Sideband Injection Locking in Microresonator Frequency Combs

Frequency combs from continuous-wave-driven Kerr-nonlinear microresonators have evolved into a key photonic technology with applications from optical communication to precision spectroscopy. Essential to many of these applications is the control of the comb's defining parameters, i.e., carrier-envelope offset frequency and repetition rate. An elegant and all-optical approach to controlling both degrees of freedom is the suitable injection of a secondary continuous-wave laser into the resonator onto which one of the comb lines locks. Here, we study experimentally such sideband injection locking in microresonator soliton combs across a wide optical bandwidth and derive analytic scaling laws for the locking range and repetition rate control. As an application example, we demonstrate optical frequency division and repetition rate phase-noise reduction to three orders of magnitude below the noise of a free-running system. The presented results can guide the design of sideband injection-locked, parametrically generated frequency combs with opportunities for low-noise microwave generation, compact optical clocks with simplified locking schemes and more generally, all-optically stabilized frequency combs from Kerr-nonlinear resonators.


I. INTRODUCTION
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][2][3][4][5] , i.e. a large set of laser frequencies spaced by the repetition rate f rep = T −1 rep .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 microcombs 10 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][14][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 + µ f 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 numerically 22 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 ′ )/(2π µ ′ ), with µ ′ denoting the index of the closest resonance to which the secondary laser couples, cf.Fig. 1a.This frequency division 24 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 lownoise 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-effect 38 and avoided mode-crossings 39 , as well as the respective interplay between co-40 and counterpropagating solitons [41][42][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 non-linear filtering of microwave signals 47,48 via DKS.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 highresolution 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 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 with numeric 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 demonstrate experimentally 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.

