Modal Expansion of the Scattered Field: Causality, Non-Divergence and Non-Resonant Contribution

TL;DR

This paper develops a rigorous modal expansion for the scattered field in optical cavities, addressing divergence and non-resonant contributions, and establishing a causality-based approach for non-divergent, non-resonant field representation.

Contribution

It derives a pole expansion of the T-matrix that includes non-resonant terms and uses causality to create a non-divergent modal expansion of the scattered field.

Findings

Derived a pole expansion of T-matrix with non-resonant terms

Established a causality-based method for non-divergent field expansion

Confirmed the existence of non-resonant contributions in scattering

Abstract

Modal analysis based on the quasi-normal modes (QNM), also called resonant states, has emerged as a promising way for modeling the resonant interaction of light with open optical cavities. However, the fields associated with QNM in open photonic cavities diverge far away from the scatterer and the possibility of expanding the scattered field with resonant contributions only has not been established. Here, we address these two issues while restricting our study to the case of a dispersionless spherical scatterer. First, we derive the rigorous pole expansion of the $T$-matrix coefficients that link the scattered to the incident fields associated with an optical resonator. This expansion evinces the existence of a non-resonant term. Second, in the time domain, the causality principle allows us to solve the problem of divergence and to derive a modal expansion of the scattered field that does not diverge far from the scatterer.