Abstract

Optical frequency combs based on solitons in nonlinear microresonators open up new regimes for optical metrology and signal processing across a range of expanding and emerging applications. In this work, we advance these combs toward applications by demonstrating protected single-soliton formation and operation in a Kerr-nonlinear microresonator using a phase-modulated pump laser. Phase modulation gives rise to spatially/temporally varying effective loss and detuning parameters, leading to an operation regime in which multi-soliton degeneracy is lifted and a single soliton is the only observable behavior. We achieve direct, on-demand excitation of single solitons as indicated by reversal of the characteristic “soliton step.” Phase modulation also enables precise, high bandwidth control of the soliton pulse train’s properties, and we measure dynamics that agree closely with simulations. We show that the technique can be extended to high-repetition-frequency Kerr solitons through subharmonic phase modulation. These results will facilitate straightforward generation and control of Kerr-soliton microcombs for integrated photonics systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Dissipative temporal cavity solitons in Kerr microresonators [14] have the potential to provide the revolutionary capabilities of frequency combs in a chip-integrable platform. This would extend the reach of frequency combs to applications in communications, computation, and sensing with low size, weight, and power. Progress has come rapidly in the field of microresonator-soliton-based frequency combs, but for these combs to reach applications, simple, repeatable, and platform-independent methods of soliton generation and control are needed. The basic challenge is that solitons in microresonators are independent excitations, and a resonator can host zero, one, or many co-circulating solitons at a given pump-laser power and frequency. Further, under normal conditions these solitons can be generated only by condensation from extended modulation-instability (MI) patterns (primary comb/Turing patterns, or noisy comb/spatiotemporal chaos) that provide appropriate initial perturbations. Thermal stability must be maintained as the intracavity power drops during the transition from a high-duty-cycle MI pattern to a low-duty-cycle soliton. A variety of schemes have been demonstrated to address these challenges and obtain single solitons [59], and many achieve excellent performance. However, in general these schemes increase procedural complexity by exploiting non-adiabatic variations in pump-laser power and frequency, and/or involve at least some amount of stochastic fluctuation in the output.

One notable possibility is modulation of the pump laser at a frequency near the resonator free-spectral range (FSR) [1013], which can enable deterministic condensation of either one or zero solitons from an MI pattern. Further, it has been demonstrated that phase modulation (PM) can facilitate generation and control of single solitons [12,14,15]. In this article we theoretically describe and then experimentally demonstrate the use of a phase-modulated pump laser for deterministic excitation of single solitons directly from a chirped background that remains otherwise stable, as was proposed in Ref. [16]; the result is a train of solitons spaced by the round-trip time exiting the resonator, as shown in Fig. 1(a). Importantly, this scheme requires no transient perturbation to the system parameters for soliton generation. In addition to exploring soliton generation, we also demonstrate that PM at the FSR can be used for microsecond-level control of the pulse train’s repetition rate, and we conclude by discussing how the technique can be applied to resonators with FSR too high to be directly electronically accessible.

 figure: Fig. 1.

Fig. 1. (a) Schematic for soliton generation in a PM-pumped resonator, neglecting interference at the output. (b) Simulated energy-level diagrams for the CW- (orange) and PM-pumped (blue, δPM=π) resonator for F2=4, β2=0.0187. With PM, an interval in α exists for which the single soliton is the only available energy level. This interval is fairly narrow, but we find that it is readily accessible in experiment. We also observe non-stationary states for values of α2.7 in the PM case (red shading); whether the system exhibits one of these non-stationary states or an N=2 soliton state is determined by the initial conditions that seed the formation of the soliton ensemble.

Download Full Size | PPT Slide | PDF

2. PHYSICAL MECHANISM AND THEORETICAL EXPLORATION OF THE CONCEPT

Our results demonstrate a regime in which single-soliton operation is fundamentally protected without the degeneracy between N=0,1 and many solitons that exists for a continuous-wave (CW) pump laser. To motivate the experimental work that follows, we begin by presenting theoretical results that illustrate the utility of a PM pump. We use the nonlinear partial-differential Lugiato–Lefever equation (LLE) with modification of the driving term for phase modulation with depth δPM [1,1619]:

ψτ=(1+iα)ψ+i|ψ|2ψiβ222ψθ2+FeiδPMcosθ.
The normalized quantities used in the LLE are defined as follows [17]: ψ is the envelope for the intracavity field normalized so that |ψ|2=1 at the absolute threshold for parametric oscillation; τ=t/2τph, where t is the time and τph=1/2πΔν is the cavity photon lifetime, where Δν is the cavity resonance linewidth; α=2(ν0νpump)/Δν is the detuning between the pumped resonance with frequency ν0 and the pump laser with frequency νpump; F is the pump strength normalized so that F2=1 at the absolute threshold for parametric oscillation; and β2=2D2/2πΔν<0 is the anomalous resonator dispersion, with D2/2π=2νμ/μ2|μ=0>0, where νμ represents the set of cavity resonance frequencies and μ=0 indexes the pumped mode. The azimuthal angle θ co-rotates at the frequency fPM, which is presently assumed to be equal to the FSR. We emphasize for clarity that throughout this paper α>0, and, therefore, that the pump laser is always “red detuned” from the cavity resonance; decreasing (increasing) α corresponds to increasing (decreasing) the laser frequency νpump and moving it closer to (further from) the linear cavity resonance.

