Computational Photonics

COMPUTATIONAL\nPHOTONICS......Page 5
Contents......Page 9
Preface......Page 15
Acknowledgements......Page 17
1.1 Photonics: The Countless Possibilities of Light Propagation......Page 19
1.2.1 History of Computational Modelling......Page 21
1.2.3 Time Domain......Page 22
1.2.4 Chapter Overview......Page 23
1.2.5 Overview of Commercial Software for Photonics......Page 24
2.2 Overview of the Beam Propagation Methods......Page 27
2.4 Magnetic-Field Formulation of the Wave Equation......Page 29
2.5 Electric-Field Formulation of the Wave Equation......Page 30
2.6 Perfectly Matched Layer......Page 31
2.7.1 Types of Elements......Page 33
2.7.3 Galerkin’s Procedure......Page 34
2.8.1 Slowly Varying Envelope Approximation......Page 35
2.8.3 Discretisation by the Finite-Element Method......Page 37
2.9 Imaginary-Distance BPM: Mode Solver......Page 40
References......Page 41
3.2.1 Effect of Longitudinal Step Size......Page 43
3.2.2 Analysis of Rectangular Directional Coupler......Page 46
3.2.3 Effect of Structure Geometrical Parameters......Page 47
3.2.4 FV-BPM Versus Analytical Marcatili’s Approach......Page 48
3.3 Photonic Crystal Fibre......Page 51
3.3.1 Effective Index of Modes......Page 52
3.3.2 Losses......Page 53
3.3.3 Effective Mode Area......Page 55
3.3.4 Dispersion......Page 59
3.4.1 Design......Page 60
3.4.2 Modal Hybridness......Page 64
3.4.4 Dispersion......Page 65
3.4.5 Tunable Liquid-Crystal-Based Photonic Crystal Fibre Polarisation Converter......Page 67
3.5.1 Design of the Electro-Optical Modulator......Page 68
3.5.2 The Effect of the Core Height......Page 71
3.5.3 The Effect of the Electrode Width......Page 73
3.5.4 The Effect of the Buffer Thickness......Page 74
3.5.5 The Effect of the Al Concentration of the Buffer Layer......Page 78
3.6 Switches......Page 79
References......Page 81
4.2 Optical Waveguide Discontinuity Problem......Page 85
4.3 Finite-Element Analysis of Discontinuity Problems......Page 86
4.4 Derivation of Finite-Element Matrices......Page 88
4.5 Application of Taylor’s Series Expansion......Page 89
4.6.1 Junction Between Two Optical Waveguides......Page 90
4.6.2 Laser–Air Facet......Page 93
4.6.3 Laser–Air Facet with a Tilt Angle......Page 95
4.7 Optical Fibre Facet Problem......Page 98
4.8.1 Formulation......Page 99
4.8.2 Derivation of Finite-Element Matrices......Page 101
4.8.3 Application of Taylor’s Series Expansion......Page 102
4.9 Iterative Analysis of Multiple Discontinuities......Page 103
4.10.1 Uncoated Fibre Facet......Page 104
4.10.2 Single Antireflection Layer......Page 106
4.10.3 Slanted Interfaces......Page 110
References......Page 111
5.1 Introduction......Page 113
5.3 Brief History of the Finite-Difference Time-Domain (FDTD) Method......Page 114
5.4 Finite-Difference Time-Domain (FDTD) Method......Page 115
5.4.1 Discretisation in Space and Time: The Yee’s Algorithm and the Leapfrog Scheme......Page 116
5.4.2 Finite-Difference Notation of Maxwell’s Equations......Page 118
5.4.3 Numerical Stability......Page 121
5.4.4 Numerical Dispersion......Page 122
5.5 Alternating-Direction-Implicit FDTD (ADI-FDTD): Beyond the Courant Limit......Page 125
5.6 Complex-Envelope ADI-FDTD (CE-ADI-FDTD)......Page 128
5.7 Perfectly Matched Layer (PML) Boundary Conditions......Page 133
5.8 Uniaxial Perfectly Matched Layer (UPML) Absorbing Boundary Condition......Page 139
5.9 PML Parameters......Page 141
5.10 PML Boundary Conditions for CE-ADI-FDTD......Page 143
5.11 PhC Resonant Cavities......Page 146
