Optical Waveguide Modes: Polarization, Coupling and Symmetry

Contents......Page 6
Preface......Page 12
Acknowledgments......Page 14
1.1 Modes......Page 20
1.2 Polarization Dependence of Wave Propagation......Page 22
1.3 Weak-Guidance Approach to Vector Modes......Page 23
1.4 Group Theory for Waveguides......Page 24
1.5.1 Ray Optics Description......Page 26
1.5.2 Wave Optics Description......Page 28
1.5.3 Adiabatic Transitions and Coupling......Page 33
1.6 Outline and Major Results......Page 35
2.1 Electrically Anisotropic (and Isotropic) Media......Page 38
2.2 General Wave Equations for Electrically Anisotropic (and Isotropic) Media......Page 41
2.3 Translational Invariance and Modes......Page 43
2.4.2 Anisotropic Media with z-Aligned Principal Axis......Page 44
2.4.3 “Diagonal” Anisotropies......Page 45
2.6 Scalar Wave Equation......Page 46
2.7 Weak-Guidance Expansion for Isotropic Media......Page 47
2.8.1 First-Order Eigenvalue Correction......Page 49
2.8.3 Simplifications Due to Symmetry......Page 50
2.10 Physical Properties of Waveguide Modes......Page 51
3.1 Summary of Modal Representations......Page 54
3.1.3 Pictorial Representation and Notation Details......Page 55
3.2 Symmetry Concepts for Circular Fibers: Scalar Mode Fields and Degeneracies......Page 61
3.2.2 Scalar Wave Equation Symmetry: C[sup(S)][sub(∞υ)]......Page 65
3.2.3 Scalar Modes: Basis Functions of Irreps of C[sup(S)][sub(∞υ)]......Page 66
3.2.4 Symmetry Tutorial: Scalar Mode Transformations......Page 67
3.3 Vector Mode Field Construction and Degeneracies via Symmetry......Page 69
3.3.1 Vector Field......Page 70
3.3.3 Zeroth-Order Vector Wave Equation Symmetry: C[sup(S)][sub(∞υ)] (Omitted) C[sup(P)]sub[(∞υ)]......Page 71
3.3.4 Pseudo-Vector Modes: Basis Functions of Irreps of C[sup(S)][sub(∞V)] (Omitted) C[sup(P)]sub[(∞V)]......Page 73
3.3.5 Full Vector Wave Equation Symmetry: C[sup(S)][sub(∞V)] (Omitted) C[sup(P)]sub[(∞V)] (Omitted) C[sup(J)][sub(∞V)]......Page 74
3.3.6 True Vector Modes: Qualitative Features via C[sup(S)][sub(∞V)] (Omitted) C[sup(P)]sub[(∞V)] (Omitted) C[sup(J)][sub(∞V)]......Page 75
3.3.7 True Vector Modes via Pseudo-Modes: Basis Functions of C[sup(S)][sub(∞V)] (Omitted) C[sup(P)]sub[(∞V)] (Omitted) C[sup(J)][sub(∞V)]......Page 77
3.4.1 First-Order Eigenvalue Corrections......Page 78
3.4.2 Radial Profile-Dependent Polarization Splitting......Page 79
3.4.3 Special Degeneracies and Shifts for Particular Radial Dependence of Profile......Page 82
3.4.4 Physical Effects......Page 83
4.1.1 Branching Rules......Page 86
4.2.1 Wave Equation Symmetries and Mode-Irrep Association......Page 87
4.2.2 Mode Splittings......Page 88
4.3 C[sub(3v)] Symmetry: Equilateral Triangular Deformations......Page 91
4.4.2 Mode Splitting and Transition Consequences......Page 94
4.5.1 Irreps and Branching Rules......Page 96
4.5.2 Mode Splitting and Transition Consequences......Page 97
4.6.2 Mode Splitting and Transition Consequences......Page 99
4.7 Level Splitting Quantification and Field Corrections......Page 101
5.1.1 Wave Equations: Longitudinal Invariance......Page 102
5.1.2 Mode Transitions: Circular Symmetry......Page 104
5.1.3 Field Component Coupling......Page 106
