On the optical theorem applicability using a self-consistent wave approximation model for the grazing-incidence small-angle X-ray scattering from rough surfaces

Abstract

Based on a self-consistent wave approximation (SCWA) for describing the grazing-incidence small-angle X-ray scattering (GISAXS) from a random rough surface, the optical theorem applicability is tested. Asymptotic solutions for the specular and diffuse GISAXS waves are used to evaluate the X-ray energy flows through the planes far away from the surface interface, z → ±∞. The conventional Fresnel expressions multiplied by the corresponding Debye–Waller factors are used for the specular waves, while the diffuse X-ray energy flows are described in terms of the product of the statistical scattering factors ηR(θ, θ0) and ηT(θ, θ0) and the Fourier transform of the two-point cumulant correlation function g2(|x1 − x2|/ℓ) (θ is the grazing scattering angle with the surface, φ is the azimuth scattering angle; θ0 is the grazing-incidence angle). It is shown that the optical theorem within the SCWA does hold in the case of infinite correlation lengths ℓ → ∞ (more precisely, kℓθ02 >> 1, k is the X-ray wavenumber in a vacuum). In a general case of the typical-valued {θ0, σ, ℓ} parameters the reflected and transmitted GISAXS wave flows are numerically integrated over the scattering reciprocal space to probe the optical theorem.