1. Introduction ................................................ 1
1.1. Chaos and Lasers ....................................... 1
1.2. Historical Perspectives of Chaos in Semiconductor
Lasers ................................................. 3
1.3. Outline of This Book ................................... 6
2. Chaos in Laser Systems ..................................... 11
2.1. Laser Model and Bloch Equations ....................... 11
2.1.1. Laser Model in a Ring Resonator ................ 11
2.1.2. Light Emission and Absorption in Two-Level
Atoms .......................................... 13
2.1.3. Maxwell-Bloch Equations ........................ 14
2.2. Lorenz-Haken Equations ................................ 15
2.2.1. Lorenz-Haken Equations ......................... 15
2.2.2. First Laser Threshold .......................... 16
2.2.3. Second Laser Threshold ......................... 18
2.3. Classifications of Lasers ............................. 20
2.3.1. Classes of Lasers .............................. 20
2.3.2. Class С Lasers ................................. 30
2.3.3. Class В Lasers ................................. 23
2.3.4. Class A Lasers ................................. 24
3. Semiconductor Lasers and Theory ............................ 25
3.1. Semiconductor Lasers .................................. 25
3.2. Oscillation Conditions of Semiconductor Lasers ........ 26
3.2.1. Laser Oscillation Conditions ................... 26
3.2.2. Laser Oscillation Frequency .................... 28
3.2.3. Dependence of Oscillation Frequency on
Carrier Density ................................ 29
3.3. Derivation of Rate Equations .......................... 29
3.3.1. Gain at Laser Oscillation ...................... 29
3.3.2. Rate Equation for the Field .................... 30
3.3.3. Linewidth Enhancement Factor ................... 32
3.3.4. Laser Rate Equations ........................... 33
3.4. Linear Stability Analysis and Relaxation
Oscillation ........................................... 37
3.4.1. Linear Stability Analysis ...................... 37
3.4.2. Relaxation Oscillation ......................... 38
3.5. Langevin Noises ....................................... 40
3.5.1. Rate Equations Including Langevin Noises ....... 40
3.5.2. Langevin Noises ................................ 41
3.5.3. Noise Spectrum ................................. 43
3.5.4. Relative Intensity Noise (RIN) ................. 43
3.5.5. Phase Noise and Spectral Linewidth ............. 44
3.6. Modulation Characteristics ............................ 47
3.6.1. Injection Current Modulation ................... 47
3.6.2. Intensity Modulation Characteristics ........... 48
3.6.3. Phase Modulation Characteristics ............... 50
3.7. Waveguide Models of Semiconductor Lasers .............. 51
3.7.1. Index- and Gain-Guided Structures .............. 51
3.7.2. Waveguide Models ............................... 53
3.7.3. Spatial Modes of Gain- and Index-Guided
Lasers ......................................... 54
3.7.4. Effects of Spontaneous Emission in Gain- and
Index-Guided Lasers ............................ 56
3.7.5. Laser Types .................................... 58
4. Theory of Optical Feedback in Semiconductor Lasers ......... 63
4.1. Theory of Optical Feedback ............................ 63
4.1.1. Optical Feedback Effects and Classifications
of Optical Feedback Phenomena .................. 63
4.1.2. Theoretical Model .............................. 66
4.2. Linear Stability Analysis for Optical Feedback
Systems ............................................... 68
4.2.1. Linear Stability Analysis ...................... 68
4.2.2. Linear Mode, and Stability and Instability
in Semiconductor Lasers ........................ 73
4.2.3. Gain Reduction Due to Optical Feedback ......... 75
4.2.4. Linewidth in the Presence of Optical
Feedback ....................................... 76
4.3. Feedback from a Grating Mirror ........................ 77
4.4. Phase-Conjugate Feedback .............................. 79
4.5. Incoherent Feedback and Polarization-Rotated Optical
Feedback .............................................. 82
4.5.1. Incoherent Feedback ............................ 82
4.5.2. Polarization-Rotated Optical Feedback .......... 83
4.6. Filtered Feedback ..................................... 85
5. Dynamics of Semiconductor Lasers with Optical Feedback ..... 87
5.1. Optical Feedback from a Conventional Reflector ........ 87
5.1.1. Optical Feedback Effects ....................... 87
5.1.2. Potential Model in Feedback Induced
Instability .................................... 88
5.1.3. Optical Spectrum in Stable and Unstable
Feedback Regimes ............................... 90
5.1.4. Chaos in Semiconductor Lasers with Optical
Feedback ....................................... 93
5.1.5. Chaotic Bifurcations ........................... 95
5.1.6. Dynamics for Injection Current Variations ...... 96
5.2. Dependence of Chaotic Dynamics on the External
