Stably accessing octave-spanning microresonator frequency combs in the soliton regime.

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.

Globally, the long-haul transmission of ultra-high bandwidth data is enabled through coherent communications. Driven by the rapid pace of growth in interconnectivity over the last decade, long-haul data transmission has reached capacities on the order of tens to hundreds of terabits per second, over fiber reaches which may span thousands of kilometers. Data center communications operate in regimes featuring shorter reaches and higher cost sensitivity. While integrated microresonator frequency combs are poised to revolutionize light sources used for high-speed data transmission over fiber, recent progress has focused largely on coherent detection schemes. Furthermore, though state-of-the-art intensity modulators are advancing in speed, it has not been demonstrated in the literature if microresonator-based comb lines can accommodate higher intensity modulated direction data (IMDD) line rates in tandem with these advancements. In this manuscript, we demonstrate the use of microresonator frequency combs pumped with a single laser for the transmission of high-speed IMDD data. We demonstrate error-free transmission of 30 Gbs-1 per comb non-return-to-zero data over fiber lengths of 6 km, as well as bit error rates under the forward error correction limit for propagation through 20 km of optical fiber. 60 Gbs-1 and 42 Gbs-1 pulse modulation amplitude 4 (PAM4) data modulated on each frequency comb line is further quantified to have a bit error rate under the forward error correction limit for fiber reaches of up to 6 km and 20 km respectively. The results showcase CMOS-compatible microresonator frequency comb modulated using IMDD formats as a promising technology for high-speed transmission in the data center transceiver industry.

Dissipative temporal Kerr solitons in optical microresonators enable to convert a continuous wave laser into a train of femtosecond pulses. Of particular interest are single soliton states, whose $\mathrm{sech}^{2}$ spectral envelope provides a spectrally smooth and low noise optical frequency comb, and that recently have been generated in crystalline, silica, and silicon-nitride resonators. Here, we study the dynamics of multiple soliton states containing ${N}$ solitons and report the discovery of a novel, yet simple mechanism which makes it possible to reduce deterministically the number of solitons, one by one, i.e. ${N\! \to\! N\!-\!1\! \to\! \dots \!\to\! 1}$. By applying weak phase modulation, we directly characterize the soliton state via a double-resonance response. The dynamical probing demonstrates that transitions occur in a predictable way, and thereby enables us to map experimentally the underlying multi-stability diagram of dissipative Kerr solitons. These measurements reveal the "lifted" degeneracy of soliton states as a result of the power-dependent thermal shift of the cavity resonance (i.e. the thermal nonlinearity). The experimental results are in agreement with theoretical and numerical analysis that incorporate the thermal nonlinearity. By studying two different microresonator platforms (integrated $\mathrm{Si_{3}N_{4}}$ microresonators and crystalline $\mathrm{MgF_{2}}$ resonators) we confirm that these effects have a universal nature. Beyond elucidating the fundamental dynamical properties of dissipative Kerr solitons the observed phenomena are also of practical relevance, providing a manipulation toolbox which enables to sequentially reduce, monitor and stabilize the number ${N}$ of solitons, preventing it from decay. Achieving reliable single soliton operation and stabilization in this manner in optical resonators is imperative to applications.

It has been difficult to use the measured dispersion data to determine which transverse mode is the best to use for producing optical frequency combs in microresonators. We propose a simulation method to do so.

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.

Microcombs are optical frequency combs in microresonators [1], [2]. Laser cavity-soliton (LCS) states can be achieved in a system comprising a nonlinear Kerr microresonator nested in a fibre laser [3], resulting in a self-emergent, robust [4], and efficient [5] microcomb. Generally, the platform generally produces a broader variety of states depending on the specific parameter settings of the system. The two critical experimental parameters that need to be adjusted to achieve solitary oscillation via self-emergence [4] are the gain, which controls the state energy, and the length of the main cavity, which governs the group velocity mismatch between the two cavities. When searching for solitary oscillations, these parameters must be spanned over large ranges. While detailed information can be obtained through accurate but time-consuming methods like laser scanning spectroscopy and interferometry, real-time approaches are preferable when handling large datasets. In this work, we introduce a rapid method to distinguish typical lasing states, including soliton states, based on simple experimental properties. Specifically, our approach utilises radiofrequency (RF) data and spectral analysis to map states within nonlinear optical systems in real time, with minimal computational effort when handling large datasets.

