Nonperturbative solution to the integral equation of scattering theory

TL;DR

This paper presents a nonperturbative analytical solution to the scattering integral equation by assuming a spherical wave within the object, enabling simple, accurate calculations of scattering amplitudes and cross sections for various geometries.

Contribution

It introduces a novel approximation transforming the integral equation into an algebraic form, allowing for closed-form solutions applicable to homogeneous potentials and resonant frequencies.

Findings

Accurate scattering cross sections for spheres and cylinders at resonance.

Effective inverse scattering reconstructions using the analytical expression.

Validation of the approximation for homogeneous potentials with compact support.

Abstract

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.