Conditional robustness of propagating bound states in the continuum in structures with two-dimensional periodicity

For a periodic structure sandwiched between two homogeneous media, a bound state in the continuum (BIC) is a guided Bloch mode with a frequency in the radiation continuum. Optical BICs have found many applications, mainly because they give rise to resonances with ultrahigh-quality factors. If the periodic structure has a relevant symmetry, a BIC may have a symmetry mismatch with incoming and outgoing propagating waves of the same frequency and compatible wave vectors and is considered as protected by symmetry. Propagating BICs with a nonzero Bloch wave vector have been found in many highly symmetric periodic structures. They are not protected by symmetry in the usual sense (i.e., there is no symmetry mismatch), but some of them seem to depend on symmetry for their existence and robustness. In this paper we show that the low-frequency propagating BICs (with only one radiation channel) in two-dimensionally periodic structures with an inversion symmetry in the plane of periodicity and a reflection symmetry in the perpendicular direction are robust to symmetry-preserving structural perturbations. In other words, a propagating BIC continues its existence with a slightly different frequency and a slightly different Bloch wave vector when the periodic structure is perturbed slightly preserving the inversion and reflection symmetries. Our study enhances theoretical understanding for BICs in periodic structures and provides useful guidelines for their applications.