Transmittance of a volume grating near the total-reflection threshold of the diffracted wave

An analytic formula for the transmittance of a volume slab (planar) diffraction grating is derived taking into account the refraction of waves at the grating boundaries. The formula applies to the cases of both Laue and Bragg diffraction. An investigation is made of the frequency dependence of the transmittance near the total-reflection threshold of the diffracted wave. It is shown that in this geometry the spectral properties of the grating differ radically not only from the properties of an equivalent homogeneous plate, but also from the properties of volume diffraction gratings predicted by Kogelnik's theory.