26,971 research outputs found
Volume integral equations for electromagnetic scattering in two dimensions
We study the strongly singular volume integral equation that describes the
scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We
consider the case of a cylindrical obstacle and fields invariant along the axis
of the cylinder, which allows the reduction to two-dimensional problems. With
this simplification, we can refine the analysis of the essential spectrum of
the volume integral operator started in a previous paper (M. Costabel, E.
Darrigrand, H. Sakly: The essential spectrum of the volume integral operator in
electromagnetic scattering by a homogeneous body, Comptes Rendus Mathematique,
350 (2012), pp. 193-197) and obtain results for non-smooth domains that were
previously available only for smooth domains. It turns out that in the TE case,
the magnetic contrast has no influence on the Fredholm properties of the
problem. As a byproduct of the choice that exists between a vectorial and a
scalar volume integral equation, we discover new results about the symmetry of
the spectrum of the double layer boundary integral operator on Lipschitz
domains.Comment: 21 page
The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body
International audienceWe study the strongly singular volume integral operator that describes the scattering of time-harmonic electromagnetic waves. For the case of piecewise constant material coefficients and smooth interfaces, we determine the essential spectrum. We show that it is a finite set and that the operator is Fredholm of index zero in H(curl) if and only if the relative permeability and permittivity are both different from 0 and -1
Singular Modes of the Electromagnetic Field
We show that the mode corresponding to the point of essential spectrum of the
electromagnetic scattering operator is a vector-valued distribution
representing the square root of the three-dimensional Dirac's delta function.
An explicit expression for this singular mode in terms of the Weyl sequence is
provided and analyzed. An essential resonance thus leads to a perfect
localization (confinement) of the electromagnetic field, which in practice,
however, may result in complete absorption.Comment: 14 pages, no figure
Transverse electric scattering on inhomogeneous objects: spectrum of integral operator and preconditioning
The domain integral equation method with its FFT-based matrix-vector products
is a viable alternative to local methods in free-space scattering problems.
However, it often suffers from the extremely slow convergence of iterative
methods, especially in the transverse electric (TE) case with large or negative
permittivity. We identify the nontrivial essential spectrum of the pertaining
integral operator as partly responsible for this behavior, and the main reason
why a normally efficient deflating preconditioner does not work. We solve this
problem by applying an explicit multiplicative regularizing operator, which
transforms the system to the form `identity plus compact', yet allows the
resulting matrix-vector products to be carried out at the FFT speed. Such a
regularized system is then further preconditioned by deflating an apparently
stable set of eigenvalues with largest magnitudes, which results in a robust
acceleration of the restarted GMRES under constraint memory conditions.Comment: 20 pages, 8 figure
Classification of electromagnetic resonances in finite inhomogeneous three-dimensional structures
We present a simple and unified classification of macroscopic electromagnetic
resonances in finite arbitrarily inhomogeneous isotropic dielectric 3D
structures situated in free space. By observing the complex-plane dynamics of
the spatial spectrum of the volume integral operator as a function of angular
frequency and constitutive parameters we identify and generalize all the usual
resonances, including complex plasmons, real laser resonances in media with
gain, and real quasi-static resonances in media with negative permittivity and
gain.Comment: 4 pages, 2 figure
Stability Analysis of a Simple Discretization Method for a Class of Strongly Singular Integral Equations
Motivated by the discrete dipole approximation (DDA) for the scattering of
electromagnetic waves by a dielectric obstacle that can be considered as a
simple discretization of a Lippmann-Schwinger style volume integral equation
for time-harmonic Maxwell equations, we analyze an analogous discretization of
convolution operators with strongly singular kernels.
For a class of kernel functions that includes the finite Hilbert
transformation in 1D and the principal part of the Maxwell volume integral
operator used for DDA in dimensions 2 and 3, we show that the method, which
does not fit into known frameworks of projection methods, can nevertheless be
considered as a finite section method for an infinite block Toeplitz matrix.
The symbol of this matrix is given by a Fourier series that does not converge
absolutely. We use Ewald's method to obtain an exponentially fast convergent
series representation of this symbol and show that it is a bounded function,
thereby allowing to describe the spectrum and the numerical range of the
matrix.
It turns out that this numerical range includes the numerical range of the
integral operator, but that it is in some cases strictly larger. In these cases
the discretization method does not provide a spectrally correct approximation,
and while it is stable for a large range of the spectral parameter ,
there are values of for which the singular integral equation is well
posed, but the discretization method is unstable.Comment: 34 pages, 7 figures, In V2: added 2 new references in the
introductio
The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering
We present a new formulation for the problem of electromagnetic scattering
from perfect electric conductors. While our representation for the electric and
magnetic fields is based on the standard vector and scalar potentials in the Lorenz gauge, we establish boundary conditions on the
potentials themselves, rather than on the field quantities. This permits the
development of a well-conditioned second kind Fredholm integral equation which
has no spurious resonances, avoids low frequency breakdown, and is insensitive
to the genus of the scatterer. The equations for the vector and scalar
potentials are decoupled. That is, the unknown scalar potential defining the
scattered field, , is determined entirely by the incident scalar
potential . Likewise, the unknown vector potential defining the
scattered field, , is determined entirely by the incident vector
potential . This decoupled formulation is valid not only in the
static limit but for arbitrary .Comment: 33 pages, 7 figure
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