  A formalism is introduced for the non-perturbative, purely numerical,
solution of the reduced Rayleigh equation for the scattering of light from
two-dimensional penetrable rough surfaces. As an example, we apply this
formalism to study the scattering of p- or s-polarized light from two-
dimensional dielectric or metallic randomly rough surfaces by calculating the
full angular distribution of the co- and cross-polarized intensity of the
scattered light. In particular, we present calculations of the mean
differential reflection coefficient for glass and silver surfaces characterized
by (isotropic or anisotropic) Gaussian and cylindrical power spectra. The
proposed method is found, within the validity of the Rayleigh hypothesis, to
give reliable results. For a non-absorbing metal surface the conservation of
energy was explicitly checked, and found to be satisfied to within 0.03%, or
better, for the parameters assumed. This testifies to the accuracy of the
approach and a satisfactory discretization.
