Reconstruction of a spherical scatterer from its natural frequencies

A technique for an approximate reconstruction of the index of refraction of a spherical scatterer from natural frequencies is presented. The frequencies are treated as complex poles of the scattering amplitude in the wavenumber complex plane. A quantum mechanical inverse expression is defined for accounting for phase shifts. The index of refraction is then generated, after a Liouville transformation to remove the wavenumber dependency, from a scattering matrix at the location of its singularities. Usage of a Weierstrass product configuration for the scattering matrix is shown to yield an approximation which approaches the accuracy of an integral formulation.