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Intro
Electromagnetic and Photonic Simulation for the Beginner: Finite-Difference Frequency-Domain in MATLAB®
Contents
Foreword
Preface
Introduction
Chapter 1 MATLAB Preliminaries
1.1 Basic Structure of an FDFD Program in MATLAB
1.1.1 MATLAB Code for Ideal Structure of a Program
1.2 MATLAB and Linear Algebra
1.2.1 Special Matrices
1.2.2 Matrix Algebra
1.3 Setting Up a Grid in MATLAB
1.3.1 MATLAB Array Indexing
1.3.2 Parameters Describing a Grid in MATLAB
1.3.3 Calculating the Grid Parameters
1.4 Building Geometries onto Grids
1.4.1 Adding Rectangles to a Grid
1.4.2 The Centering Algorithm
1.4.3 The Meshgrid
1.4.4 Adding Circles and Ellipses to a Grid
1.4.5 Grid Rotation
1.4.6 Boolean Operations
1.5 Three-Dimensional Grids
1.6 Visualization Techniques
1.6.1 Visualizing Data on Grids
1.6.2 Visualizing Three-Dimensional Data
1.6.3 Visualizing Complex Data
1.6.4 Animating the Fields Calculated by FDFD
Reference
Chapter 2 Electromagnetic Preliminaries
2.1 Maxwell's Equations
2.2 The Constitutive Parameters
2.2.1 Anisotropy, Tensors, and Rotation Matrices
2.2.2 Rotation Matrices and Tensor Rotation
2.3 Expansion of Maxwell's Curl Equations in Cartesian Coordinates
2.4 The Electromagnetic Wave Equation
2.5 Electromagnetic Waves in LHI Media
2.5.1 Wave Polarization
2.6 The Dispersion Relation for LHI Media
2.7 Scattering at an Interface
2.7.1 Reflectance and Transmittance
2.8 What is a Two-Dimensional Simulation?
2.9 Diffraction from Gratings
2.9.1 The Grating Equation
2.9.2 Diffraction Efficiency
2.9.3 Generalization to Crossed Gratings
2.10 Waveguides and Transmission Lines
2.10.1 Waveguide Modes and Parameters
2.10.2 Transmission Line Parameters
2.11 Scalability of Maxwell's Equations.
2.12 Numerical Solution to Maxwell's Equations
References
Chapter 3 The Finite-Difference Method
3.1 Introduction
3.2 Finite-Difference Approximations
3.2.1 Deriving Expressions for Finite-Difference Approximations
3.2.2 Example #1-Interpolations and Derivatives from Three Points
3.2.3 Example #2-Interpolations and Derivatives from Two Points
3.2.4 Example #3-Interpolations and Derivatives from Four Points
3.3 Numerical Differentiation
3.4 Numerical Boundary Conditions
3.4.1 Dirichlet Boundary Conditions
3.4.2 Periodic Boundary Conditions
3.5 Derivative Matrices
3.6 Finite-Difference Approximation of Differential Equations
3.7 Solving Matrix Differential Equations
3.7.1 Example-Solving a Single-Variable Differential Equation
3.8 Multiple Variables and Staggered Grids
3.8.1 Example-Solving a Multivariable Problem
References
Chapter 4 Finite-Difference Approximation of Maxwell's Equations
4.1 Introduction to the Yee Grid Scheme
4.2 Preparing Maxwell's Equations for FDFD Analysis
4.3 Finite-Difference Approximation of Maxwell's Curl Equations
4.4 Finite-Difference Equations for Two-Dimensional FDFD
4.4.1 Derivation of E Mode Equations When Frequency Is Not Known
4.4.2 Derivation of H Mode Equations When Frequency Is Not Known
4.4.3 Derivation of E Mode Equations When Frequency Is Known
4.4.4 Derivation of H Mode Equations When Frequency Is Known
4.5 Derivative Matrices for Two-Dimensional FDFD
4.5.1 Derivative Matrices Incorporating Dirichlet Boundary Conditions
4.5.2 Periodic Boundary Conditions
4.5.3 Derivative Matrices Incorporating Periodic Boundary Conditions
4.5.4 Relationship Between the Derivative Matrices
4.6 Derivative Matrices for Three-Dimensional FDFD
4.6.1 Relationship Between the Derivative Matrices.
4.7 Programming the YEEDER2D() Function in MATLAB
4.7.1 Using the yeeder2d() Function
4.8 Programming the YEEDER3D() Function in MATLAB
