Lugiato-Lefever Model Research Articles

Effect of the Dispersion Orders on the Widths of the Coexistence Domain and Combs Spectra of Bright and Dark Solitons in Microresonators

Using the Lugiato–Lefever model, we base on the hysteresis approach to analyze the coexistence domain of bright and dark solitons in the zero, normal, and anomalous dispersion regimes. Our results also highlight that the fourth-order dispersion term affects the width of the frequency combs of both dark and bright solitons. It also allows the appearance of dispersive waves on the soliton spectra that disappear for high values of the fourth-order dispersion followed by the soliton destabilization into harmonic oscillations and oscillation packages.

Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially 2pi -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by textrm{i}a_tau =(zeta -textrm{i})a-d a_{x x}-|a|^2a+textrm{i}f_0+textrm{i}f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )}. The main new feature of the problem is the specific form of the source term f_0+f_1textrm{e}^{textrm{i}(k_1 x-nu _1 tau )} which describes the simultaneous pumping of two different modes with mode indices k_0=0 and k_1in mathbb {N}. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e., f_1=0, can be continued into the range f_1not =0. Our analytical findings apply both for anomalous (d>0) and normal (d<0) dispersion, and they are illustrated by numerical simulations.

Role of Eckhaus instability and pattern cracking in ultraslow dynamics of Kerr combs

The Eckhaus instability is a secondary instability of nonlinear spatiotemporal patterns in which high-wave-number periodic solutions become unstable against small-wave-number perturbations. Here we show that this instability can take place in Kerr combs corresponding to subcritical Turing patterns upon changes in the laser detuning. The development of the Eckhaus instability leads to the cracking of patterns and a long-lived transient where the peaks of the pattern rearrange in space due to spatial interactions. In the spectral domain, this results in a metastable Kerr comb dynamics with timescales that can be larger than 1 min. This time is, at least, seven orders of magnitude larger than the intracavity photon lifetime and is in sharp contrast with all the transient behaviors reported so far in cavity nonlinear optics that are typically only a few photon lifetimes long (i.e., in the picosecond to the microsecond range). This phenomenology, studied theoretically in the Lugiato-Lefever model and the observed dynamics is compatible with experimental observations in Kerr combs generated in ultra-high-$Q$ whispering-gallery mode resonators.

Elliptical solitons in the dissipative Lugiato-Lefever equation

The formation and stability of dissipative solitons in the vector Lugiato-Lefever model are considered for the case when only one of the field components (polarization) is excited by the external coherent pump. It is shown that linearly polarized solitons can be destabilized because of the growth of the excitations belonging to the orthogonal polarization. Vector (elliptically polarized) solitons bifurcating from linearly polarized ones are found, and the stability and dynamics of the vector solitons are studied.

Stability analysis of mode-coupling-assisted microcombs in normal dispersion.

We theoretically study the stability of mode-coupling-assisted frequency comb generation in normal-dispersion microresonators. With the aid of mode coupling, quantitative analysis of the modulational instability is explored in the parameter space of pump power and detuning. By exploring the coupled mode number, dispersion, and coupling strength in the normalized Lugiato-Lefever model, the modulational stability gain exists and yields extended spatial structures within the regime of eigenvalue bifurcations. Moreover, the dynamics and efficiency of microcombs are discussed, providing the accessibility of high-efficient, stable, and controllable combs. This work offers universal guidelines for operating mode-coupling-assisted combs in a normal-dispersion system.

Oscillatory motion of dissipative solitons induced by delay-feedback in inhomogeneous Kerr resonators

We investigate the destabilization mechanisms of dissipative solitons in inhomogeneous nonlinear resonators subjected to injection and to time-delayed feedback. We consider the paradigmatic Lugiato-Lefever model describing inhomogeneous driven nonlinear optical resonator. We analyze the pinning-depinning transition of dissipative solitons by introducing a potential induced by the inhomogeneity. Further, we identify conditions under which these structures are destabilized and describe different bifurcation scenarios. We show that the combined influence of inhomogeneities and delayed feedback induces an Andronov-Hopf-bifurcation that leads to oscillations of the dissipative soliton around the inhomogeneity. Finally, we show that for large values of the feedback strength, the dissipative solitons escapes from the potential well and starts to drift.

