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Table of Contents
Intro
Preface
Acknowledgements
Author biography
David Simon
Chapter 1 Topology and physics: a historical overview
1.1 Introduction: searching for holes in fields of light
1.2 Topology and physics
1.2.1 Dirac monopoles
1.2.2 Aharanov-Bohm effect
1.2.3 Topology in optics
References
Chapter 2 Electromagnetism and optics
2.1 Electromagnetic fields
2.2 Electromagnetic potentials and gauge invariance
2.3 Linear and nonlinear optical materials
2.4 Polarization and the Poincaré sphere
References
Chapter 3 Characterizing spaces
3.1 Loops, holes, and winding numbers
3.2 Homotopy classes
References
Chapter 4 Fiber bundles, curvature, and holonomy
4.1 Manifolds
4.2 Vectors and forms
4.3 Curvature
4.3.1 One-dimension: curves
4.3.2 Two-dimensions and beyond
4.4 Connections and covariant derivatives
4.5 Fiber bundles
4.6 Connection and curvature in electromagnetism and optics
4.7 The Hopf fibration and polarization
References
Chapter 5 Topological invariants
5.1 Euler characteristic
5.2 Winding number
5.3 Index of zero points of vector fields
5.4 Chern numbers
5.5 Pontrjagin index
5.6 Hopf index
5.7 Linking number and other invariants
5.8 Atiyah-Singer index theorem
References
Chapter 6 Vortices and corkscrews: singular optics
6.1 Optical singularities
6.2 Optical angular momentum
6.3 Vortices and dislocations
6.4 Polarization singularities
6.5 Optical Möbius strips
References
Chapter 7 Knotted and braided vortex lines
7.1 Knotted vortex lines
7.2 Creating and characterizing knotted vortices
7.3 Variations and applications
References
Chapter 8 Optical solitons
8.1 Solitary waves
8.2 Simple example: Sine-Gordon equation
8.3 Solitons in optics
References.
Chapter 9 Geometric and topological phases
9.1 The Pancharatnam phase
9.2 Berry phase in quantum mechanics
9.3 Geometric phase in optical fibers
9.4 Holonomy interpretation
References
Chapter 10 Topological states of matter
10.1 The quantum Hall effect
10.2 One-dimensional example: the SSH model
10.3 Topological phases and localized boundary states
10.4 The role of discrete symmetries
10.5 Varieties of topological insulators and related systems
10.6 Dirac, Majorana, and Weyl points
References
Chapter 11 Topological photonics
11.1 Overview: topological effects in photonic sytems
11.2 Photonic walks
11.3 Photonic crystals, waveguides, and coupled resonant cavities
11.4 Topologically protected waveguides and topological lasers
11.5 Topological optical computing
References
Chapter
A.1 Point-set topology: basic definitions and results
A.2 Brief review of group theory
References.
Preface
Acknowledgements
Author biography
David Simon
Chapter 1 Topology and physics: a historical overview
1.1 Introduction: searching for holes in fields of light
1.2 Topology and physics
1.2.1 Dirac monopoles
1.2.2 Aharanov-Bohm effect
1.2.3 Topology in optics
References
Chapter 2 Electromagnetism and optics
2.1 Electromagnetic fields
2.2 Electromagnetic potentials and gauge invariance
2.3 Linear and nonlinear optical materials
2.4 Polarization and the Poincaré sphere
References
Chapter 3 Characterizing spaces
3.1 Loops, holes, and winding numbers
3.2 Homotopy classes
References
Chapter 4 Fiber bundles, curvature, and holonomy
4.1 Manifolds
4.2 Vectors and forms
4.3 Curvature
4.3.1 One-dimension: curves
4.3.2 Two-dimensions and beyond
4.4 Connections and covariant derivatives
4.5 Fiber bundles
4.6 Connection and curvature in electromagnetism and optics
4.7 The Hopf fibration and polarization
References
Chapter 5 Topological invariants
5.1 Euler characteristic
5.2 Winding number
5.3 Index of zero points of vector fields
5.4 Chern numbers
5.5 Pontrjagin index
5.6 Hopf index
5.7 Linking number and other invariants
5.8 Atiyah-Singer index theorem
References
Chapter 6 Vortices and corkscrews: singular optics
6.1 Optical singularities
6.2 Optical angular momentum
6.3 Vortices and dislocations
6.4 Polarization singularities
6.5 Optical Möbius strips
References
Chapter 7 Knotted and braided vortex lines
7.1 Knotted vortex lines
7.2 Creating and characterizing knotted vortices
7.3 Variations and applications
References
Chapter 8 Optical solitons
8.1 Solitary waves
8.2 Simple example: Sine-Gordon equation
8.3 Solitons in optics
References.
Chapter 9 Geometric and topological phases
9.1 The Pancharatnam phase
9.2 Berry phase in quantum mechanics
9.3 Geometric phase in optical fibers
9.4 Holonomy interpretation
References
Chapter 10 Topological states of matter
10.1 The quantum Hall effect
10.2 One-dimensional example: the SSH model
10.3 Topological phases and localized boundary states
10.4 The role of discrete symmetries
10.5 Varieties of topological insulators and related systems
10.6 Dirac, Majorana, and Weyl points
References
Chapter 11 Topological photonics
11.1 Overview: topological effects in photonic sytems
11.2 Photonic walks
11.3 Photonic crystals, waveguides, and coupled resonant cavities
11.4 Topologically protected waveguides and topological lasers
11.5 Topological optical computing
References
Chapter
A.1 Point-set topology: basic definitions and results
A.2 Brief review of group theory
References.