Scattering of an electromagnetic plane wave by a sphere embedded in a cylinder

In this paper, we face the problem of the scattering of a plane wave by a sphere embedded in an infinitely long circular cylinder. The problems of scattering by both a sphere and a cylinder are canonical problems in optics. However, the scattering problems involving different objects with different geometries have not been solved analytically in the literature: only asymptotic or approximated solutions are available. The problem of scattering by cylinders and spheres concurrently present can be of great importance in several areas, from optical microscopy to biomedical applications, and from metamaterials to civil engineering applications. To solve the problem, the incident wave is expressed as a superposition of cylindrical harmonics. The scattered wave by the cylinder, being a cylindrical wave as well, has been expressed as a superposition of spherical harmonics in order to take into account the interaction with the sphere. The theoretical procedure returns a linear system of equations for the computation of the unknown coefficients of the series. A numerical code is presented to compute the scattered field, where a suitable truncation criterion for the series expansions has been proposed. Comparisons with a finite-element method have been presented to validate the results.