Abstract
Dissipative Kerr solitons are selfsustaining optical wavepackets in resonators. They use the Kerr nonlinearity to both compensate dispersion and offset optical loss. Besides providing insights into nonlinear resonator physics, they can be applied in frequency metrology, precision clocks, and spectroscopy. Like other optical solitons, the dissipative Kerr soliton can radiate power as a dispersive wave through a process that is the optical analogue of Cherenkov radiation. Dispersive waves typically consist of an ensemble of optical modes. Here, a limiting case is studied in which the dispersive wave is concentrated into a single cavity mode. In this limit, its interaction with the soliton induces hysteresis behaviour in the soliton’s spectral and temporal properties. Also, an operating point of enhanced repetitionrate stability occurs through balance of dispersivewave recoil and Ramaninduced solitonselffrequency shift. The singlemode dispersive wave can therefore provide quiet states of soliton comb operation useful in many applications.
Introduction
A new dissipative soliton^{1} has recently been observed in optical resonators. These dissipative Kerr solitons (DKS) have been demonstrated in fibre resonators^{2} and in various microcavity systems^{3,4,5,6,7}. In microcomb research^{8,9} soliton formation produces phaselocked spectra with reproducible envelopes, as required in frequency comb applications^{10,11,12,13,14}. Moreover, their unusual properties and interactions create a rich landscape for research in nonlinear optical phenomena^{5,15,16,17,18,19,20,21,22,23,24}. Two such phenomena, the Ramaninduced solitonselffrequencyshift (SSFS) and dispersivewave generation, are important to this work.
The Raman SSFS causes a spectral red shift of the soliton. In optical fibre systems, this shift continuously increases with propagation distance^{25,26}, however, in microresonators the shift is fixed and depends upon soliton power^{17,18,27,28}. Dispersive waves also occur in optical fibre systems^{29} where they are an important process in continuum generation^{30} and have been used to study general nonlinear phenomena^{31}. They are formed when a soliton radiates into a spectral region of normal dispersion and can be understood as the optical analog of Cherenkov radiation^{32}. In microcavities, dispersive waves provide a powerful way to spectrally broaden a soliton within a microresonator as a precursor to self referencing^{12,33}. Their formation also induces soliton recoil^{32} which, similar to SSFS, causes a frequency shift in the spectral centre of the soliton^{5,17}. Dispersive waves normally consist of an ensemble of modes that are phase matched to a soliton. This phase matching can be assisted by avoided mode crossings in microcavities^{19,34}. Avoided mode crossings can also produce zero group velocity effects^{35,36}, enable microcombs to form in regions of normal dispersion^{37}, and provide a way to induce dark solitons^{16}.
In this work, an avoidedmode crossing is used to excite a dispersive wave consisting of a single cavity mode. The coupling of this singlemode dispersive wave to the soliton is strongly influenced by the total soliton frequency shift produced by the combined Ramaninduced SSFS and the dispersivewave recoil. The combination is shown to induce hysteresis behaviour in soliton properties. Included in this behaviour, there is an operating point of improved pulserate stability (a quiet point) where the coupling of repetition rate and cavitypump detuning is greatly reduced. Pulserate stability is centrally important in many frequency comb applications^{10,13,38}. Coupling of pulse rate and cavitypump detuning through avoidedmodecrossing recoil effects has been observed in crystalline resonators^{39}. Also, the fundamental contributions to phase noise in the pulse train have been considered theoretically^{40}. However, technical noise mechanisms are also present. For example, DKS generation using onchip silica resonators exhibits phase noise that tracks in spectral profile the phase noise of the optical pump^{4}. The quiet operation point is shown to reduce technical noise contributions to the soliton pulse repetition rate. Both this regime of operation and the hysteresis behaviour are measured and modelled theoretically.
