Numerical methods in photonics:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
CRC Press
2015
|
| Schriftenreihe: | Optical sciences and applications of light
|
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XIX, 334 S. Ill., graph. Darst. |
| ISBN: | 9781466563889 |
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Datensatz im Suchindex
| _version_ | 1804152693303803904 |
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| adam_text | Numerical Methods in Photonics
Choosing the appropriate computational method for a photonics
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 on 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 basic theory is complemented by sets of exercises, which allow
you to grasp the essence of the modeling tools.
Contents
я
Series Preface.......................................................xiii
Preface................................................................xv
Authors..............................................................xvii
Acronyms..............................................................xix
Chapter 1 Introduction............................................... 1
Chapter 2 Maxwell’s Equations.........................................5
2.1 Notation.............................................. 5
2.2 Maxwell’s Equations....................................5
2.3 Material Equations.....................................6
2.4 Frequency Domain.......................................7
2.5 ID and 2D Maxwell’s Equations..........................9
2.6 Wave Equations .......................................11
2.7 Waveguides and Eigenmodes.............................13
2.7.1 Eigenvalue Problem.............................14
2.7.2 Slab Waveguides................................16
2.7.3 Boundary Conditions and Eigenmode Classes......17
2.7.4 Orthogonality..................................18
References.................................................22
Chapter 3 Finite-Difference Time-Domain Method.......................23
3.1 Introduction..........................................23
3.1.1 Finite-Difference Approximations of Derivatives.24
3.1.2 Finite-Difference Approximation of ID
Maxwell’s Equations............................27
3.1.3 Fortran, C, MATLAB®, Etc., Adaptation of the
FDTD Method....................................30
3.1.4 FDTD Method in 3D..............................31
3.1.5 FDTD Method in 2D..............................34
3.2 Numerical Dispersion and Stability Analysis of the
FDTD Method...........................................34
3.2.1 Dispersion Equation in 3D......................35
3.2.2 Numerical Stability Criteria...................37
3.2.3 Divergence-Free Character of the FDTD Method....39
3.3 Making Your Own 1D FDTD...............................42
3.3.1 Step 1: Setting Material Properties on a Grid..43
3.3.2 Step 2: Setting Sources and Detectors..........46
vii
vîii Contents
3.3.3 Step 3: Evolving Fields ............................48
3.3.4 Step 4: Postprocessing of Information...............49
3.4 Absorbing Boundary Conditions...............................50
3.4.1 Analytical Absorbing Boundary Conditions............51
3.4.2 Perfectly Matched Layer: Basic Idea.................52
3.4.3 Perfectly Matched Layer: Generalization and
Realization.........................................55
3.5 FDTD Method for Materials with Frequency Dispersion........58
3.5.1 Frequency Dispersion Models.........................58
3.5. i. 1 Debye Material............................58
3.5.1.2 Drude Model...............................59
3.5.1.3 Lorentz Model.............................60
3.5.2 Numerical Implementation of Frequency
Dispersion in FDTD through Auxiliary Equation......60
3.5.2.1 Debye Material............................61
3.5.2.2 Drude Model of Dispersion.................63
3.5.2.3 Lorentz Model of Dispersion...............63
3.5.3 Linear Polarization Model for Dispersive
Materials in FDTD...................................64
3.5.4 Piecewise Linear Recursive Convolution
Scheme..............................................66
3.6 FDTD Method for Nonlinear Materials, Materials with
Gain, and Lasing............................................67
3.6.1 Nonlinear Polarization in FDTD......................67
3.6.2 Medium with Gain: Phenomenological Approach
in FDTD ............................................69
3.6.3 Lasing in FDTD......................................69
3.7 Conclusion..................................................71
Exercises........................................................71
References.......................................................74
Chapter 4 Finite-Difference Modelling of Straight Waveguides...............77
4.1 Introduction................................................77
4.2 General Considerations......................................77
4.2.1 Time Domain versus Frequency Domain ................77
4.2.2 Finite-Difference Methods for Straight
Waveguides..........................................78
4.3 Modified Finite-Difference Operators........................80
4.3.1 Discretizing the Scalar Wave Equation...............80
4.3.2 Inclusion of Discontinuities: General Formalism....83
4.3.3 Inclusion of Discontinuities: TE Case...............86
4.3.4 Inclusion of Discontinuities: TM Case...............87
4.4 Numerical Linear Algebra in MATLAB..........................88
4.4.1 Sparse Matrices.....................................88
4.4.2 Direct and Iterative Eigensolvers...................89
Contents
IX
4.5 2D Waveguides and the Yee Mesh ........................92
4.5.1 Yee Mesh..........................................92
4.5.2 Dielectric Function Averaging.....................95
4.5.3 Use of Mirror Symmetries..........................99
