## Introduction

Dissipative Kerr solitons (DKSs) are self-organized wave packets in photonic-chip-based microcavities1,2,3,4,5,6,7, where the chromatic dispersion is balanced by Kerr nonlinear shift, and the cavity dissipation is offset by Kerr parametric gain. DKSs produce ultrashort pulse trains and equally spaced comb lines8,9,10,11,12,13,14,15, and have attracted significant interest for applications, including optical communications16,17, spectroscopy18,19, ultrafast ranging20,21,22,23, low-noise microwave generation24, frequency synthesis25/division26, and quantum key distribution (QKD)27, etc. Up to date, various soliton states have been theoretically and experimentally demonstrated, such as Stokes solitons28, Brillouin-Kerr solitons29, breather solitons30,31,32,33,34, laser cavity solitons35 as well as soliton crystals (SC)36,37,38,39. Particularly, perfect soliton crystals (PSC)38,39, where temporal solitons are equally spaced in the microresonator, are of great interest. While preserving many characteristics from the single-soliton state, i.e., smooth spectral envelope and highly ordered temporal soliton distribution, they offer higher comb power per line and flexible soliton repetition rate, which are critical for microcomb applications. Previously, the generation and stabilization of SC rely on avoided mode crossing (AMX) induced background potential field36,38 or delicate control of the pump condition39. However, the above schemes both have complex dynamics processes and require extreme fabrication precision to reproduce or demand material-specific properties, which cannot offer a wide-range, universal switching for SC states.

In this article, we propose and demonstrate on-demand synthesis and manipulation of a library of PSCs through a controlled potential field. The N-period modulated background potential field is constructed by pumping the cavity with an additional control light at the frequency which is N-mode spaced from the primary pump light. As the period and intensity of the potential field can be conveniently tuned by the control light, our method offers full reconfigurable capability to the PSCs generation. We demonstrate synthesized SCs with a given soliton number N from 1 to 32 on demand, equivalent to a reconfigurable soliton generator with a repetition rate ranging from ~49 GHz to ~1.57  THz. Furthermore, the group velocity mismatch between the SC and potential field40,41 are investigated both theoretically and experimentally, which induces forced oscillation in the power and intracavity position of the synthesized SCs42. Finally, we show that the repetition rate of the synthesized SC changes linearly with the relative velocity between the potential field and the SC. The mechanism shown in this work can be extended to other platforms to greatly enhance the flexibility of on-chip SC for practical applications.

## Results

### Synthesized SCs

In a conventional monochromatic pumping scheme, the cavity soliton is initialized from a flat CW background field (Fig. 1a). The uniformly distributed gain makes the positions and quantity of the generated solitons unpredictable43, resulting in irregular multiple soliton pulse train and unsmooth spectral envelope (Fig. 1b). In our approach, a second CW laser is introduced as a control light to provide a periodically modulated CW background from the two-beam beating and control the SC generation process. The periodic number of potential field N is equal to the mode number spacing between the primary pump light and control light. Figure 2a shows the experimental setup for SC synthesis. The pump laser and the control laser are amplified and coupled to a single high-Q microcavity resonance from two counter-propagating directions via two fiber circulators. The role of the control laser is twofold. On the one hand, a small portion of the control light is back-scattered to the opposite direction and beats with the co-propagating clockwise pump light, constructing a periodically modulated background field. On the other hand, the main part of the control light propagating in the counter-clockwise direction is adopted to stabilize the cavity temperature when the pump light transitions into the red-detuned regime40,44. The add-drop type microcavity is fabricated on a CMOS-compatible high-index doped silica glass platform45,46. The Q factor and free spectral range (FSR) of the microresonator are ~2.63 million and ~48.9 GHz, respectively. To couple the light onto the chip, a standard 250 μm pitch fiber array glued to the on-chip bus waveguide is used47,48. The microcavity is butterfly-packaged with a thermo-electric cooler (TEC) which is controlled by an external controller12,37.

