Stekloff Eigenvalues in Inverse Scattering

Abstract

We consider a problem in nondestructive testing in which small changes in the (possibly complex valued) refractive index $n(x)$ of an inhomogeneous medium of compact support are to be determined from changes in measured far field data due to incident plane waves. The problem is studied by considering a modified far field operator ${\cal F}$ whose kernel is the difference of the measured far field pattern due to the scattering object and the far field pattern of an auxiliary scattering problem with the Stekloff boundary condition imposed on the boundary of a domain $B$, where $B$ is either the support of the scattering object or a ball containing the scattering object in its interior. It is shown that ${\cal F}$ can be used to determine the Stekloff eigenvalues corresponding to $B$, where, if $B\not=D$, the refractive index is set equal to one in $B\setminus\overline{D}$. A formula is obtained relating changes in $n(x)$ to changes in the Stekloff eigenvalues and numerical examples are given showing the effectiveness of determining changes to the refractive index in this way.