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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
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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