Ultrashort Laser Pulse Phenomena, 2nd Edition

Contents

Preface xv

Preface to the First Edition xvii

Chapter 1 Fundamentals 1

1.1. Characteristics of Femtosecond Light Pulses 1

1.1.1. Complex Representation of the Electric Field 1

1.1.2. Power, Energy, and Related Quantities 6

1.1.3. Pulse Duration and Spectral Width 9

1.1.4. Wigner Distribution, Second-Order Moments,

Uncertainty Relations 12

1.2. Pulse Propagation 20

1.2.1. The Reduced Wave Equation 21

1.2.2. Retarded Frame of Reference 26

1.2.3. Dispersion 30

1.2.4. Gaussian Pulse Propagation 33

1.2.5. Complex Dielectric Constant 38

1.3. Interaction of Light Pulses with Linear Optical Elements 42

1.4. Generation of Phase Modulation 44

1.5. Beam Propagation 46

1.5.1. General 46

1.5.2. Analogy between Pulse and Beam Propagation 49

1.5.3. Analogy between Spatial and Temporal

Imaging 50

1.6. Numerical Modeling of Pulse Propagation 53

1.7. Space–Time Effects 56

1.8. Problems 57

Bibliography 58

Chapter 2 Femtosecond Optics 61

2.1. Introduction 61

2.2. White Light and Short Pulse Interferometry 64

v

vi Contents

2.3. Dispersion of Interferometric Structures 70

2.3.1. Mirror Dispersion 70

2.3.2. Fabry–Perot and Gires–Tournois Interferometer 73

2.3.3. Chirped Mirrors 80

2.4. Focusing Elements 82

2.4.1. Singlet Lenses 82

2.4.2. Space–Time Distribution of the Pulse

Intensity at the Focus of a Lens 86

2.4.3. Achromatic Doublets 91

2.4.4. Focusing Mirrors 92

2.5. Elements with Angular Dispersion 94

2.5.1. Introduction 94

2.5.2. Tilting of Pulse Fronts 95

2.5.3. GVD through Angular Dispersion—General 100

2.5.4. GVD of a Cavity Containing a Single Prism 102

2.5.5. Group Velocity Control with Pairs of Prisms 105

2.5.6. GVD Introduced by Gratings 117

2.5.7. Grating Pairs for Pulse Compressors 120

2.5.8. Combination of Focusing and Angular

Dispersive Elements 122

2.6. Wave-Optical Description of Angular Dispersive

Elements 124

2.7. Optical Matrices for Dispersive Systems 130

2.8. Numerical Approaches 136

2.9. Problems 136

Bibliography 140

Chapter 3 Light–Matter Interaction 143

3.1. Density Matrix Equations 144

3.2. Pulse Shaping with Resonant Particles 154

3.2.1. General 154

3.2.2. Pulses Much Longer Than the Phase

Relaxation Time (τ

p

 T

2

) 156

3.2.3. Phase Modulation by Quasi-Resonant

Interactions 161

3.2.4. Pulse Durations Comparable with or Longer

Than the Phase Relaxation Time (τ

p

≥ T

2

) 165

3.3. Nonlinear, Nonresonant Optical Processes 166

3.3.1. General 166

3.3.2. Noninstantaneous Response 168

3.3.3. Pulse Propagation 170

Contents vii

3.4. Second Harmonic Generation (SHG) 172

3.4.1. Type I Second Harmonic Generation 173

3.4.2. Second Harmonic Type II: Equations for

Arbitrary Phase Mismatch and

Conversion Efficiencies 180

3.4.3. Pulse Shaping in Second Harmonic

Generation (Type II) 183

3.4.4. Group Velocity Control in SHG through

Pulse Front Tilt 185

3.5. Optical Parametric Interaction 188

3.5.1. Coupled Field Equations 188

3.5.2. Synchronous Pumping 190

3.5.3. Chirp Amplification 190

3.6. Third-Order Susceptibility 192

3.6.1. Fundamentals 192

3.6.2. Short Samples with Instantaneous Response 195

3.6.3. Short Samples and Noninstantaneous

Response 197

3.6.4. Counter-Propagating Pulses and Third-Order

Susceptibility 199

3.7. Continuum Generation 202

3.8. Self-Focusing 205

3.8.1. Critical Power 205

3.8.2. The Nonlinear Schrödinger Equation 208

3.9. Beam Trapping and Filaments 209

3.9.1. Beam Trapping 209

3.9.2. Ultrashort Pulse Self-Focusing 212

3.10. Problems 213

Bibliography 215

Chapter 4 Coherent Phenomena 221

4.1. From Coherent to Incoherent Interactions 221

4.2. Coherent Interactions with Two-Level Systems 225

4.2.1. Maxwell–Bloch Equations 225

4.2.2. Rate Equations 229

4.2.3. Evolution Equations 230

4.2.4. Steady-State Pulses 239

4.3. Multiphoton Coherent Interaction 243

4.3.1. Introduction 243

4.3.2. Multiphoton Multilevel Transitions 245

4.3.3. Simplifying a N-Level System to a

Two-Level Transition 258

viii Contents

4.3.4. Four Photon Resonant Coherent Interaction 262

4.3.5. Miscellaneous Applications 268

4.4. Problems 272

Bibliography 273

Chapter 5 Ultrashort Sources I: Fundamentals 277

5.1. Introduction 277

5.1.1. Superposition of Cavity Modes 277

5.1.2. Cavity Modes and Modes of a

Mode-Locked Laser 280

5.1.3. The “Perfect” Mode-Locked Laser 283

5.1.4. The “Common” Mode-Locked Laser 285

5.1.5. Basic Elements and Operation of a fs Laser 291

5.2. Circulating Pulse Model 293

5.2.1. General Round-Trip Model 293

5.2.2. Continuous Model 295

5.2.3. Elements of a Numerical Treatment 298

5.2.4. Elements of an Analytical Treatment 300

5.3. Evolution of the Pulse Energy 303

5.3.1. Rate Equations for the Evolution of the

Pulse Energy 304

5.3.2. Connection of the Model to Microscopic

Parameters 311

5.4. Pulse Shaping in Intracavity Elements 314

5.4.1. Saturation 315

5.4.2. Nonlinear Nonresonant Elements 317

5.4.3. Self-Lensing 320

5.4.4. Summary of Compression Mechanisms 323

5.4.5. Dispersion 323

5.5. Cavities 325

5.5.1. Cavity Modes and ABCD Matrix Analysis 325

5.5.2. Astigmatism and Its Compensation 328

5.5.3. Cavity with a Kerr Lens 332

5.6. Problems 335

Bibliography 337

Chapter 6 Ultrashort Sources II: Examples 341

6.1. Synchronous Mode-Locking 341

6.2. Hybrid Mode-Locking 345

6.3. Additive Pulse Mode-Locking 346

6.3.1. Generalities 346

6.3.2. Analysis of APML 348