Propagating bound states in the continuum in dielectric gratings

We consider propagating bound states in the continuum in dielectric gratings. The gratings consist of a slab with ridges periodically arranged either on top or on both sides of the slab. Based on the Fourier modal approach, we recover the leaky zones above the line of light to identify the geometries of the gratings supporting Bloch bound states propagating in the direction perpendicular to the ridges. Most importantly, it is demonstrated that if a two-sided grating possesses either mirror or glide symmetry, the Bloch bound states are stable to variation of parameters as long as the above symmetries are preserved.