Preface ............................................................ iii 1 Optical Propagation in Linear Media ............................... 1 1.1 Maxwell's Equations ........................................... 1 1.2 Linear Wave Propagation in Homogeneous, Linear Isotropic Media ......................................................... 4 1.3 Wave Propagation in Anisotropic Media ........................ 12 1.4 Diffraction .................................................. 25 1.5 Dispersion ................................................... 31 1.6 Problems ..................................................... 34 References ................................................... 35 2 The Origin and Modeling of Optical Nonlinearity .................. 37 2.1 A Simple Physical Model for Optical Nonlinearity ............. 38 2.2 Physical Effects of Nonlinear Polarization ................... 42 2.3 Mathematical Modeling of Optical Nonlinearities .............. 43 2.4 An Alternative Approach ...................................... 54 2.5 Summary ...................................................... 56 2.6 Problems ..................................................... 57 References ................................................... 57 3 Second Harmonic Generation ....................................... 59 3.1 The Physics of Second Harmonic Generation .................... 59 3.2 SHG in Crystals—Mathematical Formulation ..................... 62 3.3 Phase Matching in Anisotropic Crystal ........................ 70 3.4 Nonlinear Transverse Effects in Second Harmonic Generation ... 71 3.5 SHG in a Medium with Second-Order and Third-Order Nonlinear Susceptibilities ....................... 73 3.6 Summary ...................................................... 76 3.7 Problems ..................................................... 77 References ................................................... 77 4 Self-Refraction of Optical Beams ................................. 79 4.1 The Nonlinear Refractive Index Coefficient n2 ................ 80 4.2 Self-Refraction of Gaussian Beams ............................ 81 4.3 Focused Gaussian Beam in a Kerr-Slice: The z-Scan Method ..... 90 4.4 Self-Bending of Optical Beams ................................ 96 4.5 Self-Phase Modulation Due to χ(2): Effective χ(3) .............. 97 4.6 Summary ...................................................... 98 4.7 Problems ..................................................... 99 References .................................................. 100 5 Optical Bistability ............................................. 101 5.1 The Nonlinear Ring Cavity ................................... 101 5.2 Transverse Optical Bistability .............................. 109 5.3 Linear-Nonlinear Interface .................................. 110 5.4 Conclusion .................................................. 119 5.5 Problems .................................................... 120 References .................................................. 120 6 Optical Phase Conjugation ....................................... 123 6.1 Comparison with Holography .................................. 124 6.2 The k-Space Formalism ....................................... 125 6.3 Semiclassical Analysis: CW Case ............................. 132 6.4 Phase Conjugation of Pulses and Transient Phase Conjugate Response ........................................... 137 6.5 Discussion .................................................. 142 6.6 Problems .................................................... 142 References .................................................. 144 7 Stimulated Brillouin and Raman Scattering ....................... 145 7.1 Stimulated Brillouin Scattering ............................. 145 7.2 Stimulated Raman Scattering ................................. 151 7.3 Problems .................................................... 153 References .................................................. 154 8 Solitons in Optical Fibers ...................................... 155 8.1 Background on Linear Optical Fibers ......................... 156 8.2 Fiber Dispersion and Nonlinearity ........................... 158 8.3 Fiber-Optic Solitons and the NLS Equation ................... 161 8.4 Dispersion Managed Soliton Communication Systems ............ 164 8.5 Dark Solitons ............................................... 166 8.6 Optical Shocks and Self-Steepemiag of Pulses ................ 171 8.7 Problems .................................................... 173 References .................................................. 173 9 Photorefractive Nonlinear Optics ................................ 175 9.1 Introduction ................................................ 175 9.2 The Kukhtarev Equations and Their Simplification ............ 177 9.3 Beam Fanning and Distortion in Photorefractive Materials .... 181 9.4 Two-Wave Mixing in Photorefractive Materials ................ 194 9.5 Four-Wave Mixing and Phase Conjugation in Photorefractive Materials ................................................... 196 9.6 Self-Phase Conjugation and Edge Enhancement ................. 200 9.7 Organic Photorefractive Materials ........................... 202 9.8 Problems .................................................... 206 References .................................................. 207 10 Nonlinear Optical Properties of Nematic Liquid Crystals ......... 209 10.1 The Liquid Crystalline State of Matter ..................... 210 10.2 Classification of Liquid Crystals .......................... 210 10.3 Liquid Crystal Alignment ................................... 211 10.4 Principles of the Continuum Theory ......................... 213 10.5 Director Distribution of Homogeneously Aligned Nematic Liquid Crystal Under an External Electrical Field .......... 215 10.6 Nonlinear Optical Properties from Optically Induced Molecular Reorientation .................................... 224 10.7 Optically Induced Reorientational Nonlinearity with External Voltage ...................................... 228 10.8 Analysis of Beam Propagation in Liquid Crystals and z-Scan ................................................. 235 10.9 Problems ................................................... 240 References ................................................. 241 11 Self-Organization in Nonlinear Optical Systems .................. 243 11.1 Introduction ............................................... 243 11.2 Nonlinear Two-Dimensional Systems with Feedback ............ 245 11.3 Self-Organization in Photorefractive Materials ............. 249 11.4 Theory of Self-Organization in Photorefractive Materials ... 255 11.5 Instability Criterion and the Dispersion Relation .......... 257 11.6 Nonlinear Eigenmodes in the Steady State ................... 259 11.7 Model of Hexagonal Formation Based on Transverse Electrical Instability ..................................... 264 11.8 Potential Applications ..................................... 265 11.9 Problems ................................................... 267 References ................................................. 268 12 Nonlinear Optics of Photonic Bandgap Structures ................. 271 12.1 Introduction ............................................... 271 12.2 The Linear Dispersion Relation ............................. 272 12.3 The Effect of Cubic Nonlinearity ........................... 276 12.4 Second Harmonic Generation in Photonic Bandgap Structures ................................................. 282 12.5 Problems ................................................... 287 References ................................................. 288 Appendix A The Split Step Beam Propagation Method ................. 289 A.1 Examples ........................................... 290 References ......................................... 294 Appendix B Wavelet Transforms and Application to Solution of Partial Differential Equations ...................... 295 B.1 Introduction to Wavelets ........................... 295 B.2 Wavelet Properties and Scaling Functions ........... 296 B.3 Digital Filters and Multi-Resolution Analysis ...... 298 B.4 Application of Wavelet Transform to the NLS Equation ........................................... 302 References ......................................... 306 Index .............................................................. 307