A general perturbation formula for electromagnetic fields in presence of low volume scatterers

Abstract

Abstract.In several practically interesting applications of electromagnetic scattering theory like, e.g., scattering from small point-like objects such as buried artifacts or small inclusions in non-destructive testing, scattering from thin curve-like objects such as wires or tubes, or scattering from thin sheet-like objects such as cracks, the volume of the scatterers is small relative to the volume of the surrounding medium and with respect to the wave length of the applied electromagnetic fields. This smallness typically causes problems when solving direct scattering problems due to the need to discretize the objects and also when solving inverse scattering problems because small objects have very little effect on electromagnetic fields. In this paper we consider an asymptotic representation formula for scattered electromagnetic waves caused by low volume objects contained in some otherwise homogeneous threedimensional bounded domain, assuming only that the scatterers are measurable and well-separated from the boundary of the domain. The formula yields a very general asymptotic model for electromagnetic scattering due to low volume objects that can either be used to simulate the corresponding electromagnetic fields or as the foundation of efficient reconstruction methods for inverse scattering problems with low volume scatterers.

An Asymptotic Representation Formula for Scattering by Thin Tubular Structures and an Application in Inverse Scattering

“…Various low volume expansions for electrostatic potentials, as well as elastic and electromagnetic fields are available in the literature, (see, e.g., [1,5,7,8,12,14,18,22,24,25,28]). The framework we use in this work was first introduced in [18,19,21] for electrostatic potentials.…”

“…The framework we use in this work was first introduced in [18,19,21] for electrostatic potentials. The very general low volume perturbation formula for time-harmonic Maxwell's equations in bounded domains from [1,28] can be extended to the electromagnetic scattering problem in unbounded free space as considered in this work using an integral equation technique developed in [4,9]. Applying this result to the special case of thin tubular scattering objects, the first observation is that the scattered field away from the scatterer converges to zero as the diameter of its cross-section tends to zero.…”

“…The outline of this paper is as follows. After providing the mathematical model for electromagnetic scattering by a thin tubular scattering object in the next section, we summarize the results on the general asymptotic analysis from [1,28] for the special case of thin tubular scattering objects in Section 3. In Section 4 we state and prove our main theoretical result concerning the explicit characterization of the polarization tensor of a thin tubular scattering object.…”

“…We derive an asymptotic perturbation formula for the scattered electric field E s ρ and the electric far field pattern E ∞ ρ as the radius ρ of the cross-section D ′ ρ of the scattering object D ρ in (2.6) tends to zero relative to the wave length λ = 2π/k. Expansions of this type are available in the literature for time-harmonic electromagnetic fields (see, e.g., [1,8,9,28]). However, the existing results for Maxwell's equations are either formulated on bounded domains, or for scattering problems on unbounded domains but with different geometrical assumptions on the scattering objects than considered in this work.…”

“…However, the existing results for Maxwell's equations are either formulated on bounded domains, or for scattering problems on unbounded domains but with different geometrical assumptions on the scattering objects than considered in this work. In the following we combine a result for boundary value problems with scatterers of very general geometries from [1,28] and an integral equation technique developed in [4,9] to arrive at an asymptotic perturbation formula that applies to our setup.…”