Lecture Notes on Topological Photonics

I offered this lecture as a 16-week regular graduate course in the Department of Physics at Hanyang University during the Fall 2023 semester.

This lecture explores the topological aspects of wave physics that have emerged at the intersection of optics and condensed matter physics. Beginning with the concept of the geometric phase discovered in experiments by Pancharatnam and Aharonov-Bohm in the 1950s, the course covers M. Berry’s generalization of the idea and its application to band theory. Students further study models of topological band theory such as the SSH model, Dirac equation, and Chern insulator, as well as more recent concepts like higher-order topology and semimetals, with discussions on their photonic realizations.

[Topological Photonics] Lecture 01 Introduction to Photonics.pdf

Lecture #01 Introduction to Nanophotonics & Topological Metasurfaces

In this class, we will start by learning the basics of nanophotonics, which are essential for understanding topological photonics. We'll cover the fundamental ideas in nanophotonics, helping us understand the challenges in both theory and real-world applications in the field of photonics. Furthermore, in a seminar featuring the lecturer's recent research, we will explore the exciting ways for manipulating light, made possible by new concepts in topological photonics.

[Topological Photonics] Lecture 02 Geometric Phase.pdf

Lecture #02 Geometric Phase from Pancharatnam to Aharonov-Bohm Effect

In this class, we will discuss the concept of Geometric Phase. First, we will explore Pancharatnam's geometric phase, which arises in polarization states of light. Next, we will discuss the geometric phase for charged particles that is the Aharonov-Bohm effect in electron dynamics influenced by the vector potential. Lastly, we will examine the implication of the additional phases that occur in these two distinct physical systems.

[Topological Photonics] Lecture 03 Berry Phase.pdf

Lecture #03 Berry Phase: Generalization of the Geometric Phases

In this class, we will focus on the Berry phase, a fascinating topic in physics. We'll start by discussing M. Berry's insights into the geometric phase that occurs in different systems, building upon what we learned in the previous class. To grasp these concepts fully, we will study the Adiabatic theorem, a fundamental principle that underpins the understanding of Berry phase. We'll also explore the historical context that led to the recognition of Berry's contributions, emphasizing their significance in the scientific community. In the realm of spin systems, we will work through practical examples, calculating non-zero Berry phases to deepen our understanding. Furthermore, we will delve into related concepts such as Berry connection, curvature, and their magnetic analogy that is the beginning of linking topological concepts and empirical physical systems.

[Topological Photonics] Lecture 04 Zak Phase & Photonic Interface States.pdf

Lecture #04 Zak Phase for labeling the bands and predicting the interface states

In this class, we will investigate the 1D photonic crystals, focusing on the Berry phase defined within the Brillouin zone also known as Zak phase. Specifically, we will explore the quantization of Zak phase in periodic systems possessing inversion symmetry and investigate its implications on band labeling. The course will also scrutinize the correlation between non-zero Zak phase and the surface impedance properties of the bandgap crystals. Through this knowledge, we will discuss the emergence of interface localized states within the bandgap at the junction of two photonic crystal mirrors under the band inversion relation.

[Topological Photonics] Lecture05 Su-Schrieffer-Heeger Model and Jackiw-Rebbi State in Dirac Equation.pdf

Lecture #05 Su-Schrieffer-Heeger model & Jackiw-Rebbi soliton in Dirac equation

In this class, we will explore the Su-Schrieffer-Heeger (SSH) model and the Dirac equation. The SSH model serves as a fundamental two-band toy model for understanding topological concepts, providing insights into topological invariants, topological phases of matter, and the basics of topological band theory. Additionally, we will discuss the mathematical structure of Jackiw-Rebbi soliton, arising from the hypothetical domain-wall problem of the Dirac equation, and its relevance in understanding topological states. The discussion will extend to research in photonic systems related to these concepts.

[Topological Photonics] Lecture06 Photonic Topological Insulators.pdf

Lecture #06 Topological phases of matters I: A chronological overview of the concept of topological insulators and their photonic implementations

In this class, we will explore topological insulators as novel phases of matter. Starting from the classical Hall effect, we will delve into the historical context of phenomena such as integer quantum Hall, quantum anomalous Hall, and quantum spin Hall effects. The goal is to understand the chronological development leading to an understanding of topological phases of matter and the concept of bulk boundary correspondence. In addition, the course covers recent research on photonic versions of topological insulators.

[Topological Photonics] Lecture07 Topological Phases of Matters II.pdf

Lecture #07 Topological phases of matters II: Recent developments beyond topological insulators from higher-order topology to topological semimetals

In this class, we will explore the recent developments in topological physics beyond the topological insulators, including higher-order topological corner states and topological semimetals. We will study the Benalcazar-Bernevig-Huges model which is the first proposal for higher-order topological phases and the Dirac equation approach for describing corner states. We will see how photonics systems simply realize these subtle concepts. Furthermore, we will discuss the topological nodal phases in 2D and 3D systems such as Weyl semimetals. We will deal with the gapless structures with momentum dependent topological invariants and edge states connecting two nodal points on the Fermi-arc. We will also discuss the possible schemes for demonstration of high dimensional topological semimetals by using the notion of the synthetic dimension in low dimensional photonics systems.

[Topological Photonics] Lecture 08 Recent Advances in Topological Photonics.pdf

Lecture #08 Recent advances in topological photonics: Hermitian counterparts

In this class, we will review recent developments in topological photonics from the perspective of the Hermitian counterpart. We will learn about photonic platforms including coupled resonantors, waveguides, and photonic crystals that can be used to implement the topological concepts such as Pancharatnam-Berry phase, SSH model, Jackiw-Rebbi soliton, and 2D topological phases of matter. We will also discuss the major milestones and challenges of these cutting-edge research. Finally, we will discuss the implications of these research and further think about the manifestation of the unique physical natures of photonic systems.

[Topological Photonics] Lecture 09 Non-Hermitian extension.pdf

Lecture #09 Non-Hermitian extension: Non-Hermiticity induced by radiative loss and balenced gain-loss

In this class, we will discuss the intriguing realm of non-Hermitian systems. Unlike Hermitian systems, these systems exhibit unique phenomena such as unidirectionality, exceptional points, and time-asymmetric dynamics. Our exploration will focus on understanding these key peculiarities. We will begin by examining parity-time symmetry in systems that maintain a balance between gain and loss. Additionally, we'll explore chirally symmetric radiative non-Hermitian systems and the primary research conducted in these areas. A particular emphasis will be placed on understanding thin-film photonic crystal systems as non-Hermitian systems affected by leakage radiation. We will study the Hamiltonian describing these systems and the temporal coupled mode theory for their resonace spectral response. Moreover, we will investigate new topological invariants, such as polarization vortex, and non-Hermitian skin effects, for which counterparts do not exist in Hermitian systems.

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