Mirror Position ...................................... 101
5.2.1. Periodic Stability Enhancement for
Variations of the External Cavity Length ...... 101
5.2.2. Origin of Periodic Stability Enhancement ...... 103
5.2.3. Effects of Linewidth Enhancement Factor ....... 105
5.2.4. Sensitivity of the Optical Phase .............. 107
5.2.5. Chaotic Dynamics for a Small Change of the
External Cavity Length ........................ 109
5.3. Low-Frequency Fluctuations (LFFs) .................... 112
5.3.1. Low-Frequency Fluctuation Phenomena ........... 112
5.3.2. LFF Characteristics ........................... 115
5.3.3. Origin of LFFs ................................ 117
5.4. Chaotic Dynamics in Short External Cavity Limit ...... 120
5.4.1. Stable and Unstable Conditions in Short
External Cavity ............................... 120
5.4.2. Regular Pulse Package Oscillations in Short
External Cavity ............................... 122
5.4.3. Bifurcations of Regular Pulse Package ......... 124
5.5. Dynamics in Semiconductor Lasers with Grating
Mirror Feedback ...................................... 126
5.6. Dynamics in Semiconductor Lasers with Phase-
Conjugate Mirror Feedback ............................ 129
5.6.1. Linear Stability Analysis ..................... 129
5.6.2. Dynamics Induced by Phase-Conjugate
Feedback ...................................... 131
5.6.3. Dynamics in the Presence of Frequency
Detuning ...................................... 132
5.6.4. Finite and Slow Response Phase-Conjugate
Feedback ...................................... 132
5.7. Dynamics of Semiconductor Lasers with Incoherent
Optical Feedback ..................................... 134
5.7.1. Dynamics of Incoherent Optical Feedback ....... 134
5.7.2. Dynamics of Polarization-Rotated Optical
Feedback ...................................... 138
5.8. Dynamics of Filtered Optical Feedback ................ 140
5.8.1. Filtered Optical Feedback ..................... 140
5.8.2. External Cavity Modes ......................... 142
5.8.3. Frequency Oscillations and Chaotic Dynamics ... 144
6. Dynamics in Semiconductor Lasers with Optical Injection ... 147
6.1. Optical Injection .................................... 147
6.1.1. Optical Injection Locking ..................... 147
6.1.2. Injection Locking Condition ................... 150
6.2. Stability and Instability in Optical Injection
Systems .............................................. 152
6.2.1. Rate Equations ................................ 152
6.2.2. Chaotic Bifurcations by Optical Injection ..... 153
6.2.3. Chaos Map in the Phase Space of Frequency
Detuning and Injection ........................ 157
6.2.4. Coexistence of Chaotic Attractors in
Optically Injected Semiconductor Lasers ....... 161
6.3. Enhancement of Modulation Bandwidth and Generation
of High Frequency Chaotic Oscillation by Strong
Optical Injection .................................... 164
6.3.1. Enhancement of Modulation Bandwidth by
Strong Optical Injection ...................... 164
6.3.2. Origin of Modulation Bandwidth Enhancement .... 168
6.3.3. Modulation Response by Strong Optical
Injection ..................................... 170
6.3.4. Suppression of Frequency Chirping by Strong
Optical Injection ............................. 172
6.3.5. Generation of High Frequency Chaotic
Oscillation by Strong Optical Injection ....... 174
7. Dynamics of Semiconductor Lasers with Optoelectronic
Feedback and Modulation ................................... 177
7.1. Theory of Optoelectronic Feedback .................... 177
7.1.1. Optoelectronic Feedback Systems ............... 177
7.1.2. Pulsation Oscillations in Optoelectronic
Feedback Systems .............................. 179
7.2. Linear Stability Analysis for Optoelectronic
Feedback Systems ..................................... 182
7.2.1. Linear Stability Analysis ..................... 182
7.2.2. Characteristics of Semiconductor Lasers with
Optoelectronic Feedback ....................... 185
7.3. Dynamics and Chaos in Semiconductor Lasers with
Optoelectronic Feedback .............................. 187
7.3.1. Chaotic Dynamics in Negative Optoelectronic
Feedback ...................................... 187
7.3.2. Chaotic Dynamics in Positive Optoelectronic
Feedback ...................................... 189
7.4. Optoelectronic Feedback with Wavelength Filter ....... 193
7.4.1. System of Optoelectronic Feedback with
Wavelength Filter ............................. 193
7.4.2. Dynamics of Optoelectronic Feedback with
Wavelength Filter ............................. 195
7.5. Chaotic Dynamics of Semiconductor Lasers Induced
by Injection Current Modulation ...................... 198
7.5.1. Instabilities of a Modulated Semiconductor
Laser ......................................... 198
7.5.2. Linear Stability Analysis ..................... 200
7.5.3. Chaotic Dynamics in Modulated Semiconductor
Lasers ........................................ 204
7.6. Nonlinear Dynamics of Various Combinations of
