Light Propagation in Gain Media: Optical Amplifiers

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1 Introduction......Page 17
1.1.1 Maxwell\'s equations in the time domain......Page 18
1.1.2 Maxwell\'s equations in the frequency domain......Page 20
1.1.3 Maxwell\'s equations in the momentum domain......Page 22
1.1.4 Constitutive relations for different optical media......Page 24
1.2.1 Debye-type permittivity and its extensions......Page 25
1.2.2 Lorentz dielectric function......Page 27
1.3 Dispersion relations......Page 28
1.3.1 Dispersion relation in free space......Page 29
1.3.2 Dispersion relation in isotropic materials......Page 30
1.3.3 Refractive index and phase velocity......Page 31
1.3.4 Instabilities associated with a gain medium......Page 33
1.4.1 Hilbert transform......Page 35
1.4.2 Kramers--Kronig relations......Page 37
1.5.1 Continuous-wave plane waves......Page 38
1.5.2 Pulsed plane waves......Page 40
References......Page 42
2 Light propagation through dispersive dielectric slabs......Page 44
2.1 State of polarization of optical waves......Page 45
2.2 Impedance and refractive index......Page 47
2.3.1 Case of plane s-waves......Page 49
2.3.2 Case of plane p-waves......Page 51
2.3.3 Fresnel equations in lossy dielectrics......Page 52
2.4 Propagation of optical pulses......Page 53
2.4.1 One-dimensional propagation model......Page 54
2.4.2 Case of a Gaussian pulse......Page 55
2.4.3 Numerical approach......Page 57
2.5.1 FDTD algorithm in one spatial dimension......Page 58
2.5.2 Total and scattered electromagnetic fields......Page 61
2.5.3 Inclusion of material dispersion......Page 62
2.5.4 Perfectly matched layer for dispersive optical media......Page 64
2.6 Phase and group velocities......Page 69
2.7 Pulse propagation through a dielectric slab......Page 72
2.7.1 Fabry--Perot resonators......Page 73
2.7.2 Transfer function of a dielectric slab......Page 75
References......Page 76
3 Interaction of light with generic active media......Page 79
3.1.1 Total internal reflection and the Goos--Hänchen effect......Page 80
3.1.2 Total internal reflection from an active cladding......Page 81
3.2.1 SPP dispersion relation and other properties......Page 85
3.2.2 Propagation loss and its control......Page 86
3.2.3 Gain-assisted propagation of SPPs......Page 88
3.3 Gain-assisted management of group velocity......Page 91
3.3.1 Dispersion and group velocity......Page 92
3.3.2 Gain-assisted superluminal propagation of light......Page 93
3.4.1 Group velocity of a multilayer cylindrical nanowaveguide......Page 95
3.4.2 Silver nanorod immersed in an active medium......Page 97
References......Page 101
4 Optical Bloch equations......Page 104
4.1 The bra and ket vectors......Page 105
4.2 Density operator......Page 107
4.3 Density-matrix equations for two-level atoms......Page 109
4.3.1 Atomic states with even or odd parity......Page 110
4.3.2 Interaction of a two-level atom with an electromagnetic field......Page 111
4.3.3 Feynman--Bloch vector......Page 113
4.3.4 Rotating-wave approximation......Page 115
4.4 Optical Bloch equations......Page 117
4.5 Maxwell--Bloch equations......Page 120
4.6 Numerical integration of Maxwell--Bloch equations......Page 123
References......Page 127
5 Fiber amplifiers......Page 129
5.1 Erbium-doped fiber amplifiers......Page 130
5.2 Amplifier gain and its bandwidth......Page 132
5.3 Rate equations for EDFAs......Page 135
5.4 Amplification under CW conditions......Page 138
5.5 Amplification of picosecond pulses......Page 140
5.5.1 Inclusion of dopant\'s susceptibility......Page 141
5.5.2 Derivation of the Ginzburg--Landau equation......Page 142
5.5.3 Numerical solution of Ginzburg--Landau equation......Page 145
5.6 Autosolitons and similaritons......Page 147
5.6.1 Autosolitons......Page 148
5.6.2 Similaritons......Page 151
5.7 Amplification of femtosecond pulses......Page 154
References......Page 156
6 Semiconductor optical amplifiers......Page 159
6.1 Material aspects of SOAs......Page 160
6.2 Carrier density and optical gain......Page 163
6.2.1 Rate equation for carrier density......Page 164
6.2.2 Gain characteristics of SOAs......Page 165
6.3.1 General theory of pulse amplification......Page 167
6.3.2 Method of multiple scales......Page 169
6.3.3 Pulse distortion and spectral broadening......Page 173
6.3.4 Impact of waveguide losses......Page 175
6.4 Femtosecond pulse amplification......Page 180
References......Page 187
7 Raman amplifiers......Page 189
7.1.1 Spontaneous Raman scattering......Page 190
7.1.2 Stimulated Raman scattering......Page 191
7.2 Raman gain spectrum of optical fibers......Page 193
7.3.1 CW operation of a Raman amplifier......Page 199
7.3.2 Amplification of ultrashort pulses......Page 201
7.4 Silicon Raman amplifiers......Page 204
7.4.1 Coupled pump and signal equations......Page 205
7.4.2 CW operation of silicon Raman amplifiers......Page 207
7.4.3 Amplification of picosecond pulses......Page 211
References......Page 221
8.1 Physics behind parametric amplification......Page 224
8.2 Phase-matching condition......Page 227
8.3.1 Coupled amplitude equations......Page 228
8.3.2 Impact of nonlinear effects on phase matching......Page 230
8.3.3 Parametric gain......Page 231
8.3.4 Signal gain in parametric amplifiers......Page 233
8.3.5 Amplifier bandwidth......Page 235
8.3.6 Dual-pump parametric amplifiers......Page 237
8.4 Three-wave mixing in birefringent crystals......Page 240
8.4.1 Uniaxial nonlinear crystals......Page 241
8.4.2 Ordinary and extraordinary waves......Page 242
8.4.3 Phase-matching condition......Page 244
8.5 Phase matching in birefringent fibers......Page 245
8.5.1 Polarization-maintaining fibers......Page 246
References......Page 251
9 Gain in optical metamaterials......Page 253
9.1 Classification of metamaterials......Page 254
9.2.1 Review of loss-compensation schemes......Page 257
9.2.2 Modeling of spaser-based amplifiers......Page 259
9.3.1 Configuration and basic equations for three-wave mixing......Page 262
9.3.2 Solution with undepleted-pump approximation......Page 263
9.4 Resonant four-wave mixing using dopants......Page 266
9.4.1 Density-matrix equations for the dopants......Page 267
9.4.2 Iterative solution for the induced polarization......Page 268
9.4.3 Coupled signal and idler equations......Page 270
9.5 Backward self-induced transparency......Page 271
9.5.1 Mathematical approach......Page 272
9.5.2 Bloch equations and Rabi frequencies......Page 274
9.5.3 Solution in the form of a solitary wave......Page 276
References......Page 278
Index......Page 281