Results
Mode family dispersion
A silica whisperinggallery resonator^{41} is used for soliton generation. The devices feature a freespectralrange (FSR) of ∼22 GHz (3 mm diameter resonator) and have intrinsic Qfactors around 250 million. Specific details on soliton formation in these resonators are given elsewhere^{4,42}. The resonators support multiple, transverse mode families. It is essential that the solitonforming mode family feature dispersion that is primarily secondorder and anomalous^{43}. To characterize the frequency spectrum of the resonator, mode frequencies were measured from 190.95 THz (1,570 nm) to 195.94 THz (1,530 nm) using an externalcavity diode laser calibrated by a fibre Mach–Zehnder interferometer^{4}. This provides a set of mode frequencies {ω_{μ,s}} for each spatial mode family ‘s’ with μ as the mode index.
The mode family frequency data are presented in Fig. 1a by plotting the relativemodefrequency, Δω_{μ,s}≡ω_{μ,s}−ω_{0}−μD_{1} versus mode index μ where ω_{0} and D_{1} are specific to the solitonforming mode family. ω_{0} is the frequency of the mode (set to have index μ=0) that is optically pumped to produce the soliton, and D_{1} is the FSR of the solitonforming mode family at μ=0 (note: μ is a relative and not an absolute mode index). By plotting the data in this way the second and higherorder dispersion of the solitonforming mode family become manifest. To illustrate, the relativemodefrequency of the solitonmode family is fit with a green, dashed parabolic curve of positive curvature in Fig. 1a showing that it features anomalous secondorder dispersion over a wide range of mode numbers.
A second mode family also appears in Fig. 1a and causes an avoidedmodecrossing near μ=72. Hybridization of this ‘crossingmode’ family with the solitonmode family occurs near the avoided crossing^{19,44}. The relativemodefrequencies of the unperturbed solitonforming mode family and crossingmode family are denoted as Δω_{μA} and Δω_{μB}. Over the range of mode indices measured Δω_{μA}=D_{2}μ^{2} where D_{2} is the secondorder dispersion at μ=0. The lower (upper) branch of the hybrid mode family is denoted by Δω_{μ−} (Δω_{μ+}). The spatial modes associated with the soliton and crossing mode families are identified in the Supplementary Note 3. Avoided mode crossing behaviour has been intensively studied in the context of DKS formation and can interfere with soliton generation by creation of distortions in the dispersion spectrum^{43,45,46}. In the present system, the avoided modecrossing induces only minimal distortion in the otherwise parabolic shape of the solitonforming mode family. Soliton spectra produced on this mode family by pumping at μ=0 are shown in Fig. 1b along with theoretical sech^{2} spectral envelopes predicted for DKSs. As an aside, the horizontal scales in Fig. 1a,b are identical and the location of the μ=0 pumping mode is indicated by a vertical dashed line in Fig. 1b.
Singlemode dispersivewave formation
Also shown in Fig. 1a are the comb frequencies associated with a hypothetical soliton spectrum plotted in the relative frequency frame. This comb line is given by,
where ω_{μ,comb}=μω_{rep}+ω_{p} is the frequency of μth comb line, ω_{rep} is the soliton repetition frequency, ω_{p} is the pump frequency, and δω≡ω_{0}−ω_{p} is the cavitypump detuning frequency. It is necessary to distinguish between relative frequencies for the soliton comb and the resonator modes because the frequency components of the soliton comb are strongly reddetuned relative to the coldcavity mode frequencies by the Kerr nonlinearity. Indeed, dispersive waves typically form when a set of modes break this rule and become resonant with a set of comb lines. A limiting case of this condition is shown in Fig. 1a, where the occurrence of an isolated resonance between a hybrid mode with relative frequency Δω_{r−} and a comb line at Δω_{r,comb} is illustrated. The equation of motion for the hybrid mode field amplitude h_{r−} is shown in the Methods to have the following form,
where κ_{r−} is its loss rate and f_{r} is an effective pumping term associated with the soliton comb line. The pumping term is given by f_{r}=iΓ(Δω_{rA}−Δω_{r,comb})a_{r}, where Γ is the fraction of the family A mode in the hybrid mode, and a_{r} is the field amplitude of the unperturbed soliton hyperbolic solution at μ=r. Also, the Kerreffect shift of h_{r−} is of order 10 kHz and is therefore negligible in comparison to κ_{r−}.
