Laser Dynamics

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents\n......Page 9
Preface......Page 13
Abbreviations......Page 17
Part I Basic tools......Page 19
1 Rate equations......Page 21
1.1 Dimensionless equations......Page 22
1.2 Steady states and linear stability......Page 24
1.2.1 Steady states......Page 25
1.2.2 Linear stability......Page 26
1.2.3 Damped relaxation oscillations......Page 27
1.3 Turn-on transients......Page 28
1.3.1 Typical turn-on experiment......Page 29
1.3.2 Switching-on or turn-on time......Page 32
1.4 Transfer function......Page 35
1.5 Dynamical system......Page 37
1.5.1 Laser bifurcation......Page 38
1.5.2 Normal form......Page 39
Derivation......Page 40
Validity......Page 42
1.5.3 Phase space......Page 43
1.6.1 Imperfect bifurcation and ghosts......Page 44
1.6.2 Dynamical effects......Page 47
1.7 Semiconductor lasers......Page 49
1.8.3 Linear stability analysis......Page 52
1.8.7 Turn-on experiment with a pump square pulse......Page 53
1.8.8 RO frequency near threshold......Page 54
1.8.9 RO frequency and the design of high-speed SLs......Page 55
1.8.10 Two-time analysis of the laser rate equations......Page 56
2 Three- and four-level lasers......Page 57
2.1 Energy level schemes in lasers......Page 58
2.2 Three-level lasers......Page 59
2.2.1 Ruby laser......Page 60
2.2.2 CO2 laser......Page 62
Model......Page 63
Linear stability analysis......Page 65
2.3.1 Model......Page 67
2.3.2 Connection with the two-level model......Page 69
2.3.3 Modified four-level model for CO2 lasers......Page 70
2.3.4 Dimensionless equations......Page 73
2.3.5 Long time solution......Page 74
2.4.2 Two-photon laser......Page 75
2.4.4 Dimensionless formulation......Page 76
3.1 Phase-locking in laser dynamics......Page 77
3.2.1 Theoretical model......Page 78
3.2.2 Experiments......Page 80
3.3.2 Phase drift......Page 82
3.3.3 Long period motion......Page 83
3.4 Laser with an injected signal......Page 84
3.4.1 Experiments......Page 85
3.4.2 Theory......Page 86
3.5 Counterpropagating waves in ring class A lasers......Page 88
3.5.1 Dither control of ring laser gyro......Page 91
3.5.2 High-frequency asymptotics......Page 92
3.6.1 Experiments......Page 93
3.6.2 Theory......Page 95
3.7.1 Rotation induced by loss anisotropy......Page 97
Exact solution......Page 98
Solution of Adler’s equation close to locking......Page 99
3.7.4 Ring laser with diffraction locking......Page 100
4 Hopf bifurcation dynamics......Page 102
4.1 Electrical feedback......Page 105
4.1.1 Steady-state solutions......Page 108
4.1.3 The Hopf bifurcation......Page 110
4.1.4 Bifurcation diagrams......Page 112
4.2 Ikeda system......Page 114
4.2.1 Limit and Hopf bifurcation points......Page 117
4.2.2 Hopf bifurcation approximation......Page 119
4.3 From harmonic to pulsating oscillations......Page 121
4.4.1 Saddle-node bifurcation......Page 122
4.4.4 Multiple steady states......Page 123
4.4.5 Laser with feedback on the cavity length......Page 124
4.4.6 Double Hopf bifurcation and the eye bifurcation diagram......Page 125
Part II Driven laser systems......Page 127
5 Weakly modulated lasers......Page 129
5.1.1 Weakly nonlinear and arbitrary modulation......Page 130
5.1.2 Strongly nonlinear and weak modulation......Page 133
A change of variable......Page 134
Resonance......Page 135
5.2.1 Nearly conservative oscillations......Page 137
5.2.2 Pump and loss modulations......Page 140
Analysis of the successive orders......Page 142
Bifurcations......Page 145
5.2.4 Subharmonic modulation and period doubling bifurcation......Page 148
Bifurcations......Page 150
5.3.3 A change of variables in the rate equations......Page 152
5.3.4 Failure of the regular perturbation method......Page 153
6 Strongly modulated lasers......Page 154
6.1.1 Experiments and simulations......Page 155
6.1.2 Branches of subharmonic periodic solutions......Page 158
6.2.1 Exploring the large oscillation regimes......Page 161
6.2.2 Building the map......Page 162
6.2.3 The first period doubling bifurcation......Page 165
6.3.1 Period-doubling lasers as small-signal detectors......Page 167
6.3.2 Two-tone modulation of a class B laser......Page 169
7 Slow passage......Page 173
7.1 Dynamical hysteresis......Page 174
7.2 Slow passage through a bifurcation point......Page 176
7.2.1 The limit epsilon rarrow 0 and the role of noise......Page 178