Exercises.................................................... 102
References................................................... 106
Chapter 5 Modelling of Nonlinear Propagation in Waveguides.............. 107
5.1 Introduction........................................... 107
5.2 Formalism.............................................. 108
5.2.1 General Propagation Equation.................... 108
5.2.2 Pulse Power and Pulse Energy.................... 110
5.3 Nonlinear Polarization................................. Ill
5.3.1 Nonlinear Processes............................. 112
5.3.2 x(3) Nonlinear Processes ....................... 114
5.3.3 Single-Mode Propagation Model................... 115
5.4 Nonlinear Schrodinger Equation......................... 120
5.4.1 Derivation of the NLS Equation.................. 120
5.4.2 Dispersion and Self-Phase Modulation ........... 122
5.4.3 Optical Soli tons............................... 124
5.4.4 Solitons and Raman Effects...................... 125
5.4.5 Self-Steepening................................. 126
5.4.6 Conservation Laws............................... 127
5.5 Numerical Implementation .............................. 129
5.5.1 Fourier Method.................................. 129
5.5.2 Stepping Techniques............................. 130
5.5.3 Discrete Fourier Grids.......................... 132
5.5.4 Implementation in MATLAB........................ 134
Exercises.................................................... 136
References................................................... 137
Chapter 6 The Modal Method............................................ 139
6.1 Introduction........................................... 139
6.2 Eigenmodes............................................. 140
6.3 ID Geometry............................................ 142
6.3.1 Recursive Matrix Formalism...................... 143
6.3.2 ID Interface.................................... 145
6.3.3 Multilayer Structure............................ 146
6.3.4 ID Cavity....................................... 149
6.4 2D Geometry.......................................... 150
6.4.1 Plane-Wave Expansion............................ 151
6.4.1.1 Li’s Factorization Rules.............. 151
6.4.1.2 Eigenvalue Problem.................... 153
6.4.2 Semi-Analytical Approach........................ 157
X
Contents
6.4.3 Interface.......................................... 162
6.4.4 5 Matrix Theory.................................... 166
6.4.5 Absorbing Boundary Conditions...................... 171
6.5 Periodic Structures....................................... 176
6.5.1 Bloch Modes........................................ 177
6.5.2 Classification..................................... 180
6.5.3 Interface.......................................... 182
6.5.4 Field Profile in a Periodic Element................ 184
6.6 Current Sources .......................................... 185
6.6.1 Uniform Layer...................................... 186
6.6.2 Multilayer Geometry................................ 188
6.7 3D Geometries ............................................ 190
Exercises....................................................... 191
References...................................................... 194
Chapter 7 Green’s Function Integral Equation Methods for
Electromagnetic Scattering Problems........................... 197
7.3 Introduction............................................ 197
7.2 Theoretical Foundation ................................. 198
7.3 Green’s Function Area Integral Equation Method.......... 198
7.4 Green’s Function Volume Integral Equation Method........ 204
7.5 Green’s Function Surface Integral Equation
Method (2D)............................................. 209
7.5.1 Surface Integral Equations....................... 209
7.5.2 Calculating the Field and Normal Derivative
at the Boundary.................................. 212
7.6 Construction of 2D Green’s Functions for
Layered Structures.................................... 218
7.6.1 Plane-Wave Expansion of the Free-Space
Green’s Function ................................ 219
7.6.2 2D TE-Polarized Scalar Green’s Function for a
Layered Structure................................ 222
7.6.3 2D TM-Polarized Scalar Green’s Function for a
Layered Structure................................ 224
7.6.4 Fresnel Reflection and Transmission Coefficients
for a Few Simple Geometries...................... 224
7.6.5 Calculating the Sommerfeld Integral.............. 226
7.6.6 Far-Field Approximation.......................... 228
7.6.7 Excitation of Bound Waveguide Modes ............. 230
7.7 Construction of the Periodic Green’s Function........... 233
7.7.1 ID Periodic Scalar Green’s Function for a
Layered Structure................................ 234
7.8 Reflection from a Periodic Surface Microstructure........234
7.8.1 Calculating Reflection and Transmission.......... 237
Contents xi
7.9 Iterative Solution Scheme Taking Advantage of the
Fast Fourier Transform....................................240
7.9.1 2D Discrete Convolution............................242
7.10 Further Reading...........................................245
Exercises.......................................................245
References..................................................... 247
Chapter 8 Finite Element Method............................................. 251
8.1 Introduction: Helmholtz Equation in ID.....................252
8.1.1 Variational Formulation............................252
8.1.2 Weak Form .........................................254
8.1.3 Galerkin Method ...................................255
8.1.4 Discrete Problem...................................256
