Nonlinear OpticsISBN: 978-3-527-40923-5
Paperback
259 pages
April 2010
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1 Historical Background 1
1.1 Introduction 1
1.2 Kirchhoff (1859)1
1.2.1 The Birth of Spectroscopy 1
1.2.2 The First Law 3
1.2.3 The Second Law 5
1.3 Stefan (1879) and Boltzmann (1884) 7
1.3.1 Experimental Background 7
1.3.2 Maxwell’s Theory and Thermodynamics 8
1.4 Wien (1893) 10
1.5 Planck (1900) 15
1.5.1 ModelingMatter 15
1.5.2 The QuantumHypothesis 16
1.6 Einstein (1905, 1907, and 1916) 19
1.6.1 Quantization of Light Absorption (1905) 19
1.6.2 Specific Heat of Solids (1907) 20
1.6.3 Spontaneous Emission (1916) 20
I Quantized Matter 23
2 Two-Level Medium 25
2.1 Electric Field Equation 25
2.2 Material Equations 26
2.3 Phenomenology: Incoherent Pumping and Decay 30
3 Propagation Regimes 33
3.1 Linear Propagation Regime 33
3.2 Nonlinear Susceptibility 37
3.3 Nonlinear Steady Propagation 39
3.4 Group and Phase Velocity 40
4 Coherence and Atomic Interference 43
4.1 Atomic Interference 43
4.2 Semiclassical Formulation 46
4.3 Electromagnetically Induced Transparency 49
4.4 Slow Light 52
II Sine–Gordon Solitons 55
5 Self-Induced Transparency 57
5.1 Derivation of the Area Theorem 57
5.2 Properties of the Area 61
6 Sine–Gordon Equation 65
6.1 The Bäcklund Transformation 65
6.2 2π Solitons 67
6.3 Short Bibliography on Solitons 70
7 Higher-Order Sine–Gordon Solitons 73
7.1 The Bianchi Theorem 73
7.2 4π Solitons 74
7.3 Unstable 0π Solitons 79
7.4 Stable 0π Solitons 79
7.5 Solutions in an Amplifier 82
III Cavity Nonlinear Optics 85
8 Laser Theory 87
8.1 Introduction 87
8.2 Single-Mode Ring Laser 91
8.2.1 Steady States 94
8.2.2 Rate Equations 95
8.2.3 Good Cavity Limit 97
8.3 Single-Mode Fabry–Perot Laser 98
8.3.1 Semiclassical Equations 98
8.3.2 Population Gratings in Steady State 101
8.3.3 Rate Equations 102
8.4 Warning 104
8.5 Short Bibliography on Cavity Optics 105
9 Optical Bistability I 113
9.1 Introduction 113
9.2 Steady-State Solutions 114
9.3 Optical Devices 115
9.4 Generic Description 118
9.5 Nonlinear Stability 119
9.6 Address Pulses 121
9.7 Pulse Area Law 124
9.8 Appendix: the Schmitt Trigger 126
10 Optical Bistability II 129
10.1 Delay–Differential Equations 129
10.2 DiscreteMaps 130
10.3 Deterministic Chaos 133
10.4 Bibliography 136
10.4.1 Ikeda Delay–Differential Equations 136
10.4.2 Deterministic Chaos 137
IV Weakly Nonlinear Systems: χ(2) Media 141
11 Frequency Mixing 143
11.1 Tensor → Vector → Scalar Description 143
11.2 Multiple Time-Scales 144
11.3 χ(2) Media 146
11.4 Bibliography 148
11.4.1 Founding Papers of Nonlinear Optics 148
11.4.2 Nonlinear Optics (χ(2) and χ(3) Media) 148
12 Second Harmonic Generation 153
12.1 Formulation 153
12.2 Free-Running Second Harmonic Generation 155
12.2.1 Perfect Matching and G = 0 158
12.2.2 Perfect Matching but G = 0 160
12.2.3 Imperfect PhaseMatching 161
12.2.4 PhaseMatching 162
12.3 Intra-Cavity Second Harmonic Generation 164
13 Sum and Difference Frequency Generation 171
13.1 Sum Frequency Generation 171
13.1.1 Formulation 171
13.1.2 Free-Running Sum Frequency Generation 172
13.2 Difference Frequency Generation 176
13.2.1 Two Intense Input Fields 177
13.2.2 One Intense Input Field 178
14 Optical Parametric Oscillator 181
14.1 Formulation 181
14.2 Threshold Condition 182
14.3 Degenerate Optical Parametric Oscillator 184
14.4 Ring and Fabry–Perot Cavities 186
V Weakly Nonlinear Systems: χ(3) Media 189
15 Multiwave Frequency Mixing 191
15.1 Introduction 191
15.2 Monochromatic Input Field 192
15.3 Three-WaveMixing 193
15.4 Optical Phase Conjugation 194
15.4.1 Fresnel’s Laws 194
15.4.2 Optical Phase Conjugation 195
15.5 Degenerate Four-WaveMixing 200
16 Nonlinear Schrödinger Solitons 203
16.1 Introduction 203
16.2 Formulation 203
16.3 MathematicalDigression 205
16.3.1 The Δ Operator 206
16.3.2 The ∇(∇·) Operator 206
16.3.3 The Operator ∂2/∂t2 206
16.3.4 The Integral Operator 207
16.3.5 Expansion of the Operator ϕ(ω) 208
16.3.6 Third-Order Polarization 209
16.3.7 Summing Up 209
16.4 Perturbation Expansion 210
16.4.1 The O(μ0) Equation 210
16.4.2 The O(μ) Equation 211
16.4.3 The O(μ2) Equation 211
16.5 The Nonlinear Schrödinger Equation 213
16.6 Basic Solutions of the Nonlinear Schrödinger Equation 214
16.7 Comments 216
17 Higher-Order Nonlinear Schrödinger Solitons 219
17.1 Inverse ScatteringMethod 219
17.2 First-Order Solutions 221
17.2.1 Turing/Modulational Instability (Time-Periodic Solutions)221
17.2.2 Aperiodic Solution 221
17.2.3 Space-Periodic Solutions 224
17.2.4 Space- and Time-Periodic Solutions 225
17.3 Second-Order Solutions 225
17.3.1 Colliding Bright Solitons 225
17.3.2 Breathers (Space-Periodic) 230
17.3.3 Aperiodic Solution 231
17.4 Colliding Dark Solitons 231
17.5 Existence and Stability 233
17.6 Short Bibliography on Patterns 235
References 237
Index 241
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