Modal Method Based on Subsectional Gegenbauer Polynomial Expansion for Lamellar Gratings: Weighting Function, Convergence and Stability

Abstract

Abstract-The Modal Method by Gegenbauer polynomials Expansion (MMGE) has been recently introduced for lamellar gratings by Edee [8]. This method shows a promising potential of outstanding convergence but still suffers 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.

Electromagnetic modelization of spherical focusing on a one-dimensional grating thanks to a conical B-spline modal method

“…Instead, one has to completely solve the electromagnetic problem, which can prove burdensome, and all the more when optimizing a great number of parameters. In contrast to finite elements methods or finite difference time domain methods, modal methods, such as rigorous coupled-wave analysis [3][4][5] or the modal method based on Gegenbauer polynomial expansions [6,7], permit us to reduce the numerical complexity of an electromagnetic problem for layered metallic gratings.…”

Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating

“…In earlier papers [1][2][3][4], we present a modal method based on the use of Gegenbauer polynomial [5] expansion for the analysis of a 1D structure with piecewise homogeneous media. As with other modal methods, the computed eigenmodes and propagation constants are obtained by searching the eigenvalues and eigenvectors of a matrix, which are derived from Maxwell's equations by using the method of moments.…”

Efficient implementation of B-spline modal method for lamellar gratings

“…On the contrary, using subdomain expansions within each domain of the lamellar gratings allows one to express rigorously the different continuity relations that determine the eigenvalue problem and thus leads to exponential convergence for the eigenvalues and eigenvectors. Indeed, polynomials modal methods [8][9][10] and pseudo spectral modal methods [11][12][13] outperform the Fourier modal method in term of speed of convergence. Another alternative to Fourier expansion consists in B-spline expansion.…”

“…Let us illustrate the use of multiple knots on a particular example. Consider function f x defined on interval [01] such that f x j cosπxj (10) and its derivative …”