Because the damping rate κ_{r−} is low (that is, the mode has a high optical Qfactor) slight shifts in the slope of the comb frequency line (equivalently, shifts of Δω_{r,comb} relative to Δω_{r−}) will cause large changes in the power coupled to the hybrid mode. These changes are observable in Fig. 1b where a strong spectral line appears in the case of the blue soliton spectrum. Note that scattering from the soliton into the spectral line is strong enough so that the power in the line is greater than the comb line power near the spectral centre of the soliton, itself. The strong spectral line can be understood as a singlemode dispersive wave and it induces a recoil in the spectral centre of the soliton. This recoil contribution is indicated for the blue soliton spectrum in the figure. In the case of the red soliton spectrum, the operating point was changed and the resonance between the soliton and the mode is diminished. Accordingly, most of the spectral shift in this case results from the Raman SSFS.
Soliton recoil and hysteresis
A change in the slope of the soliton comb line will occur when the soliton repetition frequency, ω_{rep}, is changed (equation (1)). On account of secondorder dispersion ω_{rep} depends linearly on the frequency offset, Ω, of the soliton spectral maximum relative to the pump frequency^{19,40}. This frequency offset has contributions from both the Raman SSFS, Ω_{Raman}, and the dispersivewave recoil, Ω_{Recoil} (that is, Ω=Ω_{Raman}+Ω_{Recoil}). Accordingly, the soliton repetition rate is given by,
where D_{2} (the secondorder dispersion of solitonforming mode family at μ=0) is measured to be 17 kHz from Fig. 1a. Substituting for the repetition rate in the comb line expression (equation (1)) gives,
It is shown in the Methods (equation (25)) that the soliton recoil frequency has a linear dependence on the hybrid mode power,
where κ_{A} and κ_{B} denote the power loss rates of the family A and family B modes, respectively; and E is the circulating soliton energy.
Solving equation (2) for the steadystate power in the hybrid mode at the soliton comb line frequency and using equations (4 and 5) gives the following result,
Equation (6) suggests that a bistable state and hysteresis behaviour in the dispersivewave power is possible when varying the soliton operating point. Consistent with this possibility, it is noted that the two soliton spectra in Fig. 1b (blue and red), which show very different dispersivewave powers, were produced at nearly identical detuning frequencies, δω. A more detailed survey of the dispersivewave power behaviour is provided in Fig. 1c and is again consistent with a hysteresis behaviour versus detuning. Moreover, since the total spectral shift of the soliton is given by Ω=Ω_{Raman}+Ω_{Recoil}=Ω_{Raman}+γh_{r−}^{2}, a corresponding behaviour is observed in the overall soliton spectral shift (Fig. 1d). Theoretical fits are provided in Fig. 1c,d using equation (6). The fitting procedure and parameter values are provided in the Methods.
In plotting the data, the detuning frequency, δω/2π, was determined from the measured total soliton spectral shift (Ω) and pulse width (τ_{s}) using the relation δω=(D_{2}/2)(1/+Ω^{2}). This expression is a generalization of a relationship derived elsewhere^{18}. The generalization extends the shift Ω to include both the SSFS and the recoil and is derived as equation (33) in the Methods. As an aside, the pulse width is determined by fitting the soliton optical spectrum^{4}.
Likewise, the recoil frequency, Ω_{Recoil}, can also be extracted from the data as Ω−Ω_{Raman} by first using the soliton pulse width to determine the Raman shift using Ω_{Raman}=−8τ_{R}D_{2}/15κ_{A}. A plot of the recoil shift determined this way versus the dispersivewave power is given as the inset in Fig. 1c and verifies the linear dependence (equation (5)). Equation (5) is also plotted for comparison using parameters given in the Methods. As an aside, the Raman shift formula noted above is also a generalization of a result proven elsewhere^{18}. Curiously, as shown in the Methods, this formula maintains its previous form in the presence of the dispersive wave.
