Self-organization in Kerr-cavity-soliton formation in parametric frequency combs

The Lugiato-Lefever Equation (LLE), first developed to provide a description of spatial dissipative structures in optical systems, has recently made a significant impact in the integrated photonics community, where it has been adopted to help understand and predict Kerr-mediated nonlinear optical phenomena such as parametric frequency comb generation inside microresonators. The LLE is essentially an application of the nonlinear Schrodinger equation (NLSE) to a damped, driven Kerr nonlinear resonator, so that a periodic boundary condition is applied. Importantly, a slow-varying time envelope is stipulated, resulting in a mean-field solution in which the field does not vary within a round trip. This constraint, which differentiates the LLE from the more general Ikeda map, significantly simplifies calculations while still providing excellent physical representation for a wide variety of systems. In particular, simulations based on the LLE formalism have enabled modeling that quantitatively agrees with reported experimental results on microcomb generation (e.g., in terms of spectral bandwidth), and have also been central to theoretical studies that have provided better insight into novel nonlinear dynamics that can be supported by Kerr nonlinear microresonators. The great potential of microresonator frequency combs (microcombs) in a wide variety of applications suggests the need for efficient and widely accessible computational tools to more rapidly further their development. Although LLE simulations are commonly performed by research groups working in the field, to our knowledge no free software package for solving this equation in an easy and fast way is currently available. Here, we introduce pyLLE, an open-source LLE solver for microcomb modeling. It combines the user-friendliness of the Python programming language and the computational power of the Julia programming language.

The Lugiato-Lefever equation (LLE) has been extensively studied since its derivation in 1987, when this meanfield model was introduced to describe nonlinear optical cavities. The LLE was originally derived to describe a ring cavity or a Fabry-Perot resonator with a transverse spatial extension and partially filled with a nonlinear medium but it has also been shown to be applicable to other types of cavities, such as fiber resonators and microresonators. Depending on the parameters used, the LLE can present a monostable or bistable input-output response curve. A large number of theoretical studies have been done in the monostable regime, but the bistable regime has remained widely unexplored. One of the reasons for this was that previous experimental setups were not able to works in such regimes of the parameter space. Nowadays the possibility of reaching such parameter regimes experimentally has renewed the interest in the LLE. In this contribution, we present an in-depth theoretical study of the different dynamical regimes that can appear in parameter space, focusing on the dynamics of localized solutions, also known as cavity solitons (CSs). We show that time-periodic oscillations of a 1D CS appear naturally in a broad region of parameter space. More than this oscillatory regime, which has been recently demonstrated experimentally,1 we theoretically report on several kinds of chaotic dynamics. We show that the existence of CSs and their dynamics is related with the spatial dynamics of the system and with the presence of a codimension-2 point known as a Fold-Hopf bifurcation point. These dynamical regimes can become accessible by using devices such as microresonators, for instance widely used for creating optical frequency combs.

Kerr resonator is one of the most popular platforms to produce optical frequency comb and temporal cavity soliton. As an essential method for investigating the nonlinear dynamics of Kerr resonators, traditional numerical simulations rely on solving the Lugiato-Lefever equation (LLE) using the split-step Fourier method (SSFM), which is computationally intensive and time-consuming. To address this challenge, this study proposes a recurrent neural network model with prior information feedback, enabling efficient and accurate prediction of soliton dynamics in Kerr resonator. With the acceleration of graphics processing unit (GPU), the computational efficiency improved by 20 times. We compared various recurrent neural networks and found that the gated recurrent unit (GRU) network demonstrated superior performance in this task. This work highlights the potential of artificial intelligence (AI) for modeling nonlinear optical dynamics in Kerr resonator, paving the way for designing optical frequency comb and generating ultrafast pulse.

Microresonator frequency combs can be an enabling technology for optical frequency synthesis and timekeeping in low size, weight, and power architectures. Such systems require comb operation in low-noise, phase-coherent states such as solitons, with broad spectral bandwidths (e.g., octave-spanning) for self-referencing to detect the carrier-envelope offset frequency. However, accessing such states is complicated by thermo-optic dispersion. For example, in the Si3N4 platform, precisely dispersion-engineered structures can support broadband operation, but microsecond thermal time constants often require fast pump power or frequency control to stabilize the solitons. In contrast, here we consider how broadband soliton states can be accessed with simple pump laser frequency tuning, at a rate much slower than the thermal dynamics. We demonstrate octave-spanning soliton frequency combs in Si3N4 microresonators, including the generation of a multi-soliton state with a pump power near 40 mW and a single-soliton state with a pump power near 120 mW. We also develop a simplified two-step analysis to explain how these states are accessed without fast control of the pump laser, and outline the required thermal properties for such operation. Our model agrees with experimental results as well as numerical simulations based on a Lugiato-Lefever equation that incorporates thermo-optic dispersion. Moreover, it also explains an experimental observation that a member of an adjacent mode family on the red-detuned side of the pump mode can mitigate the thermal requirements for accessing soliton states.

