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.

We analyze dark pulse Kerr frequency combs in optical resonators with normal group-velocity dispersion using the Lugiato-Lefever model. We show that in the time domain these correspond to interlocked switching waves between the upper and lower homogeneous states, and explain how this fact accounts for many of their experimentally observed properties. Modulational instability does not play any role in their existence. Furthermore, we provide a detailed map indicating where stable dark pulse Kerr combs can be found in parameter space, and how they are destabilized for increasing values of frequency detuning.Optical frequency combs generated in passive Kerr microresonators have attracted a lot of interest in recent years for their potential for on-chip integration of frequency comb technology [1][2][3][4][5][6]. Their applications span arbitrary waveform synthesis [5], telecommunications [7], and ultra-accurate clocks [4]. These so-called "Kerr frequency combs" (KFCs) are obtained by driving a high-Q microresonator with a continuous-wave (cw) laser. This leads to the generation of spectral sidebands through modulational instability (MI) and subsequent cascaded four-wave mixing. Interestingly, much can been learned about KFCs by considering a time domain representation. In fact, most reported KFCs correspond either to extended periodic patterns or to ultrashort pulses known as temporal cavity solitons (CSs), stable or fluctuating [8][9][10][11][12][13][14]. These studies have benefitted from the fact that KFCs can be modeled using a simple mean-field equation, the Lugiato-Lefever equation (LLE) [9,15].The bulk of KFC studies so far deals with microresonators exhibiting anomalous second-order group velocity dispersion (GVD) at the pump wavelength. However, due to the difficulty in obtaining anomalous GVD in some spectral ranges, generation of KFCs from normal GVD microresonators is now also being sought and has recently been achieved experimentally by several groups [16][17][18]. In [18], a full time-domain characterization is reported: the field is found to consist of square dark pulses of different widths -low intensity dips embedded in a high intensity homogeneous background -with a complex temporal structure. These observations match several previous numerical predictions [14,16,[19][20][21] and are in stark contrast with the isolated ultrashort bright localized structures observed with anomalous GVD [8,12].There has been some speculation as to the physical * lendert.gelens@kuleuven.be origin of the temporal structures observed in normal GVD KFCs, which have been called platicons, dark pulse KFCs, or dark CSs [18,21]. To clarify this issue, we present here a detailed bifurcation analysis of dark structures in the LLE with normal GVD, and predict their region of existence and stability. In particular, we clearly show that they are intimately related to so-called switching waves (SWs) -traveling front solutions of the LLE that connect the upper and lower homogenous state solutions. These SWs were studied theoretica...

The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.

Using numerical simulations of an extended Lugiato-Lefever equation we analyze the stability and nonlinear dynamics of Kerr frequency combs generated in microresonators and fiber resonators, taking into account third-order dispersion effects. We show that cavity solitons underlying Kerr frequency combs, normally sensitive to oscillatory and chaotic instabilities, are stabilized in a wide range of parameter space by third-order dispersion. Moreover, we demonstrate how the snaking structure organizing compound states of multiple cavity solitons is qualitatively changed by third-order dispersion, promoting an increased stability of Kerr combs underlined by a single cavity soliton.

The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.

Using the Lugiato-Lefever model, we analyze the effects of third-order chromatic dispersion on the existence and stability of dark-and bright-soliton Kerr frequency combs in the normal dispersion regime. While in the absence of third-order dispersion only dark solitons exist over an extended parameter range, we find that third-order dispersion allows for stable dark and bright solitons to coexist. Reversibility is broken and the shape of the switching waves connecting the top and bottom homogeneous solutions is modified. Bright solitons come into existence thanks to the generation of oscillations in the switching-wave profiles. Temporal oscillatory instabilities of dark solitons are suppressed in the presence of sufficiently strong third-order dispersion, while bright solitons are never found to oscillate in time. As a result of third-order dispersion both bright and dark solitons are found to move with a velocity that depends on their width.

We report a theoretical investigation of quadratic frequency combs in a dispersive second-harmonic generation cavity system. We identify different dynamical regimes and demonstrate that the same system can exhibit both bright and dark localized cavity solitons in the absence of a temporal walk-off.

We study temporally localized structures in doubly resonant degenerate optical parametric oscillators in the absence of temporal walk-off. We focus on states formed through the locking of domain walls between the zero and a nonzero continuous-wave solution. We show that these states undergo collapsed snaking and we characterize their dynamics in the parameter space.