4.8.1 Using the yeeder3d() Function
4.9 The 2× Grid Technique
4.10 Numerical Dispersion
References
Chapter 5 The Perfectly Matched Layer Absorbing Boundary
5.1 The Absorbing Boundary
5.2 Derivation of the UPML Absorbing Boundary
5.3 Incorporating the UPML into Maxwell's Equations
5.4 Calculating the UPML Parameters
5.5 Implementation of the UPML in MATLAB
5.5.1 Using the addupml2d() Function
5.6 The SCPML Absorbing Boundary
5.6.1 MATLAB Implementation of calcpml3d()
5.6.2 Using the calcpml3d() Function
References
Chapter 6 FDFD for Calculating Guided Modes
6.1 Formulation for Rigorous Hybrid Mode Calculation
6.2 Formulation for Rigorous Slab Waveguide Mode Calculation
6.2.1 Formulation of E Mode Slab Waveguide Analysis
6.2.2 Formulation of H Mode Slab Waveguide Analysis
6.2.3 Formulations for Slab Waveguides in Other Orientations
6.2.4 The Effective Index Method
6.3 Implementation of Waveguide Mode Calculations
6.3.1 MATLAB Implementation of Rib Waveguide Analysis
6.3.2 MATLAB Implementation of Slab Waveguide Analysis
6.3.3 Animating the Slab Waveguide Mode
6.3.4 Convergence
6.3.5 MATLAB Implementation for Calculating SPPs
6.4 Implementation of Transmission Line Analysis
References
Chapter 7 FDFD for Calculating Photonic Bands
7.1 Photonic Bands for Rectangular Lattices
7.2 Formulation for Rectangular Lattices
7.3 Implementation of Photonic Band Calculation
7.3.1 Description of MATLAB Code for Calculating Photonic Band Diagrams
7.3.2 Description of MATLAB Code for Calculating IFCs
References
Chapter 8 FDFD for Scattering Analysis
8.1 Formulation of FDFD for Scattering Analysis.
8.1.1 Matrix Wave Equations for Two-Dimensional Analysis
8.2 Incorporating Sources
8.2.1 Derivation of the QAAQ Equation
8.2.2 Calculating the Source Field fsrc(x,y)
8.2.3 Calculating the SF Masking Matrix Q
8.2.4 Compensating for Numerical Dispersion
8.3 Calculating Reflection and Transmission for Periodic Structures
8.4 Implementation of the FDFD Method for Scattering Analysis
8.4.1 Standard Sequence of Simulations for a Newly Written FDFD Code
8.4.2 FDFD Analysis of a Sawtooth Diffraction Grating
8.4.3 FDFD Analysis of a Self-Collimating Photonic Crystal
8.4.4 FDFD Analysis of an OIC Directional Coupler
References
Chapter 9 Parameter Sweeps with FDFD
9.1 Introduction to Parameter Sweeps
9.2 Modifying FDFD for Parameter Sweeps
9.2.1 Generic MATLAB Function to Simulate Periodic Structures
9.2.2 Main MATLAB Program to Simulate the GMRF
9.2.3 Main MATLAB Programs to Analyze a Metal Polarizer
9.3 Identifying Common Problems in FDFD
References
Chapter 10 FDFD Analysis of Three-Dimensional and Anisotropic Devices
10.1 Formulation of Three-Dimensional FDFD
10.1.1 Finite-Difference Approximation of Maxwell's Curl Equations
10.1.2 Maxwell's Equations in Matrix Form
10.1.3 Interpolation Matrices
10.1.4 Three-Dimensional Matrix Wave Equation
10.2 Incorporating Sources into Three-Dimensional FDFD
10.3 Iterative Solution for FDFD
10.4 Calculating Reflection and Transmission for Doubly Periodic Structures
10.5 Implementation of Three-Dimensional FDFD and Examples
10.5.1 Standard Sequence of Simulations for a Newly Written Three-Dimensional FDFD Code
10.5.2 Generic Three-Dimensional FDFD Function to Simulate Periodic Structures
10.5.3 Simulation of a Crossed-Grating GMRF
10.5.4 Simulation of a Frequency Selective Surface.
10.5.5 Parameter Retrieval for a Left-Handed Metamaterial
10.5.6 Simulation of an Invisibility Cloak
References
Appendix A
A.1 Best Practices for Building Devices onto Yee Grids
A.2 Method Summaries
List of Acronyms and Abbreviations
About the Author
Index.