We perform simulations of the LLE to investigate soliton degeneracy for the range of pump-laser detunings over which solitons exist. We use a fourth-order Runge–Kutta algorithm in the interaction picture [20] with adaptive step size [21]. The resulting soliton-energy-level diagrams for the CW case (δPM=0) and the PM case (δPM=π) are shown in Fig. 1(b). We find that PM transforms the resonator excitation spectrum from a series of N=0,1,2,,Nmax solitons to a smaller number of available energy levels. In particular, for detuning α slightly greater than the lower bound for soliton existence (below which lie extended MI patterns), a single level N=1 is the only available state, and soliton degeneracy is eliminated.

The elimination of degeneracy and emergence of the protected N=1 level occurs as a result of spatial/temporal variations of effective loss and detuning parameters that result from the phase modulation. We can obtain an approximation for these parameters by inserting the ansatz ψ(θ,τ)=φ(θ,τ)eiδPMcosθ into Eq. (1) [15]. By expanding the second-derivative term and setting derivatives of φ to zero we arrive at an equation for the quasi-CW background in the PM-pumped resonator:

F=(γ(θ)+iαeff(θ))φi|φ|2φ.
The effective loss and detuning terms are
γ(θ)=1+β22δPMcosθ,
αeff(θ)=αβ22δPM2sin2θ.
The field ψ can be approximated using these local parameters:
ψ=FeiδPMcosθγ(θ)+i(αeff(θ)ρ(θ)),
where ρ(θ)=|φ(θ)|2 is the (smallest real) solution to the cubic polynomial in ρ obtained by taking the modulus-square of Eq. (2):
F2=(γ(θ)2+(αeff(θ)ρ(θ))2)ρ(θ).
In neglecting the spatial derivatives of φ but retaining the derivatives of the term eiδPMcosθ, we make the approximation that the dominant effect of dispersion is its action on the existing phase-modulation spectrum. We note that a full analysis of the rich behavior of Eq. (1) without this approximation remains a promising avenue for future research.

Figure 2(a) compares the predictions of simulations of the LLE (color) and the analytical model discussed immediately above (black) for the quasi-CW background in the resonator in the presence of phase modulation. The two agree quantitatively for weak modulation (δPM=π2, blue) and qualitatively for larger depth (δPM=4π, green); both indicate that the field ψ exhibits amplitude variations due to spatially varying effective loss and detuning. These parameters determine whether the quasi-CW background locally (as a function of θ) exhibits the bistability that is well known in the case of a CW pump laser [19,22], which suggests a mechanism for spontaneous single-soliton generation: as α is decreased, the stable effectively red-detuned branch of the resonance locally vanishes at the larger peak of the quasi-CW background at θ=0, leading to the formation of a soliton. If α is decreased further, the stable effectively red-detuned branch vanishes at the smaller peak at θ=π and a second soliton is formed. By following the analysis in, e.g., Ref. [19] [Eqs. (11)–(14)], we can approximate the detunings α1 and α2 where the first and second solitons are generated, respectively, by determining at what detuning the red-detuned branch vanishes for θ=0 and θ=π. For parameters matching the level diagram shown in Fig. 1(b), this predicts generation of the first soliton at α1=2.741 and generation of the second at α2=2.705, in excellent agreement with the values α1=2.729 and α2=2.699 obtained in numerical simulations. A simulation of single-soliton generation enabled by pump phase modulation is shown in Fig. 2(b).

 figure: Fig. 2.

Fig. 2. (a) Simulated quasi-CW background intensity without (orange) and with PM of depth π/2 (blue) and 4π (green), with F2=3 and β2=0.02, with analytical approximations in black. Here α is slightly larger than α1, the critical value for soliton formation. Dashed traces show the simulated phase profile of the field ψ (green) and of the driving term FeiδPMcosθ (red) with modulation depth 4π. The phase of the field is very nearly the phase of the drive plus a constant offset. (b) A simulation of spontaneous single-soliton generation using a phase-modulated pump laser with δPM=π, F2=4, and β=0.0187. Left: α is decreased smoothly as a function of time τ and then held constant after a soliton is generated. Middle: false-color plot of the intensity in the cavity as a function of time, with the final intensity shown above. Right: false-color plot of the comb spectrum on a logarithmic scale as a function of time, assuming a repetition rate of 22 GHz. Final spectrum is shown above. (c) A simulation of the collapse of a soliton ensemble to N=1 with initial N=5 for α=2.8, F2=4, and β=0.0187. To slow down the initial dynamics for clarity, the modulation depth is initialized at δPM=0, and then linearly ramped to δPM=π from τ=50 to τ=550. We note that the simulations presented in parts (b) and (c) are similar to simulations presented in Ref. [16]; we present these for clarity without claiming novelty. (d) Plots of approximations (obtained as described in the text, β2=0.0187) to the values α1 and α2 of detuning at which the first soliton and a second soliton are generated, respectively, with a zoomed plot depicting details in the upper right. Several values of PM depth are plotted: δPM=0 (blue), δPM=π (yellow), and δPM=2π (green). For each the line to the right is α1 and the line to the left is α2; the single line for δPM=0 represents the line at which the bistability of the CW background vanishes everywhere in this case. Horizontal lines indicate the interval α2<α<α1 obtained in the corresponding LLE simulations with F2=5.1.

