Temporal solitons in optical microresonators

We demonstrate soliton mode-locking in continuously pumped, non-linear optical MgF2 microresonators, resulting in low noise frequency comb spectra and ultra-short pulses of 200 fs duration with a repetition rate of 35.2 GHz.

<p indent="0mm">A soliton is a stationary local structure that keeps its waveform and spreading speed during propagation. Recent research shows that chip-scale optical microresonators can support dissipative solitons with surprisingly high energy efficiency and stability. The formation of optical microresonator dissipative solitons requires two balances, gain-loss balance and dispersion-nonlinearity balance. The gain-loss balance maintains the soliton’s amplitude while the dispersion-nonlinearity balance keeps its width. In this paper, we review the formation, development, and applications of dissipative optical solitons, as well as dissipative mechanical solitons in optical microresonators, and analyze the balances in different solitons. This review mainly consists of two parts: Optical microresonator dissipative solitons and opto-mechanical microresonator dissipative solitons. The first part discusses a typical kind of solitons named dissipative Kerr solitons. Dissipative Kerr solitons balance the propagation dispersion through Kerr nonlinearity in optical microresonators and are widely studied due to their easy implementation in silicon-based microcavities. In order to improve the practical performance of dissipative Kerr solitons, researchers have proposed many schemes to improve their stability and efficiency. Meanwhile, various high-precision sensing applications based on dissipative Kerr solitons, such as photonic radar, range measurement, and absorption spectrum detection, have drawn extensive attention. The second part introduces the dissipative solitons in optomechanical microresonators. A recently-discovered type of solitons, optomechanical dissipative solitons, is introduced. Unlike the dissipative Kerr solitons, the optomechanical dissipative solitons gain their power from phonon lasing, and compensate the propagation dispersion by optomechanical nonlinearity. The dynamics of the optomechanical dissipative solitons are described by the modified Korteweg-de Vries equation. Low-frequency optomechanical dissipative solitons can achieve kHz-accuracy acoustic signal measurement, which can be used in acoustic detection and communication. Finally, we summarize the formation, development, and applications of dissipative solitons in optomechanical microresonators. We also provide an outlook for future applications like radio-frequency calibration, radio-frequency communications, and underwater tomography.

The development of compact, chip-scale optical frequency comb sources (microcombs) based on parametric frequency conversion in microresonators has seen applications in terabit optical coherent communications, atomic clocks, ultrafast distance measurements, dual-comb spectroscopy, and the calibration of astophysical spectrometers and have enabled the creation of photonic-chip integrated frequency synthesizers. Underlying these recent advances has been the observation of temporal dissipative Kerr solitons in microresonators, which represent self-enforcing, stationary, and localized solutions of a damped, driven, and detuned nonlinear Schrödinger equation, which was first introduced to describe spatial self-organization phenomena. The generation of dissipative Kerr solitons provide a mechanism by which coherent optical combs with bandwidth exceeding one octave can be synthesized and have given rise to a host of phenomena, such as the Stokes soliton, soliton crystals, soliton switching, or dispersive waves. Soliton microcombs are compact, are compatible with wafer-scale processing, operate at low power, can operate with gigahertz to terahertz line spacing, and can enable the implementation of frequency combs in remote and mobile environments outside the laboratory environment, on Earth, airborne, or in outer space.

The observation of temporal dissipative Kerr solitons in optical\nmicroresonators provides, on the applied side, compact sources of coherent\noptical frequency combs that have already been applied in coherent\ncommunications, dual comb spectroscopy and metrology. On a fundamental level,\nit enables the study of soliton physics in driven nonlinear cavities.\nMicroresonators are commonly multimode and, as a result, inter-mode\ninteractions inherently occur among mode families - a condition referred to as\n"avoided mode crossings". Avoided mode crossings can cause soliton decay, but\ncan also modify the soliton spectrum, leading to e.g. the formation of\ndispersive wave and inducing a spectral recoil. Yet, to date, the entailing\ntemporal soliton dynamics from inter-mode interactions has rarely been studied,\nbut is critical to understand regimes of soliton-stability. Here we report the\ndiscovery of an inter-mode breather soliton. Such breathing dynamics occurs\nwithin a laser detuning range where conventionally stationary dissipative\nsolitons are expected. We demonstrate experimentally the phenomenon in two\nmicroresonator platforms (crystalline magnesium fluoride and photonic\nchip-based silicon nitride microresonators), and theoretically describe the\ndynamics based on a pair of coupled Lugiato-Lefever equations. We demonstrate\nexperimentally that the breathing is associated with a periodic energy exchange\nbetween the soliton and another optical mode family. We further show that\ninter-mode interactions can be modeled by a response function acting on\ndissipative solitons. The observation of breathing dynamics in the\nconventionally stable soliton regime is critical to applications, ranging from\nlow-noise microwave generation, frequency synthesis to spectroscopy. On a\nfundamental level, our results provide new understandings of the rich\ndissipative soliton dynamics in multimode nonlinear cavities.\n

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.

