### Spie Press Book • on sale

Computing the Flow of Light: Nonstandard FDTD Methodologies for Photonics DesignFormat | Member Price | Non-Member Price |
---|---|---|

Pages: 430

ISBN: 9781510604810

Volume: PM272

### 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
- Appendix 8.1 Properties of the Fourier Transform
- A8.1.1 Definitions
- A8.1.2 Representation of the
*δ*function - A8.1.3 Convolution theorem
- A8.1.4 Fourier transform of the derivative
- References
**9 The Yee Algorithm in One Dimension**- 9.1 Basic Solution of Maxwell's Equations
- 9.2 Standard Yee Algorithm
- 9.2.1 Zero conductivity
- 9.2.2 Including conductivity
- 9.2.3 Including source current and conductivity
- 9.2.4 Alternative configurations of the Yee algorithm
- 9.3 Computational Boundaries
- 9.4 The Nonstandard Yee Algorithm
- 9.4.1 Zero conductivity
- 9.4.2 Including conductivity
- 9.4.3 Including a source current
- 9.4.4 Computational boundaries
- Appendix 9.1 Analytic Solution of Maxwell's 1D Equations for a Point Source Current
- Appendix 9.2 Computer Program in Pseudocode
- References
**10 The Yee Algorithm in Two and Three Dimensions**- 10.1 General Development
- 10.2 Implementation in Two Dimensions
- 10.2.1 TM mode
- 10.2.2 TE mode
- 10.3 The Scattered Field
- 10.4 Grid Representations for the Yee Algorithm
- 10.5 Algorithm Validation via Mie Scattering
- Appendix 10.1 Mie Scattering off of a Cylinder in the TE Mode
- Appendix 10.2 Mie Scattering off of a Cylinder in the TM Mode
- Appendix 10.3 Cylindrical Whispering Gallery Modes
- References
**11 Example Applications of FDTD**- 11.1 Beam Formation
- 11.1.1 Gaussian beam
- 11.1.2 Bessel beam
- 11.1.3 Airy–Gauss beam
- 11.2 Whispering Gallery Modes and Devices
- References
**12 FDTD for Dispersive Materials**- 12.1 Material Dispersion and Dispersive FDTD
- 12.1.1 Recursive convolution FDTD and its monochromatic cousin
- 12.2 Mathematical Description of MRC-FDTD
- 12.3 MRC-FDTD for the Wave Equation in the TE Mode
- 12.4 Stability of MRC-FDTD
- 12.5 S-FDTD for Materials Described by Normal Dispersion
- 12.6 Numerical Evaluation of Extinction and Scattering Cross-Sections
- 12.7 Reflection/Transmission of Thin Metal Films: MRC-FDTD versus Analytical Calculation
- 12.8 Summary
- Appendix 12.1 Eigenvalues and Eigenvectors of Matrices
- Appendix 12.2 MRC-FDTD Stability Analysis
- References
**13 Photonics Problems**- 13.1 Simulation of Metal Nanoparticles
- 13.1.1 Simulations of pairs or arrays of nanocylinders
- 13.2 Iimulation of Metal–Insulator–Metal Nanoresonators for Color Filtering
- 13.2.1 Proposed structure
- 13.2.2 Simulation results
- 13.3 Summary
- References
**14 Photonics Design**- 14.1 Designing Low-Reflection, Multilayer Wire-Grid Polarizers for LCD Applications
- 14.1.1
*Ab initio*design - 14.1.2 Results
- 14.1.3 Discussion
- 14.2 Designing Wavelength-Selective, Polarizing Reflector Films for LCD Applications
- 14.2.1 Method of design
- 14.2.2 Multilayer films
- 14.2.3 Application in LCD devices
- 14.2.4 Discussion
- 14.3 The Scattered Field
- References
**Appendix A Supplemental Topics****Appendix B List of Programs and Instructions**- B.1 Instructions for Mathcad Programs
- B.2 List of Mathcad Programs
- B.3 Pseudocode Programs
*Index*

## Preface

People often speak of *the* finite difference time domain (FDTD) method, as if
there were only one such method, formulated long ago by wise sages and fixed
for all time. This is not so. FDTD is a topic of active research, and its
methodology is constantly evolving. 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.

FDTD is particularly useful for investigating time-dependent phenomena. As the name suggests, the time evolution of a system is computed at discrete time steps, and periodic visualizations can show its time evolution—information that is not available in a frequency domain technique. This yields useful physical insights into transient processes as well as an intuitive feel for what is happening; often, one can see at a glance if something is wrong with a calculation—saving not only computer time, but more importantly, human time.

