A perturbative expression up to fourth order in k0σ (k0 = 2π/λ; σ being the rms of the surface height and λ the wavelength) has been obtained for the mean diffuse intensity from a one-dimensional random rough surface that has normal statistics and a Gaussian correlation function for s polarization. For p polarization it is not possible to obtain this expression because of the existence of certain resonances; thus the calculations must be restricted to second order in k0σ. Perturbative calculations were derived from the Rayleigh hypothesis and also from the extinction theorem. The expression for the diffuse component of the mean scattered intensity was the same for p waves in both cases up to second order in k0σ. For s waves the equality was obtained up to fourth order in k0σ. Comparisons with exact numerical results and with those obtained by using the Kirchhoff approximation are made. This comparison allows us to establish assessments on the validity of the perturbative solution and to obtain some new interesting facts. In addition, the behavior of the diffuse halo at small σ/λ as a function of the correlation length T, the angle of incidence θ0, and the polarization is discussed. The validity of the Rayleigh hypothesis is also studied.
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