Green's Function Integral Equation Methods in Nano-Optics: 1st Edition (Hardback) book cover

Green's Function Integral Equation Methods in Nano-Optics

1st Edition

By Thomas M. Søndergaard

CRC Press

418 pages | 205 B/W Illus.

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pub: 2019-01-28
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Description

This book gives a comprehensive introduction to Green’s function integral equation methods (GFIEMs) for scattering problems in the field of nano-optics. First, a brief review is given of the most important theoretical foundations from electromagnetics, optics, and scattering theory, including theory of waveguides, Fresnel reflection, and scattering, extinction, and absorption cross sections. This is followed by a presentation of different types of GFIEMs of increasing complexity for one-, two-, and three-dimensional scattering problems. In GFIEMs, the electromagnetic field at any position is directly related to the field at either the inside or the surface of a scattering object placed in a reference structure. The properties of the reference structure, and radiating or periodic boundary conditions, are automatically taken care of via the choice of Green’s function. This book discusses in detail how to solve the integral equations using either simple or higher-order finite-element-based methods; how to calculate the relevant Green’s function for different reference structures and choices of boundary conditions; and how to calculate near-fields, optical cross sections, and the power emitted by a local source. Solution strategies for large structures are discussed based on either transfer-matrix-approaches or the conjugate gradient algorithm combined with the Fast Fourier Transform. Special attention is given to reducing the computational problem for three-dimensional structures with cylindrical symmetry by using cylindrical harmonic expansions.

Each presented method is accompanied by examples from nano-optics, including: resonant metal nano-particles placed in a homogeneous medium or on a surface or waveguide; a microstructured gradient-index-lens; the Purcell effect for an emitter in a photonic crystal; the excitation of surface plasmon polaritons by second-harmonic generation in a polymer fiber placed on a thin metal film; and anti-reflective, broadband absorbing or resonant surface microstructures. Each presented method is also accompanied by guidelines for software implementation and exercises.

Features

  • Comprehensive introduction to Green’s function integral equation methods for scattering problems in the field of nano-optics
  • Detailed explanation of how to discretize and solve integral equations using simple and higher-order finite-element approaches
  • Solution strategies for large structures
  • Guidelines for software implementation and exercises
  • Broad selection of examples of scattering problems in nano-optics

