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In recent years, a new approach has been proposed in the study of the inverse scattering problem for electromagnetic waves. In particular, a study is made of the analytic properties of the scattering operator, and the results of this study are used to design target signatures that respond to changes in the electromagnetic parameters of the scattering medium. These target signatures are characterized by novel eigenvalue problems such that the eigenvalues can be determined from measured scattering data. Changes in the structural properties of the material or the presence of flaws cause changes in the measured eigenvalues. In this article, we provide a general framework for developing target signatures and numerical evidence of the efficacy of new target signatures based on recently introduced eigenvalue problems arising in electromagnetic scattering theory for anisotropic media. 

We provide a graduate student accessible survey of the fascinating topic of transmission eigenvalues 

This short note was motivated by our eorts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justi cation of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21, 23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our ndings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically strati ed media. We hope that these results can provide reasonable conjectures for general electromagnetic media. 

In this paper, we develop a conceptually unified approach for characterizing and determining scattering poles and interior eigenvalues for a given scattering problem. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Our discussion includes the scattering problem for a Dirichlet obstacle where duality is between scattering poles and Dirichlet eigenvalues, and the inhomogeneous scattering problem where the duality is between scattering poles and transmission eigenvalues. Our new characterization of the scattering poles suggests a numerical method for their computation in terms of scattering data for the corresponding interior scattering problem. 

The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al (2013 Inverse Problems 29 075012) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (2017 J. Integral Equ. Appl. 29 271–95). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region.