46 research outputs found
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 electromagnetic version of Kirchhoff
imaging and prove, in the Fraunhofer asymptotic regime, that range and
cross-range resolution estimates are identical to those in acoustics. Our
asymptotic analysis provides error estimates for the cross-range dipole
orientation reconstruction and shows that the range component of the dipole
orientation is lost in this regime. A naive generalization of the Kirchhoff
imaging function is afflicted by oscillatory artifacts in range, that we
characterize and correct. We also consider the active imaging problem which
consists in imaging both the position and polarizability tensors of small
scatterers in the medium using an array of collocated sources and receivers. As
in the passive array case, we provide resolution estimates that are consistent
with the acoustic case and give error estimates for the cross-range entries of
the polarizability tensor. Our theoretical results are illustrated by numerical
experiments.Comment: 35 pages, 18 figure
Mathematical models for dispersive electromagnetic waves: an overview
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 media and 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.Comment: 46 pages, 16 figure
Truth and Consequences
International audienc
High contrast elliptic operators in honeycomb structures
We study the band structure of self-adjoint elliptic operators , where has the symmetries of a
honeycomb tiling of . We focus on the case where is
a real-valued scalar: within identical, disjoint "inclusions",
centered at vertices of a honeycomb lattice, and (high
contrast) in the complement of the inclusion set (bulk). Such operators govern,
e.g. transverse electric (TE) modes in photonic crystal media consisting of
high dielectric constant inclusions (semi-conductor pillars) within a
homogeneous lower contrast bulk (air), a configuration used in many physical
studies. Our approach, which is based on monotonicity properties of the
associated energy form, extends to a class of high contrast elliptic operators
that model heterogeneous and anisotropic honeycomb media.
Our results concern the global behavior of dispersion surfaces, and the
existence of conical crossings (Dirac points) occurring in the lowest two
energy bands as well as in bands arbitrarily high in the spectrum. Dirac points
are the source of important phenomena in fundamental and applied physics, e.g.
graphene and its artificial analogues, and topological insulators. The key
hypotheses are the non-vanishing of the Dirac (Fermi) velocity ,
verified numerically, and a spectral isolation condition, verified analytically
in many configurations. Asymptotic expansions, to any order in , of
Dirac point eigenpairs and are derived with error bounds.
Our study illuminates differences between the high contrast behavior of
and the corresponding strong binding regime for Schroedinger
operators.Comment: 63 pages, 13 figure
Space-time focusing of acoustic waves on unknown scatterers
International audienceConsider 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 proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy-Lax model. In this context, several mathematical and numerical arguments that support our idea are explored
Active Thermal Cloaking and Mimicking
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 different one to an observer outside the
cloaked region, from the perspective of thermal measurements. Our results
assume a homogeneous isotropic bulk medium and require knowledge of the source
to cloak or mimic, but are in most cases independent of the object to cloak.Comment: 29 pages,12 figure
Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
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 electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances