Surface scattering near grazing angles: The distorted wave born approximation for rough surfaces

The scattering of radiation by mildly rough surfaces of non-absorbing materials is analysed in the distorted wave Born approximation wherein the Toughness is treated as a perturbation The basic formulae for the scattering cross-section are derived and corresponding expressions for the reflectivity and transmissivity are given. It is shown that off-specular scattering is proportional to the Fourier transform of an autocorrelation function involving phase factors produced when the actual surface deviates from a flat reference surface. In the specular direction the Fourier transform reduces to a simple surface average. The theory is then applied to periodic surfaces where the Fourier transform is reduced to a weighted sum of delta-functions. Numerical reflectivities are given for a piecewise rectangular and a sinusoidal grating. For such surfaces it is shown that plausible results me obtained even when the deviations am large. Random roughness is considered and explicit results are given for a surface with a Gaussian distribution of the height variation. It is demonstrated that the surface roughness for this type of surface must he small for our perturbative technique to he valid For off-specular scattering numerical reflectivities are given in the limit of small deviations or small wave vector transfers where the scattering is proportional to the Fourier transform of the autocorrelation function of the deviations themselves.