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Spie Press Book

Computing the Flow of Light: Nonstandard FDTD Methodologies for Photonics Design
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Book Description

The finite difference time domain (FDTD) method computes the time evolution of a system at discrete time steps, and periodically visualizing the results lets us view its time evolution, yielding valuable physical insights. FDTD and FDTD-like methods can be used to solve a wide variety of problems, including—but not limited to—the wave equation, Maxwell’s equations, and the Schrödinger equation. In addition to introducing useful new methodologies, this book provides readers with analytical background and simulation examples that will help them develop their own methodologies to solve yet-to-be-posed problems. The book is written for students, engineers, and researchers grappling with problems that cannot be solved analytically. It could also be used as a textbook for a mathematical physics or engineering class. An accompanying CD provides supplemental Mathcad and pseudocode programs.

Book Details

Date Published: 8 June 2017
Pages: 430
ISBN: 9781510604810
Volume: PM272

Table of Contents
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Table of Contents

Preface

1 Finite Difference Approximations
1.1 Basic Finite Difference Expressions
      1.1.1 Higher-order finite difference approximations
      1.1.2 Computational molecules
1.2 Nonstandard Finite Difference Expressions
      1.2.1 Exact nonstandard finite difference expressions
      1.2.2 Terminology
1.3 Standard Finite Difference Expressions for the Laplacian
      1.3.1 Two dimensions
      1.3.2 Three dimensions
1.4 Nonstandard Finite Difference Expressions for the Laplacian
      1.4.1 Two dimensions
      1.4.2 Three dimensions
1.5 Factoring the Nonstandard Finite Difference Laplacian
      1.5.1 Two dimensions
      1.5.2 Three dimensions
Appendix 1.1 Mathematical Properties of Finite Difference Operators
      A1.1.1 Forward finite difference operator
      A1.1.2 Central finite difference operator
      A1.1.3 Multiple variables
Appendix 1.2 Noninteger-Order Sums and Differences
      A1.2.1 Noninteger-order summation
      A1.2.2 Noninteger-order differences
References

2 Accuracy, Stability, and Convergence of Numerical Algorithms
2.1 Introduction
      2.1.1 Stable versus unstable algorithms
      2.1.2 Truncation and round-off error
2.2 Bessel Recursion
2.3 Recursive Iteration
2.4 Decay Equation
      2.4.1 Standard forward finite difference model
      2.4.2 Standard backward finite difference model
      2.4.3 Standard central finite difference model
      2.4.4 Nonstandard central finite difference model
      2.4.5 Nonstandard forward finite difference model
2.5 Malthus Model of Population Growth
2.6 Finite Difference Models of Nonlinear Differential Equations
      2.6.1 Standard forward finite difference model
      2.6.2 Nonstandard forward finite difference model
2.7 Relaxation Algorithm for the Laplace Equation and Poisson's Equation
      2.7.1 Basic relaxation algorithm
      2.7.2 Relaxation and diffusion
      2.7.3 Analytical solutions
      2.7.4 Parallel versus serial computing
      2.7.5 Over-relaxation
2.8 The Random Walk Model and the Diffusion Equation
      2.8.1 Random walk model
      2.8.2 Diffusion equation
      2.8.3 1D random walk
Appendix 2.1 Matrix Formulations of Relaxation
      A2.1.1 Parallel relaxation
      A2.1.2 Serial relaxation
      A2.1.3 Over-relaxation
      A2.1.4 Convergence
      A2.1.5 Two dimensions
Appendix 2.2 Noninteger-Order Integrals and Derivatives
      A2.2.1 Noninteger-order integration
      A2.2.2 Noninteger-order differentiation
References

3 Finite Difference Models of the Simple Harmonic Oscillator
3.1 Analytic Solution of the Simple Harmonic Oscillator
3.2 Second-Order Finite Difference Model of the Simple Harmonic Oscillator
3.3 Fourth-Order Finite Difference Model of the Simple Harmonic Oscillator
3.4 Nonstandard Finite Difference Model of the Simple Harmonic Oscillator
3.5 Analytical Solutions of the Forced Damped Simple Harmonic Oscillator
      3.5.1 Free damped oscillator
      3.5.2 Forced damped oscillator
3.6 Forced Damped Harmonic Oscillator: Standard Finite Difference Models
3.7 Forced Damped Harmonic Oscillator: Nonstandard Finite Difference Models
      3.7.1 No driving force
      3.7.2 NS-FD model for the forced damped harmonic oscillator
Appendix 3.1 Stability Analysis of the Second-Order Finite Difference Model
Appendix 3.2 Stability Analysis of the Fourth-Order Finite Difference Model
Appendix 3.3 Green's Function for the Damped Harmonic Oscillator
      A3.3.1 Simple harmonic oscillator without damping
      A3.3.2 Simple harmonic oscillator with damping
Appendix 3.4 Properties of the δ Function and Step Function
      A3.4.1 δ function definition and elementary properties
      A3.4.2 Derivative of the δ function
      A3.4.3 Representations of the δ function
      A3.4.4 The step function
      A3.4.5 Derivatives of |x|
      A3.4.6 sng(x)
Appendix 3.5 Discrete Green's Function for the Finite Difference Model of the Damped Harmonic Oscillator
References