Download Full Size | PPT Slide | PDF

If solitons exist at θ0 in a PM-pumped cavity, they drift to the intracavity intensity maximum at θ=0 [15], making superpositions of N>1 solitons unstable and practically forbidden for values of detuning α>α2 (for αα2, a second soliton is spontaneously generated at θ=π and persists there). We depict the simulated collapse to N=1 of such a superposition in Fig. 2(c). Thus, over the range of detuning α2<α<α1, application of PM to the pump laser removes the degeneracy between N=1 and N=0 and also between N=1 and N>1 solitons. Single-soliton generation and operation then simply require tuning the pump power and frequency to appropriate values, regardless of initial conditions.

Quantitative determination of the full soliton level structure plotted in Fig. 1(b) is numerically quite involved. However, a qualitative estimate of the dependence of the detuning interval α2<α<α1 for protected single-soliton generation and operation on pump power F2 and modulation depth δPM can be obtained by generalizing the process described above: as a function of F2 and δPM, one determines at what values of detuning the bistability vanishes for θ=0 (defining α1) and θ=π (defining α2). We present example approximations in Fig. 2(d). The basic observation is that the interval between α1 and α2 over which single-soliton operation is protected increases in size as δPM increases, as a result of the scaling of the θ-dependence of γ(θ) with δPM in Eq. (3). In a sample comparison of these approximations with a determination of α1 and α2 in an LLE simulation we find that this basic observation holds, but that the prediction for the absolute location of the interval α2<α<α1 becomes less accurate for larger δPM; this is consistent with the greater deviation between approximation and simulation at higher δPM seen in Fig. 2(a).

Before discussing our experimental implementation of the approach described above, we note that it is natural to consider whether a similar technique can be employed using amplitude modulation (AM) outside of the pulsed-pumping limit [13] (this limit requires prior generation of a train of temporally short input pulses, which brings additional complexity). While we have not conducted an exhaustive study of soliton generation with AM, as that is not the primary subject of this work, our preliminary simulations show that AM can indeed be used to spontaneously generate single solitons. Specifically, simulations indicate that one implementation with complexity comparable to our PM technique, in which a single Mach–Zehnder-type modulator with sinusoidal modulation is used to generate a 50% duty-cycle train of nearly flat-topped pulses (see Ref. [23]), is likely to be successful under a somewhat smaller range of parameters than the PM scheme we discuss here. A full quantitative study of AM is, however, beyond the scope of the present work, and is accordingly left as a promising avenue for future research.

3. EXPERIMENTAL RESULTS: SOLITON GENERATION

We implement the approach described above to realize deterministic generation of single solitons without condensation from an extended pattern. Our approach is summarized in Fig. 3, and results are depicted in Fig. 4. We use a 22 GHz FSR silica wedge resonator with Δν1.5MHz linewidth [24] (loaded Q129 million, see also Ref. [25]), pumped by a laser with normalized power F2 between 2 and 6, phase modulated at a rate fPM22GHz with relatively small depth δPMπ. The pump laser is derived from a seed CW laser using a single-sideband modulator driven by a voltage-controlled oscillator (VCO) [25], yielding high frequency-control bandwidth. To overcome the challenges presented by thermal instabilities [26], we also address the resonance with a counter-propagating probe beam that is frequency shifted in an acousto-optic modulator (AOM) by fAOM=55MHz. We phase modulate this probe beam at a variable frequency fPDH<55MHz to enable Pound–Drever–Hall (PDH) locking of the red-detuned PDH sideband of the probe beam to the resonance [25]; the lock is achieved by feeding back to the pump-laser frequency using the VCO. This enables real-time measurement and control of the detuning ν0νpump according to ν0νpump=fAOMfPDH. A frequency-domain depiction of our detuning-control system is presented in Fig. 3(a), and a schematic depiction of the components used to realize this scheme is shown in Fig. 3(b). Future work could simplify the apparatus while maintaining useful detuning stabilization and control.

 figure: Fig. 3.

Fig. 3. (a) Frequency-domain depiction of the experiment, with cavity modes shown in blue and laser frequencies shown in red. Modulation of a counter-propagating probe beam at fPDH after shifting by fAOM and subsequent locking of the red PDH sideband to the resonance allows detuning control via ν0νpump=fAOMfPDH, with adjustments implemented by changing fPDH. (b) Schematic depiction of the components of the experiment used for frequency control of the system.

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

Fig. 4. (a) Measurement of a reversed soliton step in the comb power associated with soliton generation from the background. Inset shows qualitatively similar dynamics observed in an LLE-simulated comb-power trace. (b) Measured optical spectrum of the soliton generated with a phase-modulated pump laser. The spectrum of the pump, which contains phase-modulation sidebands, is visible in the center. The soliton’s spectral envelope closely matches the sech2 envelope that has been overlaid in dashed red. (c) Plot of the out-coupled soliton pulse train’s repetition rate as recorded by a photodetector, exhibiting a characteristically high signal-to-noise ratio. We emphasize that this data is obtained with the phase modulation turned off; otherwise the recorded RF signal is dominated by through-coupled phase-modulation sidebands at fPMfFSRfrep. (d) Histogram of measured offset in detuning from a reference value at which a soliton is generated over 160 successive trials, with a Gaussian fit shown in red. The width of the interval over which solitons are generated is larger than the calculated width of the protected N=1 level shown in Fig. 1(b), but the model does not include laser fluctuations and other experimental effects.

