# Gaussian regularization for resonant states: open and dispersive optical systems

## TL;DR

This paper introduces Gaussian regularization techniques to analytically compute inner products of resonant states in open, dispersive optical systems, enabling accurate spectral analysis aligned with Mie theory results.

## Contribution

It presents a novel Gaussian regularization approach for calculating resonant state inner products in open systems with dispersive media, bridging the gap with exact Mie theory.

## Key findings

- Gaussian regularization yields analytical inner product results.
- Method accurately reproduces Mie theory calculations.
- Applicable to systems with electric and magnetic dispersion.

## Abstract

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.