Far-field patterns for electromagnetic waves in an inhomogeneous medium

This paper considers the scattering of time-harmonic electromagnetic waves by an inhomogeneous medium of compact support, i.e., the permittivity $\varepsilon = \varepsilon (x)$ and the conductivity $\sigma = \sigma (x)$ are functions of $x \in R^3 $. If $\sigma > 0$, it is shown that the set of far-field patterns of the electric fields corresponding to incident plane waves propagating in arbitrary directions with arbitrary polarization is complete in the space of square integrable tangential vector fields defined on the unit sphere. On the other hand, if $\sigma = 0$ it is shown that for the case of a spherically stratified medium there exist values of the frequency such that the set of far-field patterns is not complete. Finally, it is shown that, if from each far-field pattern is subtracted the electric far-field pattern corresponding to an electromagnetic field satisfying an impedance boundary condition on the boundary of a ball containing the inhomogeneity, then the resulting class is complete for $\s...