Comb Dynamics Research Articles

A stochastic approach to phase noise analysis for microwaves generated with Kerr optical frequency combs

Kerr optical frequency combs are expected to play a major role in photonic technology, with applications related to spectroscopy, sensing, aerospace, and communication engineering. Most of these applications are related to the metrological performance of Kerr combs, which is ultimately limited by their noise-driven fluctuations. For this reason, it is of high importance to understand the influence of random noise on the comb dynamics. In this communication, we theoretically investigate a model where Gaussian white noise is added to the coupled-mode equations governing the comb dynamics. This stochastic model allows us to characterize the noise-induced broadening of the spectral lines. Moreover, this study permits to determine the phase noise spectra of the microwaves generated via comb photodetection. In this latter case, our analysis indicates that the low-frequency part of the spectra is dominated by pattern drift while the high-frequency part is dominated by pattern deformation. The theoretical results are found to be in excellent agreement with numerical simulations.

Near-zero-dispersion soliton and broadband modulational instability Kerr microcombs in anomalous dispersion

The developing advances of microresonator-based Kerr cavity solitons have enabled versatile applications ranging from communication, signal processing to high-precision measurements. Resonator dispersion is the key factor determining the Kerr comb dynamics. Near the zero group-velocity-dispersion (GVD) regime, low-noise and broadband microcomb sources are achievable, which is crucial to the application of the Kerr soliton. When the GVD is almost vanished, higher-order dispersion can significantly affect the Kerr comb dynamics. Although many studies have investigated the Kerr comb dynamics near the zero-dispersion regime in microresonator or fiber ring system, limited by dispersion profiles and dispersion perturbations, the near-zero-dispersion soliton structure pumped in the anomalous dispersion side is still elusive so far. Here, we theoretically and experimentally investigate the microcomb dynamics in fiber-based Fabry-Perot microresonator with ultra-small anomalous GVD. We obtain 2/3-octave-spaning microcombs with ~10 GHz spacing, >84 THz span, and >8400 comb lines in the modulational instability (MI) state, without any external nonlinear spectral broadening. Such widely-spanned MI combs are also able to enter the soliton state. Moreover, we report the first observation of anomalous-dispersion based near-zero-dispersion solitons, which exhibits a local repetition rate up to 8.6 THz, an individual pulse duration <100 fs, a span >32 THz and >3200 comb lines. These two distinct comb states have their own advantages. The broadband MI combs possess high conversion efficiency and wide existing range, while the near-zero-dispersion soliton exhibits relatively low phase noise and ultra-high local repetition rate. This work complements the dynamics of Kerr cavity soliton near the zero-dispersion regime, and may stimulate cross-disciplinary inspirations ranging from dispersion-controlled microresonators to broadband coherent comb devices.

Optical frequency combs in dispersion-controlled doubly resonant second-harmonic generation.

We report on the experimental realization and a systematic study of optical frequency comb generation in doubly resonant intracavity second harmonic generation (SHG). The efficiency of intracavity nonlinear processes usually benefits from the increasing number of resonating fields. Yet, achieving the simultaneous resonance of different fields may be technically complicated, all the more when a phase matching condition must be fulfilled as well. In our cavity we can separately control the resonance condition for the fundamental and its second harmonic, by simultaneously acting on an intracavity dispersive element and on a piezo-mounted cavity mirror, without affecting the quasi-phase matching condition. In addition, by finely adjusting the laser-to-cavity detuning, we are able to observe steady comb emission across the whole resonance profile, revealing the multiplicity of comb structures, and the substantial role of thermal effects on their dynamics. Lastly, we report the results of numerical simulations of comb dynamics, which include photothermal effects, finding a good agreement with the experimental observations. Our system provides a framework for exploring the richness of comb dynamics in doubly resonant SHG systems, assisting the design of chip-scale quadratic comb generators.

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.

Two Polarization Comb Dynamics in VCSELs Subject to Optical Injection

Optical frequency comb technologies have received intense attention due to their numerous promising applications ranging from optical communications to optical comb spectroscopy. In this study, we experimentally demonstrate a new approach of broadband comb generation based on the polarization mode competition in single-mode VCSELs. More specifically, we analyze nonlinear dynamics and polarization properties in VCSELs when subject of optical injection from a frequency comb. When varying injection parameters (injection strength and detuning frequency) and comb properties (comb spacing), we unveil several bifurcation sequences enabling the excitation of free-running depressed polarization mode. Interestingly, for some injection parameters, the polarization mode competition induces a single or a two polarization comb with controllable properties (repetition rate and power per line). We also show that the performance of the two polarization combs depends crucially on the injection current and on the injected comb spacing. We explain our experimental findings by utilizing the spin-flip VCSEL model (SFM) supplemented with terms for parallel optical injection of frequency comb. We provide a comparison between parallel and orthogonal optical injection in the VCSEL when varying injection parameters and SFM parameters. We show that orthogonal comb dynamics can be observed in a wide range of parameters, as for example dichroism linear dichroism (γa=−0.1 ns−1 to γa=−0.8 ns−1), injection current (μ=2.29 to μ=5.29) and spin-flip relaxation rate (γs=50 ns−1 to γs=2300 ns−1).