II. RESULTS
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 million (linewidth κ/(2π) ≈ 100 MHz), a free-spectral range (FSR) of D 1 /(2π) = 300 GHz and exhibits anomalous group velocity dispersion D 2 /(2π) = 9.7 MHz so that the resonance frequencies are well-described by (1.6 × 0.8 µm 2 cross-section, 76 µm radius).To achieve 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 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 approximately 5.2 THz (cf.Fig. 2a) corresponding to a bandwidth limited pulsed of ∼60 fs duration.The soliton spectrum closely follows a sech 2 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 towards longer wavelengths due to the Raman self-frequency shift 51,52 .
A secondary CW laser (∼300 kHz linewidth), tunable both in power and 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. 2a.The spectrogram of the repetition rate signal recorded during this process is shown in Fig. 2b, for µ ′ = −13, and exhibits the canonical signature of locking oscillators 53 (cf.Supplementary Information (SI), Section 2 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 f rep = 1 2π ω p −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 FIG. 2. Soliton sideband injection locking.a, Single DKS comb spectrum, following a sech 2 envelope, with a full-width-at-halfmaximum (FWHM) of 5.2 THz, corresponding to a ∼60 fs pulse.The secondary laser is introduced in the spectral wing of the soliton and scanned across the µ ′ th sideband.b, Repetition rate beatnote observed while the secondary laser is scanned across the µ ′ th sideband.The locking bandwidth corresponds to the region of linear evolution of the repetition rate beatnote.c, Spectra of two sideband injectionlocked DKS states from either end of the locking range, exhibiting a differential spectral shift of 860 GHz.Note that a filter blocks the central pump component ω p .FIG. 3. DKS sideband injection locking dynamics.a, Transmission obtained when the secondary laser frequency ω ′ is scanned in the vicinity of comb line µ ′ = −3.The trace contains features indicating the position of the microresonator resonance frequency ω −3 /(2π) and of the soliton comb line frequency f −3 as well as the sideband injection locking range (see main text for details).b, Similar to a but for all µ ′ that can be reached by the scanning laser frequency ω ′ .In this representation, the resonance frequencies form a quadratic integrated dispersion profile (due to anomalous dispersion) while the equidistant soliton microcomb lines (highlighted in gray and expanded in panel b) form a straight line, enabling retrieval of pump laser detuning and microcomb repetition rate (see main text for details).c, Zoom into b, focusing on the vicinity of the comb lines.The spectral dependence of the locking range can be observed (cf.panel a and see main text for details).d, Locking range as a function of the relative mode number µ ′ .The measured data closely follows the predicted scaling (cf.main text).The grey area indicates the uncertainty we expect from 10% detuning fluctuations during the recording procedure.e, Locking range in terms of the repetition rate f rep for µ ′ = −13 as a function of secondary pump power (estimated on-chip power).Analogous to d, the uncertainty is approx.±4%.
no hysteresis is observed.Figure 2c 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 will discuss below and in the SI, Section 3 b, 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 scan 54 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 3a 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 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. 3b as a horizontal stack.For plotting these segments on a joint vertical axis, ω p + µ ′ D 1 has been subtracted from ω ′ .In this representation, the parabolic curve (blue line in Fig. 3b) connecting the C-resonances signifies the anomalous dispersion of the resonator modes ω µ .In contrast, the equidistant comb lines form a straight feature (grey highlight), of which a magnified view is presented in Fig. 3c.Due to the Raman selffrequency shift, the free-running repetition rate of the DKS comb f 0 rep is smaller than the cavity's FSR D 1 /(2π), resulting in the negative tilt of the line.Here, to obtain a horizontal arrangement of the features, ω p + µ ′ 2π f 0 rep has been subtracted from ω ′ .The locking range δ lock corresponds to the vertical extent of the characteristic locking feature in Fig. 3c.Its value is plotted as a function of the mode number in Fig- ure 3d, 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 (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. 3e, we observe an almost perfect squareroot scaling δ lock ∝ √ P ′ , revealing the proportionality of the locking range to the strength of the injected field.
The observed scaling of the locking range may be understood in both the time and frequency domain.In the time domain, the beating between the two driving lasers creates a modulated background field inside the resonator, forming an optical lattice trap for DKS pulses 22,28 .Here, to derive the injection locking range δ lock , we extend the approach proposed by Taheri et al. 26 , which is based on the momentum p = ∑ µ µ|a µ | 2 = μ ∑ µ |a µ | 2 of the waveform (in a co-moving frame), where a µ is the complex field amplitude in the mode with index µ, normalized such that |a µ | 2 scales with the photon number and μ the photonic center of mass in mode number/photon momentum space.As we show in the SI, Section 3 c, the secondary driving laser modifies the waveform's momentum, thereby its propagation speed and repetition rate.For the locking range of the secondary laser, we find and for the repetition rate tuning range where η is the coupling ratio, and the P µ refer to the spectral power levels of the comb lines with index µ measured outside the resonator.The spectral shift of the spectrum in units of mode number µ is 2πδ f rep /D 2 .In the SI, Section 1, we recast Eq. 2 in terms of the injection ratio IR = P ′ /P µ ′ to enable comparison with CW laser injection locking 57 .The results in Eqs. 2 and 1 may also be obtained in a frequency domain picture (see SI, Section 3 d), realizing that the waveform's momentum is invariant under Kerr-nonlinear interaction (neglecting the Raman effect) and hence entirely defined by the driving lasers and the rate with which they inject photons of specific momentum into the cavity (balancing the cavity loss).If only the main pump laser is present, then μ = 0.However, in an injection-locked state, depending on phase, the secondary pump laser can coherently inject (extract) photons from the resonator, shifting μ towards (away from) µ ′ .This is equivalent to a spectral translation of the intracavity field, consistent with the experimental evidence in Fig. 2c.
To verify the validity of Eq. 1 and 2, we perform numeric simulation (SI, Section 4) based on the Lugiato-Lefever Equation (LLE) (see SI, Section 3 a).We find excellent agreement between the analytic model and the simulated locking range.We note that Eq. 1 and 2 are derived in the limit of low injection power, which we assume is the most relevant case.For 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 δ lock ∝ µ ′2 P ′ P µ ′ to the measured locking range in Fig. 3d, where we assume P µ ′ to follow an offset (Ramanshifted) sech 2 -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 we show numerically in the SI, Section 4, 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 spectral shift).Therefore, although the underlying mechanism is the same as in DKS combs, Eq. 1 and Eq. 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 FIG. 4. Optical frequency division.a, Repetition rate phase noise of the soliton microcomb in the free-running and locked states, with values of µ ′ ranging from 1 to 42.At higher offset frequencies (>100 kHz), the phase noise of the electro-optic modulation used to down-mix the 300 GHz repetition rate signal to detectable frequencies (cf.SI) limits the measurement.b, Repetition rate beat note recorded in the free-running state.c, Repetition rate beatnote recorded in the locked state (µ ′ = 42).The sidebands at approx.±300 kHz are an artifact of the electro-optic modulation-based repetition rate detection scheme.locking, we demonstrate optical frequency division, similar to previous work 34 , and measure the noise reduction in f rep (Fig. 4a).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 3 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 secondary laser.Fig. 4b and c compare the repetition rate beatnote of the free-running and injection-locked cases.

