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Numerical Methods in Photonics [Mīkstie vāki]

(Zuse Institute, Berlin, Germany), (Technical University of Denmark, Denmark), (Aalborg University, Aalborg East, Denmark), (Technical University of Denmark, Kongens Lyngby), (Technical University of Denmark, Kongens Lyngby)
  • Formāts: Paperback / softback, 364 pages, height x width: 234x156 mm, weight: 453 g, 3 Tables, black and white; 96 Illustrations, black and white
  • Sērija : Optical Sciences and Applications of Light
  • Izdošanas datums: 22-Nov-2017
  • Izdevniecība: CRC Press
  • ISBN-10: 1138074691
  • ISBN-13: 9781138074699
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Numerical Methods in PhotonicsPalielināt
  • Formāts: Paperback / softback, 364 pages, height x width: 234x156 mm, weight: 453 g, 3 Tables, black and white; 96 Illustrations, black and white
  • Sērija : Optical Sciences and Applications of Light
  • Izdošanas datums: 22-Nov-2017
  • Izdevniecība: CRC Press
  • ISBN-10: 1138074691
  • ISBN-13: 9781138074699
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781138074699.html
Keywords:

Simulation and modeling using numerical methods is one of the key instruments in any scientific work. In the field of photonics, a wide range of numerical methods are used for studying both fundamental optics and applications such as design, development, and optimization of photonic components. Modeling is key for developing improved photonic devices and reducing development time and cost.

Choosing the appropriate computational method for a photonics modeling problem requires a clear understanding of the pros and cons of the available numerical methods. Numerical Methods in Photonics presents six of the most frequently used methods: FDTD, FDFD, 1+1D nonlinear propagation, modal method, Green’s function, and FEM.

After an introductory chapter outlining the basics of Maxwell’s equations, the book includes self-contained chapters that focus on each of the methods. Each method is accompanied by a review of the mathematical principles in which it is based, along with sample scripts, illustrative examples of characteristic problem solving, and exercises. MATLAB® is used throughout the text.

This book provides a solid basis to practice writing your own codes. The theoretical formulation is complemented by sets of exercises, which allow you to grasp the essence of the modeling tools.

Recenzijas

" useful to students and researchers who want to have a deeper understanding of the methods commonly used in computational electromagnetics. After addressing the basic principles, this book provides the readers with the details and mathematical/numerical framework of commonly used methods including FDTD, finite element, Greens function, and modal. It then goes on to more advanced topics such as modelling nonlinear materials and materials with gain. This book is a useful addition to the library of any research university." C T Chan, Hong Kong University of Science and Technology

