## 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 [1–4] 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 [5–9], 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) [10–13], 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.

## 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 ${\delta}_{\mathrm{PM}}$ [1,16–19]:

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 (${\delta}_{\mathrm{PM}}=0$) and the PM case (${\delta}_{\mathrm{PM}}=\pi $) are shown in Fig. 1(b). We find that PM transforms the resonator excitation spectrum from a series of $N=0,1,2,\dots ,{N}_{\mathrm{max}}$ solitons to a smaller number of available energy levels. In particular, for detuning $\alpha $ 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 $\psi (\theta ,\tau )=\phi (\theta ,\tau ){e}^{i{\delta}_{\mathrm{PM}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}$ into Eq. (1) [15]. By expanding the second-derivative term and setting derivatives of $\phi $ to zero we arrive at an equation for the quasi-CW background in the PM-pumped resonator:

The effective loss and detuning terms areFigure 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 (${\delta}_{\mathrm{PM}}=\frac{\pi}{2}$, blue) and qualitatively for larger depth (${\delta}_{\mathrm{PM}}=4\pi $, green); both indicate that the field $\psi $ exhibits amplitude variations due to spatially varying effective loss and detuning. These parameters determine whether the quasi-CW background locally (as a function of $\theta $) 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 $\alpha $ is decreased, the stable effectively red-detuned branch of the resonance locally vanishes at the larger peak of the quasi-CW background at $\theta =0$, leading to the formation of a soliton. If $\alpha $ is decreased further, the stable effectively red-detuned branch vanishes at the smaller peak at $\theta =\pi $ and a second soliton is formed. By following the analysis in, e.g., Ref. [19] [Eqs. (11)–(14)], we can approximate the detunings ${\alpha}_{1}$ and ${\alpha}_{2}$ where the first and second solitons are generated, respectively, by determining at what detuning the red-detuned branch vanishes for $\theta =0$ and $\theta =\pi $. For parameters matching the level diagram shown in Fig. 1(b), this predicts generation of the first soliton at ${\alpha}_{1}=2.741$ and generation of the second at ${\alpha}_{2}=2.705$, in excellent agreement with the values ${\alpha}_{1}=2.729$ and ${\alpha}_{2}=2.699$ obtained in numerical simulations. A simulation of single-soliton generation enabled by pump phase modulation is shown in Fig. 2(b).

If solitons exist at $\theta \ne 0$ in a PM-pumped cavity, they drift to the intracavity intensity maximum at $\theta =0$ [15], making superpositions of $N>1$ solitons unstable and practically forbidden for values of detuning $\alpha >{\alpha}_{2}$ (for $\alpha \le {\alpha}_{2}$, a second soliton is spontaneously generated at $\theta =\pi $ 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 ${\alpha}_{2}<\alpha <{\alpha}_{1}$, 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 ${\alpha}_{2}<\alpha <{\alpha}_{1}$ for protected single-soliton generation and operation on pump power ${F}^{2}$ and modulation depth ${\delta}_{\mathrm{PM}}$ can be obtained by generalizing the process described above: as a function of ${F}^{2}$ and ${\delta}_{\mathrm{PM}}$, one determines at what values of detuning the bistability vanishes for $\theta =0$ (defining ${\alpha}_{1}$) and $\theta =\pi $ (defining ${\alpha}_{2}$). We present example approximations in Fig. 2(d). The basic observation is that the interval between ${\alpha}_{1}$ and ${\alpha}_{2}$ over which single-soliton operation is protected increases in size as ${\delta}_{\mathrm{PM}}$ increases, as a result of the scaling of the $\theta $-dependence of $\gamma (\theta )$ with ${\delta}_{\mathrm{PM}}$ in Eq. (3). In a sample comparison of these approximations with a determination of ${\alpha}_{1}$ and ${\alpha}_{2}$ in an LLE simulation we find that this basic observation holds, but that the prediction for the absolute location of the interval ${\alpha}_{2}<\alpha <{\alpha}_{1}$ becomes less accurate for larger ${\delta}_{\mathrm{PM}}$; this is consistent with the greater deviation between approximation and simulation at higher ${\delta}_{\mathrm{PM}}$ 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 $\mathrm{\Delta}\nu \sim 1.5\text{\hspace{0.17em}}\mathrm{MHz}$ linewidth [24] (loaded $Q\sim 129$ million, see also Ref. [25]), pumped by a laser with normalized power ${F}^{2}$ between 2 and 6, phase modulated at a rate ${f}_{\mathrm{PM}}\sim 22\text{\hspace{0.17em}}\mathrm{GHz}$ with relatively small depth ${\delta}_{\mathrm{PM}}\sim \pi $. 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 ${f}_{\mathrm{AOM}}=55\text{\hspace{0.17em}}\mathrm{MHz}$. We phase modulate this probe beam at a variable frequency ${f}_{\mathrm{PDH}}<55\text{\hspace{0.17em}}\mathrm{MHz}$ 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 ${\nu}_{0}-{\nu}_{\mathrm{pump}}$ according to ${\nu}_{0}-{\nu}_{\mathrm{pump}}={f}_{\mathrm{AOM}}-{f}_{\mathrm{PDH}}$. 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.

