Preface ...................................................... XIII
Acknowledgments .............................................. XVII
1 A One-Dimensional Optical Cavity with Output Coupling:
Classical Analysis ........................................... 1
1.1 Boundary Conditions at Perfect Conductor and
Dielectric Surfaces ..................................... 1
1.2 Classical Cavity Analysis ............................... 2
1.2.1 One-Sided Cavity ................................. 2
1.2.2 Symmetric Two-Sided Cavity ....................... 5
1.3 Normal Mode Analysis: Orthogonal Modes .................. 7
1.3.1 One-Sided Cavity ................................. 7
1.3.2 Symmetric Two-Sided Cavity ...................... 12
1.4 Discrete versus Continuous Mode Distribution ........... 15
1.5 Expansions of the Normalization Factor ................. 17
1.6 Completeness of the Modes of the "Universe" ............ 17
2 A One-Dimensional Optical Cavity with Output Coupling:
Quantum Analysis ............................................ 23
2.1 Quantization ........................................... 23
2.2 Energy Eigenstates ..................................... 24
2.3 Field Commutation Relation ............................. 26
2.4 Thermal Radiation and the Fluctuation-Dissipation
Theorem ................................................ 28
2.4.1 The Density Operator of the Thermal Radiation
Field ........................................... 28
2.4.2 The Correlation Function and the Power
Spectrum ........................................ 29
2.4.3 The Response Function and the Fluctuation-
Dissipation Theorem ............................. 31
2.4.4 Derivation of the Langevin Noise for a Single
Cavity Resonant Mode ............................ 33
2.4.5 Excitation of the Cavity Resonant Mode by
a Current Impulse
2.5 Extension to an Arbitrarily Stratified Cavity .......... 38
2.5.1 Description of the Cavity Structure ............. 38
2.5.2 The Modes of the "Universe" ..................... 40
3 A One-Dimensional Quasimode Laser: General Formulation ...... 47
3.1 Cavity Resonant Modes .................................. 47
3.2 The Atoms .............................................. 49
3.3 The Atom-Field Interaction ............................. 49
3.4 Equations Governing the Atom-Field Interaction ......... 51
3.5 Laser Equation of Motion: Introducing the Langevin
Forces ................................................. 53
3.5.1 The Field Decay ................................. 53
3.5.2 Relaxation in Atomic Dipole and Atomic
Inversion ....................................... 55
4 A One-Dimensional Quasimode Laser: Semiclassical and
Quantum Analysis ............................................ 61
4.1 Semiclassical Linear Gain Analysis ..................... 61
4.2 Semiclassical Nonlinear Gain Analysis .................. 64
4.3 Quantum Linear Gain Analysis ........................... 67
4.4 Quantum Nonlinear Gain Analysis ........................ 74
5 A One-Dimensional Laser with Output Coupling: Derivation
of the Laser Equation of Motion ............................. 81
5.1 The Field .............................................. 81
5.2 The Atoms .............................................. 83
5.3 The Atom-Field Interaction ............................. 84
5.4 Langevin Forces for the Atoms .......................... 85
5.5 Laser Equation of Motion for a Laser with Output
Coupling ............................................... 86
6 A One-Dimensional Laser with Output Coupling:
Contour Integral Method ..................................... 91
6.1 Contour Integral Method: Semiclassical Linear Gain
Analysis ............................................... 91
6.2 Contour Integral Method: Semiclassical Nonlinear Gain
Analysis ............................................... 94
6.3 Contour Integral Method: Quantum Linear Gain
Analysis ............................................... 95
6.4 Contour Integral Method: Quantum Nonlinear Gain
Analysis .............................................. 100
7 A One-Dimensional Laser with Output Coupling:
Semiclassical Linear Cain Analysis ......................... 103
7.1 The Field Equation Inside the Cavity .................. 104
7.2 Homogeneously Broadened Atoms and Uniform Atomic
Inversion ............................................. 106
7.3 Solution of the Laser Equation of Motion .............. 108
7.3.1 The Field Equation for Inside the Cavity ....... 108
7.3.2 Laplace-Transformed Equations .................. 109
7.3.3 The Field Inside the Cavity .................... 113
7.3.4 The Field Outside the Cavity ................... 114
8 A One-Dimensional Laser with Output Coupling:
Semiclassical Nonlinear Cain Analysis ...................... 119
8.1 The Field Equation Inside the Cavity .................. 119
8.2 Homogeneously Broadened Atoms and Uniform Pumping ..... 121
8.3 The Steady State ...................................... 122
8.4 Solution of the Coupled Nonlinear Equations ........... 125
8.5 The Field Outside the Cavity .......................... 129
9 A One-Dimensional Laser with Output Coupling: Quantum
Linear Cain Analysis ....................................... 133
9.1 The Equation for the Quantum Linear Gain Analysis ..... 134
9.2 Homogeneously Broadened Atoms and Uniform Atomic
Inversion ............................................. 137
9.3 Laplace-Transformed Equations ......................... 138
9.4 Laplace-Transformed Noise Forces ...................... 140
9.5 The Field Inside the Cavity ........................... 144
9.5.1 Thermal Noise .................................. 146
9.5.2 Quantum Noise .................................. 148
9.5.3 The Total Field ................................ 151
9.6 The Field Outside the Cavity .......................... 254
9.7 The Field Correlation Function ........................ 156
9.8 The Laser Linewidth and the Correction Factor ......... 162
10 A One-Dimensional Laser with Output Coupling: Quantum
Nonlinear Cain Analysis .................................... 167
10.1 The Equation for the Quantum Nonlinear Gain
Analysis .............................................. 167
10.2 Homogeneously Broadened Atoms and Uniform Pumping ..... 170
10.3 The Steady-State and Laplace-Transformed Equations .... 171