Within narrow detuning frequency bands in the vicinity of the hysteresis both measurements and calculations show that the total cavity power (soliton and dispersivewave contributions) can decrease with increasing cavitypump detuning as opposed to increasing with detuning as is typical for a soliton. Under these special conditions, the pumpcavity detuning will no longer be dynamically stable on account of the thermal nonlinearity^{47}. Evidence of this was observable in the current work as it was not possible to completely map out the theoretically predicted hysteresis curves.
While the present results are produced using a dispersive wave that is bluedetuned relative to the soliton spectral maximum, the hysteresis behaviour is also predicted to occur for a reddetuned dispersive wave. However, in the reddetuned case, the orientation of the curve in Fig. 1c is reversed with respect to the detuning frequency. The essential feature for appearance of the hysteresis is that the recoil advances and retreats versus detuning. As a result, the existence of hysteresis behaviour predicted in equation (6) is not limited to microresonator materials having a strong Raman SSFS. It is also predicted to occur, for example, in crystalline resonators given an appropriate avoidedmode crossing. The requirements imposed on the device and mode crossing for this to occur are discussed further below.
Numerical simulation
To further investigate the singlemode dispersivewave phenomena, we perform numerical simulations based on the coupled LugiatoLefever equations^{34,48,49,50,51} involving the solitonforming mode family (family A) and the crossingmode family (family B). Additional information including parameter values is provided in the Methods, but is outlined here. The two mode families are coupled using a model studied elsewhere^{19}. The coupling is characterized by a rate constant G and is designed to induce an avoidedmodecrossing around mode index μ=72, similar to the experimental mode family dispersion. Figure 2 shows the results of the numerical simulation including 2,048 modes. The hysteresis behaviour in the soliton total frequency shift and the dispersivewave power resembles the experimental observation and is also in agreement with the analytical model (Fig. 2a,b). As predicted by equation (5) (and observed in the Fig. 1c, inset), the recoil is numerically predicted to vary linearly with the dispersivewave power (Fig. 2b, inset).
Frequency and time domain features of the soliton (blue) and dispersive wave (red) are also studied in Fig. 2c,d in units of intracavity power. They show that the dispersivewave emerges on mode family B and consists primarily of a single mode. The singlemode dispersive wave leads to a modulated background field in the resonator with a period determined by the beating between the pump and the dispersive wave. This modulation is observable in Fig. 2d. Spectral recoil of the soliton is also observable in the numerical spectra. The combined power of mode A and B spectra in Fig. 2c is the total intracavity power.
Soliton repetition rate quiet point
The nonlinear behaviour associated with soliton coupling to the singlemode dispersive wave can be used to suppress soliton repetition rate noise produced by coupling of pumplaser frequency noise. This noise source is suspected to be a significant contributor to repetitionrate noise in certain frequencyoffset regimes^{4}. From equation (3) the repetition frequency depends linearly on the total soliton spectralcentre frequency shift, Ω. However, this total shift frequency versus cavitypump detuning has a stationary point on the upper hysteresis branch (Fig. 1d). As expected from the simple dependence in equation (3), this same stationary point is observed in measurements of the repetition frequency versus detuning (Fig. 3a). To measure the repetition frequency the soliton pulse train is directly detected and an electrical spectrum analyser is used to observe the pulse train spectrum. The theoretical prediction using analysis from the Methods is also provided for comparison.
