Spectral expansions of open and dispersive optical systems: Gaussian regularization and convergence

Resonant states (RS), also known as quasi-normal modes, 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 typical in the studies of closed systems. A multipolar Gaussian regularization method for RS inner products is introduced in the context of light scattering and shown to provide analytical results for the crucial RS inner product integrals in the problematic region exterior to the scattering system. We detail the applicability of this method to arbitrary scattering geometries while providing semi-analytic benchmark results for spherical scatterers. This formulation is then used to highlight the lack of 'convergence' in directly truncated RS spectral expansions and the necessity of adding non-resonant contributions to the RS spectral expansions. Solutions to these difficulties are illustrated in the case of dispersive media spheres, but these methods should prove generalizable to arbitrary RS spectral expansions.