The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A 13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A 13, 1019 ( 1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.
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