Direct and inverse electromagnetic scattering problems for bi-anisotropic media

Abstract

This paper is concerned with the direct and inverse scattering of time-harmonic electromagnetic waves from bi-anisotropic media. For the direct problem, we establish an integro-differential equation formulation, its Fredholm property, and the uniqueness of a weak solution. Using this integro-differential formulation we study a fast spectral Galerkin method for the numerical solution to the direct problem. Numerical examples are presented and convergence of the spectral method is proved via Gårding estimates for (strongly singular) integro-differential equations. We solve the inverse problem of recovering bi-anisotropic scatterers from far-field data using orthogonality sampling methods. These methods aim to construct imaging functionals which are robust to noise, computationally cheap, and require data for only one or a few incident fields. We provide some theoretical analysis as well as numerical simulations for the proposed imaging functionals.

Orthogonality Sampling Method for the Electromagnetic Inverse Scattering Problem

“…Numerical results for three-dimensional anisotropic scatterers are presented. Together with our recent work in [19], where the OSM is justified for the electromagnetic inverse scattering with one incident plane wave, we have provided a versatile approach for solving the electromagnetic inverse scattering problem.…”

“…Just as in recent works [11,19] we will use the Funk-Hecke formula to show that Hφ ys 2 decays as dist(y s , Ω) → ∞. Indeed, using the Funk-Hecke formula for m = = 0 with the formula curl z curl z (pe…”

“…The simulations were carried on a Quad Core 3.6 GHz machine with 32GB RAM and the implementation was done using the computing software Matlab. The synthetic data are generated by numerically solving the direct problem with the spectral solver studied in [19]. We solve the direct problem ( 4)-( 5) with incident field…”

“…We also refer to [9] for the analysis of a multifrequency OSM. The OSM for Maxwell's equations and its analysis for the case of small scatterers have been recently established in [19]. Motivated by these recent works we study in this paper the OSM for anisotropic Maxwell's equations and provide a theoretical foundation of the method for scatterers with arbitrary shape as well as a resolution analysis for its imaging functional.…”

Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

“…[8,24]. These sampling methods have been studied in great detail for far-field measurements for multiple inverse scattering problems, see [8,12,13,20,23,24] and references therein. Direct sampling methods also have other variants [7,19] that are connected to the factorization method (see [17] for details for details).…”