Inverse electromagnetic scattering for radially inhomogeneous dielectric spheres

The scattering of a plane electromagnetic wave by a dielectric sphere with a radially distributed index of refraction is analyzed in terms of Debye potentials. The scattering amplitude, expanded in vector spherical harmonics, is given in terms of two sets of phase shifts. The inverse problem considered consists in reconstructing the index of refraction when one of the phase shifts is known as a function of frequency. The approach is based on the use of Liouville transformations which cast the equations for the radial parts of the Debye potentials in the form of Schroedinger-like equations with frequency independent potentials. Given the frequency dependence of one of the phase shifts, one such potential can be found by solving the quantum mechanical inverse problem by using the Marchenko formalism. The index of refraction can be reconstructed from the potential by solving a nonlinear integro-differential equation; the existence and uniqueness of its solution is discussed.