Perturbation theories for symmetry-protected bound states in the continuum on two-dimensional periodic structures
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 /δ2) , where δ 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 /δ4) or O (1 /δ6) . 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 /β2) , where β is the Bloch wave number. We show that the Q factors can be O (1 /β6) if the ASW satisfies a simple condition.