The coupling of pumplaser frequency noise into the soliton repetition rate is expected to be minimal at the stationary point. To verify this prediction, the phase noise of the detected soliton pulse train is measured at different soliton operating points on the upper and lower branches in Fig. 3a using a phase noise analyser. Phase noise spectra corresponding to operating points I, II and III in Fig. 3a are plotted in Fig. 3b. Operating points I and II correspond to nearly identical cavitypump detuning, but lie on different branches. As expected, operating point II in the upper branch has a lower phase noise level compared to operating point I on account of its reduced slope. Operating point III is close to the zeroslope detuning point in the upper branch. This quiet point has the lowest phase noise among the recorded phase noise spectra. At higher offset frequencies, the phase noise is shot noise limited, while at lower offset frequencies the phase noise indicates >0 dBc Hz^{−1} and is mainly contributed by frequency drift of the repetition rate.
For comparison, the phase noise associated with the detuning frequency δω was also measured. For this measurement, the error signal of a Pound–Drever–Hall feedback control system is operated openloop and recorded using an oscilloscope. Its power spectral density is converted into phase noise in Fig. 3b (Supplementary Note 1). The relatively high noise floor in this measurement is caused by the oscilloscope sensitivity. Nonetheless, a noise bump at 25 kHz offset frequency originates from the laser and provides a lasernoise reference point against which comparison to the soliton phase noise is possible. The soliton phase noise at 25 kHz offset frequency noise is plotted versus detuning in Fig. 3c. The soliton phase noise is calculated in the Methods and the results are presented for comparison using the cavitypump detuning noise level at 25 kHz offset. The dip of the phase noise occurs at the quiet point. One outlier point (red branch) is believed to have resulted from loss of lock of the phase noise analyser. For lower offset frequencies, the contributions to noise are believed to originate from thermal contributions within the resonator and are under investigation. Nonetheless, the measured noise contributions at these frequencies show a trend of reduction for operation at the quiet point.
An analytical study comparing the detuning response of the Raman and recoil effects was performed to determine conditions required to observe the quiet point. The quiet point occurs when the retreating soliton recoil balances the always advancing SSFS. Accordingly, Fig. 4 is a contour plot of the maximum ratio of ∂Ω_{Recoil}/∂δω to ∂Ω_{Raman}/∂δω while varying the coupling strength between the solitonmode and crossingmode families and the damping rate of the crossing mode (see Methods). The existence regime for observation of the quiet point corresponds to the ratio >1 shown in red. Stronger mode interaction and weaker dissipation are required to operate in this regime. Also, the impact of these parameters on the detuning range of the hysteresis is studied in the Supplementary Note 4.
Discussion
Microfabrication control of resonator diameter, oxide thickness and wedge angle all impact the spectral placement of mode families. Numerical simulation of these families based on scanning electron micrograph measurement of resonator cross sections provides reasonably accurate dispersion maps for prediction of resonator properties. Also, process control of the resonator fabrication is sufficient to guarantee fabrication of mode families exhibiting the features shown in Fig. 1a within the 1,530–1,570 nm band.
In summary, coupling of a dissipative Kerr soliton to a singlemode dispersive wave has been shown to produce hysteresis behaviour in both the dispersivewave power and in the soliton properties. These properties include the frequency shift of the soliton spectral centre relative to the pumping frequency and the soliton repetition frequency. The hysteresis results from the dependence of the dispersivewave phase matching condition upon the dispersivewave power. The hysteresis behaviour of the dispersive wave also leads to an operating point wherein coupling of laser pump frequency noise into the soliton repetition rate is greatly reduced. This reduction was modelled and measured, and the requirements for quiet point existence were also studied. The operating point for quiet soliton operation is of potential use for ultralownoise microwave generation.
