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Modal analysis based on the quasi-normal modes (QNM), also called resonantstates, has emerged as a promising way for modeling the resonant interaction oflight with open optical cavities. However, the fields associated with QNM inopen photonic cavities diverge far away from the scatterer and the possibilityof expanding the scattered field with resonant contributions only has not beenestablished. Here, we address these two issues while restricting our study tothe case of a dispersionless spherical scatterer. First, we derive the rigorouspole expansion of the $T$-matrix coefficients that link the scattered to theincident fields associated with an optical resonator. This expansion evincesthe existence of a non-resonant term. Second, in the time domain, the causalityprinciple allows us to solve the problem of divergence and to derive a modalexpansion of the scattered field that does not diverge far from the scatterer.

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