Sideband injection locking in microresonator frequency combs

Continuous-wave (CW) coherently driven Kerr-nonlinear resonators can create temporally structured waveforms that circulate stably without changing their temporal or spectral intensity profile. The out-coupled optical signal is periodic with the resonator roundtrip time T_rep and corresponds to an optical frequency comb,^1–5 i.e., a large set of laser frequencies spaced by the repetition rate $frep=Trep−1$⁠. One important class of such stable waveforms are CW-driven dissipative Kerr-solitons (DKSs), which have been observed in fiber-loops,⁶ traveling- and standing-wave microresonators,^7,8 and free-space cavities.⁹ In microresonators, these soliton microcombs¹⁰ provide access to low-noise frequency combs with ultra-high repetition rates up to THz frequencies, enabling novel applications in diverse fields, including optical communication,^11,12 ranging,^13–15 astronomy,^16,17 spectroscopy,¹⁸ microwave photonics,^19,20 and all-optical convolutional neural networks.²¹ 

In a CW-driven microresonator, the comb’s frequency components are defined by f_μ = f_p + μf_rep, where f_p denotes the frequency of the central comb line and μ is the index of the comb line with respect to the central line (μ is also used to index the resonances supporting the respective comb lines). For many applications,^4,5 it is essential to control both degrees of freedom in the generated frequency comb spectra, i.e., the repetition rate f_rep and the central frequency f_p (which together define the comb’s carrier-envelope offset frequency). Conveniently, for Kerr-resonator-based combs, f_p is defined by the pump laser frequency f_p = ω_p/(2π). However, the repetition rate f_rep depends on the resonator and is subject to fundamental quantum mechanical as well as environmental fluctuations.

A particularly attractive and all-optical approach to controlling f_rep is the injection of a secondary CW laser of frequency ω′ into the resonator, demonstrated numerically²² and experimentally.²³ If ω′ is sufficiently close to one of the free-running comb lines (sidebands) f_μ ≈ ω′/(2π), i.e., within locking range, the comb will lock onto the secondary laser so that f_μ → ω′/(2π). The repetition rate is then f_rep = (ω_p − ω′)/(2πμ′), with μ′ denoting the index of the closest resonance to which the secondary laser couples; cf. Fig. 1(a). This frequency division²⁴ of the frequency interval defined by the two CW lasers (as well as their relative frequency noise) by the integer μ′ can give rise to a low-noise repetition rate f_rep. In previous work, sideband injection locking has been leveraged across a large range of photonic systems, including for parametric seeding,^25,26 dichromatic pumping,²⁷ optical trapping,^22,28,29 synchronization of solitonic and non-solitonic combs,^30,31 soliton crystals,²³ soliton time crystals,³² multi-color solitons,³³ and optical clockworks by injection into a DKS dispersive wave.³⁴ Related dynamics also govern the self-synchronization of comb states,^35,36 the binding between solitons,³⁷ modified soliton dynamics in the presence of Raman effect³⁸ and avoided mode-crossings,³⁹ as well as the respective interplay between co-propagating⁴⁰ and counter-propagating solitons^41–43 and multi-soliton state-switching.⁴⁴ Moreover, sideband injection locking is related to modulated and pulsed driving for broadband stabilized combs,^17,45,46 as well as spectral purification and nonlinear filtering of microwave signals^47,48 via DKSs. Despite the significance of sideband injection locking, a broadband characterization and quantitative understanding of its dependence on the injecting laser are lacking, making the design and implementation of such systems challenging.

In this work, we study the dynamics of sideband injection locking with DKS combs. Our approach leverages high-resolution coherent spectroscopy of the microresonator under DKS operation, enabling precise mapping of locking dynamics across a large set of comb modes, including both the central region and the wing of the comb. We derive the sideband injection locking range’s dependence on experimentally accessible parameters and find excellent agreement with the experimental observation and numerical simulation. Specifically, this includes the square dependence on the mode number, the square-root dependence on injection laser and DKS spectral power, as well as the associated spectral shifts.

In addition, we experimentally demonstrate the optical frequency division and repetition rate phase-noise reduction in a DKS state to three orders of magnitude below the noise of a free-running system.