External Perturbations ............................... 206
7.6.1. Optically Injected Semiconductor Laser
Subject to Optoelectronic Feedback ............ 206
7.6.2. Semiconductor Lasers with Optical Feedback
and Modulation ................................ 209
8. Instability and Chaos in Various Laser Structures ......... 213
8.1. Multimode Lasers ..................................... 213
8.1.1. Multimode Operation of Semiconductor Lasers ... 213
8.1.2. Theoretical Model of Multimode Lasers ......... 214
8.1.3. Dynamics of Multimode Semiconductor Lasers
with Optical Feedback ......................... 217
8.2. Self-Pulsating Lasers ................................ 220
8.2.1. Theory of Self-Pulsating Lasers ............... 220
8.2.2. Instabilities at Solitary Oscillations ........ 223
8.2.3. Instability and Chaos by Optical Feedback ..... 227
8.2.4. Instability and Chaos by Injection Current
Modulation .................................... 230
8.3. Vertical-Cavity Surface-Emitting Lasers (VCSELs) ..... 232
8.3.1. Theoretical Model of Vertical-Cavity
Surface-Emitting Lasers ....................... 232
8.3.2. Spin-Flip Model ............................... 235
8.3.3. Characteristics of VCSELs in Solitary
Oscillations .................................. 239
8.3.4. Spatio-Temporal Dynamics in VCSELs ............ 242
8.3.5. Feedback Effects in VCSELs .................... 245
8.3.6. Short Optical Feedback in VCSELs .............. 250
8.3.7. Orthogonal Optical Injection Dynamics in
VCSEL ......................................... 252
8.4. Broad Area Lasers .................................... 255
8.4.1. Theoretical Model of Broad Area Lasers ........ 255
8.4.2. Dynamics of Broad Area Semiconductor Lasers
at Solitary Oscillations ...................... 258
8.4.3. Feedback Effects in Broad Area Semiconductor
Lasers ........................................ 264
8.5. Laser Arrays ......................................... 266
9. Chaos Control and Applications ............................ 269
9.1. General Methods of Chaos Control ..................... 269
9.1.1. OGY Method .................................... 269
9.1.2. Continuous Control Method ..................... 270
9.1.3. Occasional Proportional Method ................ 271
9.1.4. Sinusoidal Modulation Method .................. 272
9.2. Chaos Control in Semiconductor Lasers ................ 273
9.2.1. Continuous Control ............................ 273
9.2.2. Occasional Proportional Feedback Control ...... 275
9.2.3. Sinusoidal Modulation Control ................. 276
9.2.4. Optical Control ............................... 279
9.3. Controlling Chaos and Noise Suppression .............. 282
9.3.1. Noise Suppression by Sinusoidal Modulation .... 282
9.3.2. Stability and Instability of LFFs by
Injection Current Modulation .................. 286
9.3.3. Chaos Targeting ............................... 288
10. Stabilization of Semiconductor Lasers ..................... 291
10.1.Linewidth Narrowing by Optical Feedback .............. 291
10.1.1. Linewidth Narrowing by Strong Optical
Feedback ..................................... 291
10.1.2. Linewidth Narrowing by Grating Feedback ...... 294
10.1.3. Linewidth Narrowing by Phase-Conjugate
Optical Feedback ............................. 295
10.1.4. Linewidth Narrowing by Resonant Optical
Feedback ..................................... 299
10.2.Linewidth Narrowing by Optoelectronic Feedback ....... 301
10.3.Stabilization in Lasers with Various Structures ...... 304
10.3.1. Noise Suppression in Self-Pulsation
Semiconductor Laser .......................... 304
10.3.2. Stabilization of VCSELs ...................... 305
10.3.3. Stabilization of Broad-Area Semiconductor
Lasers ....................................... 308
10.3.4. Stabilization of Laser Arrays ................ 312
10.4.Controls in Nobel Structure Lasers ................... 313
10.4.1. Photonic VCSELs .............................. 313
10.4.2. Quantum-Dot Broad-Area Semiconductor
Lasers ....................................... 315
11. Stability and Bistability in Feedback Interferometers,
and Their Applications .................................... 319
11.1.Optical Feedback Interferometers ..................... 319
11.1.1. Bistability and Multistability in Feedback
Interferometers .............................. 319
11.1.2. Interferometric Measurement in Self-Mixing
Semiconductor Lasers ......................... 323
11.2.Applications in Feedback Interferometer .............. 325
11.2.1. Displacement and Vibration Measurement ....... 325
11.2.2. Velocity Measurement ......................... 328
11.2.3. Absolute Position Measurement ................ 329
11.2.4. Angle Measurement ............................ 330
11.2.5. Measurement of Linewidth and Linewidth
Enhancement Factor ........................... 332
11.3.Active Feedback Interferometer ....................... 334
11.3.1. Stability and Bistability in Active
Feedback Interferometer ...................... 334