8.1.5 Linear Finite Elements............................ 256
8.1.6 Domain Mapping.....................................259
8.1.7 Assembly Process...................................260
8.1.8 Algorithm: Plane-Wave Propagation..................261
8.2 General Scattering Problem in ID...........................262
8.2.1 Variational Formulation in ID with
DtN Operator.......................................263
8.2.1.1 DtN Operator..............................263
8.2.2 Variational Formulation in ID with Perfectly
Matched Layers.....................................265
8.2.2.1 Completion to a Continuous Function.....268
8.2.3 Discretization.....................................269
8.2.4 A Posteriori Error Estimation......................270
8.2.4.1 Galerkin Orthogonality....................274
8.2.4.2 A Different Viewpoint.....................275
8.2.4.3 Error Localization and Error Indicator..275
8.2.5 Adaptive Mesh Refinement.......................... 276
8.2.6 FEM Notions: Element Support, Basis Functions,
Shape Functions, Finite Elements, and Finite
Element Spaces.................................... 278
8.3 Mathematical Background: Maxwell and Helmholtz
Scattering Problems and Their Variational Forms...........280
8.3.1 Maxwell’s Scattering Problem.......................280
8.3.1.1 Discussion of the Silver-MUller
Radiation Condition......................282
8.3.2 Slight Simplification: The Helmholtz
Scattering Problem ................................283
8.3.3 Transformation Rules...............................284
8.3.3.1 Mapping of Geometric Quantities...........284
8.3.3.2 Mapping of grad, curl, and div........... 286
8.3.3.3 Mapping of Fields.........................287
8.3.3.4 Mapping of x and e ......................288
XII
Contents
8.3.4 PML in 2D and 3D.................................. 288
8.3.5 Integration by Parts.............................. 289
8.3.6 Variational Formulation for the Helmholtz
Equation with PML ................................ 290
8.3.6.1 Interior Problem.......................... 290
8.3.6.2 Exterior Problem for the
Scattered Field.......................... 291
8.3.6.3 Variational Formulation on the
Entire Domain............................ 292
8.3.7 Variational Formulation for Maxwell’s Equations
with PML.......................................... 293
8.3.7.1 Interior Problem.......................... 293
8.3.7.2 Exterior Problem for the
Scattered Field.......................... 294
8.3.7.3 Variational Formulation on the
Entire Domain............................ 295
8.4 FEM for Helmholtz Scattering in 2D and 3D............. 298
8.4.1 Rectangular Meshes................................ 298
8.4.2 Mesh and Assembly Process: General Scheme........299
8.4.3 Finite Elements for Rectangular Meshes............ 300
8.4.3.1 Rectangular Elements...................... 300
8.4.3.2 Polynomial Space ......................... 301
8.4.3.3 DOFs on a Rectangle....................... 301
8.4.3.4 Bilinear Finite Element Discretization — 304
8.4.3.5 Boundary Integral ........................ 306
8.4.4 Finite Elements for Triangular Meshes............. 308
8.4.4.1 Global Data Structure and
Connectivity Matrix...................... 310
8.5 FEM for Maxwell’s Scattering in 2D and 3D ............... 311
8.5.1 Finite Elements for Rectangular Meshes............ 311
8.5.1.1 Polynomial Space ......................... 311
8.5.1.2 DOFs on a Rectangle....................... 312
8.5.1.3 Linear Finite Element Discretization..... 313
8.5.2 Finite Elements for Triangular Meshes............. 315
8.5.2.1 Triangular Elements....................... 315
8.5.2.2 Polynomial Space ......................... 316
8.5.2.3 DOFs on a Triangle........................ 316
8.5.2.4 Linear Finite Element Discretization..... 317
Exercises....................................................... 318
References...................................................... 325
Index
327
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| illustrated | Illustrated |
| indexdate | 2024-07-10T01:14:46Z |
| institution | BVB |
| isbn | 9781466563889 |
| language | English |
| lccn | 2014009744 |
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| physical | XIX, 334 S. Ill., graph. Darst. |
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| spelling | Numerical methods in photonics Andrei V. Lavrinenko ... Boca Raton [u.a.] CRC Press 2015 XIX, 334 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Optical sciences and applications of light Includes bibliographical references and index Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Photonik (DE-588)4243979-6 gnd rswk-swf Photonik (DE-588)4243979-6 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Lavrinenko, Andrei V. Sonstige oth Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027622358&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027622358&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Numerical methods in photonics Numerisches Verfahren (DE-588)4128130-5 gnd Photonik (DE-588)4243979-6 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4243979-6 |
| title | Numerical methods in photonics |
| title_auth | Numerical methods in photonics |
| title_exact_search | Numerical methods in photonics |
| title_full | Numerical methods in photonics Andrei V. Lavrinenko ... |
| title_fullStr | Numerical methods in photonics Andrei V. Lavrinenko ... |
| title_full_unstemmed | Numerical methods in photonics Andrei V. Lavrinenko ... |
| title_short | Numerical methods in photonics |
| title_sort | numerical methods in photonics |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Photonik (DE-588)4243979-6 gnd |
| topic_facet | Numerisches Verfahren Photonik |
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