Methods
Dynamical equation of hybrid mode
Equation (2) can be derived from coupled mode equations that include dispersion, mode interaction and the Kerr nonlinearity. The intracavity field of mode μ in the solitonforming mode family A can be represented by , where A_{μ}(t) is the slowly varying amplitude, t is the time and φ is the azimuthal angle along the resonator. In the rotation frame of comb frequencies ω_{μ,comb}=ω_{0}−δω+μω_{rep}, the intracavity field can be expressed as . We denote the intracavity field in the crossingmode family B as b_{μ} and express it in the same reference frame as the solitonforming mode a_{μ}. It should be noted that the relative mode number μ is referenced to the mode that is being optically pumped, and does not represent the actual azimuthal index. The intracavity fields can be calculated using the equations of motion with Kerr nonlinearity terms^{50,52} and modalcoupling terms^{44},
where κ_{A,B}=ω_{0}/Q_{A,B} is the dissipation rate. g=ħn_{2}D_{1}/2πn_{0}A_{eff} represents the normalized Kerr nonlinear coefficient with A_{eff} the effective nonlinear mode area. g_{B} is defined similarly. G is the linear coupling coefficient between the two mode families^{19} and F is the normalized coupled laser pump field. Also, to calculate equation (2) it is not necessary to include Raman coupling terms in equations (7 and 8) since the leadingorder contribution to the forcing term, f_{r}, is from the Kerr nonlinearity.
Modal coupling causes two branches of hybrid modes to form as shown in Fig. 1a. The frequencies of the hybrid modes in the upper (+) and lower (−) branches are given by (refs 44, 53, 54),
where the corresponding field amplitude of the hybrid modes is a linear combination of a_{μ} and b_{μ}. In the fardetuned regime where , the field amplitude of the lower branch hybrid mode is approximately given by,
In this experiment, only one mode was observed to be near resonance with the soliton comb and that mode is assigned mode index μ=r. Consistent with Fig. 1a, the hybridization of mode r is assumed weak (that is, and ) so that b_{r} is the dominant contribution to . Also, since the amplitude of b_{μ} with μ≠r is small, the Kerr interaction summation term can be neglected in equation (8) in this calculation.
By taking the time derivative of equation (10) and then substituting using (7) and (8) the following dynamical equation results for ,
where f_{r} is the pumping term given by,
and where is the fraction of the family A mode in and κ_{r−}≈κ_{B} is assumed for r when . When converting equation (11) into the rotation frame of (ω_{0}+μD_{1}) with , the following expression results,
where Δω_{r−}=ω_{r−}−ω_{o}−μD_{1} is the relativemodefrequency of hybrid mode h_{r−}. Equation (13) is identical to equation (2) in the main text.
Effective pumping term
The pumping term in equation (11) can be expressed in parameters of the resonator and soliton. The soliton field envelope takes the form^{3,18}
where soliton properties are: amplitude B_{s}, angular position φ_{c}, temporal pulse width τ_{s}, spectralcentre frequency shift (relative to pump) Ω and phase relative to the pump laser . Also, this solution assumes . By applying the Fourier transform to A(φ, t), a_{μ} can be expressed in terms of the soliton properties,
The pump f_{r} can therefore be derived by inserting equation (16) into equation (12). The following expression results from simplification of the summation,
where g has been replaced using equation =D_{2}/g, which holds for DKSs^{18,40} and is also verified in a section below. Finally, by using^{18} (see derivation below), f_{r} can be further reduced to
Recoil and soliton self frequency shift
In addition to the Raman SSFS^{17,18}, the spectral centre of the DKS is also shifted by the singlemode dispersivewave recoil. The effect of the recoil and Raman shift can be calculated using the moment analysis method^{17,55}. Using the Fourier transform, equation (7) is transformed into the perturbed LugiatoLefever equation (LLE)^{50}
where the Raman shock term has been added^{17,18} and τ_{R} is the Raman time constant. The moment analysis method treats the soliton as a particle. The energy E and the spectral centre mode number μ_{c} are given by,
Taking the time derivative of equation (21) and substituting ∂A/∂t using equation (19), the following equation of motion for μ_{c} is obtained,
The second term on the righthandside corresponds to the Ramaninduced frequency shift and the third term is the frequency shift caused by recoil.
