MODAL METHOD BASED ON SUBSECTIONAL GEGENBAUER POLYNOMIAL EXPANSION FOR LAMELLAR GRATINGS: WEIGHTING FUNCTION, CONVERGENCE AND STABILITY

Abstract

The Modal Method by Gegenbauer polynomials Expan- sion (MMGE) has been recently introduced for lamellar gratings by Edee (8). This method shows a promising potential of outstanding convergence but still sufiers from instabilities when the number of polynomials is increased. In this work, we identify the origin of these instabilities and propose a way to remove them. Among the numerical methods developed for the analysis of lamellar difiraction gratings, modal methods play an important role because of their great versatility and relative efiectiveness. In the classical modal method (1,2), the eigenvalues are obtained by solving a transcendental equation. In other modal methods, the eigenmodes and propagation constants are generally obtained by searching the eigenvalues and eigenvectors (3{8) of a matrix which is derived from the Maxwell's equations by using the method of moments (9). Mathematically speaking, for one-dimensional gratings with piecewise homogeneous media, and plane wave excitation, the eigenmodes are solutions of the Helmholtz equation subject to boundary conditions at the interfaces between two media and to the pseudo-periodicity condition. Numerically, the rate of convergence of the method depends on how the matrix from which eigenvalues are sought takes into account the continuity relations. Indeed, one of the main difierences between