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Topology is the study of properties of geometrical objects that remain invariant as the object is bent, twisted, or otherwise continuously deformed. It has been an indispensable tool in particle physics and solid-state physics for decades, but in recent years it has become increasingly relevant in classical and quantum optics as well. It makes appearances through such diverse phenomena as Pancharatnam–Berry phases, optical vortices and solitons, and optical simulations of solid-state topological phenomena. The goal of this book is to provide in concise form the necessary mathematical background needed to understand these developments and to give a rapid survey of some of the optical applications where topological issues arise.
Table of Contents Contents 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 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 Poincare sphere 3 Characterizing Spaces 3.1 Loops, holes, and winding numbers 3.2 Homotopy classes 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 5 Topological invariants 5.1 Euler characteristic 5.2 Winding number 5.3 Index 5.4 Chern Numbers 5.5 Linking Number and Other Invariants 6 Vortices and Corkscrews: Singular Optics 6.1 Optical Singularities 6.2 Optical angular momentum 6.3 Vortices and Dislocations 6.4 Knotted and Braided Vortex Lines 6.5 Polarization Singularities 6.6 Optical Mobius strips 7 Optical Solitons 7.1 Solitary waves 7.2 Solitons in optics 8 Geometric and Topological Phases 8.1 The Pancharatnam Phase 8.2 Berry Phase in Quantum Mechanics 8.3 Geometric Phase in Optical Fibers 8.4 Holonomy Interpretation 9 Topological States of Matter and Light 9.1 The Quantum Hall Effect 9.2 Topological Phases and Localized Boundary States 9.3 Topological Photonics A Appendices A.1 Point-Set Topology: Basic Definitions And Results A.2 Brief Review of Group Theory