Table of Contents

Chapter 1 Introduction ………………………………………………………………………………1

1.1 Overview of methods and scattering problems………………………1

1.2 Optics versus microwaves…………………………………………………..3

1.3 Examples of nano optics…………………………………………………….4

1.4 Notation, abbreviations and symbols……………………………………6

Chapter 2 Theoretical foundation ……………………………………………………………….9

2.1 Maxwells equations …………………………………………………………..9

2.1.1 Boundary conditions …………………………………………….11

2.1.2 Wave equations ……………………………………………………11

2.1.3 Poynting vector ……………………………………………………12

2.2 Planar layered structures…………………………………………………..14

2.2.1 Fresnel reflection and transmission…………………………14

2.2.2 Planar waveguides and guided modes……………………..18

2.3 Scattering theory……………………………………………………………..22

2.3.1 Scatterer in homogeneous material (2D) …………………23

2.3.2 Scatterer on a layered structure (2D) ………………………26

2.3.3 Scatterer in homogeneous media (3D)…………………….32

2.3.4 Scatterer on a layered structure (3D) ………………………35

2.4 Exercises ………………………………………………………………………..37

Chapter 3 One-dimensional scattering problems…………………………………………39

3.1 Greens function integral equations …………………………………….39

3.2 Numerical approach…………………………………………………………41

3.3 Example of a simple barrier………………………………………………42

3.4 Iterative FFT-based approach for large structures ………………..43

3.5 Guide lines for software implementation ……………………………45

3.6 Exercises ………………………………………………………………………..46

Chapter 4 Surface integral equation method for 2D scattering problems………..49

4.1 Scatterer in a homogeneous medium………………………………….50

4.1.1 Greens function integral equations …………………………50

4.1.2 Finite-element-based discretization approaches ……….54

4.1.3 Pulse expansion and point-matching ………………………60

4.1.4 Linear-field expansion and point-matching ……………..65

4.1.5 Higher-order polynomial field expansion and point

matching……………………………………………………………..67

4.1.6 Fourier expansion methods ……………………………………75

4.1.7 Calculating electric and magnetic field distributions…77

4.1.8 Examples of metal nanostrip resonators ………………….79

4.1.9 Guidelines for software implementation………………….90

4.1.10 Exercises …………………………………………………………….92

4.2 Scatterer on or near planar surfaces……………………………………93

4.2.1 Greens function for a layered reference structure

with planar surfaces ……………………………………………..94

4.2.2 GFSIEM for a layered reference structure……………..105

4.2.3 Calculation of fields using the angular spectrum

representation…………………………………………………….107

4.2.4 Example: Gold nanostrip on a dielectric substrate ….116

4.2.5 Example: Silver nanostrip above a silver surface ……123

4.2.6 Example: Single groove in metal ………………………….129

4.2.7 Example: Silver nanostrip on a thin-film-siliconon-

silver waveguide ……………………………………………133

4.2.8 Example: Microstructured gradient-index lens for

THz photoconductive antennas…………………………….143

4.2.9 Guidelines for software implementation………………..154

4.2.10 Exercises …………………………………………………………..156

4.3 Periodic structures …………………………………………………………158

4.3.1 Bloch waves ………………………………………………………159

4.3.2 Greens function for periodic structures………………….159

4.3.3 GFSIEM for periodic structures……………………………161

4.3.4 Derivatives of periodic Greens function and tabulation

………………………………………………………………..164

4.3.5 Calculating the fields…………………………………………..166

4.3.6 Calculating reflection and transmission …………………167

4.3.7 Multilayer periodic structures ………………………………169

4.3.8 Transfer-matrix method for large structures …………..175

4.3.9 Example: Photonic crystal …………………………………..184

4.3.10 Example: Anti-reflective groove array in a dielectric 189

4.3.11 Example: broadband-absorber ultra-sharp groove

array in a metal…………………………………………………..195

4.3.12 Guidelines for software implementation………………..202

4.3.13 Exercises …………………………………………………………..203

Chapter 5 Area integral equation method for 2D scattering problems ………….205

5.1 Greens function integral equations …………………………………..206

5.1.1 s polarization……………………………………………………..206

5.1.2 p polarization …………………………………………………….207

5.2 Discretization with square-shaped elements………………………209

5.3 Discretization with triangular elements …………………………….211

5.4 Scatterer in a homogeneous medium………………………………..214

5.4.1 s-polarization……………………………………………………..214

5.4.2 p-polarization…………………………………………………….222

5.5 Scatterer on or near planar surfaces………………………………….229

5.5.1 s polarization……………………………………………………..229

5.5.2 p polarization …………………………………………………….229

5.6 Periodic surface microstructures………………………………………234

5.6.1 s-polarization……………………………………………………..235

5.6.2 p-polarization…………………………………………………….238

5.7 Fast iterative FFT-based approach for large structures ………..241

5.8 Example: Purcell factor of emitter in a photonic crystal ……..244

5.9 Example: Excitation of surface plasmon polaritons by second

harmonic generation in a single organic nanofiber……….252

5.10 Guidelines for software implementation …………………………..261

5.11 Exercises ………………………………………………………………………262

Chapter 6 Volume integral equation method for 3D scattering problems………265

6.1 Greens function integral equation…………………………………….265

6.2 Scatterer in a homogeneous medium………………………………..266

6.2.1 Discretization with cubic volume elements ……………268

6.2.2 Discrete dipole approximation (DDA)…………………..273

6.3 Scatterer on or near planar surfaces………………………………….275

6.3.1 Greens tensor for layered reference structures………..275

6.3.2 Far-field Greens tensor………………………………………..282

