Metamaterials Analysis, Modeling, and Design in the Point Dipole Approximation

266 Pages·2017·4.71 MB·English

by  Patrick T. Bowen

Metamaterials Analysis, Modeling, and Design in
the Point Dipole Approximation
by
Patrick T. Bowen
Department of Electrical and Computer Engineering
Duke University
Date:
Approved:
David R. Smith, Supervisor
Daniel Gauthier
Willie Padilla
Maiken Mikkelsen
Nathan Kundtz
Dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in the Department of Electrical and Computer Engineering
in the Graduate School of Duke University
2017
Abstract
Metamaterials Analysis, Modeling, and Design in the Point
Dipole Approximation
by
Patrick T. Bowen
Department of Electrical and Computer Engineering
Duke University
Date:
Approved:
David R. Smith, Supervisor
Daniel Gauthier
Willie Padilla
Maiken Mikkelsen
Nathan Kundtz
An abstract of a dissertation submitted in partial fulfillment of the requirements for
the degree of Doctor of Philosophy in the Department of Electrical and Computer
Engineering
in the Graduate School of Duke University
2017
Copyright (cid:13)c 2017 by Patrick T. Bowen
All rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial Licence
Abstract
This dissertation is focused on applying the discrete dipole approximation to model-
ing metamaterial structures and devices. In particular, it is focused on modeling the
linear and nonlinear behavior of one particular kind of metasurface, called a film-
coupled metasurface. Film-coupled metasurfaces are periodic structures of metama-
terial elements where the elements are placed a deeply subwavelength distance away
from a metal film. The optical nanopatch antenna is an example of a particularly
interesting film-coupled metasurface, and it is explored in depth in this dissertation.
Starting with fundamental coupled mode theory approaches, fully predictive, ana-
lytic formula are developed that solve for the polarizabilities of the elements, which
in turn are used to compute the reflective properties of the metasurface, including
the effects of spatial dispersion using the language of effective medium theory. The
theory is able to explain Wood’s anomalies of the structure from an effective medium
standpoint, again using purely analytic results that show excellent agreement with
experiments and full-wave simulations. fThe linear optical theory is extended in
later chapters to applications in nonlinear optics including bistability and lasing in
four-level systems. The final chapter is devoted to solving for surface modes of the
structure with complex eigenfrequencies, which may be useful in future work for ex-
plaining recent experiments that show lasing in modes that are spatially coherent
across the surface.
Modeling other metamaterial devices using the discrete dipole approximation,
iv
including radio frequency metamaterial antennas, is discussed in the appendices.
v
To my grandfather, Donald Lanier Bowen, who was the greatest example of
husband, father and grandfather that I’ve witnessed.
vi
Contents
Abstract iv
List of Tables xi
List of Figures xii
Acknowledgements xviii
1 Introduction 1
2 Review of Effective Medium Theory and the Dipolar Description of
Materials 7
2.1 The Point Dipole in Free Space . . . . . . . . . . . . . . . . . . . . . 12
2.2 Metamaterial Lattices in Free Space . . . . . . . . . . . . . . . . . . . 14
2.3 Applying the Dipole Approximation to Metamaterial Elements . . . . 20
3 Coupled Mode Theory Analysis of Plasmonic Patch Antennas 24
3.1 Derivation of Eigenmode Coupled Equations . . . . . . . . . . . . . . 28
3.2 Coupling to the Far-Field . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Ohmic Losses of Plasmonic Nanopatch Antennas . . . . . . . . . . . . 39
3.4 No Modal Cross-Coupling Approximation . . . . . . . . . . . . . . . 40
3.5 ModalCrossCouplingDuetoRadiationDamping: SurfaceImpedance
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Modal Cross-Coupling Due to Radiation Damping: Fourier Method . 48
4 Metamaterial Perfect Absorbers 51
4.1 Limits on the Absorption of a Homogenous, Anisotropic Material . . 53
vii
