Computational Photonics: An Introduction with MATLAB

Contents......Page 7
Preface......Page 13
1.1 What is photonics?......Page 17
1.2.1 Methods of computational photonics. Computational electromagnetics......Page 18
1.2.2 Computational nano-photonics......Page 19
1.2.3 Overview of commercial software for photonics......Page 20
1.3.2 Short history of communication......Page 21
1.3.3 Development of optical fibre......Page 24
1.3.4 Comparison with electrical transmission......Page 25
1.3.5 Governing standards......Page 26
1.3.6 Wavelength division multiplexing (WDM)......Page 27
1.5 Photonic sensors......Page 28
1.6 Silicon photonics......Page 29
References......Page 30
2.1.1 Ray theory and applications......Page 33
2.1.2 Critical angle......Page 34
2.1.3 Lenses......Page 35
2.1.4 GRIN systems......Page 36
2.2 Wave optics......Page 37
2.2.1 Phase velocity......Page 38
2.2.2 Group velocity......Page 39
2.2.3 Stokes relations......Page 40
2.2.4 Interference in dielectric film......Page 42
2.2.5 Multiple interference in a parallel plate......Page 43
2.2.6 Fabry-Perot (FP) interferometer......Page 45
2.3 Problems......Page 46
Appendix 2A: MATLAB listings......Page 47
References......Page 50
3.1 Maxwell's equations......Page 51
3.2.1 Electric boundary conditions......Page 52
3.2.2 Magnetic boundary conditions......Page 53
3.3 Wave equation......Page 54
3.4 Time-harmonic fields......Page 55
3.5.2 Circularly and elliptically polarized waves......Page 58
3.6.1 TE polarization......Page 60
3.6.2 TM polarization......Page 63
3.7 Polarization by reflection from dielectric surfaces......Page 64
3.8 Antireflection coating......Page 66
3.8.1 Transfer matrix approach......Page 67
3.9 Bragg mirrors......Page 69
3.10 Goos-Hanchen shift......Page 74
3.11 Poynting theorem......Page 75
3.13 Project......Page 76
Appendix 3A: MATLAB listings......Page 77
References......Page 79
4.1.1 Numerical aperture......Page 80
4.1.2 Guided modes......Page 81
4.1.3 Transverse resonance condition......Page 82
4.1.4 Transverse condition: normalized form......Page 83
4.2.1 General discussion......Page 85
4.2.2 Explicit form of general equations......Page 86
4.3 Wave equation for a planar wide waveguide......Page 87
4.4 Three-layer symmetrical guiding structure (TE modes)......Page 88
4.5 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D......Page 91
4.5.1 TE modes......Page 92
4.5.2 Field profiles for TE modes......Page 93
4.6 Multilayer slab waveguides: 1D approach......Page 95
4.6.1 TE mode......Page 97
4.6.2 Propagation constant......Page 99
4.6.3 Electric field......Page 100
4.7 Examples: 1D approach......Page 101
4.7.2 Six-layer lossy waveguide......Page 102
4.7.3 Structure by Visser......Page 103
4.8 Two-dimensional (2D) structures......Page 104
4.8.1 Effective index method......Page 105
4.10 Projects......Page 108
Appendix 4A: MATLAB listings......Page 109
References......Page 120
5.1.1 Numerical aperture (NA)......Page 122
5.1.4 Loss mechanisms in silica fibre......Page 124
5.1.5 Intrinsic loss......Page 125
5.2 Fibre modes in cylindrical coordinates......Page 126
5.2.1 Maxwell's equations in cylindrical coordinates......Page 127
5.2.3 Solution of the wave equation in cylindrical coordinates......Page 128
5.2.4 Boundary conditions and modal equation......Page 131
5.2.5 Mode classification......Page 132
5.2.6 Modes with m = 0......Page 133
5.2.7 Weakly guiding approximation (wga)......Page 134
5.2.8 The unified expression......Page 136
5.2.9 Universal relation for fundamental mode HE11......Page 137
5.2.10 Single-mode fibres......Page 138
5.3 Dispersion......Page 139
5.3.1 Group delay-general discussion......Page 140
5.3.2 Material dispersion: Sellmeier equation......Page 141
5.3.3 Waveguide dispersion......Page 142
5.4 Pulse dispersion during propagation......Page 143
5.5 Problems......Page 144
Appendix 5A: Some properties of Bessel functions......Page 145
Appendix 5B: Characteristic determinant......Page 146
Appendix 5C: MATLAB listings......Page 147
References......Page 153
6.1.1 Rectangular pulses......Page 154
6.1.2 Gaussian pulses......Page 155
