  We study the convergence properties of the two-dimensional Rigorous Coupled
Wave Approach (RCWA) for s-polarized monochromatic incident light. The RCWA is
widely used to solve electromagnetic boundary-value problems where the relative
permittivity varies periodically in one direction, i.e., scattering by a
grating. This semi-analytical approach expands all the electromagnetic field
phasors as well as the relative permittivity as Fourier series in the spatial
variable along the direction of periodicity, and also replaces the relative
permittivity with a stairstep approximation along the direction normal to the
direction of periodicity. Thus, there is error due to Fourier truncation and
also due to the approximation of grating permittivity. We prove that the RCWA
is a Galerkin scheme, which allows us to employ techniques borrowed from the
Finite Element Method to analyze the error. An essential tool is a Rellich
identity that shows that certain continuous problems have unique solutions that
depend continuously on the data with a continuity constant having explicit
dependence on the relative permittivity. We prove that the RCWA converges with
an increasing number of retained Fourier modes and with a finer approximation
of the grating interfaces. Numerical results show that our convergence results
for increasing the number of retained Fourier modes are seen in practice, while
our estimates of convergence in slice thickness are pessimistic.
