Eigenstate normalization for open and dispersive systems

Resonant States, also known as `Quasi-Normal' Modes, are attractive basis sets for the spectral expansions of open radiative systems. Nevertheless, the exponential divergence of the resonant states in the far-field has long been an obstacle for defining viable scalar products. This theoretical work shows that the apparent `divergences' of the resonant state scalar product integrals can be resolved analytically and that this leads to orthogonalization of the resonant states for dispersionless scatterers and finite normalizations for the resonant states of spherical scatterers with electric and/or magnetic temporal dispersion. The correctly normalized expansions are shown to decouple the power of the iconic `Mie' theory by re-expressing the exact Mie coefficients as meromorphic (spectral) functions of frequency. The methods and formulas demonstrated here can be generalized to more complex geometries, and provide new physical insight into the nature of light-matter interactions.