III. CONCLUSION
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 an 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, 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 Kerrnonlinear resonators.

DATA AVAILABILITY STATEMENT
The data supporting this study's findings are available from the corresponding author upon reasonable request.

Measuring the comb's repetition rate
The soliton comb's repetition rate, too high for direct detection, is measured by splitting off a fraction of the pump light and phase-modulating it with frequency f m = 17.68 GHz, creating an electro-optic (EO) comb spanning ∼600 GHz which is then combined with the DKS light (Fig. 5).A bandpass filter extracts the 17 th line of the EO comb and the first sideband of the DKS comb, resulting in a beatnote at a frequency f s = f rep − 17 f m from which the soliton repetition rate f rep can be recovered, similar to 59 .FIG. 5. Experimental setup.A single-DKS state is generated inside a silicon nitride microring resonator using approximately 200 mW of pump power.A secondary continuously tunable laser is combined with the pump light before the cavity in order to investigate the sideband injection locking dynamics.In order to monitor the ∼300 GHz DKS repetition rate, we use a hybrid electro-optic detection scheme, similar to 59 (see.supplementary information, section 1).EDFA: erbium-doped fiber amplifier; CW: continuous wave laser; CTL: continuously tunable laser; PD: photodetector; OSC: oscilloscope; ESA: electrical spectrum analyzer; OSA: optical spectrum analyzer; EOM: electro-optic modulator; BPF: band-pass filter.

Analytic description of sideband injection locking a. Definitions
We start from the dimensionless form of the Lugiato-Lefever equation (LLE) 60,61 , describing the dynamics in a frame moving with the (angular) velocity d 1 (the angular interval [0, 2π] corresponds to one resonator round-trip): where θ is the azimuthal coordinate, τ = κt/2 denotes the normalized time (κ being the cavity decay rate/total linewidth), Ψ(θ , τ) is the waveform, ζ 0 = 2(ω 0 − ω p )/κ the normalized pump detuning, d n = 2D n /κ the normalized dispersion coefficients and f = s 8ηg/κ 2 a pump field where |s| 2 = P p hω p is the pump photon flux.The microresonator's coupling rate to the bus waveguide is κ ext and η = κ ext /κ is the microresonator's coupling coefficient.Let a µ (τ) be the normalized complex mode amplitudes such that: where µ is the relative mode number and a µ is related to the circulating intracavity power Here g = hω 2 0 cn 2 /(n 2 0 V eff ) is the nonlinear coupling coefficient , where c is the speed of light, n 0 the refractive index, n 2 the nonlinear index, and V eff the mode volume.
The momentum p of the intra-cavity field Ψ(θ , τ) is defined as: where we used Eq.6, μ denotes the photonic center of mass and N = ∑ µ |a µ | 2 scales with the number of photons in the cavity.

