book

Numerical Methods in Photonics

by Andrei V. Lavrinenko, Jesper Lægsgaard, Niels Gregersen, Frank Schmidt, Thomas Søndergaard

September 2018

Intermediate to advanced

364 pages

9h 35m

English

CRC Press

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Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Series Preface
Preface
Authors
Acronyms

Chapter 1 Introduction
Chapter 2 Maxwell’s Equations
2.1 Notation2.2 Maxwell’s Equations2.3 Material Equations2.4 Frequency Domain2.5 1D and 2D Maxwell’s Equations2.6 Wave Equations2.7 Waveguides and Eigenmodes2.7.1 Eigenvalue Problem2.7.2 Slab Waveguides2.7.3 Boundary Conditions and Eigenmode Classes2.7.4 OrthogonalityReferences
Chapter 3 Finite-Difference Time-Domain Method
3.1 Introduction3.1.1 Finite-Difference Approximations of Derivatives3.1.2 Finite-Difference Approximation of 1D Maxwell’s Equations3.1.3 Fortran, C, MATLAB®, Etc., Adaptation of the FDTD Method3.1.4 FDTD Method in 3D3.1.5 FDTD Method in 2D3.2 Numerical Dispersion and Stability Analysis of the FDTD Method3.2.1 Dispersion Equation in 3D
3.2.2 Numerical Stability Criteria
3.2.3 Divergence-Free Character of the FDTD Method3.3 Making Your Own 1D FDTD3.3.1 Step 1: Setting Material Properties on a Grid3.3.2 Step 2: Setting Sources and Detectors3.3.3 Step 3: Evolving Fields3.3.4 Step 4: Postprocessing of Information3.4 Absorbing Boundary Conditions3.4.1 Analytical Absorbing Boundary Conditions3.4.2 Perfectly Matched Layer: Basic Idea3.4.3 Perfectly Matched Layer: Generalization and Realization3.5 FDTD Method for Materials with Frequency Dispersion3.5.1 Frequency Dispersion Models3.5.1.1 Debye Material3.5.1.2 Drude Model3.5.1.3 Lorentz Model3.5.2 Numerical Implementation of Frequency Dispersion in FDTD through Auxiliary Equation3.5.2.1 Debye Material3.5.2.2 Drude Model of Dispersion3.5.2.3 Lorentz Model of Dispersion3.5.3 Linear Polarization Model for Dispersive Materials in FDTD3.5.4 Piecewise Linear Recursive Convolution Scheme3.6 FDTD Method for Nonlinear Materials, Materials with Gain, and Lasing3.6.1 Nonlinear Polarization in FDTD3.6.2 Medium with Gain: Phenomenological Approach in FDTD3.6.3 Lasing in FDTD3.7 ConclusionExercisesReferences
Chapter 4 Finite-Difference Modelling of Straight Waveguides
4.1 Introduction4.2 General Considerations4.2.1 Time Domain versus Frequency Domain4.2.2 Finite-Difference Methods for Straight Waveguides4.3 Modified Finite-Difference Operators4.3.1 Discretizing the Scalar Wave Equation4.3.2 Inclusion of Discontinuities: General Formalism4.3.3 Inclusion of Discontinuities: TE Case4.3.4 Inclusion of Discontinuities: TM Case4.4 Numerical Linear Algebra in MATLAB4.4.1 Sparse Matrices4.4.2 Direct and Iterative Eigensolvers4.5 2D Waveguides and the Yee Mesh4.5.1 Yee Mesh4.5.2 Dielectric Function Averaging4.5.3 Use of Mirror SymmetriesExercisesReferences
Chapter 5 Modelling of Nonlinear Propagation in Waveguides
5.1 Introduction5.2 Formalism5.2.1 General Propagation Equation5.2.2 Pulse Power and Pulse Energy5.3 Nonlinear Polarization5.3.1 Nonlinear Processes5.3.2 χ(3) Nonlinear Processes5.3.3 Single-Mode Propagation Model5.4 Nonlinear Schrödinger Equation5.4.1 Derivation of the NLS Equation5.4.2 Dispersion and Self-Phase Modulation5.4.3 Optical Solitons5.4.4 Solitons and Raman Effects5.4.5 Self-Steepening5.4.6 Conservation Laws5.5 Numerical Implementation5.5.1 Fourier Method5.5.2 Stepping Techniques5.5.3 Discrete Fourier Grids5.5.4 Implementation in MATLABExercisesReferences
Chapter 6 The Modal Method
6.1 Introduction6.2 Eigenmodes6.3 1D Geometry6.3.1 Recursive Matrix Formalism6.3.2 1D Interface6.3.3 Multilayer Structure6.3.4 1D Cavity6.4 2D Geometry6.4.1 Plane-Wave Expansion6.4.1.1 Li’s Factorization Rules6.4.1.2 Eigenvalue Problem
6.4.2 Semi-Analytical Approach
6.4.3 Interface6.4.4 S Matrix Theory
6.4.5 Absorbing Boundary Conditions
6.5 Periodic Structures6.5.1 Bloch Modes6.5.2 Classification6.5.3 Interface6.5.4 Field Profile in a Periodic Element6.6 Current Sources6.6.1 Uniform Layer6.6.2 Multilayer Geometry6.7 3D GeometriesExercisesReferences
Chapter 7 Green’s Function Integral Equation Methods for Electromagnetic Scattering Problems