11.3.2. Chaos Control in Active Feedback
Interferometers .............................. 338
12. Chaos Synchronization in Semiconductor Lasers ............. 341
12.1.Concept of Chaos Synchronization ..................... 341
12.1.1. Chaos Synchronization ........................ 341
12.1.2. Generalized and Complete Chaos
Synchronization .............................. 344
12.2.Theory of Chaos Synchronization in Semiconductor
Lasers with Optical Feedback ......................... 347
12.2.1. Model of Synchronization Systems ............. 347
12.2.2. Rate Equations in Unidirectional Coupling
Systems ...................................... 349
12.2.3. Generalized Chaos Synchronization ............ 350
12.2.4. Complete Chaos Synchronization ............... 351
12.2.5. Mutual Coupling Systems ...................... 351
12.3.Chaos Synchronization in Semiconductor Lasers with
an Optical Feedback System ........................... 353
12.3.1. Chaos Synchronization - Numerical Examples ... 353
12.3.2. Chaos Synchronization - Experimental
Examples ..................................... 357
12.3.3. Anticipating Chaos Synchronization ........... 359
12.3.4. Bandwidth Enhanced Chaos Synchronization ..... 360
12.3.5. Incoherent Synchronization Systems ........... 362
12.3.6. Polarization Rotated Chaos Synchronization ... 364
12.4.Chaos Synchronization in Injected Lasers ............. 367
12.4.1. Theory of Chaos Synchronization in
Injected Lasers .............................. 367
12.4.2. Examples of Chaos Synchronization in
Injected Lasers .............................. 369
12.5.Chaos Synchronization in Optoelectronic Feedback
Systems .............................................. 370
12.5.1. Theory of Chaos Synchronization in
Optoelectronic Feedback Systems .............. 370
12.5.2. Examples of Chaos Synchronization in
Optoelectronic Feedback Systems .............. 372
12.6.Chaos Synchronization in Injection Current
Modulated Systems .................................... 373
12.7.Chaos Synchronization in Mutually Coupled Lasers ..... 374
12.7.1. Chaos Synchronization of Semiconductor
Lasers with Mutual Optical Coupling .......... 374
12.7.2. Chaos Synchronization of Semiconductor
Lasers with Mutual Optoelectronic Coupling ... 375
13. Chaotic Communications in Semiconductor Lasers ............ 379
13.1.Message Encryption in a Chaotic Carrier and
Its Decryption ....................................... 379
13.1.1. Chaotic Communications ....................... 379
13.1.2. Chaos Masking ................................ 381
13.1.3. Chaos Modulation ............................. 383
13.1.4. Chaos Shift Keying ........................... 383
13.1.5. Chaotic Data Communications in Laser
Systems ...................................... 384
13.2.Cryptographic Applications in Optical Feedback
Systems .............................................. 385
13.2.1. Chaotic Communications in Optical Feedback
Systems ...................................... 385
13.2.2. Chaos Masking in Optical Feedback Systems .... 388
13.2.3. Chaos Modulation in Optical Feedback
Systems ...................................... 393
13.2.4. Chaos Shift Keying in Optical Feedback
Systems ...................................... 394
13.2.5. Chaotic Communications in Incoherent
Optical Feedback Systems ..................... 396
13.2.6. Chaos Pass Filtering Effects ................. 396
13.3.Cryptographic Applications in Optical Injection
Systems .............................................. 399
13.4.Cryptographic Applications in Optoelectonic
Systems .............................................. 401
13.5.Performance of Chaotic Communications ................ 404
13.6.Security of Chaotic Communications ................... 408
13.7.Chaotic Carrier and Bandwidth of Communications ...... 410
13.8.Chaos Communications in the Real World ............... 412
13.8.1. Chaos Masking Video Signal Transmissions ..... 412
13.8.2. Chaotic Signal Transmissions through Public
Data Link .................................... 414
A Appendix: Chaos ........................................... 419
A.l. Nonlinear Chaotic Systems ............................ 420
A.l.1. Discrete Systems .............................. 420
A.1.2. Continuous Systems ............................ 422
A.1.3. Delay Differential Systems .................... 424
A.2. Analysis and Characteristic Descriptions for
Chaotic Data ......................................... 425
A.2.1. Phase Space, Attractor, and Poincare Map ...... 425
A.2.2. Steady State Behaviors ........................ 427
A.2.3. Fractal Dimension and Correlation Dimension ... 430
A.2.4. Lyapunov Exponent ............................. 431
А.З. Chaos Control ........................................ 432
A.4. Chaos Synchronization ................................ 437
References .................................................... 441
Index ......................................................... 469
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