Mathematical theory for topological photonic materials in one dimension

Abstract

This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence and stability of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, the associated Zak phase (or Berry phase) may not be quantized. We investigate the existence of an interface mode which is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both timereversal and inversion symmetry, the Zak phase is quantized, taking only two values 0, π. We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm-Liouville operators. The methods and results can be extended to general topological Sturm-Liouville systems in one dimension.

Asymptotic characterisation of localised defect modes: Su-Schrieffer-Heeger and related models

“…It also extends previous homogenisation results for homogeneous background media [23] and generalises results for local perturbations to periodic potentials [15]. The results presented here build on work characterising the existence of topologically protected edge modes [11,12,19] by providing a means to quantify their most important properties.…”

“…These studies have been based, for example, on multiple scattering formulations [30] and asymptotic expansions in terms of material parameters [3]. There is also a growing body of theory providing rigorous foundations for the existence of topologically protected defect modes in terms of multi-scale expansions [11,12,19].…”

Functional analytic methods for discrete approximations of subwavelength resonator systems

“…With this formulation, scattering resonances can be characterised as the poles of meromorphic operator-valued functions [39]. In some settings, this can be paired with a scattering matrix [11] or transfer matrix [52] formulation to give a concise description of the response of the system.…”

Traveling edge states in massive Dirac equations along slowly varying edges

“…A variety of rigorous research has studied the existence of Dirac points and the local band gap brought after a time reversal symmetry breaking perturbation in domain wall and tight binding models [6,19,20,29,30,44]. Meanwhile, one dimensional topologically protected edge state (or namely interface mode) always occur at the band gap when two adjacent medium state the distinct topological invariant corresponding to Zak phase and Chern number [4,5,9,16,18,25,30,31]. Instead of dealing with the highly oscillated interface mode directly, a canonical way is to exploit the essentially homogenized envelope emerged by the time-harmonic massive Dirac equation which expresses the topological protected properties more clearly and intuitively.…”

“…Instead of dealing with the highly oscillated interface mode directly, a canonical way is to exploit the essentially homogenized envelope emerged by the time-harmonic massive Dirac equation which expresses the topological protected properties more clearly and intuitively. Studies about the existence of edge states or the derivation of governed envelopes-Dirac equations are carried out in many settings, such as microlocal analysis, transfer matrix method, Fredholm operator index, K-theory, and so on [7,8,9,16,31,40].…”