All-optical azimuthal trapping of dissipative Kerr multi-solitons for relative noise suppression

The prospect of generating cavity solitons (CSs) was extensively theoretically explored in the 1990s,^1–4 with the focus being on spatially diffractive resonators where the solitons would manifest themselves as self-localized, individually addressable spots on a two-dimensional spatial plane.⁵ Research on spatial CSs targeted the realization of all-optical memories⁶ and optical computing,⁷ leveraging the ability to controllably trap the solitons in specific spatial locations, with experimental demonstrations achieved in semiconductor microcavities.^8–10 Despite such results, the limit in cavity size to house a large number of solitons, together with different spurious effects (cavity defects, free carrier processes, etc.), hindered robust development toward market applications.

In part to overcome issues relating to spatial CSs, the early 2010s saw significant efforts to explore temporal CSs: pulses of light that can persist in passive waveguide resonators. With the object of addressing similar applications as their spatial counterparts, temporal CSs were first experimentally demonstrated in a macroscopic fiber ring resonator,¹¹ and it was shown that they could be trapped in the time domain by appropriately modulating the coherent field driving the resonator.^12–14 While such trapping has enabled a temporal CS-based optical buffer operating at 10 GHz,¹⁵ technical requirements to sustain the solitons make the application impractical as of at present.

Temporal CSs have also been realized in integrated optical microresonators, where they are commonly referred to as dissipative Kerr solitons (DKSs).¹⁶ In that context, the solitons have enabled the creation of optical frequency combs (OFCs), unlocking altogether new applications such as integrated optical clockworks^17,18 and low-noise microwave generation^19,20 while maintaining compatibility with low size, weight, power, and cost (SWAP-C) requirements.^21,22 Therefore, temporal DKS trapping, beyond applications originally envisaged for spatial DKSs, is of use in OFC systems. For example, it can be used to discipline the repetition rate for improved noise performance.²³ Yet, established methods relying on direct modulation of the monochromatic field driving the resonator or the cavity detuning^13,14,23 are not well suited for on-chip microring resonators since the DKS repetition rate is too high to be compatible with electronics that create the trapping potential. Hence, an all-optical technique is preferable, which can be achieved using bichromatic pumping.^24,25 To minimize the repetition rate noise, it is further preferable to operate with large bichromatic modal spacing, hence maximizing the resultant optical frequency division (OFD) factor.

Recently, a novel scheme known as Kerr-induced synchronization (KIS) has been experimentally demonstrated,^18,26 whereby a reference laser whose frequency is tuned close to a comb tooth makes it possible to synchronize the DKS comb to the reference, corresponding to phase trapping in the azimuthal domain.¹⁸ KIS efficiency can be further improved through on-resonance comb-tooth operation^18,27 at a cavity mode close to the phase-matched dispersive wave frequency.^28–30 While the efficacy of this scheme has been demonstrated for single-DKS operation, its metrological impact on multi-DKS states remains completely unstudied. Addressing this shortcoming is particularly important, given that multi-DKS states have been shown to exhibit higher conversion efficiency³¹ and enhanced robustness against thermal effects³² than single-DKS states. Yet, multi-DKS states are usually not desirable since they exhibit a relative jitter between each DKS [Fig. 1(a)], leading to higher microcomb repetition rate noise.³³ Such multi-soliton relative jitter may be mitigated in the soliton-molecule regime, where two or more solitons interact directly with one another,³⁴ potentially in a long-range fashion through their tails.³⁵ Soliton molecules rely on the asymptotically stable balance between attractive and repulsive interactions and have been demonstrated in bulky fiber systems^36,37 and crystalline millimeter cavities.^38,39 However, various sources of noise can still influence the solitons’ characteristics (and hence mutual interactions), leaving residual relative jitter.⁴⁰ In small resonators, molecule formation has relied on modal interaction (defect or avoided mode crossing) for different solitons to interact, resulting in their binding into so-called soliton crystals.⁴¹ In their more robust form, this leads to perfect soliton crystal formation⁴² akin to cnoidal waves.⁴³ However, this sturdiness comes at the cost of a frequency comb whose frequency spacing is large (multiple times that of a single soliton state), making it impractical for repetition rate detection. In addition, the soliton crystal behavior does not prevent the entire molecule from being sensitive to noise, particularly thermo-refractive noise, which is a dominant noise source in small microring resonators.^44–46 To this extent, it is paramount to determine if a coherent temporal DKS trapping compatible with integrated photonics technology can exist, enabling predictable and deterministic low-noise microcomb operation independent of the microring design and multi-DKS state.

