Comparison of perturbation theories for rough-surface scattering

Rayleigh–Rice perturbation theory is compared with the more rigorously derived perturbation theory based on the extinction theorem (or extended boundary condition) for the case of Dirichlet boundary conditions. Numerical calculations for a sinusoidal surface profile show that the two theories give identical results when carried out to high order. This is true even when the two series do not seem to converge numerically. Numerical convergence of the two series is found in several cases, including some cases in which the Rayleigh hypothesis is invalid. For surfaces of arbitrary shape, the two perturbation series appear superficially to differ at third and higher order. It is shown, however, that a reciprocity transformation of one series makes it identical to the other through fifth order. Finally, it is shown that reciprocity holds in both perturbation theories, establishing analytically the equivalence of the two theories at least through fifth order. These results are surprising given the limited regime of validity for the Rayleigh hypothesis compared with the extinction theorem. They are useful in that they justify the use of perturbation formulas obtained from the Rayleigh hypothesis in lieu of the more complicated formulas obtained from the extinction theorem. While these results do not establish the identity of the two perturbation methods to all orders or settle questions as to their convergence, they do provide some theoretical clues to the nature of these problems.