Mathematical and numerical modeling for wave propagation in the presence of metamaterials – METAMATH

Recent discoveries have shown the feasibility of weakly dissipative electromagnetic materials, whose effective dielectric and magnetic constants have negative real parts. These "metamaterials," which have a complex multiscale structure, lead to extraordinary phenomena as regards the propagation of electromagnetic waves (negative refraction, 'sub wavelength' resonance cavities etc ...) and thus arouse great interest in many potential applications (super lenses, stealth coatings, miniaturization of antennas etc.).

The optimization of devices using metamaterials requires the development of appropriate simulation tools. Yet it is not conceivable to simulate these materials in all their complexity. An attractive alternative is to model the metamaterial by a homogeneous material, with physical constants of negative real part. This approach is now widely used by physicists and is the subject of an active mathematical research in the community of homogenization.

However, when the dissipation in the metamaterial is negligible (which is desired for applications), the homogenized model is not standard (the dielectric permittivity and magnetic permeability are assumed to be real negative in a certain range of frequencies) and the use of conventional numerical methods is not obvious. There are even simple configurations such that the model in the frequency domain is clearly not correct. Thus, when considering an interface between a dielectric and a metamaterial, with a permittivity and / or permeability contrast equal to -1, it appears an accumulation of energy at the interface that is not compatible with the usual mathematical/physical framework. Surprising phenomena also occur for other contrasts when the interface is not smooth.

With this ANR project, our ambition is to contribute to the development of models for metamaterials that are both physically relevant and accessible to numerical computation.

We will combine three types of approaches:
-- In the continuation of the work already done by POEMS, we will precise the range of validity of the simpler homogenized model, in frequency and time domains, both theoretically and numerically.
-- In the continuation of the work of the University of Toulon in homogenization and of other partners in the field of asymptotic methods, we will try to correct the existing models to extend their range of validity.
-- These models will be validated by numerical simulations that take into account the microstructure of the metamaterial by exploiting the periodicity of the medium. We intend for this to generalize the method developed by POEMS for a planar interface and to develop a method of numerical homogenization.
With powerful analytical and numerical methods at hand, we tackle questions related to non-destructive testing and shielding of metamaterials: how can one detect defects inside or on top of such a structure ? Reciprocally, can one design coatings of metamaterials to hide objects behind ?