Fundamentals of Optical Fibers (Wiley Series in Pure and Applied Optics)/ John A.Buck,

Chapter 1. Selected Topics in Electromagnetic Wave Propagation
1.1. Maxwell's Equations and the Fundamental Fields 1
1.2. Electromagnetic Wave Propagation in Sourceless Media 2
1.2.1. Wave Equations in Simple Media 3
1.2.2. Time-Harmonic Field Solutions 3
1.2.3. Vector Helmholtz Equations and the Uniform
Plane Wave
1.2.4. E and H as Related Through the Intrinsic Impedance 5
1.3. Power Transmission ^
1.3.1. Computation of the Time-Average Power Density 6
1.3.2. Standing Wave Power 7
1.4. Group Velocity ^
1.4.1. Propagation of a Wave Containing Two Frequencies 8
1.4.2. Group Velocity Definition 9
1.5. Reflection and Transmission of Waves at Plane Interfaces 10
1.5.1. Reflection Geometry 10
1.5.2. Applying the Field Boundary Conditions 11
1.5.3. Special Cases: Total Transmission and Total Reflection 13
1.6. Material Resonances and Their Effects on Wave Propagation 13
1.6.1. The Classical Electron Oscillator Model and the Electric
Susceptibility
1.6.2. Wave Propagation in Media with Complex
Susceptibilities
1.6.3. Off-Resonance Behavior and the Sellmeier Equation 18
1.6.4. Time Domain Analysis 18
Problems
91
References
Chapter2. Symmetric Dielectric Slab Waveguides 22
2.1. Ray Analysis of the Slab Waveguide 22
2.1.1. Guided Mode Requirements and Mode Types 23
2.1.2. Plane Wave Field Representations 25
2.1.3. Surface Waves and the Reflective Phase Shift 26
2.1.4. Transverse Resonance and the Eigenvalue Equations 28CONTENTS
2.2. Field Analy.si.s of the Slab Waveguide 29
2.2.1. Solving for the Longitudinal Field Components 29
2.2.2. Obtaining the Transverse Field Components 31
2.3. Solutions of the Eigenvalue Equations 32
2.3.1. Graphical Solution Method 32
2.3.2. Interpreting the Graphical Solution 33
2.4. Power Transmission and Confinement 34
2.4.1. Power Computation and the Confinement Factor 35
2.4.2. Mode Orthogonality 35
2.5. Leaky Waves 35
2.5.1. TE and TM Polarization 38
2.5.2. Power Loss in Leaky Wave Transmission 38
2.6. Radiation Modes 40
2.6.1. Physical Description of Radiation Modes 40
2.6.2. Summary of Wave Types 41
2.7. Wave Propagation in Curved Slab Waveguides 42
2.7.1. Basic Concepts of Curved Guiding 42
2.7.2. Criteria for Small Curvature Loss 43
2.7.3. Analysis of Curved Slab Waveguides Through
Conformal Transformation 44
Problems 47
References 50
Chapter3. Weakly-Guiding Fibers with Step Index Profiles 51
3.1. Rays and Fields in the Step Index Fiber 53
3.1.1. Ray Trajectories and Transverse Resonance 53
3.1.2. Relations Between Ray Paths and Mode Field Patterns 55
3.1.3. Weakly Guiding Fibers and the LP Modes 55
3.2. Field Analysis of the Weakly Guiding Fiber 56
3.2.1. Assumed Field Solutions and Wave Equations for LP
Modes 56
3.2.2. Solving the Wave Equation 57
3.2.3. Evalifating the Coefficients 59
3.3. Eigenvalue Equation for LP Modes 60
3.3.1. Derivation of the Eigenvalue Equation 60
3.3.2. Graphical Solution Method 62
3.3.3. Cutoff Conditions and Mode Designations 65
3.4. LP Mode Characteristics 66
3.4.1. Intensity Patterns and Polarizations 66
3.4.2. Parameter Computation 69
3.4.3. Power Confinement 71
3.5. Single-Mode Fiber Parameters 73
3.5.1. Cutoff Wavelength 73
3.5.2. Gaussian Approximation for the LPoi Mode Field 75CONTENTS ix
3.6. Derivation of the General Step Index Fiber Fields 79
3.6.1. Mode Field Derivation 80
3.6.2. Mode Classification and the Eigenvalue Equation 81
3.6.3. The Eigenvalue Equation Under the Weak-Guidance
Approximation 82
3.6.4. General Mode Fields Under the Weak-Guidance
Approximation 84
3.6.5. LP Modes as Superpositions of General Modes 85
Problems 87
References 90
Chapter4. Loss Mechanisms in Silica Fiber 92
4.1. Basic Loss Effects in Transmission 93
4.2. Fabrication of Silica Fibers 94
4.2.1. Perform Manufacturing Using MCVD 94
4.2.2. Dopants for Control of Refractive Index 95
4.2.3. Perform Completion and Fiber Drawing 96
4.3. Intrinsic Loss 97
4.3.1. Ultraviolet Absorption 97
4.3.2. Infrared Absorption 97
4.3.3. Rayleigh Scattering 98
4.3.4. Combined Intrinsic Losses 100
