Nonlinear Optics

GEORGE I. STEGEMAN, PhD, is Chair Professor in the College of Engineering at KFUPM, Saudi Arabia, and Emeritus Professor at the College of Optics and Photonics (CREOL) of the University of Central Florida (UCF). He is the first recipient of the Cobb Family Eminent Chair in Optical Sciences and Engineering at UCF. Dr. Stegeman is a Fellow of the Optical Society of America and has received the Canadian Association of Physicists's Herzberg Medal for achievement in physics and the Optical Society of America's R.W. Wood Prize. ROBERT A. STEGEMAN, PhD, has held professional positions at the College of Optical Sciences at The University of Arizona, as well as various industrial companies.

Preface xi

1. Introduction 1

1.1 What is Nonlinear Optics and What is it Good for? 1

1.2 Notation 2

1.3 Classical Nonlinear Optics Expansion 4

1.4 Simple Model: Electron on a Spring and its Application to Linear Optics 6

1.5 Local Field Correction 10

Suggested Further Reading 13

Part A: Second-order Phenomena 15

2. Second-Order Susceptibility and Nonlinear Coupled Wave Equations 17

2.1 Anharmonic Oscillator Derivation of Second-Order Susceptibilities 18

2.2 Input Eigenmodes, Permutation Symmetry, and Properties of χ (2) 23

2.3 Slowly Varying Envelope Approximation 25

2.4 Coupled Wave Equations 26

2.5 Manley–Rowe Relations and Energy Conservation 31

Suggested Further Reading 38

3. Optimization and Limitations of Second-Order Parametric Processes 39

3.1 Wave-Vector Matching 39

3.2 Optimizing d(2)eff 53

3.3 Numerical Examples 59

References 67

Suggested Further Reading 67

4. Solutions for Plane-Wave Parametric Conversion Processes 69

4.1 Solutions of the Type 1 SHG Coupled Wave Equations 69

4.2 Solutions of the Three-Wave Coupled Equations 77

4.3 Characteristic Lengths 80

4.4 Nonlinear Modes 81

References 84

Suggested Further Reading 85

5. Second Harmonic Generation with Finite Beams and Applications 86

5.1 SHG with Gaussian Beams 86

5.2 Unique and Performance-Enhanced Applications of Periodically Poled LiNbO3 (PPLN) 98

References 107

Suggested Further Reading 107

6. Three-Wave Mixing, Optical Amplifiers, and Generators 108

6.1 Three-Wave Mixing Processes 108

6.2 Manley–Rowe Relations 110

6.3 Sum Frequency Generation 111

6.4 Optical Parametric Amplifiers 113

6.5 Optical Parametric Oscillator 119

6.6 Mid-Infrared Quasi-Phase Matching Parametric Devices 128

References 139

Selected Further Reading 140

7. χ (2) Materials and Their Characterization 141

7.1 Survey of Materials 141

7.2 Oxide-Based Dielectric Crystals 143

7.3 Organic Materials 144

7.4 Measurement Techniques 149

Appendix 7.1: Quantum Mechanical Model for Charge Transfer Molecular Nonlinearities 153

References 157

Suggested Further Reading 158

Part B: Nonlinear Susceptibilities 159

8. Second- and Third-Order Susceptibilities: Quantum Mechanical Formulation 161

8.1 Perturbation Theory of Field Interaction with Molecules 162

8.2 Optical Susceptibilities 169

Appendix 8.1: χ (3)ijk‘

Symmetry Properties for Different Crystal Classes 192

Reference 196

Suggested Further Reading 196

9. Molecular Nonlinear Optics 197

9.1 Two-Level Model 198

9.2 Symmetric Molecules 210

9.3 Density Matrix Formalism 215

Appendix 9.1: Two-Level Model for Asymmetric Molecules—Exact Solution 216

Appendix 9.2: Three-Level Model for Symmetric Molecules—Exact Solution 218

References 222

Suggested Further Reading 223

Part C: Third-order Phenomena 225

10. Kerr Nonlinear Absorption and Refraction 227

10.1 Nonlinear Absorption 228

10.2 Nonlinear Refraction 238

10.3 Useful NLR Formulas and Examples (Isotropic Media) 243

Suggested Further Reading 250

11. Condensed Matter Third-Order Nonlinearities due to Electronic Transitions 251

11.1 Device-Based Nonlinear Material Figures of Merit 252

11.2 Local Versus Nonlocal Nonlinearities in Space and Time 253

11.3 Survey of Nonlinear Refraction and Absorption Measurements 255

11.4 Electronic Nonlinearities Involving Discrete States 256

11.5 Overview of Semiconductor Nonlinearities 266

11.6 Glass Nonlinearities 281

Appendix 11.1: Expressions for the Kerr, Raman, and Quadratic Stark Effects 284

References 286

Suggested Further Reading 289

12. Miscellaneous Third-Order Nonlinearities 290

12.1 Molecular Reorientation Effects in Liquids and Liquid Crystals 291

12.2 Photorefractive Nonlinearities 300

12.3 Nuclear (Vibrational) Contributions to n2|| (-ω; ω) 306

12.4 Electrostriction 310

12.5 Thermo-Optic Effect 312

12.6 χ(3) via Cascaded χ(2) Nonlinear Processes: Nonlocal 314

Appendix 12.1: Spontaneous Raman Scattering 317

References 328

Suggested Further Reading 329

13. Techniques for Measuring Third-Order Nonlinearities 330

13.1 Z-Scan 332

13.2 Third Harmonic Generation 339

13.3 Optical Kerr Effect Measurements 343

13.4 Nonlinear Optical Interferometry 344

13.5 Degenerate Four-Wave Mixing 345

References 346

Suggested Further Reading 346

14. Ramifications and Applications of Nonlinear Refraction 347

14.1 Self-Focusing and Defocusing of Beams 348

14.2 Self-Phase Modulation and Spectral Broadening in Time 352

14.3 Instabilities 354

14.4 Solitons (Nonlinear Modes) 363

14.5 Optical Bistability 372

14.6 All-Optical Signal Processing and Switching 375

References 382

Suggested Further Reading 383

15. Multiwave Mixing 384

15.1 Degenerate Four-Wave Mixing 385

15.2 Degenerate Three-Wave Mixing 397

15.3 Nondegenerate Wave Mixing 399

Reference 413

Suggested Further Reading 413

16. Stimulated Scattering 414

16.1 Stimulated Raman Scattering 415

16.2 Stimulated Brillouin Scattering 431

References 441

Suggested Further Reading 442

17. Ultrafast and Ultrahigh Intensity Processes 443

17.1 Extended Nonlinear Wave Equation 444

17.2 Formalism for Ultrafast Fiber Nonlinear Optics 448

17.3 Examples of Nonlinear Optics in Fibers 452

17.4 High Harmonic Generation 460

References 462

Suggested Further Reading 463

Appendix: Units, Notation, and Physical Constants 465

A.1 Units of Third-Order Nonlinearity 465

A.2 Values of Useful Constants 467

Reference 467

Index 469