7.2.2 Moderately small epsilon......Page 180
7.2.3 Experiments......Page 181
Forward transition......Page 182
Changing the losses......Page 183
7.3 Period-doubling bifurcation......Page 186
7.5 Exercise......Page 188
Part III Particular laser systems......Page 191
8 Laser with a saturable absorber......Page 193
8.1 LSA parameters......Page 195
8.2.1 Optical bistability......Page 196
8.2.2 Passive Q-switching......Page 198
8.3 Rate equations......Page 200
8.3.1 Steady state solutions......Page 201
8.3.2 Two-variable reduction and PQS......Page 202
Linear stability......Page 204
Pulsating solutions......Page 206
Stability of the non-zero intensity solution......Page 209
Hopf bifurcation......Page 210
Bifurcation diagrams......Page 213
8.4 PQS in CO2 lasers......Page 215
Hopf bifurcation transition......Page 216
Complex oscillations in the CO2 LSA......Page 217
Model equations for a CO2 LSA......Page 219
Steady states, stability, and bifurcation diagram......Page 221
Bursting oscillations......Page 223
Comparison with experimental results......Page 224
8.5.1 Asymptotic analysis of the pulsating solutions......Page 226
8.5.3 Symmetric pulse and pulse width......Page 228
8.5.4 The LSA with two-photon absorber......Page 229
9 Optically injected semiconductor lasers......Page 231
9.1 Semiconductor lasers......Page 232
9.2 Injection-locking......Page 234
9.3 Adler\'s equation......Page 235
9.4 Experiments and numerical simulations......Page 238
9.5.1 Multiple steady states......Page 240
9.5.2 Linear stability analysis......Page 242
9.5.3 Limit point or saddle-node bifurcation......Page 243
9.5.5 Approximations of SN and Hopf bifurcation points......Page 244
9.6 Nonlinear studies......Page 247
9.6.1 Formulation......Page 248
9.6.2 Delta is arbitrary......Page 249
9.6.3 |Delta|  is small......Page 250
9.7 A third order Adler\'s equation......Page 252
9.8.1 Hopf bifurcation close to the laser threshold......Page 255
9.8.3 Bogdanov--Takens bifurcation......Page 256
9.8.4 Injected solid state laser......Page 257
10.1 History......Page 259
10.1.1 Low frequency fluctuations......Page 261
10.1.2 ECM solutions......Page 264
10.1.3 ECM ellipse, maximum gain mode, and LFF......Page 266
10.1.4 LFF experimental results......Page 268
10.1.5 Numerical simulations and bridges......Page 270
10.2 Imaging using OFB......Page 273
10.2.1 Stability analysis......Page 274
10.2.2 Low feedback rate approximation......Page 276
10.3 Optoelectronic oscillator......Page 279
10.3.1 Slowly varying oscillations…......Page 282
10.3.2 Fast bursting oscillations…......Page 283
10.4.1 Optoelectronic feedback......Page 286
10.4.3 Adler\'s equation with delay......Page 287
10.4.4 Phase plane analysis......Page 289
11 Far-infrared lasers......Page 290
11.1 Vibrational bottleneck......Page 291
11.2 Lorenz chaos in the FIR laser......Page 293
11.3.1 Experiments......Page 299
11.3.2 Model......Page 300
Steady-state intensity solutions......Page 301
Linear stability......Page 304
11.3.3 Comparison with the experiments......Page 305
11.3.4 FIR laser dynamics in the \"bad cavity limit\"......Page 307
11.4.2 Complex Haken–Lorenz equations......Page 310
12.1 Parametric processes......Page 312
12.1.1 Optical parametric amplification......Page 314
12.1.2 Second harmonic generation......Page 315
12.2.1 Steady state solutions and bistability......Page 316
12.2.2 Hopf bifurcation......Page 318
12.3 Experiments on TROPO-DOPO......Page 319
12.3.1 Power 1/2 law for the output power......Page 320
12.3.3 Relaxation oscillations......Page 321
12.4.1 Experimental results......Page 324
12.4.2 Model for thermally induced cavity drift......Page 325
12.4.3 Thermal cycles in the single-mode OPO......Page 326
12.5 Intracavity singly resonant parametric oscillator......Page 330
12.6.1 Intracavity SHG model......Page 334
12.7 Antiphase dynamics in intracavity SHG......Page 336
12.7.1 Antiphase dynamics in YAG/KTP lasers......Page 337
12.7.2 Analysis of the two-mode case......Page 338
Out-of-phase regimes......Page 340
12.8 Frequencies......Page 343
12.9 Antiphase dynamics in a fiber laser......Page 346
12.9.1 Steady state solutions......Page 348
12.9.2 Stability analysis......Page 349
12.10.3 Rescaling the SHG equations......Page 352
12.10.5 SHG inside the laser cavity......Page 353
References......Page 354
Index......Page 376