Asymptotic model for finite-element calculations of diffraction by shallow metallic surface-relief gratings

Abstract

We have formulated an asymptotic model for implementation in the finite-element method to calculate diffraction from a planar multilayered structure having a shallow surface-relief grating. The thin grating layer containing the shallow grating is replaced by a planar interface with transmission conditions that differ from the standard continuity conditions, thereby eliminating the necessity of representing the grating layer by a fine mesh. The parameters defining the shallow surface-relief grating are thereby removed from the geometry to the transmission conditions. Adoption of the asymptotic model will considerably reduce the computational cost of optimizing the grating shape because there is no need to re-mesh at every optimization step.

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

“…We have solved problem (5.3) using standard Lagrange FEM with thirddegree polynomials. For the Dirichlet-to-Neumann operators + and − , we have used a truncated Fourier expansion approach similar to that in [23].…”

“…We have fixed the angle of incidence to = 0, since most solar cells are illuminated normally to maximize photonic absorption. The physical parameters have been taken as in [23], where further details can be found: + = 3.6876 and − = −5.8828 + 0.6650 i.…”

“…For each polarization state, = or = , let̂︀ ,ℎ ,1 denote the approximate values of the solution to problem (2.7), delivered by the FEM solution of the asymptotic model (5.3) for the -polarization or (5.4) for the -polarization, for a specific choice of ℎ and . We have also used the approximate solution̂︀ ,ℎ ,1 to compute a physical quantity of interest: the so called absorptance ,ℎ (see for instance [23]).…”

“…As an alternative, it is possible to devise an asymptotic model that approximates the electromagnetic field in the structure by replacing the thin layer by an interface where suitable transmission conditions are imposed [7-9, 19, 22]. In this direction, an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surfacerelief grating has been devised in [23]. The numerical results reported in this reference show that when the asymptotic expansion of the solution is truncated to second order terms, third-order convergence with respect to the thickness of the thin grating layer is obtained for transverse electric polarization ( -polarization), and at least second-order convergence for the transverse magnetic polarization ( -polarization).…”