In essence, *an* FDTD algorithm is derived from a difference equation
model of the differential equation to be solved by replacing the derivatives
with finite difference (FD) expressions. (We use indefinite articles to indicate
that there can be more than one FD model and more than one FDTD
algorithm.) As we shall demonstrate in this book, the accuracy and stability of
FDTD algorithms can be greatly improved by using what are called
*nonstandard* FD models from which nonstandard FDTD derives.

High-precision FDTD is the primary—but not the only—subject of this book. Although the basic FDTD algorithm is simple, you will encounter many 'devils in the details' when you actually try to use it to solve a problem. Throughout this book we address these devilish details.

Green's functions, although an elegant analytical construct, are of limited practical use to solve differential equations because they can be difficult to find, but the discrete Green's function (DGF) of a difference equation model of a differential equation can be found using FDTD. The DGF methodology is a useful alternative to conventional FDTD for certain problems.

Besides introducing useful new methodologies, we have written this book to give our readers new insight, along with the analytical background necessary to develop their own methodologies to solve new problems on the leading edges of photonics and electromagnetics research. We explain not only how FDTD works, but why it works. We delve into the details of a few analytical solutions against which the numerical solutions can be compared to validate FDTD algorithms and to elucidate their limitations.

We make no attempt to be encyclopedic; rather, we delve deeply (devils in the details) into a few of the most important and basic methodologies. We hope that our readers can use this book to write their own working programs and improve on our methods, and that they will share what they have learned with the community.

Our experience in solving real-life applications with FDTD has taught us that there are two distinct communities: those who see FDTD from the algorithmic–computational point of view and those who see themselves primarily as users of black-box software. We believe that FDTD simulations must be guided by an understanding of Maxwell's equations and how these equations incorporate material properties. This premise is an important focus of this book. Through the simulation examples presented, we have attempted to show that FDTD and related techniques can be very useful in solving practical problems.

## Summary of the Contents

•**Chapter 1**introduces FD expressions and develops the notation that is used throughout this book. The appendices give some supplementary advanced topics.

• **Chapter 2** introduces algorithms in general. We analyze a few example
algorithms in detail and discuss their accuracy and numerical stability. In the
appendices we introduce additional advanced topics. Working programs
illustrate some of the main ideas.

• **Chapter 3** develops the basic concepts of the FDTD methodology using
the simple harmonic oscillator as a vehicle. We then introduce the
nonstandard FDTD methodology. We go on to develop analytic solutions
for both the differential equations and the corresponding difference equations
using Green's functions. In the appendices we review some basic mathematical
concepts and derive Green's functions for differential and difference
equations. We also analyze the accuracy and stability of a few simple FDTD
algorithms.

• **Chapters 4–7** present FDTD for the wave equation. We follow a strategy
of stepwise increasing complexity. We start with the one-dimensional wave
equation and develop the machinery needed to solve useful problems using
FDTD. The appendices contain advanced material (Green's function
solutions, including the development of a Green's function for the finite
difference form of the wave equation) and deal with various devilish details.
We then extend these developments to two and three dimensions, and give
some working example programs.

• **Chapter 8** provides a brief review of electromagnetic theory.

• **Chapters 9 and 10** present FDTD for Maxwell's equations. We first develop
standard and nonstandard FDTD (the Yee algorithm) in one dimension, and
then extend the methodology to two and three dimensions. This completes the development of conventional and nonstandard FDTD. Working programs illustrate some the main ideas.

• **Chapter 11** provides example problems.

• **Chapters 12–14** present FDTD for the dispersive case and provide
example problems in photonics design. We introduce some of our latest
research results on how to improve accuracy in the dispersive case. We solve
some interesting photonics problems and discuss photonics design for
engineering a subwavelength structure to have desired optical properties.

## Audience

Our intended audience includes intelligent beginners such as students, experimental scientists who want to model their experiments, practical engineers, and theoretical researchers grappling with problems that cannot be solved analytically.We introduce our nonstandard FDTD methodology as a useful tool for computational professionals as well as for beginners.

Very few members of the physics community are aware of the wide utility of FDTD methods, and we hope that our book will inform this group.

We include working FDTD programs that bring wave and electromagnetic phenomena to life. Our analytic solutions motivate the study of mathematical physics as a practical tool. Indeed, this book could be useful for teaching a mathematical physics, applied mathematics, or engineering class. We also hope that advanced practitioners of FDTD can use this book to extend the nonstandard approach to other FDTD methodologies not covered in this book.

Finally, this book is written for researchers who want to develop new methodologies that go beyond those we have presented.

## Acknowledgments

We thank Professor Ronald Mickens, one of the great pioneers of the nonstandard methodology, as well as our colleagues, collaborators, and students, who are too numerous to list.

## Author Contact

We welcome comments and suggestions from our readers. James B. Cole and Sasawatee Banerjee can be contacted at cole.banerjee.book@gmail.com.
**James B. Cole**

**Saswatee Banerjee**

May 2017

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