Download Full Size | PPT Slide | PDF

To generate solitons we decrease the detuning from a large initial value (40MHz) by increasing the frequency fPDH of the PDH modulation—this is done by simply turning the frequency knob on the function generator by hand. A soliton is generated when the detuning is near 5 MHz (dependent upon the pump power and coupling condition). Measuring the power converted through four-wave mixing to new frequencies, the “comb power,” reveals a step upon soliton formation, shown in Fig. 4(a). This indicates direct generation of a soliton from the background, and represents a reversal of the characteristic “soliton step” that typically signals condensation of solitons from an extended pattern. After soliton generation the comb exhibits the sech2 spectral envelope and quiet repetition-rate tone characteristic of single-soliton operation, shown in Figs. 4(b) and 4(c). Once single-soliton operation is achieved, α may be increased again without loss of the soliton, consistent with the level diagram shown in Fig. 1(b). We have verified that it is possible to turn off the PM while preserving the soliton if the PM is turned off by first decreasing its amplitude slowly (i.e., on the timescale of 1 s) before turning it off entirely (see also Ref. [16]).

To investigate the repeatability of our technique, we automate soliton generation by repeatedly decreasing α from a large initial value and then increasing it again—control of α is achieved through adjustment of fPDH. The termination points of the scan are chosen such that at the lower end of the scan a soliton is generated before α begins to increase again, and at the upper end of the scan the soliton has been lost (because the maximum detuning for soliton existence has been exceeded) before α begins to decrease. Specifically, the scan runs linearly between detuning values of approximately 5 and 20 MHz. We observe generation and extinction of 1000 solitons in 1000 trials over 100 s with this automated scan, with a 100% measured success rate indicated by the comb-power trace recorded during the repeated sweep. Our probe-laser setup enables measurement of the detuning at which soliton generation occurs, which changes little from run to run. Figure 4(d) presents a histogram of detuning measurements for the generation of 160 solitons.

4. EXPERIMENTAL RESULTS: SOLITON CONTROL

Besides enabling protected single-soliton operation, PM pumping also naturally provides timing and repetition-rate control, because the solitons are pushed toward the intracavity phase maximum [15]. We explore this control, with results summarized in Fig. 5. In our experiments, the repetition rate of the out-coupled pulse train (frep) remains locked to fPM over a bandwidth of ±40kHz. In Fig. 5(a), we show a spectrogram of frep measured as fPM is swept sinusoidally over ±50kHz. The repetition rate follows the PM except for glitches near the peaks of the sweep. In the inset of Fig. 5(a) we overlay the results of LLE simulations (see below) that qualitatively match the observed behavior. These simulations indicate that the periodic nature of the glitches is due to the residual pulling of the phase modulation on the soliton when the latter periodically cycles through the pump’s phase maximum. Our observed locking range of ±40kHz agrees well with an estimate δPM×D2/2π44kHz [15] using the approximate measured value D2/2π=14kHz per mode and modulation depth δPM=π.

 figure: Fig. 5.

Fig. 5. (a) Measured spectrogram of frep as fPM is swept over ±50kHz, with glitches where the locking range is exceeded. Inset: qualitative agreement with simulations, shown in red, when fPM is outside of the locking bandwidth. As the soliton and the pump phase evolve at different frequencies frep and fPM, the soliton periodically approaches the maximum of the phase profile. The soliton’s group velocity changes, nearly locking to the phase modulation, before becoming clearly unlocked again. (b) Measured eye diagram of frep as fPM is switched ±40kHz with 10 μs transition time. (c) The same with 60 ns transition time, and an LLE simulation of the dynamics (red) with depth δPM=0.9π and resonator linewidth Δν=1.5MHz. (d) Simulated switching dynamics for various linewidths and modulation depths. The theory trace from (c) is reproduced in solid red in both panels. Top, solid black: modulation depth 0.9π as in (c), and Δν=10MHz (fastest, left), 3 MHz, and 1.4 MHz. Bottom, dashed red: linewidth Δν=1.5MHz as in (c), with modulation depths of 2π and 6π (curves nearly overlap).

Download Full Size | PPT Slide | PDF

 figure: Fig. 6.

Fig. 6. (a) Simulated spectra of PM at fFSR with depth 0.15π (blue) and at fFSR/21 with depth 8.3π (red). The relationships between the fields that address resonator modes 1, 0, and 1 (as indicated by the gray Lorentzian curves) are the same in both cases. (b) LLE simulation of single-soliton generation using the subharmonic phase-modulation spectrum shown in red in panel (a). Only modes n=0,±21,±42, of the phase-modulated driving field are coupled into the resonator and affect the LLE dynamics, with modes |n|>21 having negligible power. As α is decreased from a large initial value, a soliton is spontaneously generated, as in the case of phase modulation near the FSR.

Download Full Size | PPT Slide | PDF

To investigate fast control of the repetition rate, we measure frep as fPM is rapidly switched by ±40kHz around the soliton’s natural repetition rate. We plot the resulting data as eye diagrams in Figs. 5(b) and 5(c). In Fig. 5(b), fPM is switched with 200 μs period and 10 μs transition time; in Fig. 5(c) it is switched with 100 μs period and 60 ns transition time. This data is obtained by detecting a portion of the pulse train’s spectrum that excludes the pump laser with a sufficiently fast photodetector and passing the resulting frep signal through two paths, one with an element that induces a frequency-dependent phase shift. After calibration, the repetition rate can be obtained from the resulting phase shift in real time. These eye diagrams show that the PM enables exquisite control of the soliton pulse train.