The Raman term can be calculated by substituting equation (14) into the integral. When calculating the recoil term, B is simplified to as the power in mode B is dominated by the near resonance mode r. In addition, because the integral of φ is over 2π, only has nonzero contribution. Furthermore, equation (8) is used to relate Ga_{r} to b_{r} and finally leads to,
The steadystate spectral centre mode number is therefore given by,
where (equivalent to ) is assumed and the recoil and Raman shifts are,
In the main text, is assumed. Equation (25) is equation (5) in the main text. The form for the Raman SSFS, Ω_{Raman}, is identical to the form previously derived in the absence of the dispersivewave coupling^{18}.
Soliton parameters with Raman and modecoupling effects
In the presence of recoil and Raman, the relations between soliton parameters in equation (14) can be derived from the Lagrangian approach^{3,18,40}. In addition, the Lagrangian approach verifies the expression for Ω_{Recoil} obtained above as well as providing a path for calculation of the repetitionrate phase noise^{40}. As detailed in previous literature^{18,40}, the perturbation Lagrangian method is applied to the LLE equation of A (equation 19). However, now an additional perturbation term is added to account for the mode coupling to the crossingmode family. Taking , produces the following equations of motion,
where we have assumed the mode r is far from the mode centre μ_{c}=Ω/D_{1} and the coupling coefficient G is smaller than or around the same order of magnitude with δω. Also, higher order terms are neglected (Supplementary Note 2). Subtracting equation (28) from equation (27) yields
This equation was previously verified in the presence of Ramanonly interactions^{18}.
An additional relation between δω, τ_{s} and Ω is derived for steady state by substituting equations (30) and (32) into equation (27)
where Ω can be obtained from (29) and (32),
This result provides an independent confirmation of equation (24). Also, equation (33) is identical in form to an expression, which included only the Raman SSFS^{18}. Significantly, however, equation (33) is more general since Ω is the total spectral centre shift provided by the combined effects of Raman SSFS and dispersivewave recoil.
Phase noise transfer function
The repetition rate of the soliton can be expressed as follows^{19},
The variation in both D_{1} and Ω contribute to fluctuations in the repetition rate. While D_{1} is subject to thermorefractive noise and fluctuations from the environment, a significant contributor to fluctuations in Ω results from fluctuations in the pumplaser frequency detuning frequency, δω. The noise conversion from cavitypump detuning to repetition rate can be calculated by linearizing equations (27)–(31), , , , using the smallsignal approximation^{40}. Accordingly, all soliton parameters (X) can be expressed as X=X_{0}+ΔX, where X_{0} is the steadystate value and ΔX is a smallsignal fluctuation. For simplicity, we further denote the Raman and recoil terms in equation (29) as −8gτ_{R}/15τ_{s}−κ_{B}πrb_{r}^{2}≡κ_{A}τ_{s}F(δω) so that Ω=F(δω) is the function of detuning measured in Fig. 1d (that is, steadystate Ω versus δω). For simplicity, we assume this steadystate holds in the dynamical equations below. This can be shown to be true when offset frequencies (see definition below) are small compared to the cavity decay rate.
In the following derivation, τ_{s} in equations (27)–(31), , , , is eliminated using equation (32). Equation (29) can therefore be expressed as
Applying the smallsignal approximation and Fourier transform to equation (36) gives the result,
where is the Fourier transform of ΔX, ω is the Fourier frequency (that is, offset frequency in the phase or frequencynoise spectrum) and where the Fourier transform of ∂ΔX/∂t equals . represents the cavitypump detuning noise. Similarly, the smallsignal approximation applied to equation equation (27) yields,
where the contribution from is neglected as it is of order (ω/δω) smaller compared to the leadingorder terms.
In the limit of and , equations (37 and 38) are solved for in terms of . The result is substituted into the Fourier transform of equation (35) to give the following result,
where sources of noise associated with D_{1} in equation (35) are ignored.
The soliton repetition rate noise can be expressed as where α(ω), the noise transfer function, is the coefficient of in equation (39). Accordingly, the phase noise of repetition rate is S_{φ}(ω)=α(ω)^{2}S_{φ,δω}(ω).
