Table Of ContentN u m e r i c a l
M e t h o d s
P h o t o n i c s
in
Andrei V. Lavrinenko
Technical University of Denmark, Kongens Lyngby
Jesper Lægsgaard
Technical University of Denmark, Kongens Lyngby
Niels Gregersen
Technical University of Denmark, Kongens Lyngby
Frank Schmidt
Zuse Institute, Berlin, Germany
Thomas Søndergaard
Aalborg University, Aalborg, Denmark
Boca Raton London New York
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OPTICAL SCIENCES AND APPLICATIONS OF LIGHT
Series Editor
James C. Wyant
Numerical Methods in Photonics, Andrei V. Lavrinenko, Jesper Lægsgaard,
Niels Gregersen, Frank Schmidt, and Thomas Søndergaard
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Dedication
Toourfamiliesandparents
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Contents
SeriesPreface.........................................................................xiii
Preface .................................................................................xv
Authors............................................................................... xvii
Acronyms.............................................................................xix
Chapter1 Introduction..............................................................1
Chapter2 Maxwell’sEquations ....................................................5
2.1 Notation............................................................5
2.2 Maxwell’sEquations..............................................5
2.3 MaterialEquations................................................6
2.4 FrequencyDomain................................................7
2.5 1Dand2DMaxwell’sEquations.................................9
2.6 WaveEquations ..................................................11
2.7 WaveguidesandEigenmodes....................................13
2.7.1 EigenvalueProblem.....................................14
2.7.2 SlabWaveguides ........................................16
2.7.3 BoundaryConditionsandEigenmodeClasses.........17
2.7.4 Orthogonality............................................18
References...............................................................22
Chapter3 Finite-DifferenceTime-DomainMethod..............................23
3.1 Introduction.......................................................23
3.1.1 Finite-DifferenceApproximationsofDerivatives .....24
3.1.2 Finite-DifferenceApproximationof1D
Maxwell’sEquations....................................27
3.1.3 Fortran,C,MATLAB(cid:2),Etc.,Adaptationofthe
FDTDMethod...........................................30
3.1.4 FDTDMethodin3D....................................31
3.1.5 FDTDMethodin2D....................................34
3.2 NumericalDispersionandStabilityAnalysisofthe
FDTDMethod....................................................34
3.2.1 DispersionEquationin3D..............................35
3.2.2 NumericalStabilityCriteria.............................37
3.2.3 Divergence-FreeCharacteroftheFDTDMethod .....39
3.3 MakingYourOwn1DFDTD....................................42
3.3.1 Step1:SettingMaterialPropertiesonaGrid..........43
3.3.2 Step2:SettingSourcesandDetectors..................46
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3.3.3 Step3:EvolvingFields .................................48
3.3.4 Step4:PostprocessingofInformation .................49
3.4 AbsorbingBoundaryConditions................................50
3.4.1 AnalyticalAbsorbingBoundaryConditions...........51
3.4.2 PerfectlyMatchedLayer:BasicIdea...................52
3.4.3 PerfectlyMatchedLayer:Generalizationand
Realization...............................................55
3.5 FDTDMethodforMaterialswithFrequencyDispersion......58
3.5.1 FrequencyDispersionModels..........................58
3.5.1.1 DebyeMaterial ...............................58
3.5.1.2 DrudeModel..................................59
3.5.1.3 LorentzModel................................60
3.5.2 NumericalImplementationofFrequency
DispersioninFDTDthroughAuxiliaryEquation .....60
3.5.2.1 DebyeMaterial ...............................61
3.5.2.2 DrudeModelofDispersion..................63
3.5.2.3 LorentzModelofDispersion ................63
3.5.3 LinearPolarizationModelforDispersive
MaterialsinFDTD ......................................64
3.5.4 PiecewiseLinearRecursiveConvolution
Scheme...................................................66
3.6 FDTDMethodforNonlinearMaterials,Materialswith
Gain,andLasing .................................................67
3.6.1 NonlinearPolarizationinFDTD........................67
3.6.2 MediumwithGain:PhenomenologicalApproach
inFDTD .................................................69
3.6.3 LasinginFDTD.........................................69
3.7 Conclusion........................................................71
Exercises ................................................................71
References...............................................................74
Chapter4 Finite-DifferenceModellingofStraightWaveguides.................77
4.1 Introduction.......................................................77
4.2 GeneralConsiderations ..........................................77
4.2.1 TimeDomainversusFrequencyDomain ..............77
4.2.2 Finite-DifferenceMethodsforStraight
Waveguides..............................................78
4.3 ModifiedFinite-DifferenceOperators...........................80
4.3.1 DiscretizingtheScalarWaveEquation.................80
4.3.2 InclusionofDiscontinuities:GeneralFormalism......83
4.3.3 InclusionofDiscontinuities:TECase..................86
4.3.4 InclusionofDiscontinuities:TMCase.................87
4.4 NumericalLinearAlgebrainMATLAB........................88
