Stability and Dynamics of Microring Combs: Elliptic function solutions of the Lugiato-Lefever equation

TL;DR

This paper analyzes the stability and dynamics of elliptic function solutions in the Lugiato-Lefever equation, revealing how microresonator pumping stabilizes certain solutions and providing a perturbation theory for practical resonator design.

Contribution

It introduces a rigorous characterization of Jacobi elliptic function solutions to the Lugiato-Lefever equation and develops a perturbation theory for real-world effects on microcomb stability.

Findings

The dn solution is stabilized by microresonator pumping.

The perturbation theory models effects like Raman scattering and stimulated emission.

Numerical simulations verify the analytical results.

Abstract

We consider a new class of periodic solutions to the Lugiato-Lefever equations (LLE) that govern the electromagnetic field in a microresonator cavity. Specifically, we rigorously characterize the stability and dynamics of the Jacobi elliptic function solutions of LLE and show that the dn solution is stabilized by the pumping of the microresonator. In analogy with soliton perturbation theory, we also derive a microcomb perturbation theory that allows one to consider the effects of physically realizable perturbations on the comb line stability, including effects of Raman scattering and stimulated emission. Our results are verified through full numerical simulations of the LLE cavity dynamics. The perturbation theory gives a simple analytic platform for potentially engineering new resonator designs.