We perform LLE simulations to further explore the dynamics of repetition-rate switching. We introduce the term +β1ψθ to the right-hand side of Eq. (1), where β1=2(fFSRfPM)/Δν represents a difference between the modulation frequency and the FSR of the resonator near the pump wavelength [15,16]; β1 may be varied in time. In Fig. 5(c) we overlay a simulation of switching conducted for parameters (Δν=1.5MHz, δPM=0.9π) near the experimental values, and the agreement between measurements and simulation indicates that the measurements are consistent with fundamental LLE dynamics. We present the results of additional simulations in Fig. 5(d); the basic observation is that the switching speed of frep is limited by the resonator linewidth, and can be modestly improved by increasing δPM.

5. APPLICATION OF THE TECHNIQUE TO HIGH FREE-SPECTRAL RANGE RESONATORS VIA SUBHARMONIC PHASE MODULATION

One apparent barrier to the use of PM for protected single-soliton operation is the electronically inaccessible FSRs of some microcomb resonators. However, this challenge can be overcome by applying PM at a subharmonic of the FSR. Simulations indicate that solitons can be generated with small modulation depth, e.g., δPM=0.15π. In this limit only the first-order PM sidebands are relevant, and their amplitude and phase relative to the carrier control the dynamics. A small desired modulation depth δPM,eff defined by the relationship between the carrier and the lowest order sidebands that are coupled into the resonator can be obtained by modulating with depth δPM at a frequency fPMfFSR/N, so that the Nth-order PM sidebands and the carrier address resonator modes with relative mode numbers 1, 0, and 1, where δPM is chosen in order to achieve effective depth δPM,eff. When N is odd, PM is recovered when the sidebands of order N, 0, and N address resonator modes 1, 0, and 1. When N is even, pure AM results, with a driving term like F(1+Acosθ).

Figure 6 presents a simulated example of this technique. In general, the values δPM, N, and δPM,eff are related non-trivially through Bessel functions according to the Jacobi–Anger expansion (see, e.g., Ref. [27], Section 10.12); for this example we choose N=21 and δPM,eff=0.15π. To achieve this effective modulation depth we employ real phase modulation depth of δPM8.3π at fPM=frep/21. The phase modulation spreads the optical power into the PM sidebands, so this technique requires higher optical power for the same effective pumping strength; in this example, the power must be increased by 15.6dB. In fact, δPM,eff=0.15π can be recovered with smaller δPM; however, this comes with far lower spectral efficiency (as defined by the fraction of total optical power that is coupled into the resonator). While the required modulation depth and pump power are higher with subharmonic PM, neither is impractical. This technique could be used for protected single-soliton generation and operation in high-repetition-rate systems; the example above indicates that it could be immediately applied to deterministic single-soliton generation in a 630 GHz FSR resonator with 30 GHz phase modulation. In principle, the ratio N=fFSR/fPM can be increased, allowing smaller phase-modulation frequencies to be applied to a given resonator; however, this comes at the cost of lower spectral efficiency and correspondingly higher required total optical power.

6. FINAL REMARKS

In this work, we have shown that phase modulation of the pump laser fundamentally changes a resonator’s excitation spectrum and enables an interesting new regime of protected single-soliton operation. In our experiments phase modulation of the pump laser, combined with our approach for detuning control, enabled deterministic, on-demand soliton generation with an observed 100% success rate. While our proof-of-concept experiments made use of sophisticated frequency- and detuning-control techniques, we expect simplification of the approach to be possible in the future, and this will facilitate its implementation in systems for photonics applications.

This technique is applicable to resonators with electronically accessible frep, which are important components of proposals for photonic integration of Kerr solitons [28,29], and can reach higher repetition-rate systems via subharmonic modulation. After soliton generation, the PM can optionally be turned off, recovering the properties of the non-PM soliton. We expect this technique to enable new experiments. For example, in principle, PM-pumped solitons are generated with known absolute timing, enabling immediate transduction of the modulation phase onto the pulse train; this is impossible with solitons stochastically condensed from an extended pattern. Our work brings microresonator solitons closer to applications.

Funding

National Aeronautics and Space Administration (NASA); National Institute of Standards and Technology (NIST); Defense Advanced Research Projects Agency (DARPA) (DODOS); National Science Foundation (NSF) (DGE 1144083); Air Force Office of Scientific Research (AFOSR) (FA9550-16-1-0016); RSNZ.

Acknowledgment

We thank Su-Peng Yu and Hojoong Jung for comments on the manuscript, and Andrew Weiner for helpful discussions.

