# Computing resonant modes of circular cylindrical resonators by vertical mode expansions

## TL;DR

This paper introduces a new numerical method for efficiently computing the complex resonant modes of layered circular cylindrical resonators, applicable to nanophotonics and capable of handling metallic and dielectric materials.

## Contribution

A novel vertical mode expansion approach combined with Chebyshev pseudospectral methods and an iterative scheme for solving nonlinear eigenvalue problems in cylindrical resonators.

## Key findings

- Validated method against existing results
- Enabled analysis of high-Q dielectric resonators
- Explored gold nanocylinder resonances

## Abstract

Open subwavelength cylindrical resonators of finite height are widely used in various photonics applications. Circular cylindrical resonators are particularly important in nanophotonics, since they are relatively easy to fabricate and can be designed to exhibit different resonance effects. In this paper, an efficient and robust numerical method is developed for computing resonant modes of circular cylinders which may have a few layers and may be embedded in a layered background. The resonant modes are complex-frequency outgoing solutions of the Maxwell's equations with no sources or incident waves. The method uses field expansions in one-dimensional (1D) ``vertical'' modes to reduce the original three-dimensional eigenvalue problem to 1D problems, and uses Chebyshev pseudospectral method to compute the 1D modes and set up the discretized eigenvalue problem. In addition, a new iterative scheme is developed so that the 1D nonlinear eigenvalue problems can be reliably solved. For metallic cylinders, the resonant modes are calculated based on analytic models for the dielectric functions of metals. The method is validated by comparisons with existing numerical results, and it is also used to explore subwavelength dielectric cylinders with high-$Q$ resonances and analyze gold nanocylinders.