Typically, for the resonators in this study ω<κ_{A} so that the first term in equation (39) expresses the trivial result that the slope of the plot in Fig. 3a, acts as a transfer function of fluctuations in δω into repetitionrate fluctutations. However, when ∂ω_{rep}/∂δω appoaches zero (the quiet point), the first term in equation (39) vanishes and the noise transfer function reaches a minimum determined by the second term. The phase noise plots in Fig. 3c were fitted using the same parameters as in analytical fitting in Figs 1c,d and 3a, and ∂ω_{rep}/∂δω extracted numerically from the fitting curves in Fig. 3a.
Analytical model fitting and parameters
Measurements are compared with the analytical model in Figs 1c,d and 3a. Measured parameters used for the analytical model are: κ_{A}/2π=2.12 MHz, D_{1}/2π=22 GHz, D_{2}/2π=17 kHz, G/2π=42.4 MHz. τ_{R}=2.49 fs can be extracted from the measured Ω in the regime without the mode recoil effect (δω/2π<30 MHz and δω/2π>40 MHz). Two free parameters are used to optimize the fitting in Figs 1 and 3 and they are in reasonable agreement with the measurement: Δω_{r−}=−62.2 MHz (−75±7 MHz in measurement) and κ_{r−}/2π=3.6 MHz (6 MHz in measurement). The procedure for fitting is as follows: a detuning frequency, δω, (horizontal axis in Figs 1c,d and 3a plots) is selected. By eliminating Ω in equations (33 and 34) a single condition relating τ_{s} and h_{r−}^{2} results. Likewise, with δω selected a second condition relating τ_{s} and h_{r−}^{2} results from equation (6) by replacing Ω_{Raman} using equation (26). This pair of equations is solved for τ_{s} and h_{r−}^{2} from which Ω is determined by equation (34) and ω_{rep} is determined by equation (3).
Numerical simulations
Numerical simulations based on the coupled LugiatoLefever equation of mode family A and B (equation (19) and Fourier transform of equation (8)) are implemented to further validate the analytical model. The Raman term in mode family B is ignored since the power in mode family B is too small to induce Ramanrelated effects. Dispersion of third order and higher as well as the selfsteepening effect^{56} are neglected. The simulations are implemented with the splitstep Fourier method^{56} where 2,048 modes in the frequency domain are taken into account. The parameters for two mode families used in Figs 2 and 4 are κ_{A}/2π=2.12 MHz, κ_{B}/2π=3.4 MHz, D_{1}/2π=22 GHz for mode A, D_{1B}/2π=D_{1}/2π+50.9 MHz for mode B, D_{2}/2π =17 kHz for both mode A and B, τ_{R}=2.489 fs, g=g_{B}=9.8 × 10^{−4} rad s^{−1} and G/2π=42.4 MHz.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Yi, X. et al. Singlemode dispersive waves and soliton microcomb dynamics. Nat. Commun. 8, 14869 doi: 10.1038/ncomms14869 (2017).
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Acknowledgements
We gratefully acknowledge the Defense Advanced Research Projects Agency under the QuASAR and PULSE programs, and the Kavli Nanoscience Institute. Xueyue Zhang gratefully thanks the Caltech SURF program and the Tsinghua University Top Open Program, Spark Program and Initiative Scientific Research Program (No. 20161080166).
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Author notes
 Xu Yi
 , QiFan Yang
 & Xueyue Zhang
These authors contributed equally to this work.
Affiliations
T.J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
 Xu Yi
 , QiFan Yang
 , Xueyue Zhang
 , Ki Youl Yang
 , Xinbai Li
 & Kerry Vahala
Department of Microelectronics and Nanoelectronics, Tsinghua University, Beijing 100084, China
 Xueyue Zhang
State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
 Xinbai Li
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Contributions
Experiments were designed by all authors. X.Y. and Q.F.Y. performed the measuerments. X.Y., Q.F.Y., X. Z., X.L. performed modelling. K.Y.Y. fabricated devices. Analysis of results was conducted by all authors. All authors participated in preparing the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Kerry Vahala.
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