Resonant-state expansion of three-dimensional open optical systems: Light scattering

A rigorous method of calculating the electromagnetic field, the scattering matrix, and scattering cross-sections of an arbitrary finite three-dimensional optical system described by its permittivity distribution is presented. The method is based on the expansion of the Green's function into the resonant states of the system. These can be calculated by any means, including the popular finite element and finite-difference time-domain methods. However, using the resonant-state expansion with a spherically-symmetric analytical basis, such as that of a homogeneous sphere, allows us to determine a complete set of the resonant states of the system within a given frequency range. Furthermore, it allows us to take full advantage of the expansion of the field outside the system into vector spherical harmonics, resulting in simple analytic expressions. We verify and illustrate the developed approach on an example of a dielectric sphere in vacuum which has an exact analytic solution known as Mie scattering.