A discussion of the properties of the Rayleigh perturbative solution in diffraction theory

The problem of two-dimensional plane-wave scattering from an infinite periodic profile with a Dirichlet boundary condition is considered. It is shown that the Rayleigh theory yields the same perturbation series for the reflection coefficients as the extinction-theorem formalism. This identity holds up to any order if the boundary profile is infinitely continuously differentiable, and at least up to some finite order if the boundary profile is finitely continuously differentiable. The nature of the derivation proposed allows the extension of these results to all linear homogeneous media encountered in electromagnetism and elastodynamics, to all usual linear boundary conditions, and to the case of scattering from finite bodies. It is indicated that, for interfaces which are periodic finite linear combinations of sinusoids, the Rayleigh perturbative solution is identical to the perturbative solutions obtained with any other approach, and verifies the reciprocity relationships.