To generate solitons we decrease the detuning from a large initial value ($\sim 40\text{\hspace{0.17em}}\mathrm{MHz}$) by increasing the frequency ${f}_{\mathrm{PDH}}$ 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 ${\mathrm{sech}}^{2}$ 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, $\alpha $ 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 $\alpha $ from a large initial value and then increasing it again—control of $\alpha $ is achieved through adjustment of ${f}_{\mathrm{PDH}}$. The termination points of the scan are chosen such that at the lower end of the scan a soliton is generated before $\alpha $ 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 $\alpha $ 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 $({f}_{\mathrm{rep}})$ remains locked to ${f}_{\mathrm{PM}}$ over a bandwidth of $\sim \pm 40\text{\hspace{0.17em}}\mathrm{kHz}$. In Fig. 5(a), we show a spectrogram of ${f}_{\mathrm{rep}}$ measured as ${f}_{\mathrm{PM}}$ is swept sinusoidally over $\pm 50\text{\hspace{0.17em}}\mathrm{kHz}$. 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 $\sim \pm 40\text{\hspace{0.17em}}\mathrm{kHz}$ agrees well with an estimate ${\delta}_{\mathrm{PM}}\times {D}_{2}/2\pi \sim 44\text{\hspace{0.17em}}\mathrm{kHz}$ [15] using the approximate measured value ${D}_{2}/2\pi =14\text{\hspace{0.17em}}\mathrm{kHz}$ per mode and modulation depth ${\delta}_{\mathrm{PM}}=\pi $.

To investigate fast control of the repetition rate, we measure ${f}_{\mathrm{rep}}$ as ${f}_{\mathrm{PM}}$ is rapidly switched by $\pm 40\text{\hspace{0.17em}}\mathrm{kHz}$ 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), ${f}_{\mathrm{PM}}$ 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 ${f}_{\mathrm{rep}}$ 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 $+{\beta}_{1}\frac{\partial \psi}{\partial \theta}$ to the right-hand side of Eq. (1), where ${\beta}_{1}=-2({f}_{\mathrm{FSR}}-{f}_{\mathrm{PM}})/\mathrm{\Delta}\nu $ represents a difference between the modulation frequency and the FSR of the resonator near the pump wavelength [15,16]; ${\beta}_{1}$ may be varied in time. In Fig. 5(c) we overlay a simulation of switching conducted for parameters ($\mathrm{\Delta}\nu =1.5\text{\hspace{0.17em}}\mathrm{MHz}$, ${\delta}_{\mathrm{PM}}=0.9\pi $) 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 ${f}_{\mathrm{rep}}$ is limited by the resonator linewidth, and can be modestly improved by increasing ${\delta}_{\mathrm{PM}}$.

## 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., ${\delta}_{\mathrm{PM}}=0.15\pi $. 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 ${\delta}_{\mathrm{PM},\mathrm{eff}}$ 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 ${\delta}_{\mathrm{PM}}$ at a frequency ${f}_{\mathrm{PM}}\sim {f}_{\mathrm{FSR}}/N$, so that the $N$th-order PM sidebands and the carrier address resonator modes with relative mode numbers $-1$, 0, and 1, where ${\delta}_{\mathrm{PM}}$ is chosen in order to achieve effective depth ${\delta}_{\mathrm{PM},\mathrm{eff}}$. 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 $F(1+A\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta )$.

Figure 6 presents a simulated example of this technique. In general, the values ${\delta}_{\mathrm{PM}}$, $N$, and ${\delta}_{\mathrm{PM},\mathrm{eff}}$ 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 ${\delta}_{\mathrm{PM},\mathrm{eff}}=0.15\pi $. To achieve this effective modulation depth we employ real phase modulation depth of ${\delta}_{\mathrm{PM}}\sim 8.3\pi $ at ${f}_{\mathrm{PM}}={f}_{\mathrm{rep}}/21$. 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 $\sim 15.6\text{\hspace{0.17em}}\mathrm{dB}$. In fact, ${\delta}_{\mathrm{PM},\mathrm{eff}}=0.15\pi $ can be recovered with smaller ${\delta}_{\mathrm{PM}}$; 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 $N={f}_{\mathrm{FSR}}/{f}_{\mathrm{PM}}$ 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 ${f}_{\mathrm{rep}}$, 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.

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