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Description
The scattering matrix, which connects the incoming and outgoing waves of a scatterer, plays a central role in describing light-matter interaction and its pole expansion provides a powerful framework for replacing full-wave simulations with resonance-based models. In this work, we investigate the analytic continuation of the optical scattering matrix and its pole expansion in terms of optical resonances for spherical particles. While it is well established that resonance frequencies appear as poles of the scattering matrix in the complex frequency plane, obtaining a pole expansion using the Mittag-Leffler theorem remains challenging for the commonly used scattering channels from Mie theory, due to their exponential divergence in the complex frequency plane.
We demonstrate that a Mittag-Leffler expansion of the scattering matrix becomes possible through an appropriate rescaling of the basis, i.e., the incoming and outgoing waves, and provide a rigorous framework on how this rescaling of the basis affects the scattering matrix. We show that certain rescalings remove exponential divergences in the scattering matrix, which we refer to as regularization. Moreover, we demonstrate that the choice of rescaling is not unique and that certain choices can fundamentally alter the analytic continuation of the scattering matrix. We identify the emergence of additional, non-physical poles, referred to as channel poles, that do not originate from the optical resonances of the system.
These results suggest that regularizations of the scattering matrix can enable efficient pole expansion schemes for more complex three-dimensional scatterers, thereby broadening the range of systems accessible to modal analysis in photonics.