# Maxence Cassier's research while affiliated with Institut Fresnel and other places

## Publications (30)

Preprint
It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnet...
Article
We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same...
Article
This paper is concerned with the time-dependent Maxwell’s equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propa...
Preprint
We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same...
Preprint
This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propa...
Article
We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active, we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a...
Preprint
We study the band structure of self-adjoint elliptic operators $\mathbb{A}_g= -\nabla \cdot \sigma_{g} \nabla$, where $\sigma_g$ has the symmetries of a honeycomb tiling of $\mathbb{R}^2$. We focus on the case where $\sigma_{g}$ is a real-valued scalar: $\sigma_{g}=1$ within identical, disjoint and "inclusions", centered at vertices of a honeycomb...
Preprint
We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a d...
Preprint
Photonic crystals are periodic structures which prevent light propagation along one or more directions in certain frequency intervals. Their band spectrum is usually analyzed using Floquet-Bloch decomposition. This spectrum is located on the real axis, and it enters the complex plane when absorption and dispersion is considered in the dielectric pe...
Article
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 mat...
Book
This is a book chapter for a World Scientific Refence Set edited by Prof. Vadim Markel, a former member at Institut Fresnel, who currently works in the Department of Radiology at the University of Pennsylvania, USA.
Article
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 notion of non-dissipativity and passivity. We consider successively the case of so-called local m...
Article
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 electromagnetic version of Kirchhoff imaging and prove, in the Fraunhofer asymptotic regime, that range and cross-range resolution estimates are identical to...
Article
Using a sum rule, we derive new bounds on Herglotz functions that generalize those given in (Gustafson and Sj\"oberg 2010) and (Bernland, Luger and Gustafson 2011). 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 i...
Article
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 trans...
Book
In Chapter 1 we review many of the linear equations of physics, and write them in a canonical form appropriate to the theory of composites. We show how conservation laws, which have played a key role throughout the history of science, can be generalized to equalities which we call boundary field equalities and inequalities . Chapter 2 reviews the a...
Article
Full-text available
In this chapter of the new book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, (2016), to appear] we give a rigorous derivation of the field equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since i...
Article
Full-text available
In this chapter of the new book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, (2016), to appear] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the conn...
Article
Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit–receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer propos...
Article
This thesis consists of two independent parts. In the first one, we consider a reference medium, containing some scatterers whose positions are unknown. Using an array of transducers, our goal is to generate an acoustic wave that would focus in space and in time near one of these scatterers. The technique proposed here is based on DORT method (Fren...
Article
We are concerned with a two-dimensional problem which models the scattering of a time-harmonic acoustic wave by an arbitrary number of sound-soft circular obstacles. Assuming that their radii are small compared to the wavelength, we propose a mathematical justification of different levels of asymptotic models available in the physical literature, i...
Article
In this paper, we are concerned with focusing effects for time-dependent waves using 'time reversal mirrors'. We consider a simple two-dimensional problem which models acoutic wave propagation in a homogeneous medium which contains several unknown scatterers. We show how to construct a wave that focuses in space and time near one of these scatterer...

## Citations

... In the mathematics literature, 'cloaking by anomalous localized resonances' has been intensively studied, see e.g., [2]. Mathematically, some of the counter-intuitive spectral properties of the time-dependent Maxwell system for an interface between a metamaterial and a vacuum are investigated for the non-dissipative case, in a special geometry, in [12,13]. For the dissipative case, in the whole space and in a setting allowing dimension-reduction, we refer to the recent article [10]. ...
... The variable coefficient is due to the permittivity function that models the dielectric of the waveguides. These systems can exhibit similar properties to those found in Schrödinger operators (mentioned below) which possess Dirac points in honeycomb lattices [48,49] and localized edge states [50]. Moreover, these systems also extend to topological insulators. ...
... On the other hand, thermoelectric devices have been proposed to pump the heat flow accurately from one side of a thermal cloak to the other side by adjusting the input current, so that the background temperature field can be restored in a stationary regime [53,54]. In our former work [55], we envisioned using Peltier devices (surrounding the object to cloak) to control transient thermal fields generated by a source. The approach we present here can be viewed as a generalization of that in [56] that considered a single dipole source placed inside the object to cloak in the stationary regime. ...
... Using the above results and remembering that the integrations in eq. (15) are performed along the path of the background pulse, given by t = n p x, we get the following ...
... Since the preprocessing creates an 8 dimensional real field from a 4 dimensional one, there are very significant errors. The key is to show that these errors do not affect images of the scatterers, if we use an electromagnetic version of Kirchhoff imaging [18]. We do this with a stationary phase argument. ...
Citing article
... 3 However, because the cloak's performance highly depends upon dispersion properties of its metamaterial's constituents that approximate the required optical parameters, it seems hopeless to achieve passive broadband optical cloaking due to the light speed limit. 4 This fundamental limit can be overcome by considering other types of waves with lower velocities, such as acoustic waves, 5 hydrodynamics, 6 and water waves. 7 Such waves are governed by form-invariant equations. ...
... In the mathematics literature, 'cloaking by anomalous localized resonances' has been intensively studied, see e.g., [2]. Mathematically, some of the counter-intuitive spectral properties of the time-dependent Maxwell system for an interface between a metamaterial and a vacuum are investigated for the non-dissipative case, in a special geometry, in [12,13]. For the dissipative case, in the whole space and in a setting allowing dimension-reduction, we refer to the recent article [10]. ...
... In applications, they play an important role e.g., as the Fourier transform of the impulse response of passive systems, which can describe the (one-port) system completely [21,43] and are essential e.g., for physical limitations using so-called sum rules [7]. Another area of application is within the theory of composites [34,35], for instance. ...
... In the classical setting, i.e., with the strong assumptions that σ * = σ ≥ 0 and σ invertible, all these problems can solved 3,8,16,17 . We will review this now and include proofs using a new approach to solving these problems which emphasizes block operator methods and the use of Schur complements 28 (although the importance of it in the theory of composites was originally recognized in Ref. 29). Later in the paper we generalize our approach when we relax the invertibility hypothesis. ...
... Remark 2.1. The methods we use in this paper to reduced linear DAEs to linear ODEs is implicit in a standard and well known approach in studying the electrodynamics of layered media [7,1,2,14,38,40,32,33,39] (as opposed to other standard methods such as those in [62,59,60,61]) and has been used effectively by the second author of this paper, for instance, in [56,50,51,8]. But in this context, our approach that we've developed in the sections below, using DA operators and spectral theory, is completely new and deserves to be further studied from this perspective. ...