  The transmission of polarized light through a two-dimensional randomly rough
interface between two dielectric media has been much less studied, by any
approach, than the reflection of light from such an interface. We have derived
a reduced Rayleigh equation for the transmission amplitudes when p- or
s-polarized light is incident on this type of interface, and have obtained
rigorous, purely numerical, nonperturbative solutions of it. The solutions are
used to calculate the transmissivity and transmittance of the interface, the
mean differential transmission coefficient, and the full angular distribution
of the intensity of the transmitted light. These results are obtained for both
the case where the medium of incidence is the optically less dense medium and
in the case where it is the optically more dense medium. Optical analogues of
Yoneda peaks observed in the scattering of x-rays from metal surfaces are
present in the results obtained in the former case. For p-polarized incident
light we observe Brewster scattering angles, angles at which the diffuse
transmitted intensity is zero in a single-scattering approximation, which
depend on the angle of incidence in contrast to the Brewster angle for
flat-surface reflection.
