Optical Waveguide Modes: Polarization, Coupling and Symmetry / Edition 1

by Richard J. Black, Langis Gagnon

Overview

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

Optical Waveguide Modes covers single- and few-mode optical waveguides with an emphasis on single-core and multicore optical fibers and couplers, including a large range of geometries and anisotropies. Analysis is performed using extensions of the weak-guidance perturbation formalism together with elementary group representation theory. This definitive volume offers a detailed introduction to and classification of diverse forms of fundamental and higher-order modes and various polarization manifestations.

Coverage includes:

• Electromagnetic theory for anisotropic media
• Weak guidance for longitudinally invariant fibers
• Circular isotropic longitudinally invariant fibers
• Azimuthal symmetry breaking
• Birefringence: linear, radial, and circular
• Multicore and multifiber couplers

Product Details

ISBN-13:	9780071622967
Publisher:	McGraw Hill LLC
Publication date:	03/12/2010
Pages:	208
Product dimensions:	6.20(w) x 9.10(h) x 0.70(d)

About the Author

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 Gagnon is a principle researcher and team leader for the Vision and Imaging team at CRIM (Centre de Recherche Informatique de Montréal).

Table of Contents

Preface xi

Acknowledgments xiii

Chapter 1 Introduction 1

1.1 Modes 1

1.2 Polarization Dependence of Wave Propagation 3

1.3 Weak-Guidance Approach to Vector Modes 4

1.4 Group Theory for Waveguides 5

1.5 Optical Waveguide Modes: A Simple Introduction 7

1.5.1 Ray Optics Description 7

1.5.2 Wave Optics Description 9

1.5.3 Adiabatic Transitions and Coupling 14

1.6 Outline and Major Results 16

Chapter 2 Electromagnetic Theory for Anisotropic Media and Weak Guidance for Longitudinally Invariant Fibers 19

2.1 Electrically Anisotropic (and Isotropic) Media 19

2.2 General Wave Equations for Electrically Anisotropic (and Isotropic) Media 22

2.3 Translational Invariance and Modes 24

2.4 Wave Equations for Longitudinally Invariant Media 25

2.4.1 General Anisotropic Media 25

2.4.2 Anisotropic Media with z-Aligned Principal Axis 25

2.4 3 "Diagonal" Anisotropies 26

2.5 Transverse Field Vector Wave Equation for Isotropic Media 27

2.6 Scalar Wave Equation 27

2.7 Weak-Guidance Expansion for Isotropic Media 28

2.8 Polarization-Dependent Mode Splitting and Field Corrections 30

2.8.1 First-Order Eigenvalue Correction 30

2.8.2 First-Order Field and Higher-Order Corrections 31

2.8.3 Simplifications Due to Symmetry 31

2.9 Reciprocity Relations for Isotropic Media 32

2.10 Physical Properties of Waveguide Modes 32

Chapter 3 Circular Isotropic Longitudinally Invariant Fibers 35

3.1 Summary of Modal Representations 35

3.1.1 Scalar and Pseudo-Vector Mode Sets 36

3.1.2 True Weak-Guidance Vector Mode Set Constructions Using Pseudo-Modes 36

3.1.3 Pictorial Representation and Notation Details 36

3.2 Symmetry Concepts for Circular Fibers: Scalar Mode Fields and Degeneracies 42

3.2.1 Geometrical Symmetry; C_∞ν 46

3.2.2 Scalar Wave Equation Symmetry: C^S_∞ν 46

3.2.3 Scalar Modes: Basis Functions of Irreps of C^S_∞ν 47

3.2.4 Symmetry Tutorial: Scalar Mode Transformations 48

3.3 Vector Mode Field Construction and Degeneracies via Symmetry 50

3.3.1 Vector Field 51

3.3.2 Polarization Vector Symmetry Group: C^P_∞ν 52

3.3.3 Zerolh-Order Vector Wave Equation Symmetry: C^S_∞ν ⊗ C^P_∞ν 52

3.3.4 Pseudo-Vector Modes: Basis Functions of Irreps of C^S_∞V ⊗ C^P_∞V 54

3.3.5 Full Vector Wave Equation Symmetry: C^S_∞V ⊗ C^P_∞V ⊃ C^J_∞V 55

3.3.6 True Vector Modes: Qualitative Features via C^S_∞V ⊗ C^P_∞V ⊃ C^J_∞V 56

3.3.7 True Vector Modes via Pseudo-Modes: Basis Functions of C^S_∞V ⊗ C^P_∞V ⊃ C^J_∞V 58

3.4 Polarization-Dependent Level-Splitting 59

3.4.1 First-Order Eigenvalue Corrections 59

3.4.2 Radial Profile-Dependent Polarization Splitting 60

3 4.3 Special Degeneracies and Shifts for Particular Radial Dependence of Profile 63