REFERENCES

1. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]  

2. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015). [CrossRef]  

3. D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017). [CrossRef]  

4. B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018). [CrossRef]  

5. X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015). [CrossRef]  

6. K. Luo, J. K. Jang, S. Coen, S. G. Murdoch, and M. Erkintalo, “Spontaneous creation and annihilation of temporal cavity solitons in a coherently driven passive fiber resonator,” Opt. Lett. 40, 3735–3738 (2015). [CrossRef]  

7. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton modelocking in microresonators,” Opt. Lett. 41, 2565–2568 (2016). [CrossRef]  

8. H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017). [CrossRef]  

9. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons in microresonators,” Opt. Lett. 41, 2037–2040 (2016). [CrossRef]  

10. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express 21, 17615–17624 (2013). [CrossRef]  

11. S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014). [CrossRef]  

12. V. E. Lobanov, G. V. Lihachev, N. G. Pavlov, A. V. Cherenkov, T. J. Kippenberg, and M. L. Gorodetsky, “Harmonization of chaos into a soliton in Kerr frequency combs,” Opt. Express 24, 27382–27394 (2016). [CrossRef]  

13. E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017). [CrossRef]  

14. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Writing and erasing of temporal cavity solitons by direct phase modulation of the cavity driving field,” Opt. Lett. 40, 4755–4758 (2015). [CrossRef]  

15. J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015). [CrossRef]  

16. H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015). [CrossRef]  

17. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013). [CrossRef]  

18. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett. 38, 37–39 (2013). [CrossRef]  

19. C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014). [CrossRef]  

20. J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]  

21. A. M. Heidt, “Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers,” J. Lightwave Technol. 27, 3984–3991 (2009). [CrossRef]  

22. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. 38, 1790–1792 (2013). [CrossRef]  

23. D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016). [CrossRef]  

24. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012). [CrossRef]  

25. J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018). [CrossRef]  

26. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004). [CrossRef]  

27. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions (Cambridge University, 2010).

28. D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018). [CrossRef]  

29. T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
    [Crossref]
  2. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
    [Crossref]
  3. D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
    [Crossref]
  4. B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
    [Crossref]
  5. X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
    [Crossref]
  6. K. Luo, J. K. Jang, S. Coen, S. G. Murdoch, and M. Erkintalo, “Spontaneous creation and annihilation of temporal cavity solitons in a coherently driven passive fiber resonator,” Opt. Lett. 40, 3735–3738 (2015).
    [Crossref]
  7. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton modelocking in microresonators,” Opt. Lett. 41, 2565–2568 (2016).
    [Crossref]
  8. H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
    [Crossref]
  9. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons in microresonators,” Opt. Lett. 41, 2037–2040 (2016).
    [Crossref]
  10. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express 21, 17615–17624 (2013).
    [Crossref]
  11. S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
    [Crossref]
  12. V. E. Lobanov, G. V. Lihachev, N. G. Pavlov, A. V. Cherenkov, T. J. Kippenberg, and M. L. Gorodetsky, “Harmonization of chaos into a soliton in Kerr frequency combs,” Opt. Express 24, 27382–27394 (2016).
    [Crossref]
  13. E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
    [Crossref]
  14. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Writing and erasing of temporal cavity solitons by direct phase modulation of the cavity driving field,” Opt. Lett. 40, 4755–4758 (2015).
    [Crossref]
  15. J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
    [Crossref]
  16. H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
    [Crossref]
  17. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
    [Crossref]
  18. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett. 38, 37–39 (2013).
    [Crossref]
  19. C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
    [Crossref]
  20. J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007).
    [Crossref]
  21. A. M. Heidt, “Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers,” J. Lightwave Technol. 27, 3984–3991 (2009).
    [Crossref]
  22. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. 38, 1790–1792 (2013).
    [Crossref]
  23. D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
    [Crossref]
  24. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
    [Crossref]
  25. J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
    [Crossref]
  26. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004).
    [Crossref]
  27. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions (Cambridge University, 2010).
  28. D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
    [Crossref]
  29. T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
    [Crossref]

2018 (4)

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

2017 (3)

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
[Crossref]

2016 (4)

2015 (6)

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

K. Luo, J. K. Jang, S. Coen, S. G. Murdoch, and M. Erkintalo, “Spontaneous creation and annihilation of temporal cavity solitons in a coherently driven passive fiber resonator,” Opt. Lett. 40, 3735–3738 (2015).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Writing and erasing of temporal cavity solitons by direct phase modulation of the cavity driving field,” Opt. Lett. 40, 4755–4758 (2015).
[Crossref]

J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
[Crossref]

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

2014 (3)

S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
[Crossref]

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

2013 (4)

2012 (1)

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

2009 (1)

2007 (1)

2004 (1)

Adibi, A.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Balakireva, I. V.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Beha, K.

Beha, K. M.

D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
[Crossref]

Bluestone, A.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Bowers, J. E.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Brasch, V.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Briles, T. C.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Carlson, D. R.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

Carmon, T.

Chang, L.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Chembo, Y. K.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[Crossref]

Chen, S.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Chen, T.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Cherenkov, A. V.

Choi, C.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Coen, S.

Coillet, A.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Cole, D. C.

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
[Crossref]

Del’Haye, P.

Diddams, S. A.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
[Crossref]

S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
[Crossref]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express 21, 17615–17624 (2013).
[Crossref]

Drake, T.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Drake, T. E.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Duan, X.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Eftekhar, A. A.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Erkintalo, M.

Feng, Z.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Fredrick, C.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Gaeta, A. L.

Godey, C.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Gorodetsky, M. L.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

V. E. Lobanov, G. V. Lihachev, N. G. Pavlov, A. V. Cherenkov, T. J. Kippenberg, and M. L. Gorodetsky, “Harmonization of chaos into a soliton in Kerr frequency combs,” Opt. Express 24, 27382–27394 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Guo, H.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Heidt, A. M.

Herr, T.

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Hoff, M.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Huang, S. W.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Huang, Y.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Hult, J.

Ilic, B. R.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Jang, J. K.

Jeon, S.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Ji, X.