In this work, we demonstrate that KIS enables relative azimuthal pinning of the different DKSs in an integrated silicon nitride (Si₃N₄) microresonator through the shared reference intracavity field. Experimentally, we demonstrate that the same repetition rate noise can be measured—which is consistent with the optical division of the two pumps’ noise—regardless of the number of DKSs present in the cavity. Our work highlights the metrological capacity of transforming a multi-DKS into a low-noise coherent state through Kerr-induced synchronization.

First, we aim to experimentally investigate the trapping of a multi-DKS state. To do so, we use a similar SiO₂-embedded Si₃N₄ microring resonator as in Moille et al. (Ref. 18) and described in more detail in Sec. IV. We use a main pump at ω₀/2π ≈ 282.5 THz (1061.9 nm) with about 150 mW on-chip power, which, given the resonator dispersion, generates an octave-spanning frequency comb with dual dispersive waves at comb tooth numbers relative to the main pump (defining μ = 0) at μ = −88 and μ = 109. With this system, different multi-DKS states can be obtained [Fig. 2(a)]. For the current experimental demonstration, we select a two-DKS state, while other states that can be reached will be discussed later. To synchronize the DKS, we use a reference laser with less than 400 μW of on-chip power that is injected close to the DKS comb line (within a KIS locking window) around the dispersive wave at mode μ = −88 (194.6 THz). We verify synchronization through the repetition rate (ω_rep) disciplining¹⁸ and set the reference laser frequency so that the same ω_rep is exhibited both in and out of synchronization for a fair comparison (this occurs at the center of the KIS locking window). Our experiments show that the two-DKS state differs when in- and out-of-synchronization [Fig. 2(a)], in particular in the interference pattern between the two DKSs that leads to modulation of the comb spectral envelope. It is worth pointing out that this phenomenon is reversible from and to the KIS regime, since once the reference pump is turned off, the comb envelope pattern returns to the unsynchronized state, highlighting the reconfigurable aspect of this all-optical trapping.

We numerically solve the mLLE and verify that the resulting comb spectra match the experiment [Fig. 2(a)]. We can then extract the azimuthal profile of the two-DKS state both in- and out-of-synchronization [Fig. 2(b)]. In the single pump case (i.e., unsynchronized), the background modulation from the dispersive wave is small and insufficient to trap the two DKSs into a molecule.^25,50,51 In contrast, in the KIS regime, the background modulation is large enough to lock the two DKSs at grid points that are defined by the modulation. This modulation arises from the beating between the main and reference lasers that are separated by μ = −88 mode numbers. This behavior is reflected in the experiment by the change in the interference pattern observed in the comb envelope [Fig. 2(a)].

Hence, from the study of λ_n, we can conclude if the perturbation is amplified (Re(λ_n) > 0), persists (Re(λ_n) = 0), or is damped (Re(λ_n) < 0). Although the linearized operator presents as many eigenvalues as the number of modes used to model the system [Fig. 3], we are interested in one in particular—the so-called position-shifting eigenvalue (λ_ps), also called the neutral mode or Goldstone mode^7,52—which is responsible for group velocity noise. Since we study a multi-DKS state, here for the sake of simplicity a two-DKS state, the eigenvector (or eigenmode) projection results in studying the projected basis of the two-DKS system. This analysis can be extended to multi-soliton systems, where the eigenvalue count increases with the number of solitons. The even eigenmode with equal weights on each DKS site represents the global motion, while odd eigenmodes that function between specific DKS pairs (zero elsewhere) describe relative motions. In a two-DKS system, the only two eigenmode possible are even and odd; they directly correspond to the global and relative motion of the DKS pair,⁵⁵ informing us about the global repetition rate noise and relative DKS jitter, respectively. However, larger DKS number systems may require linear combinations of eigenmodes to determine the relative motions between DKS pairs.

In the single pump case [Fig. 3(a)], both λ_ps values exist around zero, highlighting that any intracavity noise will persist and that the two-DKSs are weakly bound so that their group velocity noise is independent and uncorrelated. In striking contrast, in KIS [Fig. 3(b)], the two λ_ps values have migrated toward −1, corresponding to the maximum damping of the intracavity noise at a photon-lifetime rate τ_phot = 1/κ that is consistent with the initial normalization in Eq. (1). Such maximization of the damping occurs when the reference laser is at the comb tooth frequency of the unsynchronized DKS, corresponding to the center of the KIS spectral window defined as $ΔΩkis=4μsωrepκEdks−1κextPrefPμs$ with κ and κ_ext being the total and coupling loss rate, respectively; P_ref and P_μs being the intracavity reference and comb tooth powers at synchronization mode μ_s, respectively; and E_dks is the total DKS energy. For different reference frequencies within the KIS window, λ_ps will exhibit continuous values from −1 (center of KIS) to 0 (edge of KIS).⁴⁶ We note that hybridization between λ_ps and the other eigenvalues whose real parts are originally at −2 and migrate to −1 leads to the splitting of the imaginary part of the eigenvalue in KIS, which does not impact the noise damping properties. Since both the global and relative position shifting eigenvalues are damped at the same rate, the relative position in time of the two-DKS system will exhibit correlated noise that is only limited by the photon lifetime, similar to the single-DKS case.⁴⁶ Therefore, using a single- or multi-DKS while synchronized to a reference pump results in the same outcome from the linear stability analysis, and hence the noise properties of the measured repetition rate are expected to be the same. This suggests that under KIS, the multi-DKS state becomes compatible with low noise metrology applications.

