Maxence Cassier

Mathematical models for dispersive electromagnetic waves: An overview

Abstract In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of non-dissipativity and passivity. We consider successively the cases of so-called local media and then of general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.

Bounds on Herglotz functions and fundamental limits of broadband passive quasistatic cloaking

Using a sum rule, we derive new bounds on Herglotz functions that generalize those given in Bernland et al. [J. Phys. A: Math. Theor. 44(14), 145205 (2011)] and Gustafsson and Sjoberg [New J. Phys. 12(4), 043046 (2010)]. These bounds apply to a wide class of linear passive systems such as electromagnetic passive materials. Among these bounds, we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window, where we exhibit sharp bounds. Then, we apply these bounds in the context of broadband passive cloaking in the quasistatic regime to refute the following challenging question: is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band? Our rigorous approach, although limited to quasistatics, gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizability tensor associated with the cloaking device. We emphasize that our resul...

Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform

ABSTRACT We explore the spectral properties of the time-dependent Maxwell’s equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill, respectively, complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles.

Imaging Polarizable Dipoles

We present a method for imaging the polarization vector of an electric dipole distribution in a homogeneous medium from measurements of the electric field made at a passive array. We study an elect...

Imaging small polarizable scatterers with polarization data

We present a method for imaging small scatterers in a homogeneous medium from polarization measurements of the electric field at an array. The electric field comes from illuminating the scatterers with a point source with known location and polarization. We view this problem as a generalized phase retrieval problem with data being the coherency matrix or Stokes parameters of the electric field at the array. We introduce a simple preprocessing of the coherency matrix data that partially recovers the ideal data where all the components of the electric field are known for different source dipole moments. We prove that the images obtained using an electromagnetic version of Kirchhoff migration applied to the partial data are, for high frequencies, asymptotically identical to the images obtained from ideal data. We analyze the image resolution and show that polarizability tensor components in an appropriate basis can be recovered from the Kirchhoff images, which are tensor fields. A time domain interpretation of this imaging problem is provided and numerical experiments are used to illustrate the theory.