Electromagnetic and Photonic Simulation for the Beginner: Finite-Difference Frequency-Domain in MATLAB (Applied Photonics Library)

355 Pages·2022·20.855 MB·English

by  Rumpf, Raymond C.

Electromagnetic and Photonic
Simulation for the Beginner
Finite-Difference Frequency-Domain in MATLAB®
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Electromagnetic and Photonic
Simulation for the Beginner
Finite-Difference Frequency-Domain in MATLAB®
Raymond C. Rumpf
artechhouse.com
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Library of Congress Cataloging-in-Publication Data
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British Library Cataloguing in Publication Data
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ISBN-13: 978-1-63081-926-2
Cover design by Charlene Stevens
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This book is dedicated to all those who are struggling to get
started in computational electromagnetics
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Contents
Foreword xiii
Preface xv
Acknowledgments xvii
Introduction xix
References xxii
CHAPTER 1
MATLAB Preliminaries 1
1.1 Basic Structure of an FDFD Program in MATLAB 1
1.1.1 MATLAB Code for Ideal Structure of a Program 2
1.2 MATLAB and Linear Algebra 3
1.2.1 Special Matrices 6
1.2.2 Matrix Algebra 8
1.3 Setting Up a Grid in MATLAB 8
1.3.1 MATLAB Array Indexing 8
1.3.2 Parameters Describing a Grid in MATLAB 10
1.3.3 Calculating the Grid Parameters 11
1.4 Building Geometries onto Grids 15
1.4.1 Adding Rectangles to a Grid 16
1.4.2 The Centering Algorithm 17
1.4.3 The Meshgrid 19
1.4.4 Adding Circles and Ellipses to a Grid 20
1.4.5 Grid Rotation 22
1.4.6 Boolean Operations 23
1.5 Three-Dimensional Grids 25
1.6 Visualization Techniques 27
1.6.1 Visualizing Data on Grids 27
1.6.2 Visualizing Three-Dimensional Data 29
1.6.3 Visualizing Complex Data 31
1.6.4 Animating the Fields Calculated by FDFD 32
Reference 32
CHAPTER 2
Electromagnetic Preliminaries 33
2.1 Maxwell’s Equations 33
2.2 The Constitutive Parameters 37
vii
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viii Contents
2.2.1 Anisotropy, Tensors, and Rotation Matrices 37
2.2.2 Rotation Matrices and Tensor Rotation 39
2.3 Expansion of Maxwell’s Curl Equations in Cartesian Coordinates 42
2.4 The Electromagnetic Wave Equation 43
2.5 Electromagnetic Waves in LHI Media 45
2.5.1 Wave Polarization 46
2.6 The Dispersion Relation for LHI Media 49
2.7 Scattering at an Interface 49
2.7.1 Reflectance and Transmittance 52
2.8 What is a Two-Dimensional Simulation? 55
2.9 Diffraction from Gratings 56
2.9.1 The Grating Equation 57
2.9.2 Diffraction Efficiency 59
2.9.3 Generalization to Crossed Gratings 60
2.10 Waveguides and Transmission Lines 62
2.10.1 Waveguide Modes and Parameters 63
2.10.2 Transmission Line Parameters 66
2.11 Scalability of Maxwell’s Equations 68
2.12 Numerical Solution to Maxwell’s Equations 69
References 70
CHAPTER 3
The Finite-Difference Method 71
3.1 Introduction 71
3.2 Finite-Difference Approximations 72
3.2.1 Deriving Expressions for Finite-Difference Approximations 73
3.2.2 Example #1—Interpolations and Derivatives from Three Points 76
3.2.3 Example #2—Interpolations and Derivatives from Two Points 78
3.2.4 Example #3—Interpolations and Derivatives from Four Points 79
3.3 Numerical Differentiation 80
3.4 Numerical Boundary Conditions 81
3.4.1 Dirichlet Boundary Conditions 81
3.4.2 Periodic Boundary Conditions 82
3.5 Derivative Matrices 82
3.6 Finite-Difference Approximation of Differential Equations 85
3.7 Solving Matrix Differential Equations 87
3.7.1 Example—Solving a Single-Variable Differential Equation 87
3.8 Multiple Variables and Staggered Grids 89
3.8.1 Example—Solving a Multivariable Problem 92
References 94
CHAPTER 4
Finite-Difference Approximation of Maxwell’s Equations 95
4.1 Introduction to the Yee Grid Scheme 95
4.2 Preparing Maxwell’s Equations for FDFD Analysis 97
4.3 Finite-Difference Approximation of Maxwell’s Curl Equations 99
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Contents ix
4.4 Finite-Difference Equations for Two-Dimensional FDFD 103
4.4.1 Derivation of E Mode Equations When Frequency Is Not
Known 105
4.4.2 Derivation of H Mode Equations When Frequency Is Not
Known 105
4.4.3 Derivation of E Mode Equations When Frequency Is Known 106
4.4.4 Derivation of H Mode Equations When Frequency Is Known 106
4.5 Derivative Matrices for Two-Dimensional FDFD 107
4.5.1 Derivative Matrices Incorporating Dirichlet Boundary
Conditions 108
4.5.2 Periodic Boundary Conditions 112
4.5.3 Derivative Matrices Incorporating Periodic Boundary
Conditions 115
4.5.4 Relationship Between the Derivative Matrices 119
4.6 Derivative Matrices for Three-Dimensional FDFD 120
4.6.1 Relationship Between the Derivative Matrices 123
4.7 Programming the yeeder2d() Function in MATLAB 124
4.7.1 Using the yeeder2d() Function 126
4.8 Programming the yeeder3d() Function in MATLAB 128
4.8.1 Using the yeeder3d() Function 129
4.9 The 2× Grid Technique 131
4.10 Numerical Dispersion 134
References 139
CHAPTER 5
The Perfectly Matched Layer Absorbing Boundary 141
5.1 The Absorbing Boundary 141
5.2 Derivation of the UPML Absorbing Boundary 143
5.3 Incorporating the UPML into Maxwell’s Equations 146
5.4 Calculating the UPML Parameters 147
5.5 Implementation of the UPML in MATLAB 149
5.5.1 Using the addupml2d() Function 150
5.6 The SCPML Absorbing Boundary 153
5.6.1 MATLAB Implementation of calcpml3d() 155
5.6.2 Using the calcpml3d() Function 155
References 159
CHAPTER 6
FDFD for Calculating Guided Modes 161
6.1 Formulation for Rigorous Hybrid Mode Calculation 161
6.2 Formulation for Rigorous Slab Waveguide Mode Calculation 166
6.2.1 Formulation of E Mode Slab Waveguide Analysis 167
6.2.2 Formulation of H Mode Slab Waveguide Analysis 168
6.2.3 Formulations for Slab Waveguides in Other Orientations 168
6.2.4 The Effective Index Method 169
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