Dispersion engineering of GaN based coupled waveguides for blue frequency combs

We propose a design platform to obtain large anomalous dispersion at short wavelengths (in vicinity of 450 nm) in vertically coupled GaN/AlN/GaN waveguides realizable on commercially available GaN template wafer technology. We present finite-difference time-domain simulation based results on dispersion engineering by tuning the waveguide geometry, to obtain anomalous dispersion as large as ps (nm-km)−1. Numerical simulations based on the Lugiato–Lefever model show efficient Kerr frequency comb generation in blue spectral range. We present results based on tailoring the geometric parameters of the waveguides to modify the dispersive wave that induces asymmetrical spectral profile in the comb on account of higher-order dispersion.

Thermal instabilities, frequency-comb formation, and temporal oscillations in Kerr microresonators

We analyze the consequences of dissipative heating in driven Kerr microresonators theoretically and numerically, using a thermal Lugiato-Lefever model. We show that thermal sensitivity modifies the stability range of continuous wave in a way that blocks direct access to broadband frequency-comb forming waveforms, and we propose a deterministic access path that bypasses the thermal instability barrier. We describe a novel thermal instability that leads to thermooptical oscillations via a Hopf bifurcation.

Time-delayed feedback with global and local contributions on spatiotemporal dynamics of waves in fiber cavity

Lugiato–Lefever model is applied through a combined global and local time-delay feedback to study the dynamic of waves in fiber cavity. Delayed feedback describes how the state of the system at a present time is influenced by its value in the past. Naturally, we carry out an analysis of the linear stability and reveal the role of the strength and the phase of the optical feedback in the presence of its two contributions on the shape of the bistable curve as well as on the regions of instability. We show that the delay feedback strongly impacts the spatiotemporal dynamical behavior resulting from modulational instability by shifting the regions with different spatial eigenvalue spectra, reducing the threshold associated with modulational instability and decreasing the critical frequency at the onset of this instability. We show that for fairly moderate input intensities, it is possible to generate drifting cavity solitons with asymmetric radiation emitted by the soliton tails. The feedback also induces the formation of periodic and superperiodic waves, and extreme events.

Dynamics of localized dissipative structures in a generalized Lugiato-Lefever model with negative quartic group-velocity dispersion.

We study localized dissipative structures in a generalized Lugiato-Lefever equation, exhibiting normal group-velocity dispersion and anomalous quartic group-velocity dispersion. In the conservative system, this parameter-regime has proven to enable generalized dispersion Kerr solitons. Here, we demonstrate via numerical simulations that our dissipative system also exhibits equivalent localized states, including special molecule-like two-color bound states recently reported. We investigate their generation, characterize the observed steady-state solution, and analyze their propagation dynamics under perturbations.

Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation

The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven ϕ 4 equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series.

Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides

We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α &amp;lt; 0, while the rest of it is a subset of the vertical line {μ:Rμ=−α}. Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Prüss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.

Impact of time-delayed feedback on optical field dynamics in cavity with nonlinear metamaterial

We studied temporal dynamics of optical field in a resonator with the nonlinear left-handed metamaterial of Kerr type and optical feedback with a time delay on the basis of the Lugiato-Lefever model. We constructed bifurcation diagrams showing the parametric regions of a stationary optical field and unstable periodic orbits formations with a different number of “steps” of the optical field. These unstable periodic orbits are similar to heterocliniccycles, which are being actively investigated in nonlinear dynamics.

Addressing temporal Kerr cavity solitons with a single pulse of intensity modulation.

We experimentally and numerically study the use of intensity modulation for the controlled addressing of temporal Kerr cavity solitons (CSs). Using a coherently driven fiber ring resonator, we demonstrate that a single temporally broad intensity modulation pulse applied on the cavity driving field permits systematic and efficient writing and erasing of ultrashort cavity solitons. We use numerical simulations based on the mean-field Lugiato-Lefever model to investigate the addressing dynamics, and present a simple physical description of the underlying physics.