The formation of temporal dissipative solitons in optical microresonators enables compact, high-repetition rate sources of ultrashort pulses as well as low noise, broadband optical frequency combs with smooth spectral envelopes. Here we study the influence of the microresonator mode spectrum on temporal soliton formation in a crystalline MgF2 microresonator. While an overall anomalous group velocity dispersion is required, it is found that higher order dispersion can be tolerated as long as it does not dominate the resonator's mode structure. Avoided mode crossings induced by linear mode coupling in the resonator mode spectrum are found to prevent soliton formation when affecting resonator modes close to the pump laser frequency. The experimental observations are in excellent agreement with numerical simulations based on the nonlinear coupled mode equations. The presented results provide for the first time design criteria for the generation of temporal solitons in optical microresonators.

Temporal-dissipative Kerr solitons are self-localized light pulses sustained in driven nonlinear optical resonators. Their realization in microresonators has enabled compact sources of coherent optical frequency combs as well as the study of dissipative solitons. A key parameter of their dynamics is the effective detuning of the pump laser to the thermally and Kerr-shifted cavity resonance. Together with the free spectral range and dispersion, it governs the soliton-pulse duration, as predicted by an approximate analytical solution of the Lugiato-Lefever equation. Yet a precise experimental verification of this relation has been lacking so far. Here, by measuring and controlling the effective detuning, we establish a way of stabilizing solitons in microresonators and demonstrate that the measured relation linking soliton width and detuning deviates by less than 1% from the approximate expression, validating its excellent predictive power. Furthermore, a detuning-dependent enhancement of specific comb lines is revealed due to linear couplings between mode families. They cause deviations from the predicted comb power evolution and induce a detuning-dependent soliton recoil that modifies the pulse repetition rate, explaining its unexpected dependence on laser detuning. Finally, we observe that detuning-dependent mode crossings can destabilize the soliton, leading to an unpredicted soliton breathing regime (oscillations of the pulse) that occurs in a normally stable regime. Our results test the approximate analytical solutions with an unprecedented degree of accuracy and provide insights into dissipative-soliton dynamics.

Temporal dissipative solitons in continuous-wave (CW) laser-driven Kerr-nonlinear microresonators have led to the generation of highly-coherent optical frequency combs and ultra-short optical pulses with repetition rates in excess of 10 GHz. Applications of such sources include optical telecommunication, microwave signal generation and optical spectroscopy. Here, a novel nonlinear optical Fabry-Perot microresonator is synchronously driven by picosecond laser pulses (instead of a CW laser) resulting in the formation of temporal dissipative solitons at 10 GHz repetition rate. As opposed to the conventional CW-driven case, single or multiple solitons form deterministically 'on-top' of the resonantly enhanced driving pulses, which significantly increases conversion efficiency. The solitons lock to the driving pulse, which enables stable operation and coherent actuation of the solitons' repetition rate and carrier-envelope offset frequency. The Fabry-Perot microresonator with 10 GHz free-spectral range is based on a short length of standard optical fiber whose end-facets are coated with dielectric Bragg mirrors. Mounted inside a fiber-optical ferrule, the resonator can be interfaced directly with other fiber optical components. While being equivalent to whispering-gallery mode and ring-type resonators regarding nonlinear optical phenomena, the Fabry-Perot microresonator allows for straightforward design of group velocity dispersion, coupling ratio and nonlinearity via choice of fiber and dielectric mirrors. In summary, the presented results links the fields of CW driven microresonators, synchronously driven optical parametric oscillators as well as pulsebased non-resonant supercontinuum generation. Amongst others, they open new perspectives for microresonator-based frequency combs generation and for nonlinear photonics driven by temporally and spectrally structured light.

Dissipative cavity solitons rely on a double balance of dispersion with nonlinearity and gain with loss. Here we report on the spontaneous formation temporal solitons in a ultra high Q crystalline resonator. We observe the formation of single temporal soliton states, which leads to an outcoupled pulse-train that corresponds to low noise optical frequency comb. Pulse durations below 200fs are observed.

We investigate the soliton pattern formation in an erbium-doped figure-of-eight double-clad fiber laser. The mode-locking is realized with a nonlinear amplifying loop mirror. Different soliton complexes have been obtained similar to those obtained when the mode-locking is achieved through the nonlinear polarization rotation technique.

