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Computational Photonics An Introduction with MATLAB

MAREK S. WARTAK Wilfrid Laurier University

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Preface

1 Introduction 1.1 What is photonics? 1.2 What is computational photonics? 1.3 Optical fibre communication 1.4 Biological and medical photonics 1.5 Photonic sensors 1.6 Silicon photonics 1. 7 Photonic quantum information science References

2 Basicfacts about optics 2.1 Geometrical optics 2.2 Wave optics 2.3 Problems Appendix 2A: MATLAB listings References

3 Basic facts from electromagnetism 3.1 Maxwell's equations

Boundary conditions Wave equation Time-harmonic fields Polarized waves

3.2 3.3 3.4

3.5 3.6 3.7 3.8

3.9 3.10 3.11 3.12

Fresnel coefficients and phases Polarization by reflection from dielectric surfaces Antireflection coating Bragg mirrors Goos-Hanchen shift Poynting theorem Problems

3.13 Project Appendix 3A: MATLAB listings References

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4 Slab waveguides 4.1 Ray optics of the slab waveguide 4.2 Fundamentals of EM theory of dielectric waveguides 4.3 Wave equation for a planar wide waveguide 4.4 Three-layer symmetrical guiding structure (TE modes) 4.5 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D 4.6 Multilayer slab waveguides: lD approach 4.7 Examples: 1D approach 4.8 Two-dimensional (2D) structures 4.9 Problems 4.10 Projects Appendix 4A: MATLAB listings References

5 linear optical fibre and signal degradation 5.1 Geometrical-optics description 5.2 Fibre modes in cylindrical coordinates 5.3 Dispersion 5.4 Pulse dispersion during propagation 5.5 Problems 5.6 Projects Appendix SA: Some properties of Bessel functions Appendix SB: Characteristic determinant Appendix SC: MATLAB listings References

6 Propagation of linear pulses 6.1 Basic pulses 6.2 Modulation of a semiconductor laser 6.3 Simple derivation of the pulse propagation equation in the presence of

dispersion 6.4 Mathematical theory of linear pulses 6.5 Propagation of pulses 6.6 Problems Appendix 6A: MATLAB listings References

7 Optical sources 7.1 Overview of lasers 7.2 Semiconductor lasers 7.3 Rate equations 7.4 Analysis based on rate equations 7.5 Problems 7.6 Project

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106 106 110 123 127 128 129 129 130 131 137

138 138 143

146 148 152 155 156 165

167 167 172 181 187 196 196

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64 Appendix 7 A: MATLAB listings 196 64 References 202

69

71 8 Optical amplifiers and EDFA 204

72 8.1 General properties 205 le in 1D 75 8.2 Erbium-doped fibre amplifiers (EDFA) 209

79 8.3 Gain characteristics of erbium-doped fibre amplifiers 213

85 8.4 Problems 215 88 8.5 Projects 215 92 Appendix 8A: MATLAB listings . 215 92 References 222 93

104 9 Semiconductor optical amplifiers (SOA) 223 9.1 General discussion 223

106 9.2 SOA rate equations for pulse propagation 228 106 9.3 Design of SOA 231 110 9.4 Some applications of SOA 233 123 9.5 Problem 236 127 9.6 Project 236 128 Appendix 9A: MATLAB listings 236 129 References 238 129

130 10 Optical receivers 240 131 10.1 Main characteristics 241 137 10.2 Photodetectors 242

10.3 Receiver analysis 251 138 10.4 Modelling of a photoelectric receiver 257 138 10.5 Problems 258 143 10.6 Projects 258

ence of Appendix lOA: MATLAB listings 258 146 References 260 148

152 11 Finite difference time domain (FDTD) formulation 262 155 11.1 General formulation 262 156 11.2 One-dimensional Yee implementation without dispersion 266 165 11.3 Boundary conditions in I D 272

11.4 Two-dimensional Yee implementation without dispersion 275 167 11.5 Absorbing boundary conditions (ABC) in 2D 277 167 11.6 Dispersion 280 172 11.7 Problems 280 181 11.8 Projects 281 187 Appendix 11A: MATLAB listings 281 196 References 286 196

viii Contents ix

12 Beam propagation method (BPM) 288 12.1 Paraxial formulation 288 I2.2 General theory 292 I2.3 The I + I dimensional FD-BPM formulation 299 12.4 Concluding remarks 306 12.5 Problems 307 12.6 Project 308 Appendix 12A: Details of derivation of the FD-BPM equation 308 Appendix 12B: MATLAB listings 3IO References 3I4

13 Some wavelength division multiplexing (WDM) devices 3I6 13.I Basics of WDM systems 3I6 13.2 Basic WDM technologies 3I7 13.3 Applications ofBPM to photonic devices 323 13.4 Projects 325 Appendix 13A: MATLAB listings 325 References 329

14 Opticallink 331 14.1 Optical communication system 331 I4.2 Design of optical link 333 I4.3 Measures of link performance 336 14.4 Optical fibre as a linear system 338 14.5 Model of optical link based on filter functions 340 14.6 Problems 344 14.7 Projects 344 Appendix I4A: MATLAB listings 345 References 348

~5 Optical solitons 35I 15.1 Nonlinear optical susceptibility 351 15.2 Main nonlinear effects 352 I5.3 Derivation of the nonlinear Schrodinger equation 353 I5.4 Split-step Fourier method 357 I5.5 Numerical results 361 15.6 A few comments about soliton-based communications 364 15.7 Problems 364 Appendix 15A: MATLAB listings 365 References 366

16 Solar cells 368 16.1 Introduction 368 I6.2 Principles of photovoltaics 370

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288 16.3 Equivalent circuit of solar cells 373

288 16.4 Multijunctions 376

292 Appendix 16A: MATLAB listings 379

299 References 381

306 307 17 Metamaterials 384

308 17.1 Introduction 384

308 17.2 Veselago approach 388

310 17.3 How to create metamaterial? 389

314 17.4 Some applications of me!amaterials 395 17.5 Metamaterials with an active element 400

316 17.6 Annotated bibliography 401

316 Appendix 17 A: MATLAB listings 401

317 References 403

323 325 Appendix A Basic MATLAB 406 325 A.1 Working session with m-files 407

329 A.2 Basic rules 409 A.3 Some rules about good programming in MATLAB 410

331 A.4 Basic graphics 412 331 A.5 Basic input-output 416 333 A.6 Numerical differentiation 417 336 A.7 Review questions 418 338 References 418 340 344 Appendix B Summary of basic numerical methods 420 344 B.1 One-variable Newton's method 420 345 B.2 Muller's method 422 348 B.3 Numerical differentiation 425

B.4 Runge-Kutta (RK) methods 432 351 B.5 Solving differential equations 433 351 B.6 Numerical integration 435 352 B.7 Symbolic integration in MATLAB 438 353 B.8 Fourier series 438 357 B.9 Fourier transform 441 361 B.10 FFT in MATLAB 443 364 B.11 Problems 446 364 References 446 365 366 Index 448

368 368 370