A taxonomy of optical dissipative structures in whispering-gallery mode resonators with Kerr nonlinearity.

In this paper, the research related to the formation of optical dissipative structures in Kerr-nonlinear whispering-gallery mode resonators pumped with continuous-wave lasers is reviewed. Pattern formation in these systems can be analysed using the paradigmatic Lugiato-Lefever model, which is a partial differential equation ruling the dynamics of the intra-cavity laser field. Various dissipative structures such as Turing rolls, solitons, breathers and spatio-temporal chaos can emerge in the resonator depending on the laser power and frequency. The bifurcation analysis enables a classification of these patterns, and also permits identification of their basins of attraction.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Effect of Cherenkov radiation on localized-state interaction

We study theoretically the interaction of temporal localized states in all fiber cavities and microresonator-based optical frequency comb generators. We show that Cherenkov radiation emitted in the presence of third order dispersion breaks the symmetry of their interaction and greatly enlarges the interaction range thus facilitating the experimental observation of the soliton bound states. Analytical derivation of the reduced equations governing slow time evolution of the positions of two interacting localized states in the Lugiato-Lefever model with third order dispersion term is performed. Numerical solutions of the model equation are in close agreement with analytical predictions.

Localized solutions of Lugiato-Lefever equations with focused pump

Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D) accurately describe the dynamics of optical fields in pumped lossy cavities with the intrinsic Kerr nonlinearity. The external pump is usually assumed to be uniform, but it can be made tightly focused too–in particular, for building small pixels. We obtain solutions of the LL equations, with both the focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes supported by the localized pump. In the 1D setting, we first develop a simple perturbation theory, based in the sech ansatz, in the case of weak pump and loss. Then, a family of exact analytical solutions for spatially confined modes is produced for the pump focused in the form of a delta-function, with a nonlinear loss (two-photon absorption) added to the LL model. Numerical findings demonstrate that these exact solutions are stable, both dynamically and structurally (the latter means that stable numerical solutions close to the exact ones are found when a specific condition, necessary for the existence of the analytical solution, does not hold). In 2D, vast families of stable confined modes are produced by means of a variational approximation and full numerical simulations.

Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities.

Time-delayed feedback plays an important role in the dynamics of spatially extended systems. In this contribution, we consider the generic Lugiato-Lefever model with delay feedback that describes Kerr optical frequency comb in all fiber cavities. We show that the delay feedback strongly impacts the spatiotemporal dynamical behavior resulting from modulational instability by (i) reducing the threshold associated with modulational instability and by (ii) decreasing the critical frequency at the onset of this instability. We show that for moderate input intensities it is possible to generate drifting cavity solitons with an asymmetric radiation emitted from the soliton tails. Finally, we characterize the formation of rogue waves induced by the delay feedback.

Multi-resonant Lugiato–Lefever model

We introduce a new model that extends the Lugiato-Lefever equation to the description of multiple resonances in Kerr optical cavities. It perfectly agrees quantitatively (in both stationary and dynamical regimes) with the exact Ikeda map, even when using a small number of resonances. Our model predicts the onset of complex phenomena such as the recently observed super-cavity solitons and the coexistence of multiple nonlinear states. It will be of crucial importance for the analytical understanding of new nonlinear phenomena in Kerr cavities when the intensities or nonlinearities are high enough to be able to excite more than one cavity resonance.

Eckhaus instability in the Lugiato-Lefever model

We study theoretically the primary and secondary instabilities undergone by the stationary periodic patterns in the Lugiato-Lefever equation in the focusing regime. Direct numerical simulations in a one-dimensional periodic domain show discrete changes of the periodicity of the patterns emerging from unstable homogeneous steady states. Through continuation methods of the steady states we reveal that the system exhibits a set of wave instability branches. The organisation of these branches suggests the existence of an Eckhaus scenario, which is characterized in detail by means of the derivation of their amplitude equation in the weakly nonlinear regime. The continuation in the highly nonlinear regime shows that the furthest branches become unstable through a Hopf bifurcation.