To first explore the sideband injection locking dynamics experimentally, we generate a single DKS state in a silicon nitride ring-microresonator. In the fundamental TE modes, the resonator is characterized by a quality factor of Q ≈ 2 × 10⁶ [linewidth κ/(2π) ≈ 100 MHz] and a free-spectral range (FSR) of D₁/(2π) = 300 GHz and exhibits an anomalous group velocity dispersion D₂/(2π) = 9.7 MHz so that the resonance frequencies are well-described by $ωμ=ω0+μD1+μ2D22$ (1.6 × 0.8 μm² cross section, 76 μm radius). To achieve a deterministic single soliton initiation, the microresonator’s inner perimeter is weakly corrugated.^49,50 The resonator is critically coupled and driven by a CW pump laser (∼300 kHz linewidth) with an on-chip power of 200 mW at 1557 nm [pump frequency ω_p/(2π) = 192.5 THz].⁷ The generated DKS has a 3 dB bandwidth of ∼5.2 THz [cf. Fig. 2(a)] corresponding to a bandwidth limited pulse of ∼60 fs duration. The soliton spectrum closely follows a sech² envelope and is free of dispersive waves or avoided mode crossings. The spectral center of the soliton does not coincide with the pump laser but is slightly shifted toward longer wavelengths due to the Raman self-frequency shift.^51,52

A secondary CW laser (∼300 kHz linewidth), tunable both in power and in frequency (and not phase-locked in any way to the first CW laser), is then combined with the pump laser upstream of the microresonator and scanned across the μ′th sideband of the soliton microcomb, as illustrated in Fig. 2(a). The spectrogram of the repetition rate signal recorded during this process is shown in Fig. 2(b), for μ′ = −13, and exhibits the canonical signature of locking oscillators⁵³ (see Sec. II of the supplementary material for details on the measurement of f_rep). Specifically, the soliton repetition rate f_rep is observed to depend linearly on the auxiliary laser frequency ω′ over a locking range δ_lock following $frep=12πωp−ω′μ′$⁠. Within δ_lock, the soliton comb latches onto the auxiliary laser such that the frequency of the comb line with index μ′ is equal to the secondary laser frequency. The locking behavior is found to be symmetric with respect to the scanning direction, and no hysteresis is observed. Figure 2(c) shows the optical spectra of two sideband injection-locked DKS states, with the secondary laser positioned close to the respective boundaries of the locking range. A marked shift of the spectrum of 860 GHz is visible when going from one state to the other. As we discuss below and in Sec. III B of the supplementary material, the spectral shift in the presence of non-zero group velocity dispersion modifies the soliton’s group velocity and provides a mechanism for the DKS to adapt to the repetition rate imposed by the driving lasers.

Having identified characteristic features of sideband injection locking in our system, we systematically study the injection locking range and its dependence on the mode number μ′ to which the secondary laser is coupled. To this end, a frequency comb calibrated scan⁵⁴ of the secondary laser’s frequency ω′ across many DKS lines is performed. The power transmitted through the resonator coupling waveguide is simultaneously recorded. It contains the ω′-dependent transmission of the secondary laser as well as the laser’s heterodyne mixing signal with the DKS comb, which permits retrieving the locking range δ_lock.

Figure 3(a) shows an example of the recorded transmission signal, where the scanning laser’s frequency ω′ is in the vicinity of the comb line with index μ′ = −3. When the laser frequency ω′ is sufficiently close to the DKS comb line, the heterodyne oscillations (blue trace) can be sampled; when ω′ is within the locking range δ_lock, the heterodyne oscillations vanish, and a linear slope is visible, indicating the changing phase between the comb line and the secondary laser across the injection locking range. In addition to the heterodyne signal between the comb line and the laser, a characteristic resonance feature, the so-called C-resonance,^55,56 representing (approximately) the resonance frequency ω_μ is observed.

The set of equivalent traces for all comb lines μ′ in the range of the secondary (scanning) laser is presented in Fig. 3(b) as a horizontal stack. For plotting these segments on a joint vertical axis, ω_p + μ′D₁ has been subtracted from ω′. In this representation, the parabolic curve [blue line in Fig. 3(b)] connecting the C-resonances signifies the anomalous dispersion of the resonator modes ω_μ. In contrast, the equidistant comb lines form a straight feature (gray highlight), of which a magnified view is presented in Fig. 3(c). Due to the Raman self-frequency shift, the free-running repetition rate of the DKS comb $frep0$ is smaller than the cavity’s FSR D₁/(2π), resulting in the negative tilt of the line. Here, to obtain a horizontal arrangement of the features, $ωp+μ′2πfrep0$ has been subtracted from ω′. The locking range δ_lock corresponds to the vertical extent of the characteristic locking feature shown in Fig. 3(c). Its value is plotted as a function of the mode number in Fig. 3(d), revealing a strong mode number dependence of the locking range with local maxima (almost) symmetrically on either side of the central mode. The asymmetry in the locking range with respect to μ′ = 0 (with a larger locking range observed for negative values of μ′) coincides with the Raman self-frequency shift of the soliton spectrum (a higher spectral intensity for negative μ). Next, we keep μ′ fixed and measure the dependence of δ_lock on the power of the injecting laser P′. As shown in Fig. 3(e), we observe an almost perfect square-root scaling $δlock∝P′$⁠, revealing the proportionality of the locking range to the strength of the injected field.