In the experiment, SC can be generated reliably by slowly decreasing the operating temperature through the TEC controller to sweep cavity resonance from blue-detuned to the red-detuned regime. The temporal and spectral profiles are measured using an autocorrelator and an optical spectrum analyzer (OSA), respectively. By switching the mode number spacing between the control and pump light, a library of 1–32 SC with repetition rates from 1× to 32 × FSR of the microcavity, are deterministically realized with high repeatability (see section 3.1 of SI for more experimental evidence). As shown in Fig. 2c, the measured spectra are all smooth with a sech2 envelope and the comb line power is enhanced by N2 (Fig. 3), thanks to the ordered distribution of the synthesized SC38. Moreover, the spatial overlap between the CW pump light and the soliton pulses becomes larger with the increasing soliton number, providing great potential to boost the soliton conversion efficiency. The demonstrated reconfigurable and high-power SC in a monolithic microcavity provides an ideal toolroom for tunable microwave/millimeter-wave photonics49,50.

We then focus on understanding the formation dynamics of the SC by taking 10-SC as an example. The wavelength of the pump and control light are 1560.2 and 1564.1986 nm (~10 FSR away from pump), respectively. The evolution of the power is recorded while the temperature of the microcavity is gradually decreased to scan the pump-cavity resonances from longer wavelength to shorter wavelength side (Fig. 4a). Three typical stages are clearly recognized: (i) CW, (ii) modulated Turing pattern (TP) and (iii) SC. These stages are characterized by simultaneously measuring the intracavity optical field in temporal (Fig. 4c) and spectral (Fig. 4d) domain. The SC formation is recognized by the characteristic “step” power trace, ordered temporal pulse distribution, as well as the smooth sech2 spectral envelope. The soliton formation is able to bypass the chaotic regime, and is directly evolved from the modulated TP stage. The modulation field causes the multiple Turing pulses turn into N equally spaced solitons, i.e., N-SC. The evolution process of sythesized SC is well reproduced by numerical simulation of dichromatic-pump Lugiato-Lefever equation (DLLE, see SI for details). When the pump detuning is gradually increased, the three stages discussed above are sequentially distinguished, as shown in the simulated power trace in the temporal evolution trace in Fig. 4b. The simulated temporal waveforms and spectra of the three representative stages are shown in Fig. 4e, f, which agree well with the experiment.

### Forced oscillation of SCs

In the course of generating synthesized SCs, a novel type of soliton oscillation is observed when the modulated CW background field has a relative velocity to SC. This relative motion of the modulated background field creates an moving offset (phase difference) between the potential field and the SC, which provides a periodic force on the soliton51. Such force is derived as (See SI for details):

$$F={F}_{0}\cos (\eta {\phi }_{c}-{{\Delta }}\widetilde{\omega }\tau )$$
(1)

Here F0 is the amplitude of F, which is proportional to the amplitude of control light. η = μc − μp is the mode number spacing between the pump light and control light. ϕc is the angular position of soliton center, τ is the normalized slow time. $${{\Delta }}\widetilde{\omega }=2({\omega }_{c}-{\omega }_{p}-\eta {D}_{1})/\kappa$$ is the normalized mismatch angular frequency. As a result, the SCs oscillate around their equilibrium positions periodically (Fig. 5a). To gain more insights into the intracavity dynamics, we first define the soliton momentum as (see SI for details): P = 2βμμψμ2, here 2βμ and ψμ2 represent the normalized relative group velocity and photon number of mode μ, respectively. Based on the momentum analysis, the dynamical equation for SCs can be derived:

$$\frac{{{\rm{d}}}^{2}{\phi }_{c}}{{\rm{d}}{\tau }^{2}}+2\frac{{\rm{d}}{\phi }_{c}}{{\rm{d}}\tau }-\frac{F}{M}=0$$
(2)

The above equation is a typical forced oscillation equation. The effective kinetic mass of soliton is derived as M = E (given by momentum analysis, see SI for details), where E is the optical pulse energy. Based on Eq. (2), we calculate the evolution trace of the SC angular position. As shown in Fig. 5b, where three adjacent lattice sites of a 10-SC oscillate in phase. The amplitude of the oscillation is calculated as ~0.001 rad (~4% of the soliton width).

