Existence and stability of soliton-based frequency combs in the Lugiato–Lefever equation

Abstract

Kerr frequency combs are optical signals consisting of a multitude of equally spaced excited modes in frequency space. They are generated in optical microresonators pumped by a continuous-wave laser. It has been experimentally observed that the interplay of Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato–Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this paper, we rigorously demonstrate that the Lugiato–Lefever equation indeed supports arbitrarily broad Kerr frequency combs by proving the first existence and stability results of periodic solutions consisting of any number of well-separated, strongly localized and highly nonlinear pulses on a single periodicity interval. We realize these periodic multi-soliton solutions as concatenations of individual bright cavity solitons by phrasing the problem as a reversible dynamical system and employing results from homoclinic bifurcation theory. The spatial dynamics formulation enables us to harness general results, based on Evans-function techniques and Lin’s method, to rigorously establish diffusive spectral stability. This, in turn, yields nonlinear stability of the periodic multi-soliton solutions against localized and subharmonic perturbations.