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

A synchronously pumped Kerr cavity is proposed and studied for power-efficient frequency comb generation in optical microring resonators. The system is modeled using the Lugiato-Lefever equation. Analytical solutions are provided for an ideal case and extended by numerical methods to account for optical loss and higher orders of dispersion. It is shown that the average power requirement is reduced by the duty cycle of the pulse with respect to the conventional continuous-wave-pumped microrings, and it is significantly lower than the pulsed pumping of straight waveguides.

Calcium fluoride (CaF2) crystalline whispering gallery mode resonators (WGMRs) are emerging as promising platforms for the generation of temporal dissipative Kerr soliton (DKS) or soliton microcomb, attributed to their ultrahigh intrinsic quality factors and favorable anomalous dispersion properties at low power, particularly within the communication and mid-infrared bands. However, their unique negative thermo-optic (TO) effects present in CaF2 WGMRs pose challenges in achieving thermal equilibrium, a critical aspect for stable soliton comb generation. Here, we report the successful demonstration of soliton microcombs in CaF2 WGMRs. Our experiments have revealed a variety of soliton formation dynamics. These encompass the generation of single and multiple solitons, as well as the observation of breathing solitons and vibrational solitons. Particularly noteworthy within our discoveries are the soliton bidirectional switching and soliton self-starting, which are induced by the negative TO effect. We also conducted theoretical analyses using the Lugiato-Lefever Equation (LLE) which incorporates the negative TO effect. The consistency between the theoretical simulations and our experimental results substantiates the authenticity of our observations. The outcomes derived from this study enrich the soliton generation platform and contribute to understanding the formation of solitons in WGMRs composed of various materials exhibiting negative TO effects.

In [Phys. Rev. Lett. 110, 064103 (2013)], using the Swift-Hohenberg equation, we introduced a mechanism that allows to generate oscillatory and excitable soliton dynamics. This mechanism was based on a competition between a pinning force at inhomogeneities and a pulling force due to drift. Here, we study the effect of such inhomogeneities and drift on temporal solitons and Kerr frequency combs in fiber cavities and microresonators, described by the Lugiato-Lefever equation with periodic boundary conditions. We demonstrate that for low values of the frequency detuning the competition between inhomogeneities and drift leads to similar dynamics at the defect location, confirming the generality of the mechanism. The intrinsic periodic nature of ring cavities and microresonators introduces, however, some interesting differences in the final global states. For higher values of the detuning we observe that the dynamics is no longer described by the same mechanism and it is considerably more complex.

We study novel soliton glass frequency combs to a modified Lugiato-Lefever Equation (LLE) that include cross-phase modulation within a Fabry-Perot resonator. Soliton glasses are characterized by stable, spatially locked, phase-locked, and randomly spaced soliton pulses.

Recently, dual-pumped microresonators have been used for spectral extension of frequency combs [1]–[2] through the formation of multi-color solitons. The stroboscopic observation of multi-color solitons leads to generation of interleaved frequency combs [3]. So far, experiments and theoretical analysis have been performed on microres-onators exhibiting higher-order dispersion. Here, we report for the first time, to our knowledge, multi-color solitons in a Si3N4 microring resonator with only second order dispersion profile (see Fig. 1.b). The central position and amplitude of the second color is dominantly dictated by the second pump's detuning, unlike the previously reported experiments where higher-order dispersion played the dominant role.

We describe 10+ GHz parametric frequency combs at 1550nm using high-Q silica micro-resonators. We realize a microcomb optical clock that divides terahertz frequencies to the microwave domain and demonstrate coherent octave-spanning spectra for self-referencing. Article not available.

Principles of asynchronous optical parametric oscillator frequency combs are introduced and their performance in dual-comb mid-infrared molecular spectroscopy is presented, including a specific demonstration of methane absorption spectroscopy with a resolution of 0.2 cm−1.

Multi-color solitons that are parametrically created in dual-pumped microresonators generate interleaved frequency combs that can be used to obtain combs at new frequencies and, when synchronized, can be used for low-noise microwave generation and potentially as an element in a chip-scale clockwork. Here, we first derive three-wave equations that describe multi-color solitons that appear in microresonators with a nearly quartic dispersion profile. These solitons are characterized by a single angular group velocity and different angular phase velocities. We then use these equations to explain the interleaved frequency combs that are observed at the output of the microresonator. Finally, we used these equations to describe the experimentally-observed soliton-OPO effect. In this effect, the pump frequency color interacts nonlinearly with a signal frequency color to create an idler frequency color in a new frequency range, somewhat analogous to an optical parametric oscillation (OPO) process, but in which the Kerr nonlinearity plays a role in matching the angular group velocity, analogous to soliton formation. These three colors then create an interleaved frequency comb in the output waveguide. We determine the conditions under which we expect this effect to occur. We anticipate that the three-wave equations and their extensions will be of use in designing new frequency comb systems and determining their stability and noise performance.

We investigate the existence and linear stability of coupled vortex solitons supported by cascaded four-wave mixing in a Raman active medium excited away from the resonance. We present a detailed analysis for the two- and three-component vortex solitons and demonstrate the formation of stable and unstable vortex solitons, and associated spatiotemporal helical beams, under the conditions of the simultaneous frequency and vortex comb generation.

Global and localised temporal structures in driven ring quantum cascade lasers