Electromagnetic and Photonic Simulation for the Beginner: Finite-Difference Frequency-Domain in MATLAB®
Contents
Foreword
Preface
Introduction
Chapter 1 MATLAB Preliminaries
1.1 Basic Structure of an FDFD Program in MATLAB
1.1.1 MATLAB Code for Ideal Structure of a Program
1.2 MATLAB and Linear Algebra
1.2.1 Special Matrices
1.2.2 Matrix Algebra
1.3 Setting Up a Grid in MATLAB
1.3.1 MATLAB Array Indexing
1.3.2 Parameters Describing a Grid in MATLAB
1.3.3 Calculating the Grid Parameters
1.4 Building Geometries onto Grids
1.4.1 Adding Rectangles to a Grid
1.4.2 The Centering Algorithm
1.4.3 The Meshgrid
1.4.4 Adding Circles and Ellipses to a Grid
1.4.5 Grid Rotation
1.4.6 Boolean Operations
1.5 Three-Dimensional Grids
1.6 Visualization Techniques
1.6.1 Visualizing Data on Grids
1.6.2 Visualizing Three-Dimensional Data
1.6.3 Visualizing Complex Data
1.6.4 Animating the Fields Calculated by FDFD
Reference
Chapter 2 Electromagnetic Preliminaries
2.1 Maxwell's Equations
2.2 The Constitutive Parameters
2.2.1 Anisotropy, Tensors, and Rotation Matrices
2.2.2 Rotation Matrices and Tensor Rotation
2.3 Expansion of Maxwell's Curl Equations in Cartesian Coordinates
2.4 The Electromagnetic Wave Equation
2.5 Electromagnetic Waves in LHI Media
2.5.1 Wave Polarization
2.6 The Dispersion Relation for LHI Media
2.7 Scattering at an Interface
2.7.1 Reflectance and Transmittance
2.8 What is a Two-Dimensional Simulation?
2.9 Diffraction from Gratings
2.9.1 The Grating Equation
2.9.2 Diffraction Efficiency
2.9.3 Generalization to Crossed Gratings
2.10 Waveguides and Transmission Lines
2.10.1 Waveguide Modes and Parameters
2.10.2 Transmission Line Parameters
2.11 Scalability of Maxwell's Equations.
2.12 Numerical Solution to Maxwell's Equations
References
Chapter 3 The Finite-Difference Method
3.1 Introduction
3.2 Finite-Difference Approximations
3.2.1 Deriving Expressions for Finite-Difference Approximations
3.2.2 Example #1-Interpolations and Derivatives from Three Points
3.2.3 Example #2-Interpolations and Derivatives from Two Points
3.2.4 Example #3-Interpolations and Derivatives from Four Points
3.3 Numerical Differentiation
3.4 Numerical Boundary Conditions
3.4.1 Dirichlet Boundary Conditions
3.4.2 Periodic Boundary Conditions
3.5 Derivative Matrices
3.6 Finite-Difference Approximation of Differential Equations
3.7 Solving Matrix Differential Equations
3.7.1 Example-Solving a Single-Variable Differential Equation
3.8 Multiple Variables and Staggered Grids
3.8.1 Example-Solving a Multivariable Problem
References
Chapter 4 Finite-Difference Approximation of Maxwell's Equations
4.1 Introduction to the Yee Grid Scheme
4.2 Preparing Maxwell's Equations for FDFD Analysis
4.3 Finite-Difference Approximation of Maxwell's Curl Equations
4.4 Finite-Difference Equations for Two-Dimensional FDFD
4.4.1 Derivation of E Mode Equations When Frequency Is Not Known
4.4.2 Derivation of H Mode Equations When Frequency Is Not Known
4.4.3 Derivation of E Mode Equations When Frequency Is Known
4.4.4 Derivation of H Mode Equations When Frequency Is Known
4.5 Derivative Matrices for Two-Dimensional FDFD
4.5.1 Derivative Matrices Incorporating Dirichlet Boundary Conditions
4.5.2 Periodic Boundary Conditions
4.5.3 Derivative Matrices Incorporating Periodic Boundary Conditions
4.5.4 Relationship Between the Derivative Matrices
4.6 Derivative Matrices for Three-Dimensional FDFD
4.6.1 Relationship Between the Derivative Matrices.
4.7 Programming the YEEDER2D() Function in MATLAB
4.7.1 Using the yeeder2d() Function
4.8 Programming the YEEDER3D() Function in MATLAB
