Gaussian regularization for resonant states: open and dispersive optical systems
Resonant States (RS), also known as Quasi-Normal Modes (QNMs), are eigenstates that arise in spectral expansions of linear response functions of open systems. Manipulation of these spatially `divergent' oscillating functions requires a departure from the usual definitions of inner product, normalization and orthogonality typically encountered in the studies of closed systems. We show that once RS fields are expanded on a multipole basis, Gaussian regularization methods provide \emph{analytical} results for crucial RS inner product integrals \added{in the problematic region exterior to the scattering system}. Our demonstrations are carried out in the context of light scattering by spatially bounded objects composed of both electrically and magnetically dispersive media, with demonstrative analytic calculations being shown to \emph{completely} retrieve the results of exact Mie theory.