Output Coupling in Optical Cavities and Lasers: A Quantum Theoretical Approach

Output Coupling in Optical Cavities and Lasers......Page 5
Contents......Page 9
Preface......Page 15
Acknowledgments......Page 19
1.1 Boundary Conditions at Perfect Conductor and Dielectric Surfaces......Page 21
1.2.1 One-Sided Cavity......Page 22
1.2.2 Symmetric Two-Sided Cavity......Page 25
1.3.1 One-Sided Cavity......Page 27
1.3.2 Symmetric Two-Sided Cavity......Page 32
1.4 Discrete versus Continuous Mode Distribution......Page 35
1.6 Completeness of the Modes of the \'\'Universe\'\'......Page 37
2.1 Quantization......Page 43
2.2 Energy Eigenstates......Page 44
2.3 Field Commutation Relation......Page 46
2.4.1 The Density Operator of the Thermal Radiation Field......Page 48
2.4.2 The Correlation Function and the Power Spectrum......Page 49
2.4.3 The Response Function and the Fluctuation–Dissipation Theorem......Page 51
2.4.4 Derivation of the Langevin Noise for a Single Cavity Resonant Mode......Page 53
2.4.5 Excitation of the Cavity Resonant Mode by a Current Impulse......Page 57
2.5.1 Description of the Cavity Structure......Page 58
2.5.2 The Modes of the \'\'Universe\'\'......Page 60
3.1 Cavity Resonant Modes......Page 67
3.3 The Atom–Field Interaction......Page 69
3.4 Equations Governing the Atom–Field Interaction......Page 71
3.5.1 The Field Decay......Page 73
3.5.2 Relaxation in Atomic Dipole and Atomic Inversion......Page 75
4.1 Semiclassical Linear Gain Analysis......Page 81
4.2 Semiclassical Nonlinear Gain Analysis......Page 84
4.3 Quantum Linear Gain Analysis......Page 87
4.4 Quantum Nonlinear Gain Analysis......Page 94
5.1 The Field......Page 101
5.2 The Atoms......Page 103
5.3 The Atom–Field Interaction......Page 104
5.4 Langevin Forces for the Atoms......Page 105
5.5 Laser Equation of Motion for a Laser with Output Coupling......Page 106
6.1 Contour Integral Method: Semiclassical Linear Gain Analysis......Page 111
6.2 Contour Integral Method: Semiclassical Nonlinear Gain Analysis......Page 114
6.3 Contour Integral Method: Quantum Linear Gain Analysis......Page 115
6.4 Contour Integral Method: Quantum Nonlinear Gain Analysis......Page 120
7 A One-Dimensional Laser with Output Coupling: Semiclassical Linear Gain Analysis......Page 123
7.1 The Field Equation Inside the Cavity......Page 124
7.2 Homogeneously Broadened Atoms and Uniform Atomic Inversion......Page 126
7.3.1 The Field Equation for Inside the Cavity......Page 128
7.3.2 Laplace-Transformed Equations......Page 129
7.3.3 The Field Inside the Cavity......Page 133
7.3.4 The Field Outside the Cavity......Page 134
8.1 The Field Equation Inside the Cavity......Page 139
8.2 Homogeneously Broadened Atoms and Uniform Pumping......Page 141
8.3 The Steady State......Page 142
8.4 Solution of the Coupled Nonlinear Equations......Page 145
8.5 The Field Outside the Cavity......Page 149
9 A One-Dimensional Laser with Output Coupling: Quantum Linear Gain Analysis......Page 153
9.1 The Equation for the Quantum Linear Gain Analysis......Page 154
9.2 Homogeneously Broadened Atoms and Uniform Atomic Inversion......Page 157
9.3 Laplace-Transformed Equations......Page 158
9.4 Laplace-Transformed Noise Forces......Page 160
9.5 The Field Inside the Cavity......Page 164
9.5.1 Thermal Noise......Page 166
9.5.2 Quantum Noise......Page 168
9.5.3 The Total Field......Page 171
9.6 The Field Outside the Cavity......Page 174
9.7 The Field Correlation Function......Page 176
9.8 The Laser Linewidth and the Correction Factor......Page 182
10.1 The Equation for the Quantum Nonlinear Gain Analysis......Page 187
10.2 Homogeneously Broadened Atoms and Uniform Pumping......Page 190
10.3 The Steady-State and Laplace-Transformed Equations......Page 191
