Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs

Abstract

It has been recently uncovered that coherent structures in microresonators such as cavity solitons and patterns are intimately related to Kerr frequency combs. In this work, we present a general analysis of the regions of existence and stability of cavity solitons and patterns in the Lugiato-Lefever equation, a mean-field model that finds applications in many different nonlinear optical cavities. We demonstrate that the rich dynamics and coexistence of multiple solutions in the Lugiato-Lefever equation are of key importance to understanding frequency comb generation. A detailed map of how and where to target stable Kerr frequency combs in the parameter space defined by the frequency detuning and the pump power is provided. Moreover, the work presented also includes the organization of various dynamical regimes in terms of bifurcation points of higher co-dimension in regions of parameter space that were previously unexplored in the Lugiato-Lefever equation. We discuss different dynamical instabilities such as oscillations and chaotic regimes.

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

“…This scenario is similar to the one regarding bright solitons in the anomalous dispersion regime [12,13]. In the anomalous dispersion regime, such temporal oscillations, also called "breathers", were experimentally first observed in fiber resonators [12] and have recently also been measured in microresonators [48][49][50].…”

“…This type of bifurcation structure is called collapsed snaking [16,17,41,42], which is significantly different from the homoclinic snaking appearing for dissipative solitons associated with subcritical patterns. Such homoclinic snaking, where many solutions coexist over a fixed parameter range around the Maxwell point, is probably better known and has been widely studied in physics [43,44] and optics [13,[45][46][47]. When TOD is taken into account, S u gradually develops oscillations when increasing d 3 , which allows bright solitons to come into existence.…”

“…In this framework a Kerr frequency comb corresponds to the frequency spectrum of a temporal dissipative structure, such as patterns or solitons, circulating inside the cavity [10,11]. While most theoretical studies have focused on the anomalous second-order group velocity dispersion (GVD) regime [12][13][14], where the typical dissipative states are bright solitons, the normal GVD regime has recently attracted interest due to the difficulty of obtaining anomalous GVD in some spectral ranges. In contrast to the anomalous regime, dark solitons are found in the normal GVD regime, i.e., low-intensity dips embedded in a highintensity homogeneous background.…”

Dynamics of microresonator frequency comb generation: models and stability

“…Although the initial growth is exponential, the sideband growth rate will eventually saturate due to pump depletion, which permits the system to reach a stable equilibrium. However, far from all comb states are stable, and the majority of the parameter space at high powers is dominated by an unstable regime of nonlinear development of MIs, where the comb spectrum is in a turbulent state of continuous fluctuations [52].…”

Nonlinear Cavity and Frequency Comb Radiations Induced by Negative Frequency Field Effects

“…There is currently an intense research activity aiming to maximise the spectral extent of the comb and its coherence, and to understand the experimentally obtained spectra from first principles. Due to the extremely complex dynamical behaviour and stability properties of the propagating CSs and patterns in the resonators, an intense theoretical activity on the mathematical properties of the traditionally used averaged propagation equation, called the temporal Lugiato-Lefever equation (LLE), has developed over the past years, with a frequent display of new and surprising results [3][4][5][6][7].…”