b. Waveform velocity, spectral shift and repetition rate
We assume a waveform Ψ with a stable non-flat temporal intensity profile inside the resonator, i.e. the shape of the intensity profile does not change.The Waveform Ψ may not be static (in the frame moving with angular velocity d 1 ), but move with an additional angular velocity component θ , so that By replacing the time derivatives with the right side of the LLE and only accounting for second-order dispersion (d n = 0, ∀n ≥ 3), one finds In the following we take const = 0, which in case of a non-zero value corresponds to a suitable (re-)definition of the moving frame.Next, integrating according to 1 2π 2π 0 dθ and using Parseval's theorem The change in the repetition rate δ the change of the repetition rate is proportional to the shift of the photonic center of gravity μ away from µ = 0.

c. Sideband injection locking: Time domain description
In the case of a pump laser at frequency ω p and a secondary laser at frequency ω ′ close to ω µ ′ , the pump field takes the form: where ζ = 2/κ(ω ′ − ω p − µ ′ D 1 ) is a term describing the mismatch between the microresonator FSR and the grid defined by the pump and secondary lasers.From ref. 22 , Eq. 26, it follows that the force on the waveform Ψ due to the presence of the secondary pump line is: We recognize that the integral term is the Fourier transform of the intracavity field, such that: We assume that Ψ is a stable waveform, moving at an angular velocity θ within the LLE reference frame, which itself moves with d 1 .In a suitable (θ ′ , τ ′ )-coordinate system, the waveform Ψ ′ (θ ′ , τ ′ ) is static (does not move), as described in SI, Section 3 b.For the relation between the Fourier transforms of Ψ and Ψ ′ we find that a µ = a ′ µ exp (−iµ θ τ) and therefore ∠a µ ′ = ∠a ′ µ ′ − µ ′ θ τ.After substitution of the angle we find We search for a steady state solution in which the momentum is constant dp/dτ = 0.As in a steady comb state ∂ a ′ µ ∂ τ = 0 (and f ′ does not depend on time), time independence is achieved when θ = ζ /µ ′ , i.e. the waveform must be moving at the velocity fixed by the pump and auxiliary laser detuning.Therefore, the momentum is purely a function of the relative phase between the secondary laser and the waveform's respective spectral component µ ′ : With Eq. 19, this means that the repetition rate range in the injection-locked state is and the locking range is given by where the power levels P ′ and P µ are the power levels measured outside the resonator.The approximation in the second line of Eq. 29 assumes that the mean frequency of the photons in the cavity is approximately ω 0 .Note that P µ ′ and P 0 can readily be measured via a drop port; however, in a through-port configuration such as the one used in our experiment, it may be buried in residual pump light.For a smooth optical spectrum, their values may also be estimated based on neighboring comb lines for a smooth optical spectrum.

d. Sideband injection locking: Frequency domain description
In the sideband injection-locked state, the nonlinear dynamics in the resonator may be described by the following set of coupled mode equations: Here, the a ′ µ represent the modes with the frequencies ω p + µω R = 2π f rep , where ω R is the actual repetition rate of the comb that may be different from D 1 .Note that the a ′ µ correspond to the a ′ µ in SI, Section 3 c.In a steady comb state ∂ a ′ µ ∂ τ = 0, so that a fixed phase relation between modes a ′ µ and the external driving fields f and f ′ exists.We now consider only the field of the waveform, which we again denote with a ′ µ for simplicity (Note that in a DKS, the separation of the DKS field at mode µ = 0 from that of the background is formally possible owing to their approximate phase shift of π/2).The rate at which photons are added to the waveform by the driving lasers is for the main driving laser, and for the secondary driving laser.Photons are added or subtracted from the respective mode depending on the relative differential phase angles ∠a ′ 0 − ∠ f p and ∠a ′ µ ′ − ∠ f ′ .While ∠a ′ 0 − ∠ f p ≈ 0 for the main driving laser, ∠a ′ µ ′ − ∠ f ′ (for the secondary laser) can take all values from 0 to π during sideband injection locking.In the steady state the cavity losses are balanced by the lasers so that The photonic center of gravity of the photons injected into the cavity is For clarity, we note that μ does not change when photons are transferred through Kerr-nonlinear parametric processes from one mode to another mode.This is a consequence of (angular) momentum and photon number conservation (implying total mode number conservation) in Kerr-nonlinear parametric processes.Hence the photonic center of gravity of the injected photons (by main and secondary pump) is the same as the photonic center of gravity for the entire spectrum of the waveform.For the injection-locked state we find so that with Eq. 19 we obtain the same result as in the time domain description for δ f rep and δ lock (Eq.28 and Eq.29).