4.2 Surface Impedance Requirements of Perfect Absorbers . . . . . . . . 55
4.3 Magnetic Polarizability of Patch Antennas . . . . . . . . . . . . . . . 56
4.4 Calculation of the Magnetic Field Radiated by an Array of Magnetic
Dipoles using Poisson’s Summation Technique . . . . . . . . . . . . . 61
4.5 Reflection of Film-Coupled Patch Antenna Arrays . . . . . . . . . . . 63
4.6 Radiation Q-Factor of a Periodic Lattice of Patch Antennas . . . . . 64
4.7 Perfect Absorption by Film-Coupled Nanopatches . . . . . . . . . . . 66
4.8 Extraction of the Effective Width . . . . . . . . . . . . . . . . . . . . 72
5 Effective Medium Theory of Film-Coupled Metasurfaces 74
5.1 The Film-Coupled Magnetic Dipole . . . . . . . . . . . . . . . . . . . 76
5.2 One Dimensional Array of Film-Coupled Magnetic Dipoles . . . . . . 79
5.3 Two Dimensional Array of Film-Coupled Magnetic Dipoles . . . . . . 82
5.4 Applications to Metamaterial Absorbers . . . . . . . . . . . . . . . . 86
6 Wood’s Anomalies in Film-Coupled Optical Nanopatch Antenna
Arrays 91
6.1 Introduction to Wood’s Anomalies . . . . . . . . . . . . . . . . . . . . 91
6.2 Historical Context of Wood’s Anomalies . . . . . . . . . . . . . . . . 94
6.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.4.1 Sample Fabrication and Characterization . . . . . . . . . . . . 105
6.4.2 Optical Measurement . . . . . . . . . . . . . . . . . . . . . . . 106
6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Optical Bistability in Film-Coupled Metasurfaces 110
8 Lasing in a Single Film-Coupled Optical Nanopatch Antenna 120
8.1 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
viii
8.2 Analytical Solution of the Interaction between an Optical Cavity and
a Homogeneous Gap Material . . . . . . . . . . . . . . . . . . . . . . 126
8.3 Finite-element implementation . . . . . . . . . . . . . . . . . . . . . . 127
8.4 Application to a Fabry-P´erot Resonator . . . . . . . . . . . . . . . . 129
8.5 Application to Single Plasmonic Nanoparticles . . . . . . . . . . . . . 131
9 Surface Modes of Film-Coupled Metasurfaces 138
9.1 Analytical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2 Comparison with Numerical Simulations . . . . . . . . . . . . . . . . 148
10 Conclusions 152
A Solutions to Maxwell’s Equations in Cylindrical Coordinates 156
A.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.2 Transfer Matrix Method for Multilayer Planar Waveguide Modes . . . 162
A.2.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B Coupled Mode Theory in a Cylindrical Hankel Basis 169
B.1 Derivation From Unconjugated Lorentz Reciprocity . . . . . . . . . . 170
B.2 Example: Electric Dipole Above a Metal Film . . . . . . . . . . . . . 174
B.3 Example: Electric Dipole at the Origin . . . . . . . . . . . . . . . . . 175
B.4 Example: Electric Dipole at an Arbitrary Position . . . . . . . . . . . 177
B.5 Example: Magnetic Dipole at the Origin . . . . . . . . . . . . . . . . 178
C Derivation of Radiated Power by a Film-Coupled Magnetic Meta-
surface 181
D Exact Green’s Functions of Film-Coupled Magnetic Dipoles from
Sommerfeld Integration 186
E Modeling Two-Dimensional Metamaterial Devices using the Dis-
crete Dipole Approximation 192
E.1 Developing the 2D Green’s Function . . . . . . . . . . . . . . . . . . 192
ix
E.2 DDA Cross-sections in Two and Three Dimensions . . . . . . . . . . 198
E.3 2D DDA Validation Against Mie Theory . . . . . . . . . . . . . . . . 200
F Modeling Metamaterial Antennas using the Discrete Dipole Ap-
proximation (DDA) 203
F.1 Planar Waveguide DDA . . . . . . . . . . . . . . . . . . . . . . . . . 203
F.2 Green’s Functions In Planar Waveguides . . . . . . . . . . . . . . . . 210
F.3 Cavity DDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
F.3.1 Modeling the Feed Structure Using an Eigenmode Expansion . 217
F.3.2 Modeling Cavities with Electric and Magnetic Dipoles . . . . . 220
F.4 Cavity Modes in the Edge-fed Antenna Structure . . . . . . . . . . . 221
F.4.1 Solving the two-compartment cavity eigenvalue problem . . . 227
F.4.2 Mode Volume Calculation . . . . . . . . . . . . . . . . . . . . 233
Bibliography 235
Biography 248
x