Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres

Abstract

Morphology-dependent resonances commonly observed in both the elastic and the inelastic scattering of light waves from dielectric spheres are in fact direct manifestations of complex frequency poles of the scattering matrix. Time-independent solutions to the wave equation at these poles are termed quasi-normal modes, which are characterized by the outgoing wave boundary condition at infinity and cannot be normalized in the usual sense. These resonances (or quasi-normal modes) are shown to form a complete set inside the dielectric sphere, provided that there is a spatial discontinuity in the refractive index, say, at the edge of the sphere. Novel definitions of norm and inner product are introduced. In addition, a time-independent perturbation method based on this completeness relation is developed to evaluate shifts in resonance frequencies when the refractive index is changed.