  We study the convergence properties of the two-dimensional Rigorous Coupled
Wave Approach (RCWA) for p-polarized monochromatic incident light. The RCWA is
a semi-analytical numerical method that is widely used to solve the
boundary-value problem of scattering by a grating. The approach requires the
expansion of all electromagnetic field phasors and the relative permittivity as
Fourier series in the spatial variable along the direction of the periodicity
of the grating. In the direction perpendicular to the grating periodicity, the
domain is discretized into thin slices and the actual relative permittivity is
replaced by an approximation. The approximate relative permittivity is chosen
so that the solution of the Maxwell equations in each slice can be computed
without further approximation. Thus, there is error due to the approximate
relative permittivity as well as the trucation of the Fourier series. We show
that the RCWA embodies a Galerkin scheme for a perturbed problem, and then we
use tools from the Finite Element Method to show that the method converges with
increasing number of retained Fourier modes and finer approximations of the
relative permittivity. Numerical examples illustrate our analysis, and suggest
further work.