3.4.4 Physical Effects 64

Chapter 4 Azimuthal Symmetry Breaking 67

4.1 Principles 67

4.2.1 Branching Rules 67

4.1.2 Anticrossing and Mode Form Transitions 68

4.2 C_2v Symmetry: Elliptical (or Rectangular) Guides: Illustration of Method 68

4.2.1 Wave Equation Symmetries and Mode-Irrep Association 68

4.2.2 Mode Splittings 69

4.2.3 Vector Mode Form Transformations for Competing Perturbations 72

4.3 C_3v Symmetry: Equilateral Triangular Deformations 72

4.4 C_4v Symmetry: Square Deformations 75

4.4.1 Irreps and Branching Rules 75

4.4.2 Mode Splitting and Transition Consequences 75

4.4.3 Square Fiber Modes and Extra Degeneracies 77

4.5 C_5v Symmetry: Pentagonal Deformations 77

4.5.1 Irreps and Branching Rules 77

4.5.2 Mode Splitting and Transition Consequences 78

4.6 C_6v Symmetry: Hexagonal Deformations 80

4.6.1 Irreps and Branching Rules 80

4.6.2 Mode Splitting and Transition Consequences 80

4.7 Level Splitting Quantification and Field Corrections 82

Chapters 5 Birefringence: Linear, Radial, and Circular 83

5.1 Linear Birefringence 83

5.1.1 Wave Equations: Longitudinal Invariance 83

5.1.2 Mode Transitions: Circular Symmetry 85

5.1.3 Field Component Coupling 87

5.1.4 Splitting by δ_xy of Isotropic Fiber Vector Modes Dominated by Δ-Splitting 88

5.1.5 Correspondence between Isotropic "True" Modes and Birefringent LP Modes 89

5.2 Radial Birefringence 89

5.2.1 Wave Equations: Longitudinal Invariance 89

5.2.2 Mode Transitions for Circular Symmetry 91

5.3 Circular Birefringence 91

5.3.1 Wave Equation 93

5.3.2 Symmetry and Mode Splittings 93

Chapter 6 Multicore Fibers and Multifiber Couplers 97

6.1 Multilightguide Structures with Discrete Rotational Symmetry 97

6.2.1 Global C_nν Rotation-Reflection Symmetric Structures: Isotropic Materials 98

6.1.2 Global C_nν Symmetry: Material and Form Birefringence 99

6.1.3 Global C_n Symmetric Structures 99

6.2 General Supermode Symmetry Analysis 101

6.2.1 Propagation Constant Degeneracies 101

6.2.2 Basis Functions for General Field Construction 104

6.3 Scalar Supermode Fields 107

6.3.1 Combinations of Fundamental Individual Core Modes 107

6.3.2 Combinations of Other Nondegenerate Individual Core Modes 108

6.3.3 Combinations of Degenerate Individual Core Modes 108

6.4 Vector Supermode Fields 109

6.4.1 Two Construction Methods 109

6.4.2 Isotropic Cores: Fundamental Mode Combination Supermodes 113

6.4.3 Isotropic Cores: Higher-Order Mode Combination Supermodes 116

6.4.4 Anisotropic Cores: Discrete Global Radial Birefringence 119

6.4.5 Other Anisotropic Structures: Global Linear and Circular Birefringence 121

6.5 General Numerical Solutions and Field Approximation Improvements 121

6.5.1 SALCs as Basis Functions in General Expansion 121

6 5.2 Variational Approach 122

6.5.3 Approximate SALC Expansions 122

6.5.4 SALC = Supermode Field with Numerical Evaluation of Sector Field Function 123

6.5.5 Harmonic Expansions for Step Profile Cores 124

6.5.6 Example of Physical Interpretation of Harmonic Expansion for the Supermodes 125

6.5.7 Modal Expansions 126

6.5.8 Relation of Modal and Harmonic Expansions to SALC Expansions 126

6.5.9 Finite Claddings and Cladding Modes 127

6.6 Propagation Constant Splitting: Quantification 127

6.6.1 Scalar Supermode Propagation Constant Corrections 127

6.6.2 Vector Supermode Propagation Constant Corrections 130

6.7 Power Transfer Characteristics 131

6.7.1 Scalar Supermode Beating 131

6.7.2 Polarization Rotation 133

Chapter 7 Conclusions and Extensions 137

7.1 Summary 137

7.2 Periodic Waveguides 138

7.3 Symmetry Analysis of Nonlinear Waveguides and Self-Guided Waves 139

7.4 Developments in the 1990s and Early Twenty-First Century 140

7.5 Photonic Computer-Aided Design (CAD) Software 141

7.6 Photonic Crystals and Quasi Crystals 142

7.7 Microstructured, Photonic Crystal, or Holey Optical Fibers 143

7.8 Fiber Bragg Gratings 144

7.8 1 General FBGs for Fiber Mode Conversion 144

7.8.2 (Short-Period) Reflection Gratings for Single-Mode Fibers 145

7.8.3 (Long-Period) Mode Conversion Transmission Gratings 146

7.8.4 Example: LP₀₁↔LP₁₁ Mode-Converting Transmission FBGs for Two-Mode Fibers (TMFs) 146

7.8.5 Example: LP₀₁↔LP₀₂ Mode-Converting Transmission FBGs 148

Appendix: Group Representation Theory 151

A.1 Preliminaries: Notation, Groups, and Matrix Representations of Them 152

A.1.1 Induced Transformations on Scalar Functions 153

A.1.2 Eigenvalue Problems: Invariance and Degeneracies 154

A.1.3 Croup Representations 155

A.1.4 Matrix Irreducible Matrix Representations 155

4.1.5 Irrep Basis Functions 155

A.1 6 Notation Conventions 155

A.2 Rotation-Reflection Groups 156

A.2.1 Symmetry Operation and Group Definitions 156

A.2.2 Irreps for C_∞ν-and C_nν 156

A.2.3 Irrep Notation 160

A 3 Reducible Representations and Branching Rule Coefficients via Characters 160

A.3.1 Example Branching Rule for C_∞ν ⊃ C_2ν 161

A.3.2 Branching Rule Coefficients via Characters 161

A.4 Clebsch-Gordan Coefficient for Changing Basis 164

A.5 Vector Field Transformation 165

References 167

Index 179