Numeric simulation of the sideband injection locking range
To complement our experimental and theoretical results, we run numerical simulations based on the coupled mode equation framework 27,62 (a frequency-domain implementation of the LLE).In order to observe the sideband injection locking dynamic, a single DKS is first initialized inside the cavity and numerically propagated.In the absence of a secondary laser, the soliton moves at the group velocity of the pump wavelength and appears fixed within the co-moving frame (Fig. 6a).The secondary laser f ′ is then injected as per Eq. 20, where ζ controls the detuning of the secondary laser with respect to the free-running comb line a µ .For 0 < | ζ | < δ lock /κ (i.e., within the locking range), we observe that soliton moves at a constant velocity with respect to the co-moving frame (Fig. 6b).Beyond the locking range, the soliton is no longer phase-locked to the pump, which can readily be identified by tracking the relative phase between a µ ′ and f ′ .We use this signature to identify the locking range from our simulations and compare it to our theoretical prediction from Eq. 29; as can be seen in Fig. 7, simulation and theory agree with striking fidelity.
We also study sideband injection locking dynamics inside normal-dispersion combs (Fig. 6c and d).Here as well, we observe the locking of the platicon to the underlying modulation, although with a significant effect on its spectrum P µ (see inset).This platicon's spectrum lower robustness against external perturbation is not captured by our model which assumes a shifted but otherwise unchanged spectrum.Therefore, Eq. 29 does not generally apply, even though we expect our model to predict the locking range within a tolerance corresponding to the relative amplitude change of the corresponding comb line a µ ′ .

Effect of thermal resonance shifts
Across the sideband-injection locking range, the power in the comb state changes by small amounts due to the secondary laser, which may add or subtract energy from the resonator (cf.SI, Section 3 d).In consequence, the temperature of the resonator changes by small amounts, impacting via the thermo-refractive effect (and to a lesser extent by thermal expansion) the effective cavity length and hence f rep .This effect occurs on top of the sideband injection locking dynamics.In a typical microresonator, the full thermal shift of the driven resonance that can be observed prior to soliton formation is within 1 to 10 GHz.In a DKS state the coupled power is usually 1 to 10%, implying a maximal thermal shift of 1 GHz.Assuming the secondary laser will change the power in the resonator by not more than 5% (theoretical maximum for the highest pump power used in our manuscript), we expect a maximal thermal resonance shift of 50 MHz.Now, dividing by the absolute mode number of > 500, we expect δ f rep <100 kHz, which is two orders of magnitude below the effect resulting from sideband injection locking.

FIG. 1 .
FIG. 1. Principles of sideband injection locking.a, In a freerunning comb, the central comb line is defined by the pump laser around which equidistant comb lines, spaced by the free-running repetition rate f 0 rep , are formed.If a secondary injection laser of frequency ω ′ is brought close to one of the comb lines (within injection locking range), then the comb locks to the injecting laser, modifying the repetition rate as indicated.b Outside the locking range, f rep = f 0 rep is unaffected by the secondary laser.Inside the locking range, it follows a characteristic tuning behavior with a linear dependence on the injecting laser frequency ω ′ .

FUNDING
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; the work was supported through the Maxwell computational resources operated at DESY.

FIG. 6 .FIG. 7 .
FIG. 6. Sideband injection locking simulation.Evolution of the intracavity intensity profile in the free running (a, c) and locked states (b, d) for both DKS (a, b) and normal-dispersion combs (c, d).Inset: corresponding intracavity spectra.