The forced oscillation also induces the periodical variation of CW, and causes the SC power to oscillate. The measured normalized soliton power trace shows the periodic modulation (middle panel of Fig. 5d), with a modulation depth of ~10%. The modulation frequency is 69 MHz, which is verified by electrical measurement, see right panel of Fig. 5d, where the second harmonic of 138 MHz is also detected. It should be noted that this power oscillation is fundamentally distinct from the power breathing in the breather soliton, as its optical spectrum maintains the stable sech2 envelope instead of the typical triangle envelope of breather31,32. These experimental observations are further validated by DLLE-based numerical simulations (Fig. 5e). The forced oscillation corresponds to sidebands generation around the comb line in the forms of cross-phase modulation (XPM) and subsequent FWM comb lines caused by XPM and SC comb, the experimental and numerical characterization of XPM comb are included in SI. The forced oscillation of the SC may be eliminated once the control beam is switched off. However, the process of switching off the control beam while maintaining the soliton states is complicated by the thermal effect in the microresonator. Further investigation to reduce the thermal effect is required to eliminate the control beam after the formation of SC states, and this is possible through optimizing the fabrication process of microresonators52,53.

### Soliton oscillation frequency and repetition rate tuning

Furthermore, due to the photothermal effect, the oscillation frequency and the repetition rate of synthesized SC can be tuned by adjusting the frequency of the control light. As the single-soliton state has an electronic-detectable repetition rate, we study the tuning process of a single-soliton state as an example. Experimentally, the oscillation frequency and repetition rate are directly measured using two high-speed ESAs to monitor the tuning process. The frequency of the control laser is tuned finely by gradually changing the driven current of ECDL. For each driven current of the control light, we record both the oscillation frequency and the repetition rate of the single-soliton comb, as shown in Fig. 6a, b, respectively.

In our measurement, the oscillation frequency of the soliton microcomb is tuned over 20.6 MHz (Fig. 6a), and the repetition rate changes about 60 kHz (Fig. 6b). The soliton repetition rate is linearly changed along with the oscillation frequency (Fig. 6c). It provides a way to finely tune soliton repetition rate and oscillation frequency.

## Discussion

Here, we explore the SC synthesis on demand and the forced oscillation process of the SC. The artificial background potential field is potentially reconfigurable in real-time, provides a platform for studying the cavity Kerr soliton dynamics. The demonstrated reconfigurable SC microcombs in a monolithic integrated microresontors with the repetition rate covering broad microwave bands (including V band, W band, and G band) and THz band, has great potential in applications such as satellite communications54, photonic radar55, radio-frequency filter56,57, and THz technologies. The proposed approach is easy to implement, and therefore applicable to most materials and different cavity configurations.

## Methods

### Controllable artificial background field

A pre-designed background is proposed to change the soliton generation dynamics and regulate the soliton distribution and quantity. Operationally, this can be realized by adding an additional control light that is N × FSR from the primary pump light. This leads to the formation of controllable artificial background field expressed as: $$\widetilde{{E}_{b}}={E}_{p}{e}^{-i{\omega }_{p}t}{e}^{i{\mu }_{p}{\phi }_{l}}+{E}_{c}{e}^{-i{\omega }_{c}t}{e}^{i{\mu }_{c}{\phi }_{l}}$$, where Ep,c and ωp,c represent the electric field amplitude and angular frequency of the pump (control) laser, respectively. ϕl is the polar angle in lab coordinate. In the rotating angular coordinate ϕ = ϕl − D1t where D1 = 2π × FSR, the background field can be rewritten as $$\widetilde{{E}_{b}}=({E}_{p}+{E}_{c}{e}^{-i{{\Delta }}\omega t}{e}^{i\eta \phi }){e}^{-i{\omega }_{p}t}{e}^{i{\mu }_{p}{\phi }_{l}}$$, which is N-periodically modulated (N = η = μc − μp). Δω = ωc − ωp − ηD1 is the frequency mismatch. Δω/η is the traveling speed of the background field, which could be adjusted by tuning the frequency (detuning) of the control light (detailed derivation see Supplementary information). The modulated background field redistributes the refractive index of the cavity in a periodic fashion (Fig. 1c), and regularizes the soliton formation probability around the cavity. Figure 1d shows a synthesized SC where the solitons are evenly distributed with a smooth spectral envelope. Note that if there is a relative motion between the SC and the field, i.e., Δω is detuned from zero, the SC would periodically oscillate in the temporal domain, which corresponds to modulation sidebands generation in the frequency domain as shown in the inset of Fig. 1d.