7.1 Introduction7.2 Theoretical Foundation7.3 Green’s Function Area Integral Equation Method7.4 Green’s Function Volume Integral Equation Method7.5 Green’s Function Surface Integral Equation Method (2D)7.5.1 Surface Integral Equations
7.5.2 Calculating the Field and Normal Derivative at the Boundary
7.6 Construction of 2D Green’s Functions for Layered Structures7.6.1 Plane-Wave Expansion of the Free-Space Green’s Function7.6.2 2D TE-Polarized Scalar Green’s Function for a Layered Structure7.6.3 2D TM-Polarized Scalar Green’s Function for a Layered Structure7.6.4 Fresnel Reflection and Transmission Coefficients for a Few Simple Geometries
7.6.5 Calculating the Sommerfeld Integral
7.6.6 Far-Field Approximation7.6.7 Excitation of Bound Waveguide Modes7.7 Construction of the Periodic Green’s Function7.7.1 1D Periodic Scalar Green’s Function for a Layered Structure7.8 Reflection from a Periodic Surface Microstructure7.8.1 Calculating Reflection and Transmission7.9 Iterative Solution Scheme Taking Advantage of the Fast Fourier Transform7.9.1 2D Discrete Convolution7.10 Further ReadingExercisesReferences
Chapter 8 Finite Element Method
8.1 Introduction: Helmholtz Equation in 1D8.1.1 Variational Formulation8.1.2 Weak Form8.1.3 Galerkin Method8.1.4 Discrete Problem8.1.5 Linear Finite Elements8.1.6 Domain Mapping8.1.7 Assembly Process8.1.8 Algorithm: Plane-Wave Propagation8.2 General Scattering Problem in 1D8.2.1 Variational Formulation in 1D with DtN Operator8.2.1.1 DtN Operator8.2.2 Variational Formulation in 1D with Perfectly Matched Layers8.2.2.1 Completion to a Continuous Function
8.2.3 Discretization
8.2.4 A Posteriori Error Estimation8.2.4.1 Galerkin Orthogonality8.2.4.2 A Different Viewpoint8.2.4.3 Error Localization and Error Indicator8.2.5 Adaptive Mesh Refinement8.2.6 FEM Notions: Element Support, Basis Functions, Shape Functions, Finite Elements, and Finite Element Spaces8.3 Mathematical Background: Maxwell and Helmholtz Scattering Problems and Their Variational Forms8.3.1 Maxwell’s Scattering Problem8.3.1.1 Discussion of the Silver-Müller Radiation Condition8.3.2 Slight Simplification: The Helmholtz Scattering Problem8.3.3 Transformation Rules8.3.3.1 Mapping of Geometric Quantities8.3.3.2 Mapping of grad, curl, and div8.3.3.3 Mapping of Fields8.3.3.4 Mapping of and8.3.4 PML in 2D and 3D8.3.5 Integration by Parts8.3.6 Variational Formulation for the Helmholtz Equation with PML8.3.6.1 Interior Problem8.3.6.2 Exterior Problem for the Scattered Field8.3.6.3 Variational Formulation on the Entire Domain8.3.7 Variational Formulation for Maxwell’s Equations with PML8.3.7.1 Interior Problem
8.3.7.2 Exterior Problem for the Scattered Field
8.3.7.3 Variational Formulation on the Entire Domain8.4 FEM for Helmholtz Scattering in 2D and 3D8.4.1 Rectangular Meshes8.4.2 Mesh and Assembly Process: General Scheme8.4.3 Finite Elements for Rectangular Meshes8.4.3.1 Rectangular Elements8.4.3.2 Polynomial Space8.4.3.3 DOFs on a Rectangle8.4.3.4 Bilinear Finite Element Discretization8.4.3.5 Boundary Integral8.4.4 Finite Elements for Triangular Meshes8.4.4.1 Global Data Structure and Connectivity Matrix8.5 FEM for Maxwell’s Scattering in 2D and 3D8.5.1 Finite Elements for Rectangular Meshes8.5.1.1 Polynomial Space8.5.1.2 DOFs on a Rectangle8.5.1.3 Linear Finite Element Discretization8.5.2 Finite Elements for Triangular Meshes8.5.2.1 Triangular Elements8.5.2.2 Polynomial Space8.5.2.3 DOFs on a Triangle8.5.2.4 Linear Finite Element DiscretizationExercisesReferences
Index

Content preview from Numerical Methods in Photonics

Numerical Methods in Photonics

Andrei V. Lavrinenko

Technical University of Denmark, Kongens Lyngby

Jesper Lægsgaard

Technical University of Denmark, Kongens Lyngby

Niels Gregersen

Technical University of Denmark, Kongens Lyngby

Frank Schmidt

Zuse Institute, Berlin, Germany

Thomas Søndergaard

Aalborg University, Aalborg, Denmark

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Numerical Methods in Photonics

by Andrei V. Lavrinenko, Jesper Lægsgaard, Niels Gregersen, Frank Schmidt, Thomas Søndergaard

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