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 non-symmetric. The $Q$-factor of a resonant mode on a perturbed structure is typically $O(1/delta^2)$ where $delta$ is the amplitude of the perturbation, but special perturbations can produce resonant modes with larger $Q$-factors. For two-dimensional (2D) periodic structures with a 1D periodicity, we derive conditions on the perturbation profile such that the $Q$-factors are $O(1/delta^4)$ or $O(1/delta^6)$. For the unperturbed structure, an ASW is surrounded by resonant modes with a nonzero Bloch wave vector. For 2D periodic structures, the $Q$-factors of nearby resonant modes are typically $O(1/beta^2)$, where $beta$ is the Bloch wavenumber. We show that the $Q$-factors can be $O(1/beta^6)$ if the ASW satisfies a simple condition.
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