Linear and nonlinear electromagnetic waves in modulated honeycomb media

Abstract

Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential

“…The proof is quite analogous to that shown in [6,22] by considering rotational symmetry and will not be reproduced here. The conclusions in above together with (3.21) will be vital facts in deriving Dirac equations in the forthcoming effective dynamics.…”

“…Examples include formulations of fractional derivatives acting on quasi-periodic functions, asymptotics of the fractional Laplacian acting on a wave packet with highly oscillating Floquet-Bloch modes, homogenization of such modes with degenerate eigenvalues and so on. The results also shed some light on the rigorous analysis of topologically protected wave propagation in honeycomb-based media if additional assumptions are added to the slowly varying modulations [13,16,22,29]. This paper will be organized as follows: In section 2 and 3, we briefly review the Floquet-Bloch theory from honeycomb latticed fractional Schödinger operator and verify the existence of Dirac point.…”

“…Fefferman, Weinstein and their collaborators proved the existence of Dirac points lying on the dispersion surfaces of the honeycomb latticed standard Schödinger and divergence elliptic operators via Lyapunov-Schmidt reduction strategy [16,15,26,29]. Then, the time evolution of linear and nonlinear wave packet propagation spectrally concentrated around the Dirac point has been rigorously studied, and the effective dynamics is governed by corresponding Dirac equations [1,2,22,6,17,41]. Merging the two terminologies together in a semi-classical nonlinear evolution equation, we need to deal with mathematical challenges.…”

Traveling edge states in massive Dirac equations along slowly varying edges

“…The Dirac equation with a varying mass (1.1) arises from the effective envelopes of wave propagation in topological materials. Here, the mass m(x) determines the distinct topology such that two adherent materials are topological insulators in bulks and the current or electromagnetic wave is permitted to travel along the contact edge [13,28,37,43]. The associated edge, null domain of m(x), separates the two dimensional materials with different topology in each part.…”

Three-fold Weyl points in the Schrödinger operator with periodic potentials

“…Based on the rigorous justification of the existence of Dirac points, a lot of rigorous explanations on the related physical phenomena have been extensively investigated. For example, the effective dynamics of wave packets associated with Dirac points were studied in [6,14,20,34,35,36]. The existence of edge states and associated dynamics are studied in [8,12,15].…”