Extraction of the index of refraction by embedding multiple small inclusions

Abstract

We deal with the problem of reconstructing material coefficients from the far-fields they generate. By embedding small (single) inclusions to these media, located at points z in the support of these materials and measuring the far-fields generated by these deformations we can extract the values of the total field (or the energies) generated by these media at the points z. The second step is to extract the values of the material coefficients from these internal values of the total field. The main difficulty in using internal fields is the treatment of their possible zeros.In this work, we propose to deform the medium using multiple (precisely double) and close inclusions instead of only single ones. By doing so, we derive from the asymptotic expansions of the far-fields the internal values of the Green function, in addition to the internal values of the total fields. This is possible because of the deformation of the medium with multiple inclusions which generates scattered fields due to the multiple scattering between these inclusions. Then, the values of the index of refraction can be extracted from the singularities of the Green function. Hence, we overcome the difficulties arising from the zeros of the internal fields. We test these arguments for the acoustic scattering by a refractive index in presence of inclusions modeled by the impedance type small obstacles.