5.1.4 Splitting by δ[sub(xy)] of Isotropic Fiber Vector Modes Dominated by Δ-Splitting......Page 107
5.2.1 Wave Equations: Longitudinal Invariance......Page 108
5.3 Circular Birefringence......Page 110
5.3.2 Symmetry and Mode Splittings......Page 112
6.1 Multilightguide Structures with Discrete Rotational Symmetry......Page 116
6.1.1 Global C[sub(nv)] Rotation-Reflection Symmetric Structures: Isotropic Materials......Page 117
6.1.3 Global C[sub(n)] Symmetric Structures......Page 118
6.2.1 Propagation Constant Degeneracies......Page 120
6.2.2 Basis Functions for General Field Construction......Page 123
6.3.1 Combinations of Fundamental Individual Core Modes......Page 126
6.3.3 Combinations of Degenerate Individual Core Modes......Page 127
6.4.1 Two Construction Methods......Page 128
6.4.2 Isotropic Cores: Fundamental Mode Combination Supermodes......Page 132
6.4.3 Isotropic Cores: Higher-Order Mode Combination Supermodes......Page 135
6.4.4 Anisotropic Cores: Discrete Global Radial Birefringence......Page 138
6.5.1 SALCs as Basis Functions in General Expansion......Page 140
6.5.3 Approximate SALC Expansions......Page 141
6.5.4 SALC = Supermode Field with Numerical Evaluation of Sector Field Function......Page 142
6.5.5 Harmonic Expansions for Step Profile Cores......Page 143
6.5.6 Example of Physical Interpretation of Harmonic Expansion for the Supermodes......Page 144
6.5.8 Relation of Modal and Harmonic Expansions to SALC Expansions......Page 145
6.6.1 Scalar Supermode Propagation Constant Corrections......Page 146
6.6.2 Vector Supermode Propagation Constant Corrections......Page 149
6.7.1 Scalar Supermode Beating......Page 150
6.7.2 Polarization Rotation......Page 152
7.1 Summary......Page 156
7.2 Periodic Waveguides......Page 157
7.3 Symmetry Analysis of Nonlinear Waveguides and Self-Guided Waves......Page 158
7.4 Developments in the 1990s and Early Twenty-First Century......Page 159
7.5 Photonic Computer-Aided Design (CAD) Software......Page 160
7.6 Photonic Crystals and Quasi Crystals......Page 161
7.7 Microstructured, Photonic Crystal, or Holey Optical Fibers......Page 162
7.8.1 General FBGs for Fiber Mode Conversion......Page 163
7.8.2 (Short-Period) Reflection Gratings for Single-Mode Fibers......Page 164
7.8.4 Example: LP[sub(01)](Omitted)LP[sub(11)] Mode-Converting Transmission FBGs for Two-Mode Fibers (TMFs)......Page 165
7.8.5 Example: LP[sub(01)(Omitted)LP[sub(02)] Mode-Converting Transmission FBGs......Page 167
Appendix: Group Representation Theory......Page 170
A.1 Preliminaries: Notation, Groups, and Matrix Representations of Them......Page 171
A.1.1 Induced Transformations on Scalar Functions......Page 172
A.1.2 Eigenvalue Problems: Invariance and Degeneracies......Page 173
A.1.6 Notation Conventions......Page 174
A.2.2 Irreps for C[sub(∞υ) and C[sub(nυ)]......Page 175
A.3 Reducible Representations and Branching Rule Coefficients via Characters......Page 179
A.3.2 Branching Rule Coefficients via Characters......Page 180
A.4 Clebsch-Gordan Coefficient for Changing Basis......Page 183
A.5 Vector Field Transformation......Page 184
Reference......Page 186
C......Page 198
E......Page 199
G......Page 200
L......Page 201
P......Page 202
R......Page 203
S......Page 204
W......Page 205
Z......Page 206