10.4 The Lowest-Order Solution ............................. 176
10.5 The First-Order Solution: Temporal Evolution .......... 178
10.5.1 The Formal Temporal Differential Equation ...... 178
10.5.2 Thermal Noise .................................. 182
10.5.3 Quantum Noise .................................. 182
10.5.4 The Temporal Differential Equation ............. 186
10.5.5 Penetration of Thermal Noise into the Cavity ... 187
10.6 Phase Diffusion and the Laser Linewidth ............... 188
10.7 Phase Diffusion in the Nonlinear Gain Regime .......... 190
10.7.1 Phase Diffusion ................................ 190
10.7.2 Evaluation of the Sum ∑m(|Am|2 + |Bm|2) ......... 196
10.7.3 The Linewidth and the Correction Factors ....... 199
10.8 The Field Outside the Cavity .......................... 202
11 Analysis of a One-Dimensional Laser with Two-Side Output
Coupling: The Propagation Method ........................... 211
11.1 Model of the Laser and the Noise Sources .............. 211
11.2 The Steady State and the Threshold Condition .......... 214
11.3 The Time Rate of the Amplitude Variation .............. 218
11.4 The Phase Diffusion of the Output Field ............... 221
11.5 The Linewidth for the Nonlinear Gain Regime ........... 223
11.6 The Linewidth for the Linear Gain Regime .............. 228
12 A One-Dimensional Laser with Output Coupling: Summary
and Interpretation of the Results .......................... 235
12.1 Models of the Quasimode Laser and Continuous Mode
Laser ................................................. 235
12.2 Noise Sources ......................................... 236
12.2.1 Thermal Noise and Vacuum Fluctuation as Input
Noise .......................................... 236
12.2.2 Quantum Noise .................................. 237
12.3 Operator Orderings .................................... 238
12.4 Longitudinal Excess Noise Factor ...................... 239
12.4.1 Longitudinal Excess Noise Factor Below
Threshold ...................................... 239
12.4.2 Longitudinal Excess Noise Factor Above
Threshold ...................................... 240
12.5 Mathematical Relation between Below-Threshold and
Above-Threshold Linewidths ............................ 241
12.6 Detuning Effects ...................................... 243
12.7 Bad Cavity Effect ..................................... 245
12.8 Incomplete Inversion and Level Schemes ................ 246
12.9 The Constants of Output Coupling ...................... 247
12.10 Threshold Atomic Inversion and Steady-State Atomic
Inversion ............................................ 249
12.11 The Power-Independent Part of the Linewidth .......... 251
12.12 Linewidth and Spontaneous Emission Rate .............. 253
12.12.1 Spontaneous Emission in the Quasimode
Laser ........................................ 254
12.12.2 Spontaneous Emission in the One-Sided
Cavity Laser ................................. 256
12.12.3 Spontaneous Emission in the Two-Sided
Cavity Laser ................................. 258
12.13 Further Theoretical Problems ......................... 258
12.13.1 Filling Factor ............................... 258
12.13.2 Inhomogeneous Broadening ..................... 259
12.13.3 Amplitude-Phase Coupling ..................... 259
12.13.4 Internal Loss ................................ 260
12.13.5 Spatial Hole Burning ......................... 263
12.13.6 Transition From Below Threshold to Above
Threshold .................................... 264
13 Spontaneous Emission in a One-Dimensional Optical Cavity
with Output Coupling ....................................... 267
13.1 Equations Describing the Spontaneous Emission
Process ............................................... 267
13.2 The Perturbation Approximation ........................ 270
13.3 Wigner-Weisskopf Approximation ........................ 271
13.4 The Delay Differential Equation ....................... 272
13.5 Expansion in Terms of Resonant Modes and Single
Resonant Mode Limit ................................... 275
13.6 Spontaneous Emission Spectrum Observed Outside the
Cavity ................................................ 279
13.7 Extension to Three Dimensions ......................... 282
13.8 Experiments on Spontaneous Emission in a Fabry-Perot
Type Cavity ........................................... 289
14 Theory of Excess Noise ..................................... 293
14.1 Adjoint Mode Theory ................................... 293
14.2 Green's Function Theory ............................... 306
14.3 Propagation Theory .................................... 311
14.4 Three-Dimensional Cavity Modes and Transverse
Effects ............................................... 316
14.5 Quantum Theory of Excess Noise Factor ................. 319
14.5.1 Excess Noise Theory Based on Input-Output
Commutation Rules .............................. 319
14.5.2 Excess Noise Theory Based on Non-Orthogonal
Mode Quantization .............................. 323
14.6 Two Non-Orthogonal Modes with Nearly Equal Losses ..... 326
14.7 Multimode Theory ...................................... 329
14.8 Experiments on Excess Noise Factor .................... 329
15 Quantum Theory of the Output Coupling of an Optical
Cavity ..................................................... 335
15.1 Quantum Field Theory .................................. 336
15.1.1 Normal Mode Expansion .......................... 336
15.1.2 Natural Mode Quantization ...................... 344
15.1.3 Projection Operator Method ..................... 348
15.2 Quantum Noise Theory .................................. 349
15.2.1 The Input-Output Theory by Time Reversal ....... 349
15.2.2 The Input-Output Theory by the Boundary
Condition ...................................... 351
15.2.3 Another Quantum Noise Theory ................... 354
15.3 Green's Function Theory ............................... 355
15.4 Quasimode Theory ...................................... 355
15.5 Summary ............................................... 355
15.6 Equations for the Output Coupling and Input-Output
Relation .............................................. 356
Appendices .................................................... 359
Index ......................................................... 385
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