4.8.1 Using the yeeder3d() Function
4.9 The 2× Grid Technique
4.10 Numerical Dispersion
References
Chapter 5 The Perfectly Matched Layer Absorbing Boundary
5.1 The Absorbing Boundary
5.2 Derivation of the UPML Absorbing Boundary
5.3 Incorporating the UPML into Maxwell's Equations
5.4 Calculating the UPML Parameters
5.5 Implementation of the UPML in MATLAB
5.5.1 Using the addupml2d() Function
5.6 The SCPML Absorbing Boundary
5.6.1 MATLAB Implementation of calcpml3d()
5.6.2 Using the calcpml3d() Function
References
Chapter 6 FDFD for Calculating Guided Modes
6.1 Formulation for Rigorous Hybrid Mode Calculation
6.2 Formulation for Rigorous Slab Waveguide Mode Calculation
6.2.1 Formulation of E Mode Slab Waveguide Analysis
6.2.2 Formulation of H Mode Slab Waveguide Analysis
6.2.3 Formulations for Slab Waveguides in Other Orientations
6.2.4 The Effective Index Method
6.3 Implementation of Waveguide Mode Calculations
6.3.1 MATLAB Implementation of Rib Waveguide Analysis
6.3.2 MATLAB Implementation of Slab Waveguide Analysis
6.3.3 Animating the Slab Waveguide Mode
6.3.4 Convergence
6.3.5 MATLAB Implementation for Calculating SPPs
6.4 Implementation of Transmission Line Analysis
References
Chapter 7 FDFD for Calculating Photonic Bands
7.1 Photonic Bands for Rectangular Lattices
7.2 Formulation for Rectangular Lattices
7.3 Implementation of Photonic Band Calculation
7.3.1 Description of MATLAB Code for Calculating Photonic Band Diagrams
7.3.2 Description of MATLAB Code for Calculating IFCs
References
Chapter 8 FDFD for Scattering Analysis
8.1 Formulation of FDFD for Scattering Analysis.
8.1.1 Matrix Wave Equations for Two-Dimensional Analysis
8.2 Incorporating Sources
8.2.1 Derivation of the QAAQ Equation
8.2.2 Calculating the Source Field fsrc(x,y)
8.2.3 Calculating the SF Masking Matrix Q
8.2.4 Compensating for Numerical Dispersion
8.3 Calculating Reflection and Transmission for Periodic Structures
8.4 Implementation of the FDFD Method for Scattering Analysis
8.4.1 Standard Sequence of Simulations for a Newly Written FDFD Code
8.4.2 FDFD Analysis of a Sawtooth Diffraction Grating
8.4.3 FDFD Analysis of a Self-Collimating Photonic Crystal
8.4.4 FDFD Analysis of an OIC Directional Coupler
References
Chapter 9 Parameter Sweeps with FDFD
9.1 Introduction to Parameter Sweeps
9.2 Modifying FDFD for Parameter Sweeps
9.2.1 Generic MATLAB Function to Simulate Periodic Structures
9.2.2 Main MATLAB Program to Simulate the GMRF
9.2.3 Main MATLAB Programs to Analyze a Metal Polarizer
9.3 Identifying Common Problems in FDFD
References
Chapter 10 FDFD Analysis of Three-Dimensional and Anisotropic Devices
10.1 Formulation of Three-Dimensional FDFD
10.1.1 Finite-Difference Approximation of Maxwell's Curl Equations
10.1.2 Maxwell's Equations in Matrix Form
10.1.3 Interpolation Matrices
10.1.4 Three-Dimensional Matrix Wave Equation
10.2 Incorporating Sources into Three-Dimensional FDFD
10.3 Iterative Solution for FDFD
10.4 Calculating Reflection and Transmission for Doubly Periodic Structures
10.5 Implementation of Three-Dimensional FDFD and Examples
10.5.1 Standard Sequence of Simulations for a Newly Written Three-Dimensional FDFD Code
10.5.2 Generic Three-Dimensional FDFD Function to Simulate Periodic Structures
10.5.3 Simulation of a Crossed-Grating GMRF
10.5.4 Simulation of a Frequency Selective Surface.
10.5.5 Parameter Retrieval for a Left-Handed Metamaterial
10.5.6 Simulation of an Invisibility Cloak
References
Appendix A
A.1 Best Practices for Building Devices onto Yee Grids
A.2 Method Summaries
List of Acronyms and Abbreviations
About the Author
Index.