Institut Fresnel

Minh Duy Truong, PhD in ATHENA team, with Philippe Lalanne from LP2N, Institut d’Optique d’Aquitaine co-director won the "Best Student Paper Award" at SPIE Optics + Optoelectronics 2019, Prague for his article "The Exact Dispersive Quasi-Normal Mode (DQNM) Expansion for
Photonic Structures with Highly Dispersive Media in Unbounded Geometries.", Minh Duy Truong, Guillaume Démesy, Frédéric Zolla, André Nicolet

Abstract :
In this paper we present recent developments in our modal expansion technique for electromagnetic structures with highly dispersive media and its application in unbounded geometries. The basic idea is to use natural modes of photonic devices (i.e. structures defined by their geometry, the electromagnetic properties of the various media, and a range of working frequencies) obtained by solving spectral problems associated to the (sourceless) Maxwell’s equations.
Several problems have to be carefully solved to design an efficient method. First of all, the numerical approach requires a robust discretisation of the wave operators associated to the Maxwell’s equations. We use the Finite Element Method (FEM) based on edge elements (and their higher order generalizations) that have proved to be spurious mode free. Then a suitable representation of the permittivities as functions of the frequency is provided by rational functions. An interpolation method has been set up that is very accurate on a large range of frequencies and thrifty with the number of poles. The obtained functions are naturally causal (following Kramers-Kronig relations) and provides a natural analytic continuation of permittivities in the complex plane that is necessary for the computation of Dispersive Quasi-Normal Modes (DQNM) associated to complex frequencies (including plasmons in negative permittivity regions). The numerical solution of the Maxwell spectral problems with dispersive media requires efficient non-linear eigenproblem algorithms that are provided by the SLEPc library. In recent versions of this library, our non-linear eigenvalue problems with coefficents that are rational functions of frequency can be tackled directly. Once the DQNM are known, they can be used to perform a modal expansion of the resolvant operator (that can be truncated to a small number of modes for its practical uses). Indeed, despite the fact that the associated operator in non- self adjoint and is a non-linear function of frequency, there exists an exact dispersive quasi-normal mode (DQNM) expansion as we have recently shown using the Keldysh theorem on eigenfunctions of operators depending on a complex parameter. This expansion is the key to a fast algorithm: the knowledge of a limited number of eigenmodes allows a very fast computation of direct scattering problems and of the Green’s function, for instance, on a wide range of frequencies. In the present work, we have extended our exact dispersive modal expansion to the case of unbounded geometries. The Perfectly Matched Layers (PML), corresponding to a complex coordinate scaling, are used for the unbounded domain truncation and to unveil the resonances by rotating the continuous spectrum in the complex plane. Some compact scatterers and periodic structures (diffaction gratings) are taken as examples of application of the DQNM expansion.