Joshi, C.

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Karpov, M.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Kippenberg, T. J.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

V. E. Lobanov, G. V. Lihachev, N. G. Pavlov, A. V. Cherenkov, T. J. Kippenberg, and M. L. Gorodetsky, “Harmonization of chaos into a soliton in Kerr frequency combs,” Opt. Express 24, 27382–27394 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Klenner, A.

Komljenovic, T.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Kordts, A.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Kwong, D. L.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Lamb, E. S.

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

Leaird, D. E.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Lecomte, S.

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
[Crossref]

Lee, H.

S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
[Crossref]

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Lee, S. H.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Li, J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Li, Q.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Li, Y.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Lihachev, G.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Lihachev, G. V.

Lipson, M.

Liu, Y.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Lobanov, V. E.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

V. E. Lobanov, G. V. Lihachev, N. G. Pavlov, A. V. Cherenkov, T. J. Kippenberg, and M. L. Gorodetsky, “Harmonization of chaos into a soliton in Kerr frequency combs,” Opt. Express 24, 27382–27394 (2016).
[Crossref]

Lucas, E.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Luke, K.

Luo, K.

Menyuk, C. R.

Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[Crossref]

Miller, S. A.

Murdoch, S. G.

Newbury, N. R.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Norberg, E.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Obrzud, E.

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
[Crossref]

Oh, D. Y.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Okawachi, Y.

Painter, O.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Papp, S. B.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
[Crossref]

S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10–14 (2014).
[Crossref]

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express 21, 17615–17624 (2013).
[Crossref]

Pavlov, N. G.

Pfeiffer, M. H. P.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Qi, M.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Quinlan, F.

Randle, H. G.

Rao, Y.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Sinclair, L. C.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Spencer, D. T.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Srinivasan, K.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Stone, J.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Stone, J. R.

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Suh, M.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Suh, M.-G.

Sylvestre, T.

Taheri, H.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Theogarajan, L.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Vahala, K.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons in microresonators,” Opt. Lett. 41, 2037–2040 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

Vahala, K. J.

Vinod, A. K.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Volet, N.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Wang, J.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Wang, P.-H.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Weiner, A. M.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Westly, D.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

T. C. Briles, J. R. Stone, T. E. Drake, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis,” Opt. Lett. 43, 2933–2936 (2018).
[Crossref]

Wiesenfeld, K.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Wong, C. W.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Xuan, Y.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Xue, X.

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

Yang, K. Y.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons in microresonators,” Opt. Lett. 41, 2037–2040 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Yang, L.

Yang, Q.-F.

Yao, B.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

Yi, X.

Yu, M.

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

IEEE Photon. J. (1)

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

J. Lightwave Technol. (2)

J. Phys. Conf. Ser. (1)

D. C. Cole, K. M. Beha, S. A. Diddams, and S. B. Papp, “Octave-spanning supercontinuum generation via microwave frequency multiplication,” J. Phys. Conf. Ser. 723, 012035 (2016).
[Crossref]

Nat. Commun. (1)

J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
[Crossref]

Nat. Photonics (5)

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
[Crossref]

X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
[Crossref]

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[Crossref]

Nat. Phys. (1)

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Nature (2)

B. Yao, S. W. Huang, Y. Liu, A. K. Vinod, C. Choi, M. Hoff, Y. Li, M. Yu, Z. Feng, D. L. Kwong, Y. Huang, Y. Rao, X. Duan, and C. W. Wong, “Gate-tunable frequency combs in graphene-nitride microresonators,” Nature 558, 410–414 (2018).
[Crossref]

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Optica (2)

Phys. Rev. A (2)

Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[Crossref]

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Phys. Rev. Lett. (1)

J. R. Stone, T. C. Briles, T. E. Drake, D. T. Spencer, D. R. Carlson, S. A. Diddams, and S. B. Papp, “Thermal and nonlinear dissipative-soliton dynamics in Kerr-microresonator frequency combs,” Phys. Rev. Lett. 121, 063902 (2018).
[Crossref]

Other (1)

