On the bifurcation of a Dirac point in a photonic waveguide without band gap openning

TL;DR

The study demonstrates that a Dirac point in a two-dimensional periodic photonic waveguide can bifurcate without opening a global band gap. It combines a perturbation analysis that reveals a local gap opening and band inversion with a boundary-integral/analytic-continuation framework to construct a mode bifurcating from the Dirac point, yielding a bifurcating eigenvalue $λ^*$ with $\Im(λ^*)\le 0$. When $\Im(λ^*)=0$, the mode is an embedded interface mode (bound state in the continuum); when $\Im(λ^*)<0$, it is a resonant mode with radiation, and the coupling to continued Bloch modes determines the radiation condition. The approach relies on analytic continuation of Green functions and layer potentials, and its methodology extends to other settings where no global gap exists yet Dirac-point bifurcation can occur.

Abstract

Recent progress in topological insulators and topological phases of matter has motivated new methods for the localization of waves in photonic structures. Especially, it is established that a Dirac point of a periodic structure can bifurcate into in-gap eigenvalues if the periodic structure is perturbed differently on the two sides of an interface and if a common band gap can be opened for the two perturbed periodic structures near the Dirac point. The associated eigenmodes are localized near the interface and decay exponentially away from it. This paper addresses the less-known situation when the perturbation only lifts the degeneracy of the Dirac point without opening a band gap. Using a two-dimensional waveguide model, we constructed a wave mode bifurcated from a Dirac point of a periodic waveguide. We show that when the constructed mode couples to an analytically continued Floquet-Bloch mode near the Dirac energy, its eigenvalue acquires a strictly negative imaginary part, making the mode resonant. On the other hand, when the coupling vanishes, the imaginary part of the eigenvalue turns to zero, and the constructed mode becomes an interface mode that decays exponentially away from the interface. The developed method can be extended to other settings, thus providing a clear answer to the problem concerning the bifurcation of Dirac points.

Figures (5)

• Figure 1: A waveguide $\Omega$ with periodically arranged obstacles qiu2023mathematical. Here $\Omega=\mathbf{R}\times [0,1]\backslash \cup_{n\in\mathbf{N}}D_n$, where $D_n$ is an array of periodically arranged obstacles that are centered at $(\frac{1}{2}+n,\frac{1}{2})$. $\Omega$ is periodic with its minimal period equal to $1$. The primitive cell $Y$ is filled with blue in the figure. The interface $\Gamma:=\Omega\cap (\{0\}\times \mathbf{R})$ is also marked in the figure.
• Figure 2: Band structure of $\mathcal{L}$ near the Dirac point $(0,\lambda_*)$. (a) $\lambda=\mu_{n_*}(p)$ is plotted in blue while $\lambda=\mu_{m_*}(p)$ is plotted in red; they are smooth on $[-\pi,\pi]$ and constitute the band structure of $\mathcal{L}$ near the Dirac point. Moreover, $\mu_{n_*}(p)$ and $\mu_{m_*}(p)$ intersect with the energy level $\lambda=\lambda_*$ at $p=-q_*$ and $p=q_*$, respectively. The existence of those extra intersection points breaks the spectral no-fold condition. (b) $\lambda=\lambda_{\mathfrak{n}_*}(p)$ is plotted in blue while $\lambda=\lambda_{\mathfrak{n}_*+1}(p)$ is plotted in red; they are not differentiable at $p=0$.
• Figure 3: Perturbed band structure: $\lambda=\lambda_{\mathfrak{n}_*,\epsilon}(p)$ is plotted in blue while $\lambda=\lambda_{\mathfrak{n}_*+1,\epsilon}(p)$ is plotted in red; they are smooth near $p=0$. Compared with Figure \ref{['fig_unperturbed_band_structure']}, a "local" band gap is opened near $(0,\lambda_*)$ when we apply the perturbation. However, no "global" band gap appears, due to the failure of spectral no-fold condition ($\lambda=\lambda_{\mathfrak{n}_*+1,\epsilon}(p)$ always intersects with $\lambda=\lambda_*$ for $|\epsilon|\ll 1$).
• Figure 4: The domain $D_{\epsilon,\nu}$ consists of two disks of radius $\epsilon^{\frac{1}{3}}$ (filled with red) and a strip of width $\nu=\nu(\epsilon)$ (filled with blue). A dumbbell-shaped domain $\tilde{D}_{\epsilon}$ that contains $D_{\epsilon,\nu}$ is also drawn here. $\tilde{D}_{\epsilon}$ is used in the proof of Lemma \ref{['lem_analytic_domain']} to construct $D_{\epsilon,\nu}$.
• Figure 5: Several integral contours: (a) the contour $C_{\epsilon}$ is drawn in red lines while $\tilde{C}_{\tau,\lambda}$ is drawn in blue. The equation $\lambda_{\mathfrak{n}_*,\epsilon}(p)=\lambda$ has two roots for $\lambda\in \mathcal{I}_{\epsilon}\cap \mathbf{R}$, which are marked by the black crosses in the Figure. (b) the contour $C^{evan}_{\epsilon,\nu}$ is drawn in red lines. In the figure, the red, black and blue crosses denote the root of $\lambda_{\mathfrak{n}_*,\epsilon}(p)=\lambda$ ($\lambda\in \mathcal{I}_{\epsilon}$, $p$ near $q_*$) for $\Im(\lambda)<0$, $\Im(\lambda)=0$ and $\Im(\lambda)>0$, respectively.

Theorems & Definitions (44)

• Remark 1.4
• Remark 1.5
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Proposition 2.1
• Lemma 2.2: Reflection relations between Bloch modes
• Lemma 2.3: Energy flux between Bloch modes
• Lemma 2.4
• Proposition 2.5