In recent years, optical microresonators, including whispering gallery modes (WGM) microresonators, have found wide application in various fields of science and technology. For example, high quality factor (high-Q) optical microresonators represent the most promising platform for the creation of miniature, energy-efficient components of optoelectronics, photonics and radiophotonics, with a speed exceeding traditional electronic components. In addition, the high-quality factor and the relatively small effective volume of the localization of the optical field significantly reduce the threshold for the manifestation of various nonlinear effects, which has made WGM microresonators a unique platform for investigating various nonlinear optical effects. A significant breakthrough was the discovery of the possibility of generation of Kerr frequency combs and dissipative Kerr solitons in microresonators [1]. Besides that the coupling of high-Q microresonator with a laser contributes to the stabilization of the laser and to a decrease of the width of its generation line via the self-injection locking effect [2, 3]. Recently it was demonstrated the generation of dissipative Kerr solitons by the multi-frequency laser self-injection locked by high-Q microresonator [4]. Interestingly, that predominantly single-soliton regime was obtained without any additional efforts. However there is no full theory of this phenomenon.

The formation of temporal dissipative Kerr solitons in microresonators driven by a continuous-wave laser enables the generation of coherent, broadband, and spectrally smooth optical frequency combs as well as femtosecond pulse sources with compact form factors. Here we report the observation of a Raman-induced soliton self-frequency shift for a microresonator dissipative Kerr soliton also referred to as the frequency-locked Raman soliton. In amorphous silicon nitride microresonator-based single soliton states the Raman effect manifests itself by a spectrum that is sech^{2} in shape and whose center is spectrally redshifted from the continuous wave pump laser. The shift is theoretically described by the first-order shock term of the material's Raman response, and we infer a Raman shock time of ∼20 fs for amorphous silicon nitride. Moreover, we observe that the Raman-induced frequency shift can lead to a cancellation or overcompensation of the soliton recoil caused by the formation of a coherent dispersive wave. The observations are in agreement with numerical simulations based on the Lugiato-Lefever equation with a Raman shock term. Our results contribute to the understanding of Kerr frequency combs in the soliton regime, enable one to substantially improve the accuracy of modeling, and are relevant to the understanding of the fundamental timing jitter of microresonator solitons.

The model, that is usually called the Lugiato-Lefever equation (LLE), was introduced in 1987 with the aim of providing a paradigm for dissipative structure and pattern formation in nonlinear optics. This model, describing a driven, detuned and damped nonlinear Schroedinger equation, gives rise to dissipative spatial and temporal solitons. Recently, the rather idealized conditions, assumed in the LLE, have materialized in the form of continuous wave driven optical microresonators, with the discovery of temporal dissipative Kerr solitons (DKS). These experiments have revealed that the LLE is a perfect and exact description of Kerr frequency combs-first observed in 2007, i.e. 20 years after the original formulation of the LLE-and in particular describe soliton states. Observed to spontaneously form in Kerr frequency combs in crystalline microresonators in 2013, such DKS are preferred state of operation, offering coherent and broadband optical frequency combs, whose bandwidth can be extended exploiting soliton-induced broadening phenomena. Combined with the ability to miniaturize and integrate on-chip, microresonator-based soliton Kerr frequency combs have already found applications in self-referenced frequency combs, dual-comb spectroscopy, frequency synthesis, low noise microwave generation, laser frequency ranging, and astrophysical spectrometer calibration, and have the potential to make comb technology ubiquitous. As such, pattern formation in driven, dissipative nonlinear optical systems is becoming the central Physics of soliton micro-comb technology.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Continuous-wave laser-driven, high-Q Kerr–nonlinear optical microresonators have enabled the generation of optical frequency combs, ultralow-noise microwaves and ultrashort optical pulses at tens of gigahertz repetition rate. Here, we break with the paradigm of the continuous-wave driving and instead use periodic, picosecond optical pulses. In a fibre-based Fabry–Perot microresonator we observe the deterministic generation of stable femtosecond dissipative cavity solitons ‘on top’ of the resonantly enhanced driving pulses. The solitons lock to the driving pulse, which enables direct all-optical control of the soliton's repetition rate and tuning of its carrier-envelope offset frequency. When compared with continuous-wave-driven microresonators or non-resonant pulsed supercontinuum generation, this new approach is more efficient and can yield broadband frequency combs at an average driving power significantly below the continuous-wave parametric threshold. Bridging the fields of continuous-wave-driven resonant and pulse-driven non-resonant nonlinear optics, these results enable efficient microresonator frequency combs, resonant supercontinuum generation and microphotonic pulse compression. By driving a high-Q fibre-based Fabry–Perot microresonator with periodic, picosecond optical pulses, deterministic generation of stable femtosecond dissipative cavity solitons has been experimentally realized.

We discover a novel mechanism allowing for successive reduction of the number of dissipative Kerr solitons in optical microresonators. It is demonstrated that multiple and single soliton state can be deterministically accessed.