4 The 1D Wave Equation
4.1 The Homogeneous Wave Equation
      4.1.1 General solution in an unbound uniform medium
      4.1.2 Monochromatic solutions
      4.1.3 Reflecting cavity
      4.1.4 Boundary conditions at medium interfaces
      4.1.5 Reflection and transmission at a medium interface
      4.1.6 Reflection and transmission of a layer
4.2 The Damped Wave Equation
4.3 Wave Equations with a Source in Unbounded Space
      4.3.1 Lossless wave equation
      4.3.2 Damped wave equation
4.4 Source in a Reflecting Cavity
      4.4.1 Point source switched on instantaneously
      4.4.2 Point source switched on slowly
4.5 The Scattered Field
Appendix 4.1 The Wave Model and the Wave Equation
      A4.1.1 1D string
      A4.1.2 Two and three dimensions
      A4.1.3 Wave equations with a source
      A4.1.4 Damped wave equation with a source
Appendix 4.2 The Wave Model and the Wave Equation
      A4.2.1 Galilean transformation
      A4.2.2 General transformation of the wave equation
      A4.2.3 Galilean transformation of the wave equation
      A4.2.4 Lorenz transformation
      A4.2.5 Transformation of velocity
      A4.2.6 Transformation of acceleration
      A4.2.7 Relativistic momentum
      A4.2.8 Relativistic energy
      A4.2.9 Invariants under Lorentz transformation
Appendix 4.3 Reflection and Transmission of Layered Structures
Appendix 4.4 Green's function for the 1D Wave Equation
      A4.4.1 Green's function in unbound space
      A4.4.2 Green's function in a reflecting cavity
      A4.4.3 Green's function for the unbound damped wave equation
References

5 FDTD Algorithms for the 1D Wave Equation
5.1 Homogeneous Wave Equation
      5.1.1 Standard finite difference model
      5.1.2 Error of the standard difference model
      5.1.3 Nonstandard finite difference model
      5.1.4 Devils in the details
      5.1.5 Precursor waves: physical insights
5.2 Damped Wave Equation
      5.2.1 Standard finite difference models
      5.2.2 Nonstandard finite difference models
5.3 Wave Equation with a Source
      5.3.1 Standard finite difference model: lossless case
      5.3.2 Standard finite difference model: damped case
      5.3.3 Nonstandard finite difference model: lossless case
      5.3.4 Nonstandard finite difference model: damped case
5.4 Time Reversal
References

6 Program Development and Applications of FDTD Algorithms in One Dimension
6.1 The Computational Boundary
      6.1.1 One-way wave equations
      6.1.2 Finite difference models of the one-way wave equations
      6.1.3 Central finite difference model of the one-wave way equations
      6.1.4 Nonstandard finite difference model of the one-wave way equations
6.2 Extracting Field Intensity from a Calculation
6.3 Zero-Order Object Models on the Grid
6.4 Setting up an FDTD Calculation
      6.4.1 Parameter choice
6.5 The Scattered Field
      6.5.1 Standard finite difference model
      6.5.2 Nonstandard finite difference model
      6.5.3 Setup of a scattered-field computation
      6.5.4 Transmission/reflection spectrum of a layer
6.6 Solution using Discrete Green's Functions
Appendix 6.1 Discrete Green's Function for the 1D Wave Equation
      A6.1.1 Introduction
      A6.1.2 Derivation of the discrete Green's function
      A6.1.3 Derivation of the discrete Green's function using FDTD
      A6.1.4 Interpretation and analysis
References

7 FDTD Algorithms to Solve the Wave Equation in Two and Three Dimensions
7.1 The Homogeneous Wave Equation
7.2 FDTD for the Homogeneous Wave Equation
      7.2.1 Standard FDTD
      7.2.2 Nonstandard FDTD
      7.2.3 FDTD for the damped wave equation
7.3 Wave Equation with a Source
7.4 The Scattered Field
      7.4.1 Analytic solution
      7.4.2 The standard finite difference model
      7.4.3 Nonstandard FDTD
7.5 NAbsorbing Boundary Condition
      7.5.1 One-way wave equations
      7.5.2 Standard finite difference model
      7.5.3 Nonstandard finite difference model
      7.5.4 Numerical stability of the Mur absorbing boundary condition
      7.5.5 Comparison of S- and NS-Mur ABCs
7.6 Object Models on the Grid
      7.6.1 Model of a dielectric
      7.6.2 Model of an absorbing dielectric
      7.6.3 Generalization to two and three dimensions
      7.6.4 Numerical examples
7.7 Mie Scattering and Validation of Computations
      7.7.1 Effect of grid representation
7.8 Discrete Green's Function Solution of the Scattering Problem
      7.8.1 Discrete Green's functions
      7.8.2 Applications of discrete Green's functions
Appendix 7.1 Stability Analysis of the Wave Equation FDTD Algorithm
      A7.1.1 Standard FDTD
      A7.1.2 Nonstandard FDTD
Appendix 7.2 Stability Analysis of the Mur Absorbing Boundary Condition
References

8 Review of Electromagnetic Theory
8.1 Maxwell's Equations: General Formulation
8.2 Linear Media
      8.2.1 Maxwell's equations in a linear, nonconducting medium with no source current
      8.2.2 Maxwell's equations in a linear, conducting medium with a source current
8.3 Boundary Conditions
8.4 Linear Dispersive Materials
      8.4.1 Constitutive relations
      8.4.2 Maxwell's equations
8.5 Kramers–Kronig Relations