Table of Contents
Preface x
Acknowledgements xi
Author biography xii
1 Topology and physics: a historical overview 1-1
1.1 Introduction: searching for holes in fields of light 1-1
1.2 Topology and physics 1-3
1.2.1 Dirac monopoles 1-4
1.2.2 Aharanov-Bohm effect 1-6
1.2.3 Topology in optics 1-6
References 1-7
2 Electromagnetism and optics 2-1
2.1 Electromagnetic fields 2-1
2.2 Electromagnetic potentials and gauge invariance 2-5
2.3 Linear and nonlinear optical materials 2-8
2.4 Polarization and the Poincaré sphere 2-12
References 2-15
3 Characterizing spaces 3-1
3.1 Loops, holes, and winding numbers 3-1
3.2 Homotopy classes 3-3
References 3-7
4 Fiber bundles, curvature, and holonomy 4-1
4.1 Manifolds 4-1
4.2 Vectors and forms 4-4
4.3 Curvature 4-6
4.3.1 One-dimension: curves 4-7
4.3.2 Two-dimensions and beyond 4-9
4.4 Connections and covariant derivatives 4-13
4.5 Fiber bundles 4-17
4.6 Connection and curvature in electromagnetism and optics 4-22
4.7 The Hopf fibration and polarization 4-24
References 4-26
5 Topological invariants 5-1
5.1 Euler characteristic 5-1
5.2 Winding number 5-5
5.3 Index of zero points of vector fields 5-6
5.4 Chern numbers 5-8
5.5 Pontrjagin index 5-9
5.6 Hopf index 5-10
5.7 Linking number and other invariants 5-11
5.8 Atiyah-Singer index theorem 5-13
References 5-14
6 Vortices and corkscrews: singular optics 6-1
6.1 Optical singularities 6-1
6.2 Optical angular momentum 6-3
6.3 Vortices and dislocations 6-10
6.4 Polarization singularities 6-11
6.5 Optical Möbius strips 6-15
References 6-16
7 Knotted and braided vortex lines 7-1
7.1 Knotted vortex lines 7-1
7.2 Creating and characterizing knotted vortices 7-2
7.3 Variations and applications 7-4
References 7-6
8 Optical solitons 8-1
8.1 Solitary waves 8-1
8.2 Simple example: Sine-Gordon equation 8-2
8.3 Solitons in optics 8-3
References 8-7
9 Geometric and topological phases 9-1
9.1 The Pancharatnam phase 9-2
9.2 Berry phase in quantum mechanics 9-5
9.3 Geometric phase in optical fibers 9-8
9.4 Holonomy interpretation 9-8
References 9-9
10 Topological states of matter 10-1
10.1 The quantum Hall effect 10-1
10.2 One-dimensional example: the SSH model 10-7
10.3 Topological phases and localized boundary states 10-11
10.4 The role of discrete symmetries 10-13
10.5 Varieties of topological insulators and related systems 10-16
10.6 Dirac, Majorana, and Weyl points 10-17
References 10-19
11 Topological photonics 11-1
11.1 Overview: topological effects in photonic sytems 11-1
11.2 Photonic walks 11-2
11.3 Photonic crystals, waveguides, and coupled resonant cavities 11-5
11.4 Topologically protected waveguides and topological lasers 11-7
11.5 Topological optical computing 11-9
References 11-12
Appendix A A-1