5.12 5 × 5 Rectangular Lattice PhC Cavity......Page 149
5.13 Triangular Lattice PhC Cavity......Page 151
5.14 Wavelength Division Multiplexing......Page 156
References......Page 158
6.1 Introduction......Page 161
6.2 Numerical Analysis......Page 162
6.3 The UPWIND Scheme for Flux Calculation......Page 166
6.3.1 Dielectric Contrast......Page 169
6.3.2 Perfect Electric Conductor......Page 170
6.3.3 Perfect Magnetic Conductor......Page 171
6.4 Nondiffusive Scheme for the Flux Calculation......Page 172
6.5 2D Formulation of the FVTD Method......Page 174
6.5.1 Numerical Stability......Page 177
6.6.1 Silver–Müller Boundary Conditions......Page 178
6.6.2 PML Boundary Conditions......Page 179
6.7 Nonlinear Optics......Page 181
6.8 Nonlinear Optical Interactions......Page 182
6.8.2 Third Harmonic Generation......Page 183
6.8.3 Intensity-Dependent Refractive Index......Page 185
6.9 Extension of the FDTD Method to Nonlinear Problems......Page 186
6.10 Extension of the FVTD Method to Nonlinear Problems......Page 188
References......Page 191
7.2 FVTD Method Assessment: PhC Cavity......Page 193
7.2.1 UPML Effectiveness......Page 198
7.3 FVTD Method Assessment: PhC Waveguide......Page 202
7.4 FVTD Method Assessment: PBG T-Branch......Page 204
7.5 PhC Multimode Resonant Cavity......Page 205
7.6 FDTD Analysis of Nonlinear Devices......Page 209
7.7 FVTD Analysis of Nonlinear Photonic Crystal Wires......Page 221
References......Page 229
8.1 Introduction......Page 231
8.2.1 Multiresolution Analysis: Overview......Page 232
8.2.3 Method of Moments......Page 234
8.3.1 Approximation in Time: Testing with Pulse Functions......Page 236
8.3.2 Approximation in Space: Testing with Wavelet/Scaling Functions......Page 237
8.3.3 Media Discretisation......Page 240
8.3.4 Numerical Dispersion......Page 242
8.3.5 Numerical Stability......Page 243
8.4.1 Choice of Basis Functions: Cohen-Daubechies-Feauveau Family......Page 244
8.4.2 Derivation of Update Scheme......Page 246
8.4.3 UPML in S-MRTD......Page 250
References......Page 251
9.2.1 UPML in Planar Waveguide......Page 253
9.2.2 UPML in PhC Waveguide......Page 256
9.3.1 Bragg Resonators: Comparison with Analytical Approach......Page 257
9.3.2 PhC-Based Optical Filter......Page 262
References......Page 266
10.1 Introduction......Page 267
10.2.2 Nonlinear Polarisation Vector......Page 268
10.2.3 Physics of Second Harmonic Generation......Page 271
10.2.4 Phase Matching Through Effective Index for Periodic Structures......Page 274
10.3.1 TE/TE Coupling......Page 279
10.3.2 TE/TM Coupling (χ(2) Tensor)......Page 281
10.4.1 PBG Property: Calculating the PhC Dispersion Diagram......Page 283
10.4.2 Propagation Properties and SHG Efficiency......Page 284
10.5.1 Introduction......Page 287
10.5.2 Selective SHG: Coupling PhC Waveguides Through a Tapered Section......Page 288
10.6.1 Introduction......Page 289
10.6.2 Stopband Properties......Page 290
10.6.3 Coupling Waveguides and Cavities......Page 293
10.6.4 Enhancement of Efficiency and Selectivity Through Selective PhC Microcavities......Page 295
10.6.5 PhC-Microcavity Design......Page 296
10.6.6 Simulation Results......Page 298
References......Page 302
11.1 Introduction......Page 305
11.2.2 Dispersive Materials: Lorentz Model......Page 306
11.2.3 Auxiliary Differential Equations......Page 307
11.3.1 TE/TE Coupling......Page 309
11.3.2 TE/TM Coupling (χ(2) Tensor)......Page 310
11.4.1 SHG in a Planar Waveguide......Page 311
11.4.2 SHG in 1D Periodic Structure......Page 315
References......Page 318
Index......Page 321