Series Preface xiii
Preface xv
Authors xvii
Acronyms xix
Chapter 1 Introduction 1(4)
Chapter 2 Maxwell's Equations 5(18)
2.1 Notation
5(1)
2.2 Maxwell's Equations
5(1)
2.3 Material Equations
6(1)
2.4 Frequency Domain
7(2)
2.5 1D and 2D Maxwell's Equations
9(2)
2.6 Wave Equations
11(2)
2.7 Waveguides and Eigenmodes
13(9)
2.7.1 Eigenvalue Problem
14(2)
2.7.2 Slab Waveguides
16(1)
2.7.3 Boundary Conditions and Eigenmode Classes
17(1)
2.7.4 Orthogonality
18(4)
References
22(1)
Chapter 3 Finite-Difference Time-Domain Method 23(54)
3.1 Introduction
23(11)
3.1.1 Finite-Difference Approximations of Derivatives
24(3)
3.1.2 Finite-Difference Approximation of 1D Maxwell's Equations
27(3)
3.1.3 Fortran, C, MATLAB®, Etc., Adaptation of the FDTD Method
30(1)
3.1.4 FDTD Method in 3D
31(3)
3.1.5 FDTD Method in 2D
34(1)
3.2 Numerical Dispersion and Stability Analysis of the FDTD Method
34(8)
3.2.1 Dispersion Equation in 3D
35(2)
3.2.2 Numerical Stability Criteria
37(2)
3.2.3 Divergence-Free Character of the FDTD Method
39(3)
3.3 Making Your Own 1D FDTD
42(8)
3.3.1 Step 1: Setting Material Properties on a Grid
43(3)
3.3.2 Step 2: Setting Sources and Detectors
46(2)
3.3.3 Step 3: Evolving Fields
48(1)
3.3.4 Step 4: Postprocessing of Information
49(1)
3.4 Absorbing Boundary Conditions
50(8)
3.4.1 Analytical Absorbing Boundary Conditions
51(1)
3.4.2 Perfectly Matched Layer: Basic Idea
52(3)
3.4.3 Perfectly Matched Layer: Generalization and Realization
55(3)
3.5 FDTD Method for Materials with Frequency Dispersion
58(9)
3.5.1 Frequency Dispersion Models
58(2)
3.5.1.1 Debye Material
58(1)
3.5.1.2 Drude Model
59(1)
3.5.1.3 Lorentz Model
60(1)
3.5.2 Numerical Implementation of Frequency Dispersion in FDTD through Auxiliary Equation
60(4)
3.5.2.1 Debye Material
61(2)
3.5.2.2 Drude Model of Dispersion
63(1)
3.5.2.3 Lorentz Model of Dispersion
63(1)
3.5.3 Linear Polarization Model for Dispersive Materials in FDTD
64(2)
3.5.4 Piecewise Linear Recursive Convolution Scheme
66(1)
3.6 FDTD Method for Nonlinear Materials, Materials with Gain, and Lasing
67(4)
3.6.1 Nonlinear Polarization in FDTD
67(2)
3.6.2 Medium with Gain: Phenomenological Approach in FDTD
69(1)
3.6.3 Lasing in FDTD
69(2)
3.7 Conclusion
71(1)
Exercises
71(3)
References
74(3)
Chapter 4 Finite-Difference Modelling of Straight Waveguides 77(30)
4.1 Introduction
77(1)
4.2 General Considerations
77(3)
4.2.1 Time Domain versus Frequency Domain
77(1)
4.2.2 Finite-Difference Methods for Straight Waveguides
78(2)
4.3 Modified Finite-Difference Operators
80(8)
4.3.1 Discretizing the Scalar Wave Equation
80(3)
4.3.2 Inclusion of Discontinuities: General Formalism
83(3)
4.3.3 Inclusion of Discontinuities: TE Case
86(1)
4.3.4 Inclusion of Discontinuities: TM Case
87(1)
4.4 Numerical Linear Algebra in MATLAB
88(4)
4.4.1 Sparse Matrices
88(1)
4.4.2 Direct and Iterative Eigensolvers
89(3)
4.5 2D Waveguides and the Yee Mesh
92(10)
4.5.1 Yee Mesh
92(3)
4.5.2 Dielectric Function Averaging
95(4)
4.5.3 Use of Mirror Symmetries
99(3)
Exercises
102(4)
References
106(1)
Chapter 5 Modelling of Nonlinear Propagation in Waveguides 107(32)
