Optical Waveguide Modes: Polarization, Coupling and Symmetry

by Black, Richard; Gagnon, Langis

• ISBN13: 9780071622967
• ISBN10: 0071622969
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-03-10
• Publisher: McGraw-Hill Education

Summary

This in-depth work explains how transverse optical waveguide geometry influences modal field distribution and polarization properties. You will gain a thorough understanding of the fundamental physics of mode structure.

Author Biography

Dr. Richard J. Black is a leading authority on optical waveguide modes and applications. He is a founding member and chief scientist at Intelligent Fiber Optic Systems Corporation (www.ifos.com) and founder of OptoSapiens Design (www.optosapiens.com).

Dr. Langis GagnonAndnbsp;is a principle researcher and team leader for the Vision and Imaging team at CRIM (Centre de Recherche Informatique de MontrAndeacute;al).

Table of Contents

Preface	p. xi
Acknowledgments	p. xiii
Introduction	p. 1
Modes	p. 1
Polarization Dependence of Wave Propagation	p. 3
Weak-Guidance Approach to Vector Modes	p. 4
Group Theory for Waveguides	p. 5
Optical Waveguide Modes: A Simple Introduction	p. 7
Ray Optics Description	p. 7
Wave Optics Description	p. 9
Adiabatic Transitions and Coupling	p. 14
Outline and Major Results	p. 16
Electromagnetic Theory for Anisotropic Media and Weak Guidance for Longitudinally Invariant Fibers	p. 19
Electrically Anisotropic (and Isotropic) Media	p. 19
General Wave Equations for Electrically Anisotropic (and Isotropic) Media	p. 22
Translational Invariance and Modes	p. 24
Wave Equations for Longitudinally Invariant Media	p. 25
General Anisotropic Media	p. 25
Anisotropic Media with z-Aligned Principal Axis	p. 25
"Diagonal" Anisotropies	p. 26
Transverse Field Vector Wave Equation for Isotropic Media	p. 27
Scalar Wave Equation	p. 27
Weak-Guidance Expansion for Isotropic Media	p. 28
Polarization-Dependent Mode Splitting and Field Corrections	p. 30
First-Order Eigenvalue Correction	p. 30
First-Order Field and Higher-Order Corrections	p. 31
Simplifications Due to Symmetry	p. 31
Reciprocity Relations for Isotropic Media	p. 32
Physical Properties of Waveguide Modes	p. 32
Circular Isotropic Longitudinally Invariant Fibers	p. 35
Summary of Modal Representations	p. 35
Scalar and Pseudo-Vector Mode Sets	p. 36
True Weak-Guidance Vector Mode Set Constructions Using Pseudo-Modes	p. 36
Pictorial Representation and Notation Details	p. 36
Symmetry Concepts for Circular Fibers: Scalar Mode Fields and Degeneracies	p. 42
Geometrical Symmetry; C∞¿	p. 46
Scalar Wave Equation Symmetry: CS∞¿	p. 46
Scalar Modes: Basis Functions of Irreps of CS∞¿	p. 47
Symmetry Tutorial: Scalar Mode Transformations	p. 48
Vector Mode Field Construction and Degeneracies via Symmetry	p. 50
Vector Field	p. 51
Polarization Vector Symmetry Group: CP∞¿	p. 52
Zerolh-Order Vector Wave Equation Symmetry: CS∞¿ ⊗ CP∞¿	p. 52
Pseudo-Vector Modes: Basis Functions of Irreps of CS∞V ⊗ CP∞V	p. 54
Full Vector Wave Equation Symmetry: CS∞V ⊗ CP∞V ⊃ CJ∞V	p. 55
True Vector Modes: Qualitative Features via CS∞V ⊗ CP∞V ⊃ CJ∞V	p. 56
True Vector Modes via Pseudo-Modes: Basis Functions of CS∞V ⊗ CP∞V ⊃ CJ∞V	p. 58
Polarization-Dependent Level-Splitting	p. 59
First-Order Eigenvalue Corrections	p. 59
Radial Profile-Dependent Polarization Splitting	p. 60
Special Degeneracies and Shifts for Particular Radial Dependence of Profile	p. 63
Physical Effects	p. 64
Azimuthal Symmetry Breaking	p. 67
Principles	p. 67
Branching Rules	p. 67
Anticrossing and Mode Form Transitions	p. 68
C2v Symmetry: Elliptical (or Rectangular) Guides: Illustration of Method	p. 68
Wave Equation Symmetries and Mode-Irrep Association	p. 68
Mode Splittings	p. 69
Vector Mode Form Transformations for Competing Perturbations	p. 72
C3v Symmetry: Equilateral Triangular Deformations	p. 72
C4v Symmetry: Square Deformations	p. 75
Irreps and Branching Rules	p. 75
Mode Splitting and Transition Consequences	p. 75
Square Fiber Modes and Extra Degeneracies	p. 77
C5v Symmetry: Pentagonal Deformations	p. 77
Irreps and Branching Rules	p. 77
Mode Splitting and Transition Consequences	p. 78
