Inverse Mie problem

We apply functional analysis to the scattered electromagnetic field of a particle with spherical symmetry to obtain a pair of integral transforms for converting the Mie-scattering amplitudes S perpendicular (theta) and S parallel (theta) into the Mie coefficients an and bn. In the case of a homogeneous sphere, a simple mathematical construction is derived that uniquely inverts the Mie coefficients to find the refractive index and the radius of the particle. A more general method for construction of the refractive-index profile of an arbitrary sphere is discussed that follows from the treatment of Newton and Sabatier.