6.3.3 Greens tensor in cartesian vector form ………………….287

6.3.4 Optical cross sections………………………………………….288

6.3.5 Example: Scattering by a nanostrip on a thin metal

film …………………………………………………………………..289

6.4 Periodic surface microstructures………………………………………293

6.4.1 Greens tensor for periodic structures …………………….294

6.4.2 Calculating reflection and transmission …………………298

6.4.3 Example: 2D periodic antireflective surface microstructure

……………………………………………………….299

6.5 Guidelines for software implementation …………………………..303

vi Contents

6.6 Exercises ………………………………………………………………………303

Chapter 7 Volume integral equation method for cylindrically symmetric structures

……………………………………………………………………………………..305

7.1 Expansion of homogeneous-medium dyadic Greens tensor

in cylindrical harmonics …………………………………………………306

7.1.1 eigenfunctions ……………………………………………………306

7.1.2 Orthogonality relations and normalization …………….308

7.1.3 Constructing the direct Greens tensor……………………309

7.2 Greens tensor for a layered structure ………………………………..313

7.2.1 Indirect Greens tensor: cylindrical harmonics ………..314

7.2.2 Transmitted Greens tensor: cylindrical harmonics ….315

7.3 Out-of-plane far-field approximations of the cylindrical

Greens tensor elements …………………………………………………..316

7.3.1 Far-field direct Greens tensor……………………………….319

7.3.2 Far-field indirect Greens tensor…………………………….320

7.3.3 Far-field transmitted Greens tensor……………………….321

7.4 Guided-mode far-field approximations of the cylindrical

Greens tensor elements …………………………………………………..321

7.5 Optical cross sections …………………………………………………….324

7.6 Numerical approach: ring elements with rectangular cross

section ………………………………………………………………………….325

7.7 Example: Nanocylinder on a layered structure…………………..327

7.7.1 Cylindrical scatterer on a dielectric substrate …………328

7.7.2 Cylindrical scatterer on a thin-film silicon-onsilver

waveguide…………………………………………………330

7.8 Example: Microstructured Gradient-Index Lens ………………..332

7.8.1 Dipole reference field………………………………………….333

7.8.2 Calculation of emitted power……………………………….335

7.8.3 Emission patterns and emitted powers…………………..336

7.9 Guidelines for software implementation …………………………..339

7.10 Exercices………………………………………………………………………340

Chapter 8 Surface integral equation method for the quasi-static limit ………….341

8.1 Greens function integral equations …………………………………..341

8.2 Numerical approach: pulse expansion ………………………………344

8.3 Finite-element-approach: linear expansion………………………..347

8.4 Finite-element-approach: quadratic expansion…………………..353

8.5 Examples of absorption cross sections of 3D silver

nanoparticles …………………………………………………………………355

8.6 Guidelines for software implementation …………………………..356

8.7 Exercises ………………………………………………………………………357

Chapter 9 Surface integral equation method for 3D scattering problems………359

9.1 Surface integral equations……………………………………………….359

9.2 Calculating optical cross sections…………………………………….365

9.3 Numerical approach: general structure……………………………..366

9.4 Numerical approach: cylindrically symmetric structure………369

9.5 Example: metal nano-disc resonators ……………………………….375

9.6 Guidelines for software implementation …………………………..378

9.7 Exercises ………………………………………………………………………379

Chapter A Residue theorem…………………………………………………………………….381

Chapter B Conjugate gradient algorithm…………………………………………………..383

Chapter C Bessel functions……………………………………………………………………..385

Chapter D Analytic scattering from a circular cylinder……………………………….387

Chapter E Analytic scattering from a spherical particle………………………………391

Chapter F Calculating guided modes of planar waveguides ………………………..395

F.1 Exercises ………………………………………………………………………401

Chapter G Plane-wave-expansion theory…………………………………………………..403

G.1 Exercises ………………………………………………………………………406

References……………………………………………………………………………………………….407

About the Author

Dr. Thomas Søndergaard is currently an Associate Professor in Nano Optics, Aalborg University, Denmark. His areas of expertise include numerical methods for theoretical analysis of electromagnetic fields in micro- and nanostructures. Plasmonics: waveguiding, optical antennas, resonators and sensors based on a type of electromagnetic surface wave at metal-dielectric interfaces known as Surface Plasmon Polaritons. Photonic crystals: wavelength-scale periodic structures in which light with certain wavelengths cannot propagate, similar to electrons with certain energies not being able to progagate in semiconductors, and how this can be exploited for e.g. designing optical waveguides and cavities. Green’s function integral equation methods. Dr. Sondergaard has been awarded The Danish Independent Research Councils' Young Researcher's Award (2006) and The Danish Optical Society Award (2008). He is a board member of the Danish Optical Society and reviewer of 15-20 papers per year for such journals as Physical Review B, Physical Review Letters, Applied Physics Letters, Optics Express, IEEE Journal of Quantum Electronics, IEEE Journal of Lightwave Technology, Optics Communications, Physica status solidi (b), Nature Photonics, Optics Letters, and Journal of the Optical Society of America A/B. Dr. Sondergaard has also been published 84 papers in peerreviewed journals and holds three patents.

Subject Categories

BISAC Subject Codes/Headings:
TEC019000
TECHNOLOGY & ENGINEERING / Lasers & Photonics
TEC027000
TECHNOLOGY & ENGINEERING / Nanotechnology & MEMS