Méthodes des éléments finis

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The Finite Elements Method (FEM) has been introduced in electromagnetism in the late 60s by Peter P. Silvester. Its development for 3D problems as well as for scattering and eigenvalue problems dates back to the 80s, when Nédélec edge elements have been advocated by various authors such as A. Bossavit and R. Kotiuga. These elements impose the continuity of the tangential component of the discretised field and allow the normal component to be discontinuous at interfaces. Another important fundamental contribution for the application of FEM to wave problems is the invention of Perfectly Matched Layers (PMLs) by Bérenger in 1994. PMLs allow for a rigorous truncation of unbounded physical regions and can be used, for instance, to insure the correct radiation conditions in open problems. In the periodic cases, Floquet-Bloch quasi-periodic conditions can also be set up easily.

For instance, the modeling of diffraction gratings (see figure) involves both the PMLs and the Floquet-Bloch conditions. The CLARTE team of Institut Fresnel has applied these fundamental numerical concepts to various domains of the physics of wave optics. Rigorous electromagnetic formulations have been introduced in order to tackle various optical phenomena, both direct (such as scattering by periodic/aperiodic arbitrary 3D structures illuminated by arbitrary plane waves) and modal problems (microstructured optical fiber also named photonic crystal fibers, photonic crystals and grating modes).

We have shown that the remarkable accuracy of the FEM approach upon the determination of integrated energy-related physical values, even in the case of relatively coarse meshes, which makes it a fast tool for the design and optimization of diffractive optical components (e.g., reflection and transmission filters, polarizers, beam shapers).

Finally, its complete independence with respect to both geometries and linear constituent materials (magnetic, anisotropic, lossy, graded-indexed) of the domain elements makes it a versatile tool for the theoretical study of metamaterials, homogenization processes, photonic crystal slabs, periodic plasmonic structures, etc… We have also used FEM approach to solve nonlinear generalized propagation eigenvalue problems in different types of waveguides, obtaining a generalization of the Townes soliton.