4.4.1 SparseMatrices..........................................88
4.4.2 DirectandIterativeEigensolvers.......................89
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4.5 2DWaveguidesandtheYeeMesh ..............................92
4.5.1 YeeMesh ................................................92
4.5.2 DielectricFunctionAveraging..........................95
4.5.3 UseofMirrorSymmetries..............................99
Exercises .............................................................. 102
References............................................................. 106
Chapter5 ModellingofNonlinearPropagationinWaveguides................ 107
5.1 Introduction..................................................... 107
5.2 Formalism ...................................................... 108
5.2.1 GeneralPropagationEquation........................ 108
5.2.2 PulsePowerandPulseEnergy........................ 110
5.3 NonlinearPolarization......................................... 111
5.3.1 NonlinearProcesses................................... 112
5.3.2 χ(3)NonlinearProcesses .............................. 114
5.3.3 Single-ModePropagationModel..................... 115
5.4 NonlinearSchrödingerEquation .............................. 120
5.4.1 DerivationoftheNLSEquation...................... 120
5.4.2 DispersionandSelf-PhaseModulation .............. 122
5.4.3 OpticalSolitons........................................ 124
5.4.4 SolitonsandRamanEffects........................... 125
5.4.5 Self-Steepening........................................ 126
5.4.6 ConservationLaws.................................... 127
5.5 NumericalImplementation .................................... 129
5.5.1 FourierMethod........................................ 129
5.5.2 SteppingTechniques.................................. 130
5.5.3 DiscreteFourierGrids................................. 132
5.5.4 ImplementationinMATLAB......................... 134
Exercises .............................................................. 136
References............................................................. 137
Chapter6 TheModalMethod ................................................... 139
6.1 Introduction..................................................... 139
6.2 Eigenmodes..................................................... 140
6.3 1DGeometry................................................... 142
6.3.1 RecursiveMatrixFormalism.......................... 143
6.3.2 1DInterface............................................ 145
6.3.3 MultilayerStructure................................... 146
6.3.4 1DCavity.............................................. 149
6.4 2DGeometry................................................... 150
6.4.1 Plane-WaveExpansion................................ 151
6.4.1.1 Li’sFactorizationRules.................... 151
6.4.1.2 EigenvalueProblem ........................ 153
6.4.2 Semi-AnalyticalApproach............................ 157
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6.4.3 Interface................................................ 162
6.4.4 SMatrixTheory....................................... 166
6.4.5 AbsorbingBoundaryConditions ..................... 171
6.5 PeriodicStructures............................................. 176
6.5.1 BlochModes........................................... 177
6.5.2 Classification .......................................... 180
6.5.3 Interface................................................ 182
6.5.4 FieldProfileinaPeriodicElement................... 184
6.6 CurrentSources ................................................ 185
6.6.1 UniformLayer......................................... 186
6.6.2 MultilayerGeometry.................................. 188
6.7 3DGeometries ................................................. 190
Exercises .............................................................. 191
References............................................................. 194
Chapter7 Green’sFunctionIntegralEquationMethodsfor
ElectromagneticScatteringProblems ............................... 197
7.1 Introduction..................................................... 197
7.2 TheoreticalFoundation ........................................ 198
7.3 Green’sFunctionAreaIntegralEquationMethod............ 198
7.4 Green’sFunctionVolumeIntegralEquationMethod......... 204
7.5 Green’sFunctionSurfaceIntegralEquation
Method(2D).................................................... 209
7.5.1 SurfaceIntegralEquations............................ 209
7.5.2 CalculatingtheFieldandNormalDerivative
attheBoundary........................................ 212
7.6 Constructionof2DGreen’sFunctionsfor
LayeredStructures ............................................. 218
7.6.1 Plane-WaveExpansionoftheFree-Space
Green’sFunction ...................................... 219
7.6.2 2DTE-PolarizedScalarGreen’sFunctionfora
LayeredStructure...................................... 222
7.6.3 2DTM-PolarizedScalarGreen’sFunctionfora
LayeredStructure...................................... 224
7.6.4 FresnelReflectionandTransmissionCoefficients
foraFewSimpleGeometries......................... 224
7.6.5 CalculatingtheSommerfeldIntegral................. 226
7.6.6 Far-FieldApproximation.............................. 228
7.6.7 ExcitationofBoundWaveguideModes ............. 230
7.7 ConstructionofthePeriodicGreen’sFunction............... 233
7.7.1 1DPeriodicScalarGreen’sFunctionfora
LayeredStructure...................................... 234
7.8 ReflectionfromaPeriodicSurfaceMicrostructure........... 234
7.8.1 CalculatingReflectionandTransmission............ 237
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