  On periodic structures, a bound state in the continuum (BIC) is a standing or
propagating Bloch wave with a frequency in the radiation continuum. Some BICs
(e.g., antisymmetric standing waves) are symmetry-protected, since they have
incompatible symmetry with outgoing waves in the radiation channels. The
propagating BICs do not have this symmetry mismatch, but they still depend
crucially on the symmetry of the structure. In this Letter, a perturbation
theory is developed for propagating BICs on two-dimensional periodic
structures. The study shows that these BICs are robust against structural
perturbations that preserve the symmetry, indicating that these BICs are in
fact implicitly protected by symmetry.
