Dyadic formulation of morphology-dependent resonances I Completeness relation

Abstract

The magnetic (or electric) fields of morphology-dependent resonances of a dielectric sphere are shown to form an orthogonal complete set for expanding divergence-free vectorial functions inside the dielectric sphere, provided that there is a spatial discontinuity in its refractive index, say, at the edge of the sphere. A transverse projection dyad that picks up the divergence-free part (or its generalization) of a vector is defined and shown to be expandable in terms of the magnetic (or electric) fields of these morphology-dependent resonances. Moreover, the transverse dyadic Green’s function in these dielectric spheres is in turn expressed as a sum of tensor products of relevant morphology-dependent resonance fields. Each term in the sum manifests itself as a resonant response to external perturbations. Thus the morphology-dependent resonance expansion provides a powerful tool to analyze various optical phenomena in dielectric spheres.