4.4. Extrinsic Loss 101
4.4.1. Metallic and Rare Earth Impurities 101
4.4.2. Loss Arising from OH 102
4.5. Bending Loss 103
4.5.1. Wave Theory of Macrobending Loss 104
4.5.2. Additional Factors That Influence Macrobending Loss 108
4.5.3. Microbending Loss 109
4.6. Source-to-Fiber Coupling 112
4.6.1. Single-Mode Fiber Splicing 113
4.6.2. Gaussian Beam Input Coupling 115
4.6.3. General Source Coupling to Multimode Fiber 117
4.6.4. Imaging Methods in Extended-Source Coupling 119
Problems ' 120
References 122
Chapters. Dispersion 125
5.1. Pulse Propagation in Media Possessing Quadratic Dispersion 126
5.1.1. Propagation ofTransform-Limited Gaussian Pulses 126
5.1.2. Input Pulses with Initial Chirp 131
5.1.3. Gaussian Pulses Having Excess Bandwidth 133
5.1.4. Characterizing Arbitrarily Shaped Pulses 134
5.1.5. Cubic Dispersion 136* CONTENTS
5.2. Material Di.spersion 138
5.2.1. Group Delay and Group Index 138
5.2.2. Di.spersion Parameter I4I
5.2.3. Wavelength Domain Description of Cubic Dispersion 142
5.3. Di.spersion in Optical Fiber 145
5.3.1. Group Delay in Step-Index Fiber 145
5.3.2. Group Dispersion in Single-Mode Fiber 149
5.4. Chromatic Dispersion Compensation 153
5.4.1. Dispersion-Compensating Fiber 153
5.4.2. Gires-Tournois Interferometer 154
5.4.3. Chirped Fiber Bragg Grating 159
5.5. Polarization Dispersion 161
5.5.1. Wave Polarization in Single-Mode Fiber 162
5.5.2. Differential Group Delay and Polarization Mode
Dispersion in the Intrinsic Regime 164
5.5.3. Polarization Mode Di.sper.sion in the Coupled Regime 166
5.6. System Considerations and Di.spersion Mea.surement 172
5.6.1. Linear System Model—Fiber Bandwidth 173
5.6.2. Dispersion Limits 174
5.6.3. Dispersion Measurement 176
Problems 178
References 183
Chapter6. Special-Purpose Index Profiles 185
6.1. Multimode Graded Index Fiber 185
6.1.1. Ray Optics Picture 186
6.1.2. Field Analysis 188
6.1.3. Index Profile Optimization 194
6.2. Special Index Profiles in Single-Mode Fiber 198
6.2.1. The Equivalent Step Index Method 198
6.2.2. Index Profiles for Control of Loss and Dispersion 207
6.2.3. Polarization-Maintaining Fiber 214
6.2.4. Photonic Crystal Fiber 219
Problems 223
References 225
Chapter?. Nonlinear Effects in Fibers I: Nonresonant Processes 228
7.1. Nonlinear Optics Fundamentals 229
7.1.1. The Role of Medium Polarization in Wave Propagation 229
7.1.2. The Nonlinear Polarization 230
7.1.3. The Structure ofthe Nonlinear Susceptibility 232
7.1.4. Symmetries in the Third-Order Susceptibility Tensor 235
7.1.5. Example:Third Harmonic Generation 237
7.2. Nonlinear Phase Modulation on Pulses 241
7.2.1. Nonlinear Refractive Index . 241
7.2.2. Self-Phase Modulation 243CONTENTS XI
7.3. The Nonlinear Schrodinger Equation 245
7.3.1. Development of the Nonlinear Schrodinger Equation
from the Wave Equation 246
7.3.2. Normalized Form of the Nonlinear Schrodinger
Equation 249
7.3.3. Optical Solitons 251
7.4. Additional Nonresonant Proces.se.s 255
7.4.1. Cross-Phase Modulation 258
7.4.2. Four-Wave Mixing 260
Problems 263
References 265
Chapters. Nonlinear Effects in Fibers II: Resonant Processes
and Amplification 267
8.1. Raman Scattering 268
8.1.1. Basic Theory of Stimulated Raman Scattering 268
8.1.2. Raman Gain in Silica Fiber 274
8.1.3. Stimulated and Spontaneous Raman Scattering in Fiber 276
8.1.4. Multiple Stokes Orders and Raman Cross-Talk 279
8.1.5. Raman Fiber Amplifiers 282
8.2. Stimulated Brillouin Scattering 285
8.2.1. Stimulated Brillouin Scattering as a Third-Order
Process 286
8.2.2. The Acoustic Displacement Equation 287
8.2.3. The Nonlinear Polarizations and Coupled Equations for
Stimulated Brillouin Scattering 288
8.2.4. Brillouin Amplification 290
8.2.5. Adapting the Theory to Optical Fibers 292
8.3. Rare-Earth-Doped Fiber Amplifiers 293
8.3.1. Basic Theory of Amplification by Stimulated Emission 294
8.3.2. Absorption and Emission Characteristics of
Erbium-Doped Fiber 296
8.3.3. Erbium-Doped Fiber Amplifier Fabrication,
Configuration,and Operating Regimes 302
8.3.4. Gain Flattening and Noise 304
8.3.5. Other Rare-Earth-Doped Systems 305