Nonperturbative solution to the integral equation of scattering theory

We obtain a nonperturbative, analytical solution to integral equation of scattering theory by assuming the field within the scattering object is a spherical wave with a scattering amplitude equal to that of the far field. This approximation transforms the integral equation into a simple algebraic equation which can be readily solved to obtain a closed-form expression for the scattering amplitude. We show this approximation is valid for homogeneous potentials of compact support, namely circular and square cylinders, and that the calculated scattering cross sections for spheres and square cylinders are accurate for frequencies through the fundamental resonance. Then we apply our analytical expression to the inverse scattering problem for spheres and show that accurate reconstructions are possible even under resonance conditions. The simplicity and accuracy of our method suggest it can be a reliable and efficient tool for understanding a wide range of scattering problems in optics.