Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates

In the scattering of electromagnetic waves from a one-dimensional surface, the reduced Rayleigh equation is the single integral equation satisfied by the scattering amplitude that is obtained when the electromagnetic field in the scattering medium has been eliminated from the problem by the use of Green's second integral identity. We have solved the reduced Rayleigh equation numerically for the scattering of p- and s-polarized light from a one-dimensional, randomly rough dielectric film on a planar, perfectly conducting substrate, when the thickness of the film is such that it supports two or more guided waves of each polarization. The solution has been carried out by generating a segment of rough vacuum-dielectric interface of length L, where L is much larger than lambda, the wavelength of the incident electromagnetic field, and then replicating this segment periodically to obtain a classical grating of period L. The integral equation for the scattering amplitude is converted into a matrix equation for the amplitudes of the propagating and evanescent Bragg beams diffracted by this grating. By solving this matrix equation for an ensemble of random surfaces, and averaging the results over this ensemble, we obtain the cross section for diffuse scattering. The power spectrum of the surface roughness is nonzero only in the ranges of wave numbers that contain the wave numbers of these guided waves, but for comparison purposes we also present some results obtained by the use of a Gaussian power spectrum. Well-defined satellite peaks are present in the diffuse scattering cross section, in addition to the enhanced backscattering peak.