New method for solving the scattering of waves from a periodic hard surface: Solutions and numerical comparisons with the various formalisms

In this paper we present a formalism and a numerical method, called CG and $R{R}^{\prime }$, respectively, which lead to the exact solution of the scattering of waves from a periodic hard corrugated surface. The relation of this formalism to those previously in use [Rayleigh hypothesis and Masel, Merrill, and Miller (MMM) formalism] is also discussed. Numerical computations show that his formalism seems to give always a convergent numerical solution for any shape and strength of the corrugations. On the other hand, the other two formalisms give also convergent numerical solutions for small values of the corrugation strength. Numerical results show that the MMM formalism, although analytically exact, converges very slowly as the corrugation strength increases and it may become ill-conditioned. This formalism seems to be an inconvenient way of writing the boundary conditions on the scattered wave function. In particular, the theoretical validity of the Rayleigh hypothesis is numerically confirmed for a shape corrugation $D\left(x\right)=bcos\kappa x$ when $\kappa b\le 0.448$. When the computed diffraction probabilities satisfy unitarity, they are the same no matter what formalism is used. The $R{R}^{\prime }$ numerical method proposed here allows the calculation of diffracted intensities for large corrugation strengths.