Perturbation theories for symmetry-protected bound states in the continuum on two-dimensional periodic structures
Lijun Yuan, Ya Yan Lu
Abstract
On dielectric periodic structures with a reflection symmetry in a periodic direction, there can be antisymmetric standing waves (ASWs) that are symmetry-protected bound states in the continuum (BICs). The BICs have found many applications, mainly because they give rise to resonant modes of extremely large quality factors ($Q$ factors). The ASWs are robust to symmetric perturbations of the structure, but they become resonant modes if the perturbation is nonsymmetric. The $Q$ factor of a resonant mode on a perturbed structure is typically $O(1/{\ensuremath{\delta}}^{2})$, where $\ensuremath{\delta}$ is the amplitude of the perturbation, but special perturbations can produce resonant modes with larger $Q$ factors. For two-dimensional (2D) structures with a one-dimensional (1D) periodicity, we derive conditions on the perturbation profile such that the $Q$ factors are $O(1/{\ensuremath{\delta}}^{4})$ or $O(1/{\ensuremath{\delta}}^{6})$. For the unperturbed structure, an ASW is surrounded by resonant modes with a nonzero Bloch wave vector. For 2D structures with a 1D periodicity, the $Q$ factors of nearby resonant modes are typically $O(1/{\ensuremath{\beta}}^{2})$, where $\ensuremath{\beta}$ is the Bloch wave number. We show that the $Q$ factors can be $O(1/{\ensuremath{\beta}}^{6})$ if the ASW satisfies a simple condition.