Spectrally resolved linewidth enhancement factor of a semiconductor frequency comb

The linewidth enhancement factor (LEF) has recently moved into the spotlight of research on frequency comb generation in semiconductor lasers. Here we present a novel modulation experiment that enables direct measurement of the spectrally resolved LEF in a laser frequency comb. By utilizing a phase-sensitive technique, we are able to extract the LEF for each individual comb mode in any laser type. We first investigate and verify this universally applicable technique using Maxwell–Bloch simulations. Following, we present the experimental demonstration on a quantum cascade laser frequency comb, confirming the predicted key role of the LEF in frequency comb dynamics.

Nearly Quantum-Limited Josephson-Junction Frequency-Comb Synthesizer

While coherently-driven Kerr microcavities have rapidly matured as a platform for frequency comb formation, such microresonators generally possess weak Kerr coefficients; consequently, triggering comb generation requires millions of photons to be circulating inside the cavity. This suppresses the role of quantum fluctuations in the comb's dynamics. In this paper, we realize a minimal version of coherently-driven Kerr-mediated microwave frequency combs in the circuit QED architecture, where the quantum vacuum's fluctuations are the primary limitation on comb coherence. We achieve a comb phase coherence of up to 35~$\mu$s, approaching the theoretical device quantum limit of 55~$\mu$s, and vastly longer than the modes' inherent lifetimes of 13~ns. The ability within cQED to engineer stronger nonlinearities than optical microresonators, together with operation at cryogenic temperatures, and excellent agreement of comb dynamics with quantum theory indicates a promising platform for the study of complex dynamics of quantum nonlinear systems

Modulational instability and frequency combs in whispering-gallery-mode microresonators with backscattering

We introduce the first principle model describing frequency comb generation in a WGM microresonator with the backscattering-induced coupling between the counter-propagating waves. {Elaborated model provides deep insight and accurate description of the complex dynamics of nonlinear processes in such systems.} We analyse the backscattering impact on the splitting and reshaping of the nonlinear resonances, demonstrate backscattering-induced modulational instability in the normal dispersion regime and subsequent frequency comb generation. We present and discuss novel features of the soliton comb dynamics induced by the backward wave.

Frequency comb dynamics of a 1.3 μm hybrid-silicon quantum dot semiconductor laser with optical injection.

This work reports on the influence of bias voltage applied on a saturable absorber (SA) on a subthreshold linewidth enhancement factor (LEF) in hybrid-silicon quantum dot optical frequency comb lasers. Results show that the reverse bias voltage on SA contributes to enlarge the LEF and improve the comb dynamics. Optical injection is also found to be able to improve the comb spectrum in terms of 3dB bandwidth and its flatness. Such novel findings are promising for the development of high-speed dense wavelength-division multiplexing photonic integrated circuits in optical interconnects and datacom applications.

Resonance tracking in a micromechanical device using phononic frequency combs

Micro and nanomechanical resonators have been extensively researched in recent decades for applications to time and frequency references, as well as highly sensitive sensors. Conventionally, the operation of these resonant sensors is implemented using a feedback oscillator to dynamically track variations in the resonant frequency. However, this approach places limitations on the frequency stability of the output response, particularly owing to near-carrier noise shaping, limiting measurement stabilities at short-to-moderate integration times. Here, in this paper, utilizing the recent experimental demonstration of phononic frequency combs, we demonstrate an alternative resonance tracking approach with the potential to provide significant improvements in near-carrier phase noise and long-term stability. In addition, we also showcase comb dynamics mediated resonant frequency modulation which indirectly points to the possible control of inevitable noise processes including thermomechanical fluctuations. This resonant tracking approach may also have general applicability to a number of other physical oscillators.

Soliton bursts and deterministic dissipative Kerr soliton generation in auxiliary-assisted microcavities

Dissipative Kerr solitons in resonant frequency combs offer a promising route for ultrafast mode-locking, precision spectroscopy and time-frequency standards. The dynamics for the dissipative soliton generation, however, are intrinsically intertwined with thermal nonlinearities, limiting the soliton generation parameter map and statistical success probabilities of the solitary state. Here, via use of an auxiliary laser heating approach to suppress thermal dragging dynamics in dissipative soliton comb formation, we demonstrate stable Kerr soliton singlet formation and soliton bursts. First, we access a new soliton existence range with an inverse-sloped Kerr soliton evolution—diminishing soliton energy with increasing pump detuning. Second, we achieve deterministic transitions from Turing-like comb patterns directly into the dissipative Kerr soliton singlet pulse bypassing the chaotic states. This is achieved by avoiding subcomb overlaps at lower pump power, with near-identical singlet soliton comb generation over twenty instances. Third, with the red-detuned pump entrance route enabled, we uncover unique spontaneous soliton bursts in the direct formation of low-noise optical frequency combs from continuum background noise. The burst dynamics are due to the rapid entry and mutual attraction of the pump laser into the cavity mode, aided by the auxiliary laser and matching well with our numerical simulations. Enabled by the auxiliary-assisted frequency comb dynamics, we demonstrate an application of automatic soliton comb recovery and long-term stabilization against strong external perturbations. Our findings hold potential to expand the parameter space for ultrafast nonlinear dynamics and precision optical frequency comb stabilization.

