Transmission eigenvalues with artificial background for explicit material index identification

We are interested in the problem of retrieving information on the refractive index n of a penetrable inclusion embedded in a reference medium from farfield data associated with incident plane waves. Our approach relies on the use of transmission eigenvalues (TEs) that carry information on n and that can be determined from the knowledge of the farfield operator F. In this note, we explain how to modify F into a farfield operator $F^{\mathrm{art}}=F?\stackrel{?}{F}$, where $\stackrel{?}{F}$ is computed numerically, corresponding to well-chosen artificial background and for which the associated TEs provide more accessible information on n.