We estimate the electromagnetic fields generated by a cluster of dielectric nanoparticles which are small scaled but enjoy high contrast of their relative permittivity, embedded into a background made of a vacuum. Under certain ratio between their size and contrast, these nanoparticles generate resonances, called dielectric resonances. We first characterize the dominant field generated by a cluster of such dielectric-resonating nanoparticles. In this point-interaction approximation, the nanoparticles can be distributed to occupy volume-like domains or low dimensional hypersurfaces where periodicity is not required. Then we investigate the corresponding effective electromagnetic medium with periodic distribution under some mild assumptions. We show that even though the dielectric nanoparticles are merely generated by the contrasts of their permittivity (and not their permeability), the effective medium is a perturbation of the permeability and not the permittivity. Both of the cases for the effective permeability being positive and negative are studied.

Foldy-Lax models are asymptotic models of wave scattering by multiple obstacles, in the regime when their characteristic size tends to zero. While frequency-domain Foldy-Lax models are now fairly well-studied (see e.g. [1,2,3]), their time-domain counterparts were considered only very recently, see [4,5].

This talk is dedicated to the derivation, stability and convergence analysis of a time-domain Foldy-Lax model for sound-soft scattering by small obstacles. We start with the analysis of the time-domain counterpart of the respective frequency-domain model for circular scatterers from [6] and show that it is unstable for some geometric configurations. To stabilize it, we propose its reinterpretation as a perturbed Galerkin discretization of a single layer boundary integral equation.

Its unperturbed Galerkin discretization is then automatically stable due to a coercivity-like property of the underlying operator and thus serves as a basis to derive the stabilized model. Let us remark that this reinterpretation provides us with an alternative way to derive asymptotic models as Galerkin discretizations of boundary integral formulations with well-chosen basis functions.

We will present the convergence analysis of the new model, discuss its numerical implementation, and illustrate our findings with numerical experiments.

[1] P. Martin. Multiple scattering, vol. 107 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2006.

[2] D.P. Challa, M. Sini. On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 , pp. 55–108, 2014.

[3] D.P. Challa, M. Sini. The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system, Math. Nachr., 288, pp. 1834–1872, 2015.

[4] H. Barucq, J. Diaz, V. Mattesi, S. Tordeux. Asymptotic behavior of acoustic waves scattered by very small obstacles, ESAIM Math. Model. Numer. Anal., 55, pp. S705–S731, 2021.

[5] M. Sini, H. Wang, Q. Yao. Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density, Multiscale Model. Simul., 19, pp. 1083–1114, 2021.

[6] M. Cassier, C. Hazard. Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50, pp. 18–28, 2013.

In this talk, we will consider the electromagntic waves generated by an in- clusion moving at a uniformly constant speed. We will discuss the direct and inverse scattering problem for the corresponding transmission problem. We first show that the scattering electromagnetic fields satisfy a related Lippmann-Schwinger system of equations and the solutions of this system of integral equations can be written in terms of the Neumann series under certain assumption on the ratio between the speed of the moving object and the speed of light in the vaccum. Finally, as an application of this result, we will prove that the far-field map uniquely determines the unknown moving object.