Free-carrier-driven Kerr frequency comb in optical microcavities: Steady state, bistability, self-pulsation, and modulation instability

Continuous-wave pumped optical microresonators have been vastly exploited to generate a frequency comb (FC) utilizing the Kerr nonlinearity. Most of the nonlinear materials used to build photonic platforms exhibit nonlinear losses such as multiphoton absorption, free-carrier absorption, and free-carrier dispersion which can strongly affect their nonlinear performances. In this work, we model the Kerr FC based on a modified Lugiato-Lefever equation (LLE) along with the rate equation and develop analytical formulations to make a quick estimation of the steady state, bistability, self-pulsation, and modulation instability (MI) gain and bandwidth in the presence of nonlinear losses. The analytical model is valid over a broad wavelength range as it includes the effects of all nonlinear losses. Higher-order $(g3)$ characteristic polynomials of intracavity power describing the steady-state homogeneous solution of the modified LLE are discussed in detail. We derive the generalized analytical expressions for the threshold (normalized) pump detuning that initiates the optical bistability when nonlinear losses are present. Free-carrier dispersion-led nonlinear cavity detuning is observed through the reverse Kerr tilt of the resonant peaks. We further deduce the expressions of the threshold pump intensity and the range of possible cavity detuning for the initiation of the MI considering the presence of nonlinear losses. The proposed model will be helpful in explaining several numerical and experimental results which have been previously reported and thereby will be able to provide a better understanding of the comb dynamics.

The time-dependent Schrödinger equation in three dimensions under geometric constraints

We consider a quantum motion governed by the time-dependent Schrödinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrödinger equations for the reduced probability density functions by projection of the three dimensional comb dynamics in the two- and one-dimensional configuration space. This represents another physical example of a system where fractional calculus emerges. We give closed-form solutions of the corresponding equations for the reduced probability density functions in terms of the Fox H-function, by using the Green’s function approach.

Frequency comb generation in SNAP bottle resonators.

We develop a theory of optical frequency comb generation in ultra-compact surface nanoscale axial photonic (SNAP) bottle microresonators, employing the nonlinear interaction of whispering gallery modes which are confined along an optical fiber with nanoscale radius variation. We predict that a SNAP microresonator with a radius of a few micrometers can generate a frequency comb with an ultra-fine sub-gigahertz spectral spacing, which would require traditional ring resonators of centimeter radius. We identify regimes of stable or quasi-periodic comb dynamics due to soliton excitation, and show that special engineering of the SNAP radius profile can be used to compensate for nonlinearity-induced dispersion.

Spectral noise correlations of an ultrafast frequency comb.

Cavity-based noise detection schemes are combined with ultrafast pulse shaping as a means to diagnose the spectral correlations of both the amplitude and phase noise of an ultrafast frequency comb. The comb is divided into ten spectral regions, and the distribution of noise as well as the correlations between all pairs of spectral regions are measured against the quantum limit. These correlations are then represented in the form of classical noise matrices, which furnish a complete description of the underlying comb dynamics. Their eigendecomposition reveals a set of theoretically predicted, decoupled noise modes that govern the dynamics of the comb. These matrices also contain the information necessary to deduce macroscopic noise properties of the comb.

Measurement of the comb dynamics for feedback control of an etalon-based coupled optoelectronic oscillator

The response of an optical frequency comb from an etalon-based coupled optoelectronic oscillator to changes in drive current, optoelectronic loop phase, modulator bias, and laser cavity length has been measured. It is found that controlling the phase of the optoelectronic loop is best suited for control of the pulse repetition rate, whereas controlling the laser cavity length is best for stabilization of the optical carrier frequency. Moreover, by measuring the instabilities of the carrier frequency at the fixed-point frequency of the optoelectronic phase, changes to the optoelectronic phase can be decoupled from changes to the laser cavity.

Orthogonal control of the frequency comb dynamics of a mode-locked laser diode.

We have performed detailed studies on the dynamics of a frequency comb produced by a mode-locked laser diode (MLLD). Orthogonal control of the pulse repetition rate and the pulse-to-pulse carrier-envelope phase slippage is achieved by appropriate combinations of the respective error signals to actuate the diode injection current and the saturable absorber bias voltage. Phase coherence is established between the MLLD at 1550 nm and a 775-nm mode-locked Ti:sapphire laser working as part of an optical atomic clock.