Use of Fourier series in the analysis of discontinuous periodic structures

TLDR

The recent reformulation of the coupled‑wave method by Lalanne and Morris and Granet and Guizal dramatically improves convergence for metallic gratings in TM polarization. The paper provides a firm mathematical foundation for these reformulations, enabling their application to any numerical work requiring Fourier analysis of products of discontinuous periodic functions. The new formulation achieves faster convergence by uniformly satisfying boundary conditions in the grating region, unlike earlier non‑uniform.

Abstract

The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 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.