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We study the convergence properties of the two-dimensional Rigorous CoupledWave Approach (RCWA) for s-polarized monochromatic incident light. The RCWA iswidely used to solve electromagnetic boundary-value problems where the relativepermittivity varies periodically in one direction, i.e., scattering by agrating. This semi-analytical approach expands all the electromagnetic fieldphasors as well as the relative permittivity as Fourier series in the spatialvariable along the direction of periodicity, and also replaces the relativepermittivity with a stairstep approximation along the direction normal to thedirection of periodicity. Thus, there is error due to Fourier truncation andalso due to the approximation of grating permittivity. We prove that the RCWAis a Galerkin scheme, which allows us to employ techniques borrowed from theFinite Element Method to analyze the error. An essential tool is a Rellichidentity that shows that certain continuous problems have unique solutions thatdepend continuously on the data with a continuity constant having explicitdependence on the relative permittivity. We prove that the RCWA converges withan increasing number of retained Fourier modes and with a finer approximationof the grating interfaces. Numerical results show that our convergence resultsfor increasing the number of retained Fourier modes are seen in practice, whileour estimates of convergence in slice thickness are pessimistic.

Analysis of the Rigorous Coupled Wave Approach for s-Polarized Light in Gratings