Impact of mode regularization for quasinormal mode perturbation theories

TL;DR

This paper investigates the limitations of current quasinormal mode perturbation theories for open resonators, introduces a regularization approach to address divergences, and compares analytical and numerical results to improve understanding of mode perturbations.

Contribution

It identifies fundamental issues with existing QNM perturbation formulas for external perturbations and proposes a regularization method to mitigate divergences, supported by analytical and numerical examples.

Findings

Regularization prevents spatial divergence in QNM perturbations.

QNM expansion matches analytical solutions in simple cases.

Higher-order effects and multimodes influence scattering in complex scenarios.

Abstract

We give insight into the critical problem of an open resonator that is subject to a perturbation outside of its cavity region. We utilize the framework of quasinormal modes (QNMs), which are the natural mode solutions to the open boundary problem with complex eigenfrequencies. We first highlight some fundamental problems with currently adopted formulas using QNM perturbation theory, when perturbations are added outside the resonator structure and present a first potential step for solving this problem connected to a regularization of the QNMs. We then show an example for a full three-dimensional plasmonic resonator of arbitrary shape and complex dispersion and loss, clearly displaying the divergent nature of the first-order mode change predicted from QNM perturbation theory. Subsequently, we concentrate on the illustrative case of a one-dimensional dielectric barrier, where analytical QNM solutions are possible. We inspect the change of the mode frequency as function of distance between the cavity and another smaller barrier structure. The results obtained from a few QNM expansion are compared with exact analytical solutions from a transfer matrix approach. We show explicitly how regularization prevents a problematic spatial divergence for QNM perturbations in the far field, though eventually higher-order effects and multimodes can also play a role in the full scattering solution, and retaining a pure discrete QNM picture becomes questionable in such situations, since the input-output coupling ultimately involves reservoir modes.