On the convergence of the Rayleigh ansatz for hard-wall scattering on arbitrary periodic surface profiles

The plane-wave ansatz for scattered waves is convergent only if, roughly speaking, the surface is (i) not too deep and (ii) sufficiently smooth. This convergence is investigated for general one-dimensional corrugations by means of an asymptotic analysis. The results are explicitly given for a three-term Fourier series and for analytic approximations to linear profiles with discontinuous slopes. A relation between the convergence and the maximal curvature is established for triangular corrugations, and used for the definition of a pseudoinvariant which is slowly varying for various corrugation profiles.