the definite integral in Eq. (C.1)∫ π/2

0


)dθ (C.1)

We start by rewriting the integrand with τ1n and π1n defined in terms of the

associated Legendre function P 1n based based on Eq. (4.4)∫ π/2

0

J1(ka sin θ)

sin θ

(dP 1

n

dθ+ cos θ

P 1n

sin θ

)sin θdθ (C.2)

It can easily be shown that(dP 1

n

dθ+ cos θ

P 1n

sin θ

)sin θ =

d

dθ

(sin θP 1

n

)(C.3)

Substituting the result from Eq. (C.3) back into Eq. (C.2) and using Legendre’s

differential equation in Eq. (B.3) and dPn

dθ= −P 1

n , the original integral in Eq. (4.29)

can be written as

n(n+ 1)

∫ π/2

0

J1(ka sin θ)Pndθ ≡ n(n+ 1)Ψn(ka) (C.4)

where the auxiliary function Ψn(ka) has been defined as∫ π/2

0

J1(ka sin θ)Pndθ ≡ Ψn(ka) (C.5)

119

Evaluating the integral in Eq. (C.5) follows a very similar procedure to that already

presented in Appendix B for Ωn(ka). We start by rewriting the integrand as

a finite summation and reversing the order of summation and integration. As

before, Lommel functions of the form sν+1,ν(z) are encountered in the results,

which can be evaluated using Eq. (4.32). Combining all the results and upon

further simplifications, we arrive at

Ψn(ka) =1

ka− 1

2n

bn2c∑

t=0

(−1)t(2n− 2t)!

t!(n− t)!(n− 2t)!

Γ(n+1

2− t)

2(n−12−t)J(n−1

2−t)(ka)

(ka)(n+12−t)

(C.6)

Appendix DDerivation for sν+1,ν(z)

The Lommel function sµ,ν(z) is defined as [41]

sµ,ν(z) = zµ−1

∞∑m=0

(−1)m(z2

)2m+2Γ(

12µ− 1

2ν + 1

2

)Γ(

12µ+ 1

2ν + 1

2

)Γ(

12µ− 1

2ν +m+ 3

2

)Γ(

12µ+ 1

2ν +m+ 3

2

) (D.1)

where µ± ν is not a negative odd integer and Γ is the gamma function. Alterna-

tively, sµ,ν(z) can also be represented as the following definite integral

sµ,ν(z) =π

2

[Yν(z)

∫ z

0

zµJν(z)dz − Jν(z)

∫ z

0

zµYν(z)dz

](D.2)

where Jν and Yν are the Bessel function of the first and second kind of order ν

respectively. Considering the special case of Lommel functions of the form sν+1,ν ,

based on the integral definition given in Eq. (D.2) we have

sν+1,ν(z) =π

2

[Yν(z)

∫ z

0

zν+1Jν(z)dz − Jν(z)

∫ z

0

zν+1Yν(z)dz

](D.3)

Fortunately both integrals encountered in Eq. (D.3) have closed form solutions [41]∫xp+1Zp(x)dx = xp+1Zp+1(x) (D.4)

121

where Zp represents an arbitrary Bessel function of order p. Considering the first

term in Eq. (D.3), we arrive at

Yν(z)

∫ z

0

zν+1Jν(z)dz = zν+1Yν(z)Jν(z) (D.5)

In evaluating Eq. (D.3) using Eq. (D.4), we encounter a 0 ×∞ indeterminate

form due to the term zν+1Yν+1(z) when evaluated at z = 0. In this case the small

argument limit of Yν(z) [38] can be used to show that

limz→0

zν+1Yν+1(z) =−Γ(ν + 1)2ν+1

π(D.6)

Hence, the second term in Eq. (D.3) is

Jν(z)

∫ z

0

zν+1Yν(z)dz = zν+1Jν(z)Yν+1(z) + Jν(z)Γ(ν + 1)2ν+1

π(D.7)

Substituting the results from Eq. (D.5) and Eq. (D.7) back into Eq. (D.3), and

using the following relationship for Bessel functions [38]

Jν(z)Yν+1(z)− Yν(z)Jν+1(z) =−2π

z(D.8)

after some simplifications, we arrive at:

sν+1,ν(z) = zν − 2νΓ(ν + 1)Jν(z) (D.9)

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Vita

Frank (Farhad) A. Namin

Frank (Farhad) Namin (S. 08), received the B.S. and M.S. degrees in electri-cal engineering from the University of Texas at Dallas in 2008. He joined theComputational Electromagnetics and Antennas Research Lab (CEARL) at thePennsylvania State University in 2008 to pursue his PhD. His Ph.D. research isfunded through the Exploratory and Foundational Program Fellowship from theApplied Research Laboratory (ARL). He is the winner of 2012 Dr. Nirmal K. BoseDissertation Excellence Award.

nature-inspired optimization of quasicrystalline arrays and all-dielectric optical filters and ...

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