C6v Symmetry: Hexagonal Deformations	p. 80
Irreps and Branching Rules	p. 80
Mode Splitting and Transition Consequences	p. 80
Level Splitting Quantification and Field Corrections	p. 82
Birefringence: Linear, Radial, and Circular	p. 83
Linear Birefringence	p. 83
Wave Equations: Longitudinal Invariance	p. 83
Mode Transitions: Circular Symmetry	p. 85
Field Component Coupling	p. 87
Splitting by ¿xy of Isotropic Fiber Vector Modes Dominated by ¿-Splitting	p. 88
Correspondence between Isotropic "True" Modes and Birefringent LP Modes	p. 89
Radial Birefringence	p. 89
Wave Equations: Longitudinal Invariance	p. 89
Mode Transitions for Circular Symmetry	p. 91
Circular Birefringence	p. 91
Wave Equation	p. 93
Symmetry and Mode Splittings	p. 93
Multicore Fibers and Multifiber Couplers	p. 97
Multilightguide Structures with Discrete Rotational Symmetry	p. 97
Global Cn¿ Rotation-Reflection Symmetric Structures: Isotropic Materials	p. 98
Global Cn¿ Symmetry: Material and Form Birefringence	p. 99
Global Cn Symmetric Structures	p. 99
General Supermode Symmetry Analysis	p. 101
Propagation Constant Degeneracies	p. 101
Basis Functions for General Field Construction	p. 104
Scalar Supermode Fields	p. 107
Combinations of Fundamental Individual Core Modes	p. 107
Combinations of Other Nondegenerate Individual Core Modes	p. 108
Combinations of Degenerate Individual Core Modes	p. 108
Vector Supermode Fields	p. 109
Two Construction Methods	p. 109
Isotropic Cores: Fundamental Mode Combination Supermodes	p. 113
Isotropic Cores: Higher-Order Mode Combination Supermodes	p. 116
Anisotropic Cores: Discrete Global Radial Birefringence	p. 119
Other Anisotropic Structures: Global Linear and Circular Birefringence	p. 121
General Numerical Solutions and Field Approximation Improvements	p. 121
SALCs as Basis Functions in General Expansion	p. 121
Variational Approach	p. 122
Approximate SALC Expansions	p. 122
SALC = Supermode Field with Numerical Evaluation of Sector Field Function	p. 123
Harmonic Expansions for Step Profile Cores	p. 124
Example of Physical Interpretation of Harmonic Expansion for the Supermodes	p. 125
Modal Expansions	p. 126
Relation of Modal and Harmonic Expansions to SALC Expansions	p. 126
Finite Claddings and Cladding Modes	p. 127
Propagation Constant Splitting: Quantification	p. 127
Scalar Supermode Propagation Constant Corrections	p. 127
Vector Supermode Propagation Constant Corrections	p. 130
Power Transfer Characteristics	p. 131
Scalar Supermode Beating	p. 131
Polarization Rotation	p. 133
Conclusions and Extensions	p. 137
Summary	p. 137
Periodic Waveguides	p. 138
Symmetry Analysis of Nonlinear Waveguides and Self-Guided Waves	p. 139
Developments in the 1990s and Early Twenty-First Century	p. 140
Photonic Computer-Aided Design (CAD) Software	p. 141
Photonic Crystals and Quasi Crystals	p. 142
Microstructured, Photonic Crystal, or Holey Optical Fibers	p. 143
Fiber Bragg Gratings	p. 144
General FBGs for Fiber Mode Conversion	p. 144
(Short-Period) Reflection Gratings for Single-Mode Fibers	p. 145
(Long-Period) Mode Conversion Transmission Gratings	p. 146
Example: LP01↔LP11 Mode-Converting Transmission FBGs for Two-Mode Fibers (TMFs)	p. 146
Example: LP01↔LP02 Mode-Converting Transmission FBGs	p. 148
Appendix: Group Representation Theory	p. 151
Preliminaries: Notation, Groups, and Matrix Representations of Them	p. 152
Induced Transformations on Scalar Functions	p. 153
Eigenvalue Problems: Invariance and Degeneracies	p. 154
Croup Representations	p. 155
Matrix Irreducible Matrix Representations	p. 155
Irrep Basis Functions	p. 155
Notation Conventions	p. 155
Rotation-Reflection Groups	p. 156
Symmetry Operation and Group Definitions	p. 156
Irreps for C∞¿-and Cn¿	p. 156
Irrep Notation	p. 160
Reducible Representations and Branching Rule Coefficients via Characters	p. 160
Example Branching Rule for C∞¿ ⊃ C2¿	p. 161
Branching Rule Coefficients via Characters	p. 161
Clebsch-Gordan Coefficient for Changing Basis	p. 164
Vector Field Transformation	p. 165
References	p. 167
Index	p. 179
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