An asymptotic model based on matching far and near field expansions for thin gratings problems

¹ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.
² Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
³ Center for Research in Mathematical Engineering (CI 2MA), Universidad de Concepción, Casilla 160-C, Concepción, Chile.

^* Corresponding author: msolano@ing-mat-udec.cl, manolosolano@gmail.com

Received: 16 October 2019
Accepted: 28 July 2020

Abstract

In this paper, we devise an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surface-relief grating. Far from the grating, we assume that the solution can be written as a power series in terms of the grating thickness δ, the coefficients of this expansion being smooth up to the grating. However, the expansion approximates the solution only sufficiently far from the grating (far field approximation). Near the grating, we assume that there exists another expansion in powers of δ (near field approximation). Moreover, there is an overlapping zone where both expansion are valid. The proposed model is based on matching the two expansions on this overlapping domain. Then, by truncating terms of order δ² or higher, we obtain explicitly the equations satisfied by the lowest order terms in the power series. Under appropriate assumptions, we prove second order convergence of the error with respect to δ. Finally, an alternative form, more convenient for implementation, is derived and discretized with finite elements to perform some numerical tests.