5.1 Introduction
107(1)
5.2 Formalism
108(3)
5.2.1 General Propagation Equation
108(2)
5.2.2 Pulse Power and Pulse Energy
110(1)
5.3 Nonlinear Polarization
111(9)
5.3.1 Nonlinear Processes
112(2)
5.3.2 χ(3) Nonlinear Processes
114(1)
5.3.3 Single-Mode Propagation Model
115(5)
5.4 Nonlinear Schrodinger Equation
120(9)
5.4.1 Derivation of the NLS Equation
120(2)
5.4.2 Dispersion and Self-Phase Modulation
122(2)
5.4.3 Optical Solitons
124(1)
5.4.4 Solitons and Raman Effects
125(1)
5.4.5 Self-Steepening
126(1)
5.4.6 Conservation Laws
127(2)
5.5 Numerical Implementation
129(7)
5.5.1 Fourier Method
129(1)
5.5.2 Stepping Techniques
130(2)
5.5.3 Discrete Fourier Grids
132(2)
5.5.4 Implementation in MATLAB
134(2)
Exercises
136(1)
References
137(2)
Chapter 6 The Modal Method 139(58)
6.1 Introduction
139(1)
6.2 Eigenmodes
140(2)
6.3 1D Geometry
142(8)
6.3.1 Recursive Matrix Formalism
143(2)
6.3.2 1D Interface
145(1)
6.3.3 Multilayer Structure
146(3)
6.3.4 ID Cavity
149(1)
6.4 2D Geometry
150(26)
6.4.1 Plane-Wave Expansion
151(6)
6.4.1.1 Li's Factorization Rules
151(2)
6.4.1.2 Eigenvalue Problem
153(4)
6.4.2 Semi-Analytical Approach
157(5)
6.4.3 Interface
162(4)
6.4.4 S Matrix Theory
166(5)
6.4.5 Absorbing Boundary Conditions
171(5)
6.5 Periodic Structures
176(9)
6.5.1 Bloch Modes
177(3)
6.5.2 Classification
180(2)
6.5.3 Interface
182(2)
6.5.4 Field Profile in a Periodic Element
184(1)
6.6 Current Sources
185(5)
6.6.1 Uniform Layer
186(2)
6.6.2 Multilayer Geometry
188(2)
6.7 3D Geometries
190(1)
Exercises
191(3)
References
194(3)
Chapter 7 Green's Function Integral Equation Methods for Electromagnetic Scattering Problems 197(54)
7.1 Introduction
197(1)
7.2 Theoretical Foundation
198(1)
7.3 Green's Function Area Integral Equation Method
198(6)
7.4 Green's Function Volume Integral Equation Method
204(5)
7.5 Green's Function Surface Integral Equation Method (2D)
209(9)
7.5.1 Surface Integral Equations
209(3)
7.5.2 Calculating the Field and Normal Derivative at the Boundary
212(6)
7.6 Construction of 2D Green's Functions for Layered Structures
218(15)
7.6.1 Plane-Wave Expansion of the Free-Space Green's Function
219(3)
7.6.2 2D TE-Polarized Scalar Green's Function for a Layered Structure
222(2)
7.6.3 2D TM-Polarized Scalar Green's Function for a Layered Structure
224(1)
7.6.4 Fresnel Reflection and Transmission Coefficients for a Few Simple Geometries
224(2)
7.6.5 Calculating the Sommerfeld Integral
226(2)
7.6.6 Far-Field Approximation
228(2)
7.6.7 Excitation of Bound Waveguide Modes
230(3)
7.7 Construction of the Periodic Green's Function
233(1)
7.7.1 1D Periodic Scalar Green's Function for a Layered Structure
234(1)
7.8 Reflection from a Periodic Surface Microstructure
234(6)
7.8.1 Calculating Reflection and Transmission
237(3)
7.9 Iterative Solution Scheme Taking Advantage of the Fast Fourier Transform
240(5)
7.9.1 2D Discrete Convolution
242(3)
7.10 Further Reading
245(1)
Exercises
245(2)
References
247(4)
Chapter 8 Finite Element Method 251(76)
8.1 Introduction: Helmholtz Equation in 1D
252(10)
8.1.1 Variational Formulation
252(2)