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. (a) Schematic for soliton generation in a PM-pumped resonator, neglecting interference at the output. (b) Simulated energy-level diagrams for the CW- (orange) and PM-pumped (blue, δ PM = π ) resonator for F 2 = 4 , β 2 = 0.0187 . With PM, an interval in α exists for which the single soliton is the only available energy level. This interval is fairly narrow, but we find that it is readily accessible in experiment. We also observe non-stationary states for values of α 2.7 in the PM case (red shading); whether the system exhibits one of these non-stationary states or an N = 2 soliton state is determined by the initial conditions that seed the formation of the soliton ensemble.
Fig. 2.
Fig. 2. (a) Simulated quasi-CW background intensity without (orange) and with PM of depth π / 2 (blue) and 4 π (green), with F 2 = 3 and β 2 = 0.02 , with analytical approximations in black. Here α is slightly larger than α 1 , the critical value for soliton formation. Dashed traces show the simulated phase profile of the field ψ (green) and of the driving term F e i δ PM cos θ (red) with modulation depth 4 π . The phase of the field is very nearly the phase of the drive plus a constant offset. (b) A simulation of spontaneous single-soliton generation using a phase-modulated pump laser with δ PM = π , F 2 = 4 , and β = 0.0187 . Left: α is decreased smoothly as a function of time τ and then held constant after a soliton is generated. Middle: false-color plot of the intensity in the cavity as a function of time, with the final intensity shown above. Right: false-color plot of the comb spectrum on a logarithmic scale as a function of time, assuming a repetition rate of 22 GHz. Final spectrum is shown above. (c) A simulation of the collapse of a soliton ensemble to N = 1 with initial N = 5 for α = 2.8 , F 2 = 4 , and β = 0.0187 . To slow down the initial dynamics for clarity, the modulation depth is initialized at δ PM = 0 , and then linearly ramped to δ PM = π from τ = 50 to τ = 550 . We note that the simulations presented in parts (b) and (c) are similar to simulations presented in Ref. [16]; we present these for clarity without claiming novelty. (d) Plots of approximations (obtained as described in the text, β 2 = 0.0187 ) to the values α 1 and α 2 of detuning at which the first soliton and a second soliton are generated, respectively, with a zoomed plot depicting details in the upper right. Several values of PM depth are plotted: δ PM = 0 (blue), δ PM = π (yellow), and δ PM = 2 π (green). For each the line to the right is α 1 and the line to the left is α 2 ; the single line for δ PM = 0 represents the line at which the bistability of the CW background vanishes everywhere in this case. Horizontal lines indicate the interval α 2 < α < α 1 obtained in the corresponding LLE simulations with F 2 = 5.1 .
Fig. 3.
Fig. 3. (a) Frequency-domain depiction of the experiment, with cavity modes shown in blue and laser frequencies shown in red. Modulation of a counter-propagating probe beam at f PDH after shifting by f AOM and subsequent locking of the red PDH sideband to the resonance allows detuning control via ν 0 ν pump = f AOM f PDH , with adjustments implemented by changing f PDH . (b) Schematic depiction of the components of the experiment used for frequency control of the system.
Fig. 4.
Fig. 4. (a) Measurement of a reversed soliton step in the comb power associated with soliton generation from the background. Inset shows qualitatively similar dynamics observed in an LLE-simulated comb-power trace. (b) Measured optical spectrum of the soliton generated with a phase-modulated pump laser. The spectrum of the pump, which contains phase-modulation sidebands, is visible in the center. The soliton’s spectral envelope closely matches the sech 2 envelope that has been overlaid in dashed red. (c) Plot of the out-coupled soliton pulse train’s repetition rate as recorded by a photodetector, exhibiting a characteristically high signal-to-noise ratio. We emphasize that this data is obtained with the phase modulation turned off; otherwise the recorded RF signal is dominated by through-coupled phase-modulation sidebands at f PM f FSR f rep . (d) Histogram of measured offset in detuning from a reference value at which a soliton is generated over 160 successive trials, with a Gaussian fit shown in red. The width of the interval over which solitons are generated is larger than the calculated width of the protected N = 1 level shown in Fig. 1(b), but the model does not include laser fluctuations and other experimental effects.
Fig. 5.
Fig. 5. (a) Measured spectrogram of f rep as f PM is swept over ± 50 kHz , with glitches where the locking range is exceeded. Inset: qualitative agreement with simulations, shown in red, when f PM is outside of the locking bandwidth. As the soliton and the pump phase evolve at different frequencies f rep and f PM , the soliton periodically approaches the maximum of the phase profile. The soliton’s group velocity changes, nearly locking to the phase modulation, before becoming clearly unlocked again. (b) Measured eye diagram of f rep as f PM is switched ± 40 kHz with 10 μs transition time. (c) The same with 60 ns transition time, and an LLE simulation of the dynamics (red) with depth δ PM = 0.9 π and resonator linewidth Δ ν = 1.5 MHz . (d) Simulated switching dynamics for various linewidths and modulation depths. The theory trace from (c) is reproduced in solid red in both panels. Top, solid black: modulation depth 0.9 π as in (c), and Δ ν = 10 MHz (fastest, left), 3 MHz, and 1.4 MHz. Bottom, dashed red: linewidth Δ ν = 1.5 MHz as in (c), with modulation depths of 2 π and 6 π (curves nearly overlap).
Fig. 6.
Fig. 6. (a) Simulated spectra of PM at f FSR with depth 0.15 π (blue) and at f FSR / 21 with depth 8.3 π (red). The relationships between the fields that address resonator modes 1 , 0, and 1 (as indicated by the gray Lorentzian curves) are the same in both cases. (b) LLE simulation of single-soliton generation using the subharmonic phase-modulation spectrum shown in red in panel (a). Only modes n = 0 , ± 21 , ± 42 , of the phase-modulated driving field are coupled into the resonator and affect the LLE dynamics, with modes | n | > 21 having negligible power. As α is decreased from a large initial value, a soliton is spontaneously generated, as in the case of phase modulation near the FSR.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ψ τ = ( 1 + i α ) ψ + i | ψ | 2 ψ i β 2 2 2 ψ θ 2 + F e i δ PM cos θ .
F = ( γ ( θ ) + i α eff ( θ ) ) φ i | φ | 2 φ .
γ ( θ ) = 1 + β 2 2 δ PM cos θ ,
α eff ( θ ) = α β 2 2 δ PM 2 sin 2 θ .
ψ = F e i δ PM cos θ γ ( θ ) + i ( α eff ( θ ) ρ ( θ ) ) ,
F 2 = ( γ ( θ ) 2 + ( α eff ( θ ) ρ ( θ ) ) 2 ) ρ ( θ ) .

Metrics