Finite Element Method (Chapter from "Gratings: Theory and Numeric Applications")

TL;DR

This chapter presents a comprehensive finite element formulation for calculating diffraction efficiencies in complex gratings, accommodating arbitrary geometries, multilayer stacks, and various polarization states, with applications in metamaterials and photonic structures.

Contribution

It introduces a highly general finite element approach with new boundary conditions and an adaptive PML, applicable to diverse grating geometries and material configurations.

Findings

Applicable to 2D and 3D gratings with arbitrary shapes

Handles multilayered stacks with high accuracy

Flexible for studying metamaterials and photonic crystals

Abstract

In this chapter, we demonstrate a general formulation of the Finite Element Method allowing to calculate the diffraction efficiencies from the electromagnetic field diffracted by arbitrarily shaped gratings embedded in a multilayered stack lightened by a plane wave of arbitrary incidence and polarization angle. It relies on a rigorous treatment of the plane wave sources problem through an equivalent radiation problem with localized sources. Bloch conditions and a new Adaptative Perfectly Matched Layer have been implemented in order to truncate the computational domain. We derive this formulation for both mono-dimensional gratings in TE/TM polarization cases (2D or scalar case) and for the most general bidimensional or crossed gratings (3D or vector case). The main advantage of this formulation is its complete generality with respect to the studied geometries and the material properties. Its principle remains independent of both the number of diffractive elements by period and number of stack layers. The flexibility of our approach makes it a handy and powerful tool for the study of metamaterials, finite size photonic crystals, periodic plasmonic structures.