8.1.2 Weak Form
254(1)
8.1.3 Galerkin Method
255(1)
8.1.4 Discrete Problem
256(1)
8.1.5 Linear Finite Elements
256(3)
8.1.6 Domain Mapping
259(1)
8.1.7 Assembly Process
260(1)
8.1.8 Algorithm: Plane-Wave Propagation
261(1)
8.2 General Scattering Problem in 1D
262(18)
8.2.1 Variational Formulation in 1D with DtN Operator
263(2)
8.2.1.1 DtN Operator
263(2)
8.2.2 Variational Formulation in 1D with Perfectly Matched Layers
265(4)
8.2.2.1 Completion to a Continuous Function
268(1)
8.2.3 Discretization
269(1)
8.2.4 A Posteriori Error Estimation
270(6)
8.2.4.1 Galerkin Orthogonality
274(1)
8.2.4.2 A Different Viewpoint
275(1)
8.2.4.3 Error Localization and Error Indicator
275(1)
8.2.5 Adaptive Mesh Refinement
276(2)
8.2.6 FEM Notions: Element Support, Basis Functions, Shape Functions, Finite Elements, and Finite Element Spaces
278(2)
8.3 Mathematical Background: Maxwell and Helmholtz Scattering Problems and Their Variational Forms
280(18)
8.3.1 Maxwell's Scattering Problem
280(3)
8.3.1.1 Discussion of the Silver-Muller Radiation Condition
282(1)
8.3.2 Slight Simplification: The Helmholtz Scattering Problem
283(1)
8.3.3 Transformation Rules
284(4)
8.3.3.1 Mapping of Geometric Quantities
284(2)
8.3.3.2 Mapping of grad, curl, and div
286(1)
8.3.3.3 Mapping of Fields
287(1)
8.3.3.4 Mapping of μ and epsilon
288(1)
8.3.4 PML in 2D and 3D
288(1)
8.3.5 Integration by Parts
289(1)
8.3.6 Variational Formulation for the Helmholtz Equation with PML
290(3)
8.3.6.1 Interior Problem
290(1)
8.3.6.2 Exterior Problem for the Scattered Field
291(1)
8.3.6.3 Variational Formulation on the Entire Domain
292(1)
8.3.7 Variational Formulation for Maxwell's Equations with PML
293(5)
8.3.7.1 Interior Problem
293(1)
8.3.7.2 Exterior Problem for the Scattered Field
294(1)
8.3.7.3 Variational Formulation on the Entire Domain
295(3)
8.4 FEM for Helmholtz Scattering in 2D and 3D
298(13)
8.4.1 Rectangular Meshes
298(1)
8.4.2 Mesh and Assembly Process: General Scheme
299(1)
8.4.3 Finite Elements for Rectangular Meshes
300(8)
8.4.3.1 Rectangular Elements
300(1)
8.4.3.2 Polynomial Space
301(1)
8.4.3.3 DOFs on a Rectangle
301(3)
8.4.3.4 Bilinear Finite Element Discretization
304(2)
8.4.3.5 Boundary Integral
306(2)
8.4.4 Finite Elements for Triangular Meshes
308(3)
8.4.4.1 Global Data Structure and Connectivity Matrix
310(1)
8.5 FEM for Maxwell's Scattering in 2D and 3D
311(7)
8.5.1 Finite Elements for Rectangular Meshes
311(4)
8.5.1.1 Polynomial Space
311(1)
8.5.1.2 DOFs on a Rectangle
312(1)
8.5.1.3 Linear Finite Element Discretization
313(2)
8.5.2 Finite Elements for Triangular Meshes
315(3)
8.5.2.1 Triangular Elements
315
8.5.2.2 Polynomial Space
116(200)
8.5.2.3 DOFs on a Triangle
316(1)
8.5.2.4 Linear Finite Element Discretization
317(1)
Exercises
318(7)
References
325(2)
Index 327
Andrei V. Lavrinenko, Technical University of Denmark, Kongens Lyngby Jesper Lęgsgaard, Technical University of Denmark, Kongens Lyngby Niels Gregersen, Technical University of Denmark, Kongens Lyngby Frank Schmidt, Zuse Institute, Berlin, Germany Thomas Sųndergaard, Aalborg University, Denmark