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

摘要

We present a stability analysis of the Lugiato-Lefever model for Kerr opticalfrequency combs in whispering gallery mode resonators pumped in the anomalousdispersion regime. This article is the second part of a research work whosefirst part was devoted to the regime of normal dispersion, and was presented inref. \cite{Part_I}. The case of anomalous dispersion is indeed the mostinteresting from the theoretical point of view, because of the considerablevariety of dynamical behaviors that can be observed. From a technological pointof view, it is also the most relevant because it corresponds to the regimewhere Kerr combs are predominantly generated, studied, and used for differentapplications. In this article, we analyze the connection between the spatialpatterns and the bifurcation structure of the eigenvalues associated to thevarious equilibria of the system. The bifurcation map evidences a considerablerichness from a dynamical standpoint. We study in detail the emergence ofsuper- and sub-critical Turing patterns in the system. We determine the areaswere bright isolated cavity solitons emerge, and we show that soliton moleculescan emerge as well. Very complex temporal patterns can actually be observed inthe system, where solitons (or soliton complexes) co-exist with or withoutmutual interactions. Our investigations also unveil the mechanism leading tothe phenomenon of breathing solitons. Two routes to chaos in the system areidentified, namely a route via the so called secondary combs, and another viasoliton breathers. The Kerr combs corresponding to all these temporal patternsare analyzed in detail, and a discussion is led about the possibility to gainsynthetic comprehension of the observed spectra out of the dynamical complexityof the system.