Stability Analysis of the Lugiato-Lefever Model for Kerr Optical Frequency Combs. Part I: Case of Normal Dispersion

TL;DR

This paper conducts a detailed stability analysis of the Lugiato-Lefever model in normal dispersion regimes, revealing how dark cavity solitons form and coexist, leading to Kerr frequency combs in whispering gallery resonators.

Contribution

It provides a novel bifurcation analysis of stationary states, highlighting the emergence and stability of dark cavity solitons in the normal dispersion regime.

Findings

Dark cavity solitons can form in the normal dispersion regime.

Multiple solitons can coexist without interacting.

Kerr frequency combs are generated by these solitons.

Abstract

We propose a detailed stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs in whispering gallery mode resonators pumped in the normal dispersion regime. We analyze the spatial bifurcation structure of the stationary states depending on two parameters that are experimentally tunable, namely the pump power and the cavity detuning. Our study demonstrates that the non-trivial equilibria play an important role in this bifurcation map, as their associated eigenvalues undergo critical bifurcations that are foreshadowing the existence of localized spatial structures. In particular, we show that in the normal dispersion regime, dark cavity solitons can emerge in the system, and thereby generate a Kerr comb. We also show how these solitons can coexist in the resonator as long as they do not interact with each other. The Kerr combs created by these (sets of) dark solitons are also analyzed, and their stability is discussed as well.