In this final section, we proceed to measure the power spectral density of the repetition rate frequency noise S_rep(f) for synchronized single- and multi-DKS states. Using the same exact device as the one presented in Fig. 2, we proceed to generate different DKS states. In order to do so, we slightly tune both the main pump power (within 10 mW) and the main pump detuning to adiabatically land on a multi-DKS state, from which we determine the number and spacing of solitons by fitting to the observed spectral envelope. The repetition rate is measured using an electro-optic comb (EOcomb) apparatus described in more detail in Sec. IV.

In the case of the single-DKS [Fig. 4(a)], where the comb spectrum has the signature smooth envelope, the obtained S_rep(f) exhibits the same noise as the main and reference pump noise frequency divided by the optical frequency division factor $OFD=μs2=882=7744$⁠, as expected from previous study,^18,26 since the characteristic frequency noise of the pumps is well-reproduced for frequencies below 5 kHz after accounting for the OFD. The EOcomb apparatus, which lets us frequency translate the large DKS ω_rep to a detectable bandwidth, limits our noise floor to about 10² Hz²/Hz at a Fourier frequency of 2 kHz and 10⁴ Hz^2/Hz at a Fourier frequency of 100 kHz. After generating different multi-DKS states, either a two-DKS microcomb [Fig. 4(b)] or a three-DKS one [Fig. 4(c)], the obtained S_rep(f) presents the same characteristics as the single-DKS state, with the same optically frequency divided noise from the pumps, despite the multiple cavity solitons traveling within the resonator. Such noise measurements validate the metrological capacity of multi-DKS states in the KIS regime, demonstrating that the azimuthal trapping of the DKSs by the reference field is strong enough to provide a single repetition rate noise, consistent with the theoretical linear stability analysis described earlier.

To conclude, we have demonstrated that Kerr-induced synchronization enables the trapping of general multi-DKS states, which leads to an on-demand low-noise coherent state, where all the pulses are trapped by the background modulation at a period of μ_s defined by the mode at which the reference synchronizes the DKSs. Experimentally, the spectrum of a two-DKS state provides sufficient discrepancy in and out of synchronization to highlight the azimuthal trapping of the different DKSs to the μ_s reference modulation. This enables a direct comparison with pump phase-modulation trapping and provides an experimental demonstration of previous theoretical work where the background modulation period is similar to the DKS pulse width.²⁵ We analyze the multi-DKS state and its Kerr-induced synchronization through a theoretical linear stability analysis of the mLLE, observing similar behavior as in the single-DKS state, which per definition is a signature of the molecular behavior. In the linearized system, the eigenvalue of interest λ_ps has a real part that moves from zero (no intra-cavity noise damping out of synchronization) to −1 (damping at the photon decay rate at the center of the KIS window). In the multi-DKS case, the real part of both the global multi-DKS eigenvalue and the eigenvalue corresponding to the relative jitter between the DKSs is present at −1, demonstrating the relative noise suppression between each DKS. Finally, we show experimentally that the repetition rate frequency noise ends up being identical in a single-DKS or in a multi-DKS state, which is solely determined by the pump frequency noise and the optical frequency division factor $μs2$⁠, as usual in KIS. Our work highlights that KIS brings the metrological capacity to multi-DKS states through azimuthal trapping of the DKS and on-demand predictive low-noise operation. Since the integrated frequency combs rely on low SWAP-C, the overall comb power remains largely limited but can be mitigated by increasing the number of pulses in the cavity. Usually, this leads to a trade-off between either the increase in the repetition rate noise with the multi-DKS state order or an increase in the comb tooth spacing to harmonics of ω_rep in the particular case of a multi-DKS state often referred to as a soliton crystal.⁴¹ Although control of the defects that can produce such soliton crystal states has been demonstrated with laser injection,⁵⁶ our work instead shows that KIS enables coherent azimuthal trapping of all the DKSs present in the cavity. We, therefore, demonstrate that one can leverage the benefits of multi-DKS states without being metrologically limited or being forced to work with harmonics of the repetition rate. In addition, since the arbitrary multi-state DKS orders exhibit the same noise performance as the single-DKS state, our work could be extended to larger resonator circumferences (i.e., lower ω_rep combs) where many more pulses could be fit inside the resonator, significantly increasing the otherwise poor conversion efficiency of the microcomb operation without sacrificing noise performance.

