  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 ultra-high 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
wavevectors, and is considered as protected by symmetry. Propagating BICs with
nonzero Bloch wavevectors have been found on 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) on biperiodic 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
wavevector, when the biperiodic structure is perturbed slightly preserving the
inversion and reflection symmetries. Our study enhances theoretical
understanding for BICs on periodic structures and provides useful guidelines
for their applications.
