Local Robustness of Bound States in the Continuum through Scattering-Matrix Eigenvector Continuation

Abstract

We consider the diffraction of time-harmonic plane waves by a periodic structure, governed by the Helmholtz equation. Bound states in the continuum (BICs) are quasi-periodic fields that remain $L^{2}$-bounded over one period and occur at frequencies embedded in the continuous spectrum. Perturbations that break a BIC can lead to ultra-strong resonances, enabling various applications in photonics. Employing the implicit function theorem, we demonstrate how a simple BIC continuously deforms into a propagating field as system parameters vary in a neighborhood, with the frequency adjusting accordingly. In this setting, the incident coefficients of the field persist as an eigenvector of the scattering matrix with a fixed eigenvalue. By introducing a mapping $\mathcal{P}$ from the parameters to these coefficients, the zeros of $\mathcal{P}$ correspond precisely to BICs. When such a zero is isolated and the dimensions of the domain and range coincide, the BIC can be related to the mapping degree of $\mathcal{P}$ in a small neighborhood. This perspective clarifies the phase singularity associated with BICs and provides a general topological interpretation of their local robustness with respect to the given parameters. Moreover, it yields a practical numerical criterion for detecting and verifying BICs via computation of the mapping degree of $\mathcal{P}$.

Figures (5)

• Figure 1: One period of the structure $\Omega$ is partitioned into three subdomains: $\Omega_{L}$, $\Omega_{0}$, $\Omega_{R}$, separated by interfaces $\Gamma_{L}$ and $\Gamma_{R}$. The boundaries of $\Omega$ are denoted by $\Gamma_{-}$ and $\Gamma_{+}$, respectively. A rectangular coordinate system is introduced with its origin $\bm{o}$ on the central line of $\Omega$. The dielectric function of the structure is denoted by $\epsilon(\bm{x})$ and is equal to $1$ for large $|x_{2}|$.
• Figure 2: A periodic array of circles of diameter $1.2\pi$. A rectangular coordinate system in defined at the center of a circle. The dielectric function $\epsilon(\bm{x})$ is piecewise constant, taking the value $\epsilon(\bm{x})=\epsilon_{1}$ inside the circles and $\epsilon(\bm{x})=\epsilon_{0}$ in the surrounding medium. The length of the domain $\Omega_{0}$ is set to $2\pi$.
• Figure 3: Simulation results for $C=1$, $\theta=\pi$ and varying $r$. For each $r$, the left graph depicts the frequencies $k_{0}$ and $k_{1}$, and the right graph shows $\widehat{a}_{0}$ and $\widehat{a}_{1}$. A consistent sign change between $\widehat{a}_{0}$ and $\widehat{a}_{1}$ is apparent.
• Figure 4: Simulation results for $C=1$, $\theta=\pi$ and varying $r$. For each $r$, the left graph depicts the frequencies $k_{0}$ and $k_{1}$, and the right graph shows $\widehat{a}_{0}$ and $\widehat{a}_{1}$. A consistent sign change between $\widehat{a}_{0}$ and $\widehat{a}_{1}$ is apparent.
• Figure 5: Simulation results for $C=1$, $N=24$, $\theta=\pi$ and varying $r$. For each $r$, the left graph shows the frequencies $k_{n}$ ($n=0,\ldots,23$), and the right graph displays $\widehat{a}_{n}$ at the corresponding points $(\beta_{n}/0.15,\bm{\delta}_{n})$. A consistent nontrivial winding of $\widehat{a}$ is clearly visible.

Theorems & Definitions (38)

• Definition 2.1
• Remark 2.1
• Remark 2.2
• Definition 2.2
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1