6.1.3 Super-Gaussian pulse......Page 156
6.1.4 Chirped Gaussian pulse......Page 157
6.2.1 Modulation formats......Page 159
6.2.2 Creation of waveforms......Page 161
6.3 Simple derivation of the pulse propagation equation in the presence of dispersion......Page 162
6.4 Mathematical theory of linear pulses......Page 164
6.4.1 One-dimensional approach......Page 166
6.5.1 Analytical description of the propagation of a chirp Gaussian pulse......Page 168
6.5.2 Numerical method using Fourier transform......Page 169
6.5.3 Fourier transform split-step method......Page 170
6.6 Problems......Page 171
Appendix 6A: MATLAB listings......Page 172
References......Page 181
7.1 Overview of lasers......Page 183
7.1.1 Transitions in a TLS......Page 185
7.1.2 Laser oscillations and resonant modes......Page 186
7.2 Semiconductor lasers......Page 188
7.2.1 Electron transitions in semiconductors......Page 190
7.2.2 Homogeneous p-n junction......Page 191
7.2.3 Heterostructures......Page 192
7.2.4 Optical gain......Page 193
7.2.5 Determination of optical gain......Page 196
7.3 Rate equations......Page 197
7.3.1 Carriers......Page 198
7.3.2 Photons......Page 199
7.3.4 Derivation of rate equation for electric field......Page 200
7.4.1 Steady-state analysis......Page 203
7.4.2 Small-signal analysis with the linear gain model......Page 204
7.4.3 Small-signal analysis with gain saturation......Page 205
7.4.5 Frequency chirping......Page 208
7.4.6 Equivalent circuit models......Page 209
7.4.7 Equivalent circuit model for a bulk laser......Page 210
Appendix 7A: MATLAB listings......Page 212
References......Page 218
8 Optical amplifiers and EDFA......Page 220
8.1 General properties......Page 221
8.1.1 Gain spectrum and bandwidth......Page 222
8.1.2 Gain saturation......Page 223
8.1.3 Amplifier noise......Page 224
8.2 Erbium-doped fibre amplifiers (EDFA)......Page 225
8.2.1 Steady-state analysis......Page 227
8.2.2 Effective two-level approach......Page 228
8.3.1 Typical EDFA characteristics......Page 229
Appendix 8A: MATLAB listings......Page 231
References......Page 238
9.1 General discussion......Page 239
9.1.1 Gain formula for SOA with facet reflectivities......Page 241
9.1.2 The effect of facet reflectivities......Page 243
9.2 SOA rate equations for pulse propagation......Page 244
9.3 Design of SOA......Page 247
9.4.1 Wavelength conversion......Page 249
9.4.2 All-optical logic based on interferometric principles......Page 251
Appendix 9A: MATLAB listings......Page 252
References......Page 254
10 Optical receivers......Page 256
10.1.3 Bit-rate transparency......Page 257
10.2.1 Principles of photo detection......Page 258
10.2.2 Performance parameters of photodetectors......Page 262
10.2.3 Photodetector noise......Page 264
10.2.4 Detector design......Page 266
10.3 Receiver analysis......Page 267
10.3.1 BER of an ideal optical receiver......Page 268
10.3.2 Error probability in the receiver......Page 269
10.3.3 BER and Gaussian noise......Page 271
10.4 Modelling of a photoelectric receiver......Page 273
Appendix 10A: MATLAB listings......Page 274
References......Page 276
11.1 General formulation......Page 278
11.1.2 Two-dimensional formulation......Page 279
11.1.3 One-dimensional model......Page 280
11.1.4 Gaussian pulse and modulated Gaussian pulse......Page 281
11.2.1 Lossless case......Page 282
11.2.3 Dispersion and stability......Page 285
11.2.4 Stability criterion......Page 286
11.2.5 One-dimensional model with losses......Page 287
11.3.1 Mur's first-order absorbing boundary conditions (ABC)......Page 288
11.3.2 Second-order boundary conditions in 1D......Page 289
11.4 Two-dimensional Yee implementation without dispersion......Page 291
11.5 Absorbing boundary conditions (ABC) in 2D......Page 293
11.7 Problems......Page 296
Appendix 11A: MATLAB listings......Page 297
References......Page 302
12.1.1 Introduction......Page 304
12.1.2 Operators ^D and ^W......Page 306
12.1.3 The implementation using the Fourier transform split-step method......Page 307
12.2.1 Introduction......Page 308
12.2.2 Slowly varying envelope approximation (SVEA)......Page 312
