Application of a trace formula to the spectra of flat three-dimensional dielectric resonators

TL;DR

This paper demonstrates that a semiclassical trace formula can accurately describe the spectra of flat three-dimensional dielectric resonators by using a two-dimensional approximation with an effective index of refraction, validated through microwave experiments.

Contribution

It extends the application of the trace formula to realistic 3D dielectric microcavities by incorporating the effective index of refraction and accounting for dispersion effects.

Findings

Good agreement between experiment and trace formula

Effective index dispersion is crucial for accurate modeling

Systematic errors are linked to the effective index approximation

Abstract

The length spectra of flat three-dimensional dielectric resonators of circular shape were determined from a microwave experiment. They were compared to a semiclassical trace formula obtained within a two-dimensional model based on the effective index of refraction approximation and a good agreement was found. It was necessary to take into account the dispersion of the effective index of refraction for the two-dimensional approximation. Furthermore, small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation. In summary, the methods developed in this article enable the application of the trace formula for two-dimensional dielectric resonators also to realistic, flat three-dimensional dielectric microcavities and -lasers, allowing for the interpretation of their spectra in terms of classical periodic orbits.