The photonic chips were fabricated following the process presented in Moille et al. (Ref. 18) in a commercially available foundry. We use a silicon nitride (Si₃N₄) microring resonator embedded in SiO₂ with an outer ring radius of R = 23 μm and ring width RW = 850 nm. A bus waveguide (width W_wg = 460 nm) wrapped around the ring in a pulley-like fashion⁵⁷ with coupling length L_c = 17 μm and gap G = 600 nm enables critical coupling at both the main pump frequency ω₀/2π ≈ 282.5 THz (1061.9 nm) and the reference pump frequency ω_ref/2π ≈ 194.6 THz (1549.6 nm). The reference is selected to be at the low frequency dispersive wave (DW) at μ_s = −88 comb teeth away from the main pump. To access the soliton state, we use about 150 mW of on-chip main pump power while temperature stabilizing the resonator through a cooler-pump^58,59 with about 250 mW of on-chip power in a counter-propagating and cross-polarized (transverse magnetic) mode with respect to the main pump (to minimize nonlinear interaction).

The simulation result of the Lugiato–Lefever equation (LLE) presented in Fig. 2 used the same geometric parameter of the microring resonator as presented above, with the dispersion parameter $D$ in the LLE obtained via accurate finite element method (FEM) modeling.¹⁸ The simulated intrinsic and quality factors are set to Q_i = 1 × 10⁶ and Q_c = 1 × 10⁶, respectively. The detuning of the pump is set to be (ω(μ = 0) − ω₀)/2π = 1.25 × 10⁹ GHz, with the main pump frequency ω₀/2π ≈ 282.5 THz, while the round trip time of the DKS is t_r = 2π/ω_rep ≈ 1 ps, yielding a normalized parameter of α = 4.4. The main pump is set at an in-waveguide power of P_in = 160 mW, and the effective nonlinearity γ = 3.2 W⁻¹ ⋅m⁻¹ from FEM calculations, resulting in a normalized driving force of F₀ = 2.78 and a normalized time of τ = 1.7 × 10⁹t.

In order to measure the close to 1 THz repetition rate of the microcomb, we spectrally translate two adjacent comb teeth close to one another to measure their beats, similar to Moille et al. (Ref. 18) and Stone and Papp (Ref. 44). To perform such spectral translation, we use an electro-optic (EO) comb consisting of two phase modulators driven at ω_EO = (17.839 × 2π) GHz, enabling two adjacent microcomb teeth at ≈ 271.5 THz and ≈270.5 THz to be spectrally translated by N_EO = 56 EOcomb lines. Using a bandpass grating optical spectral filter with a 0.1  nm Gaussian shape full-width half maximum passband, one can measure the beat note at ω_beat/2π = 22.75 MHz (±10 kHz), letting us determine the DKS repetition rate ω_rep /2π = (N_EOω_EO + ω_beat)/2π = 999.006 75 GHz ± 10 kHz. ω_beat is detected by a 50 MHz bandwidth avalanche photodiode and processed with a phase-noise analyzer to study the repetition rate frequency noise.

S.C.O., K.S., and G.M. acknowledge the partial funding support from the Space Vehicles Directorate of the Air Force Research Laboratory and the NIST-on-a-chip program of the National Institute of Standards and Technology. P.S. and C.M. acknowledge the support from the National Science Foundation (Grant No. ECCS-1807272), the Air Force Office of Scientific Research (Grant No. FA9550-20-1-0357), and a collaborative agreement with the National Center for Manufacturing Sciences (Grant No. 2022138-142232) as a sub-award from the US Department of Defense (Cooperative Agreement No. HQ0034-20-2-0007). M.E. acknowledges the financial support from the Marsden Fund of the Royal Society of New Zealand (Grant No. 23-UOA-071). G.M. thanks T.B.M.

G.M., C.M., and K.S. have submitted a provisional patent application based on aspects of the work presented in this paper.

P.S. and S.-C.O. contributed equally to this work.

S.-C.O. and G.M. performed the experimental work. P.S. developed the theory and performed the simulations. J.S. helped with the metrology experiment and understanding. C.M., M.E., and K.S. contributed to the understanding of the physical phenomenon. G.M. supervised the project. G.M. wrote the article with the help of P.S., S.-C.O., J.S., C.M., M.E., and K.S. All the authors contributed to and discussed the content of this article.

Pradyoth Shandilya: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Shao-Chien Ou: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jordan Stone: Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Curtis Menyuk: Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Miro Erkintalo: Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Kartik Srinivasan: Funding acquisition (lead); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Grégory Moille: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.