10.4 The Lowest-Order Solution......Page 196
10.5.1 The Formal Temporal Differential Equation......Page 198
10.5.3 Quantum Noise......Page 202
10.5.4 The Temporal Differential Equation......Page 206
10.5.5 Penetration of Thermal Noise into the Cavity......Page 207
10.6 Phase Diffusion and the Laser Linewidth......Page 208
10.7.1 Phase Diffusion......Page 210
10.7.2 Evaluation of the Sum [equations omitted]......Page 216
10.7.3 The Linewidth and the Correction Factors......Page 219
10.8 The Field Outside the Cavity......Page 222
11.1 Model of the Laser and the Noise Sources......Page 231
11.2 The Steady State and the Threshold Condition......Page 234
11.3 The Time Rate of the Amplitude Variation......Page 238
11.4 The Phase Diffusion of the Output Field......Page 241
11.5 The Linewidth for the Nonlinear Gain Regime......Page 243
11.6 The Linewidth for the Linear Gain Regime......Page 248
12.1 Models of the Quasimode Laser and Continuous Mode Laser......Page 255
12.2.1 Thermal Noise and Vacuum Fluctuation as Input Noise......Page 256
12.2.2 Quantum Noise......Page 257
12.3 Operator Orderings......Page 258
12.4.1 Longitudinal Excess Noise Factor Below Threshold......Page 259
12.4.2 Longitudinal Excess Noise Factor Above Threshold......Page 260
12.5 Mathematical Relation between Below-Threshold and Above-Threshold Linewidths......Page 261
12.6 Detuning Effects......Page 263
12.7 Bad Cavity Effect......Page 265
12.8 Incomplete Inversion and Level Schemes......Page 266
12.9 The Constants of Output Coupling......Page 267
12.10 Threshold Atomic Inversion and Steady-State Atomic Inversion......Page 269
12.11 The Power-Independent Part of the Linewidth......Page 271
12.12 Linewidth and Spontaneous Emission Rate......Page 273
12.12.1 Spontaneous Emission in the Quasimode Laser......Page 274
12.12.2 Spontaneous Emission in the One-Sided Cavity Laser......Page 276
12.13.1 Filling Factor......Page 278
12.13.3 Amplitude–Phase Coupling......Page 279
12.13.4 Internal Loss......Page 280
12.13.5 Spatial Hole Burning......Page 283
12.13.6 Transition From Below Threshold to Above Threshold......Page 284
13.1 Equations Describing the Spontaneous Emission Process......Page 287
13.2 The Perturbation Approximation......Page 290
13.3 Wigner–Weisskopf Approximation......Page 291
13.4 The Delay Differential Equation......Page 292
13.5 Expansion in Terms of Resonant Modes and Single Resonant Mode Limit......Page 295
13.6 Spontaneous Emission Spectrum Observed Outside the Cavity......Page 299
13.7 Extension to Three Dimensions......Page 302
13.8 Experiments on Spontaneous Emission in a Fabry–Perot Type Cavity......Page 309
14.1 Adjoint Mode Theory......Page 313
14.2 Green\'s Function Theory......Page 326
14.3 Propagation Theory......Page 331
14.4 Three-Dimensional Cavity Modes and Transverse Effects......Page 336
14.5.1 Excess Noise Theory Based on Input–Output Commutation Rules......Page 339
14.5.2 Excess Noise Theory Based on Non-Orthogonal Mode Quantization......Page 343
14.6 Two Non-Orthogonal Modes with Nearly Equal Losses......Page 346
14.8 Experiments on Excess Noise Factor......Page 349
15 Quantum Theory of the Output Coupling of an Optical Cavity......Page 355
15.1.1 Normal Mode Expansion......Page 356
15.1.2 Natural Mode Quantization......Page 364
15.1.3 Projection Operator Method......Page 368
15.2.1 The Input–Output Theory by Time Reversal......Page 369
15.2.2 The Input–Output Theory by the Boundary Condition......Page 371
15.2.3 Another Quantum Noise Theory......Page 374
15.5 Summary......Page 375
15.6 Equations for the Output Coupling and Input–Output Relation......Page 376
Appendices......Page 379
C......Page 405
E......Page 406
L......Page 407
O......Page 408
S......Page 409
Z......Page 410