Guided modes in a hexagonal periodic graph like domain

TL;DR

The work analyzes wave propagation in hexagonal honeycomb-like media assembled as a thin, δ-thickened graph and its graph limit, establishing Dirac points at high frequencies for small δ through asymptotic graph reductions. It then introduces zigzag-edge perturbations (Cases 1–3) and proves the existence and robustness of edge states for both Neumann and Dirichlet edge conditions, with spectral features depending on edge location and thickness. The analysis centers on a limit quantum-graph model, supported by quasi-mode constructions and detailed symmetry arguments, complemented by comprehensive numerical simulations that illustrate Dirac points, gaps, and edge-state behavior. The results unify spectral theory for graph-like domains with edge-state phenomena in perturbed honeycomb media, providing rigorous and computable insights for engineered graphene-inspired photonic and acoustic devices.

Abstract

This paper deals with the existence of guided waves and edge states in particular two-dimensional media obtained by perturbing a reference periodic medium with honeycomb symmetry. This reference medium is a thin periodic domain (the thickness is denoted $δ$ > 0) with an hexagonal structure, which is close to an honeycomb quantum graph. In a first step, we show the existence of Dirac points (conical crossings) at arbitrarily large frequencies if $δ$ is chosen small enough. We then perturbe the domain by cutting the perfectly periodic medium along the so-called zigzag direction, and we consider either Dirichlet or Neumann boundary conditions on the cut edge. In the two cases, we prove the existence of edges modes as well as their robustness with respect to some perturbations, namely the location of the cut and the thickness of the perturbed edge. In particular, we show that different locations of the cut lead to almost-non dispersive edge states, the number of locations increasing with the frequency. All the results are obtained via asymptotic analysis and semi-explicit computations done on the limit quantum graph. Numerical simulations illustrate the theoretical results.

Figures (19)

• Figure 1: The hexagonal periodic medium $\Omega_\delta$ (left), the associated quantum graph $\mathcal{G}$ (right)
• Figure 2: The periodicity cell $\mathcal{G}^\sharp$ and its three oriented edges (left); The periodicity cell ${\cal C}_\delta^\sharp$ and the three fattened edges (right)
• Figure 3: The zigzag perturbed domains: (Case 1) the half-domain with a classical zigzag edge $\Omega_\delta^0$ (left), (Case 2) the perturbed half-domain $\Omega_{\delta,\mu}^{0}$ for $\mu<1$ (second figure) and (Case 3) the half-domain with a different cut position $\Omega_{\delta}^{t}$ for $t\in (0,L)$ (third figure) and $\Omega_{\delta}^{t}$ for $t\in (L,2L)$ (right). See \ref{['eq:defalphat']} for the definition of $\alpha(t)$.
• Figure 4: The hexagonal Brillouin zone $\mathcal{B}$ and its 6 vertices.
• Figure 5: Decomposition of the periodicity cell ${\cal C}_\delta^\sharp$ into three shrunken edges $\tilde{e}_{i,\delta}, i\in \{0,1,2\}$ and four junction regions $J^A_\delta, J^B_{i,\delta}, i\in \{0,1,2\}$.

Theorems & Definitions (11)

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