12.2.4 Scalar formulation......Page 313
12.2.5 Finite-difference (FD) approximations......Page 314
12.3.1 Simple approach......Page 315
12.3.2 Propagator approach......Page 316
12.3.3 Transparent boundary conditions......Page 320
12.4 Concluding remarks......Page 322
12.5 Problems......Page 323
Appendix 12A: Details of derivation of the FD-BPM equation......Page 324
Appendix 12B: MATLAB listings......Page 326
References......Page 330
13.1 Basics of WDM systems......Page 332
13.2 Basic WDM technologies......Page 333
13.2.2 Array waveguide grating......Page 334
13.2.3 Couplers and splitters......Page 335
13.2.4 Mathematical theory of a passive coupler......Page 336
13.2.5 Optical isolators......Page 338
13.3 Applications of BPM to photonic devices......Page 339
Appendix 13A: MATLAB listings......Page 341
References......Page 345
14.1 Optical communication system......Page 347
14.2 Design of optical link......Page 349
14.2.2 Rise time budget......Page 350
14.3 Measures of link performance......Page 352
14.3.1 Eye diagram......Page 353
14.4 Optical fibre as a linear system......Page 354
14.5.1 Test analysis for a rectangular pulse......Page 356
14.5.3 Fibre......Page 358
14.5.5 Implementation of link model......Page 359
14.7 Projects......Page 360
Appendix 14A: MATLAB listings......Page 361
References......Page 364
15.1 Nonlinear optical susceptibility......Page 367
15.2.1 Kerr effect......Page 368
15.3 Derivation of the nonlinear Schrodinger equation......Page 369
15.4 Split-step Fourier method......Page 373
15.4.1 Split-step Fourier transform method......Page 375
15.4.2 Symmetrized split-step Fourier transform method (SSSFM)......Page 376
15.5.1 Single solitons......Page 377
15.5.2 Chirped solitons......Page 378
15.5.3 Two interacting solitons......Page 379
15.7 Problems......Page 380
Appendix 15A: MATLAB listings......Page 381
References......Page 382
16.1 Introduction......Page 384
16.2 Principles of photovoltaics......Page 386
16.3.1 Basic model......Page 389
16.3.2 Other models......Page 390
16.4 Multijunctions......Page 392
16.4.1 Quantum dots in multijunctions......Page 393
16.4.3 Role of simulations......Page 394
Appendix 16A: MATLAB listings......Page 395
References......Page 397
17.1 Introduction......Page 400
17.1.1 Short history of MM......Page 401
17.2.2 Left-handed materials......Page 404
17.3.1 Metamaterials with negative effective permittivity in the microwave regime......Page 405
17.3.2 Magnetic properties: split-ring resonators......Page 407
17.4 Some applications of metamaterials......Page 411
17.4.2 Stopped light in metamaterials......Page 412
17.4.3 Cloaking (invisibility)......Page 414
17.4.4 Optical black holes......Page 415
17.5 Metamaterials with an active element......Page 416
Appendix 17A: MATLAB listings......Page 417
References......Page 419
Appendix A Basic MATLAB......Page 422
A.1 Working session with m-files......Page 423
A.2 Basic rules......Page 425
A.3.1 Preallocate memory......Page 426
A.3.2 Vectorize loops......Page 427
A.4.2 Two-dimensional plots......Page 428
A.4.3 Some 3D plots......Page 431
A.5.2 Reading from a text file......Page 432
A.6 Numerical differentiation......Page 433
References......Page 434
B.1 One-variable Newton's method......Page 436
B.2 Muller's method......Page 438
B.2.1 Tests of Muller's method......Page 440
B.3 Numerical differentiation......Page 441
B.3.1 Numerical differentiation using Taylor's series expansion......Page 442
B.3.2 Numerical differentiation using interpolating polynomials......Page 443
B.3.4 Simple methods of numerical differentiation......Page 445
B.4.1 Second-order Runge-Kutta......Page 448
B.5.1 Single differential equation......Page 449
B.5.2 System of differential equations......Page 450
B.6 Numerical integration......Page 451
B.6.1 Euler's rule......Page 452
B.6.3 Simpson's rule......Page 453
B.8 Fourier series......Page 454
B.8.1 Change of interval......Page 455
B.8.2 Example......Page 456
B.9 Fourier transform......Page 457
B.10 FFT in MATLAB......Page 459
References......Page 462
Index......Page 464