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.

Optical phase-noise dynamics of Titanium:sapphire optical frequency combs

Since their discovery, microresonator frequency combs have made considerable progress [1]. Octave spanning combs [2, 3], data transmission applications [4] and mode-locking [5, 6] have been demonstrated. There have also been advances in understanding the properties and the generation process of the frequency combs inside the resonators [7].

Soliton microcombs generated by the third-order nonlinearity of microresonators exhibit high coherence, low noise, and stable spectra envelopes, which can be designed for many applications. However, conventional dispersion engineering based design methods require iteratively solving Maxwell's equations through time-consuming electromagnetic field simulations until a local optimum is obtained. Moreover, the overall inverse design from soliton microcomb to the microcavity geometry has not been systematically investigated. In this paper, we propose a high accuracy microcomb-to-geometry inverse design method based on the genetic algorithm (GA) and deep neural network (DNN), which effectively optimizes dispersive wave position and power. The method uses the Lugiato-Lefever equation and GA (LLE-GA) to obtain second- and higher-order dispersions from a target microcomb, and it utilizes a pre-trained forward DNN combined with GA (FDNN-GA) to obtain microcavity geometry. The results show that the dispersive wave position deviations of the inverse designed MgF2 and Si3N4 microresonators are less than 0.5%, and the power deviations are less than 5 dB, which demonstrates good versatility and effectiveness of our method for various materials and structures.

The aim of this work is to theoretically investigate the possibility of Kerr frequency combs in mechanical systems. In particular, whether microelectromechanical devices (MEMS) can be used to generate frequency combs in a manner that is analogous to the optical frequency combs generated in optical microresonators with Kerr-type nonlinearity. The analysis assumes a beam-like micromechanical structure, and starting from the Euler-Bernoulli beam equation derives the necessary conditions in parameter space for the comb generation. The chapter equally presents potential means for the physical implementation of mechanical Kerr combs.

We show that self-organization occurs in the phase dynamics of soliton modelocking in paramet- ric frequency combs. Reduction of the Lugiato-Lefever equation (LLE) to a simpler set of phase equations reveals that this self-organization arises via mechanisms akin to those in the Kuramoto model for synchronization of coupled oscillators. In addition, our simulations show that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE. Our simplified equations intuitively explain the origin of the pump phase offset in soliton- modelocked parametric frequency combs. They also predict that the phase of the intracavity field undergoes an anti-symmetrization that precedes phase synchronization, and they clarify the role of chaotic states in soliton formation in parametric combs.

We determine the regions in the parameter space of microresonators where cnoidal waves (Turing rolls) are stable. Solitons are included as a special limit. We identify conditions to efficiently obtain broadband frequency combs.

Dissipative Kerr solitons in microcavity systems exhibit remarkable nonlinear dynamics. The real-time measurement of soliton motion facilitates the comprehensive understanding of soliton physics. In this Letter, an all-optical radio frequency (RF) spectrum analyzer (named f-LISA) is used to characterize various stable soliton states and to track relative soliton motion in real time. By applying an inverse Fourier transform to the broadband RF spectrum, the autocorrelation traces are obtained with a temporal resolution of 373 fs and an ultrahigh frame rate of 20.6 MHz. We successfully characterize not only the stable single soliton state but also the stable multi-soliton states with different azimuthal angles between adjacent solitons. Furthermore, the dynamics of soliton switching from four-soliton state to single soliton state is observed in a temporal window of 60 µs. It is believed that the proposed scheme provides an alternative way to visualize the multi-soliton trajectories and enable the study of the soliton dynamics in integrated microcavities.

We introduce the generalized Lugiato-Lefever equation describing nonlinear effects in the bottle microresonators. We demonstrate that the nonlinear modes of these resonators can form multiple coexisting and overlapping nonlinear resonances and that their instabilities lead to the generation of the low repetition rate frequency combs.