On the generation of stable Kerr frequency combs in the Lugiato-Lefever model of periodic optical waveguides

TL;DR

This paper constructs and analyzes a family of stable Kerr frequency comb solutions within the Lugiato-Lefever model, revealing their spectral stability properties and bifurcation from cnoidal waves in optical fibers.

Contribution

It introduces a two-parameter family of steady state solutions for the LL model, identifying stable combs and analyzing their spectral stability in the context of optical waveguides.

Findings

Constructed a two-parameter family of Kerr frequency combs.

Identified spectrally stable comb solutions.

Analyzed the spectrum of the linearized operator, showing eigenvalues at 0 and -2α.

Abstract

We consider the Lugiato-Lefever (LL) model of optical fibers. We construct a two parameter family of steady state solutions, i.e. Kerr frequency combs, for small pumping parameter $h>0$ and the correspondingly (and necessarily) small detuning parameter, $\alpha>0$. These are $O(1)$ waves, as they are constructed as bifurcation from the standard cnoidal solutions of the cubic NLS. We identify the spectrally stable ones, and more precisely, we show that the spectrum of the linearized operator contains the eigenvalues $0, -2\alpha$, while the rest of it is a subset of $ \{\mu: \Re\mu=-\alpha \}$. This is in line with the expectations for effectively damped Hamiltonian systems, such as the LL model.