To verify the validity of Eqs. (1) and (2), we perform a numerical simulation based on the Lugiato–Lefever equation (see Sec. IV of the supplementary material). We find excellent agreement between the analytic model and the simulated locking range. We note that Eqs. (1) and (2) are derived in the limit of low injection power, which we assume is the most relevant case. For a large injection power, the spectrum may shift substantially and consequently affect the values of P_μ. Interestingly, while this effect leads to an asymmetric locking range, the extent of the locking range is only weakly affected as long as the spectrum can locally be approximated by a linear function across a spectral width comparable to the shift. Injection into a sharp spectral feature (dispersive wave) is studied by Moille et al.³⁴ 

The values of P_μ do not generally follow a simple analytic expression and can be influenced by the Raman effect and higher-order dispersion. While our derivation accounts for the values of P_μ [e.g., for the Raman effect, a_μ and P_μ are increased (reduced) for μ below (above) μ = 0], it does not include a physical description for Raman- or higher-order dispersion effects; these effects may further modify the locking range.

Taking into account the spectral envelope of the DKS pulse as well as the power of the injecting laser (which is not perfectly constant over its scan bandwidth), we fit the scaling $δlock∝μ′2P′Pμ′$ to the measured locking range in Fig. 3(d), where we assume P_μ′ to follow an offset (Raman-shifted) sech²-function. The fit and the measured data are in excellent agreement, supporting our analysis and suggesting that the Raman shift does not significantly change the scaling behavior. Note that the effect of the last factor in Eq. (1) is marginal, and the asymmetry in the locking range is due to the impact of the Raman effect on P_μ. It is worth emphasizing that our analysis did not assume the intracavity waveform to be a DKS state and we expect that the analytic approach can, in principle, also be applied to other stable waveforms, including those in normal dispersion combs.^31,58 Indeed, as numerically shown in Sec. IV of the supplementary material, sideband-injection locking is also possible for normal dispersion combs. Here, in contrast to a DKS, sideband laser injection is found to have a strong impact on the spectral shape (not only the spectral shift). Therefore, although the underlying mechanism is the same as in DKS combs, Eqs. (1) and (2) do not generally apply (in the derivation, it is assumed that the spectrum does not change substantially).

Finally, as an example application of sideband injection locking, we demonstrate an optical frequency division, similar to previous work,³⁴ and measure the noise reduction in f_rep [Fig. 4(a)]. With a growing separation between the two driving lasers (increasing μ′), the phase noise is lowered by a factor of μ′², resulting in a phase noise reduction of more than three orders of magnitude (with respect to the free-running case) when injecting the secondary laser into the mode with index μ′ = 42 (limited by the tuning range of the secondary laser), and this without any form of stabilization of either the pump or the secondary laser. Figures 4(b) and 4(c) show the comparison of the repetition rate beatnote of the free-running and injection-locked cases.

In conclusion, we have presented an experimental and analytic study of sideband injection locking in DKS microcombs. The presented results reveal the dependence of the locking range on the intracavity spectrum and on the injecting secondary laser, with excellent agreement between experiment and theory. While our experiments focus on the important class of DKS states, we emphasize that the theoretical framework from which we derive the presented scaling laws is not restricted to DKSs and may potentially be transferred to other stable waveforms. Our results provide a solid basis for the design of sideband injection-locked and parametrically generated Kerr-frequency combs and may, in the future, enable new approaches to low-noise microwave generation, compact optical clocks with simplified locking schemes, and, more generally, stabilized low-noise frequency comb sources from Kerr-nonlinear resonators.

See the supplementary material for more details on the experimental setup, the derivation of the locking range, the presentation of the numerical simulations, and considerations on thermal effects.

This project has received funding from the European Research Council (ERC) under the EU’s Horizon 2020 research and innovation program (Grant Agreement No. 853564) and through the Helmholtz Young Investigators Group VH-NG-1404; this work was supported through the Maxwell computational resources operated at DESY.

The authors have no conflicts to disclose.

Thibault Wildi: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Resources (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Alexander Ulanov: Methodology (equal); Resources (equal); Writing – original draft (supporting). Nicolas Englebert: Methodology (equal); Resources (equal); Writing – original draft (supporting). Thibault Voumard: Methodology (equal); Resources (equal). Tobias Herr: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.