# Resonant-state expansion applied to three-dimensional open optical systems: A complete set of static modes

## TL;DR

This paper develops and applies a complete set of static modes for the resonant-state expansion in 3D open optical systems, improving accuracy in modeling static electric fields and resonant states.

## Contribution

It introduces two alternative complete static mode sets for the RSE in 3D systems and demonstrates their effectiveness in improving perturbation results.

## Key findings

- Convergence of RSE towards exact results for size-reduction perturbations.
- Complete static modes improve agreement with finite-element simulations.
- Application to dielectric cylinder resonant states confirms method's accuracy.

## Abstract

We present two alternative complete sets of static modes of a homogeneous dielectric sphere, for their use in the resonant-state expansion (RSE), a rigorous perturbative method in electrodynamics. Physically, these modes are needed to correctly describe the static electric field of a charge redistribution within the optical system due to a perturbation of the permittivity. We demonstrate the convergence of the RSE towards the exact result for a perturbation describing a size reduction of the basis sphere. We then revisit the quarter-sphere perturbation treated in [Doost {\it et al.}, Phys. Rev. A {\bf 90}, 013834 (2014)], where only a single static mode per each angular momentum was introduced, and show that using a complete set of static modes leads to a small, though non-negligible correction of the RSE result, improving the agreement with finite-element simulations. As another example of applying the RSE with a complete set of static modes, we calculate the resonant states of a dielectric cylinder, also comparing the result with a finite-element simulation.