Cherenkov radiation induced symmetry breaking in counter propagating dissipative Kerr solitons

TL;DR

This paper reports the experimental observation of symmetry breaking in counter-propagating dissipative Kerr solitons caused by Cherenkov radiation, revealing new dynamics influenced by higher-order dispersion and enabling novel dual-comb spectrometer development.

Contribution

It demonstrates symmetry breaking in counter-propagating solitons due to Cherenkov radiation and explores its implications for microcomb applications.

Findings

Symmetry breaking observed despite symmetric pumping.

Group velocity differences induced by higher-order dispersion.

Potential for developing dual-comb spectrometers.

Abstract

The process of soliton Cherenkov radiation (also known as dispersive wave emission) in microresonator frequency combs plays a critical role in generating broadband and coherent microcomb spectra. Here, we report the observation of symmetry breaking in the group velocity of counter-propagating dissipative Kerr solitons, induced by polychromatic soliton Cherenkov radiation. Results show that in the presence of higher-order dispersion, the temporal arrangement of a multi-soliton state affects its group velocity, an effect which originates from the interference between multiple radiative tails emitted by individual solitons. Experimentally, we bidirectionally pump a microresonator with laser fields of equal power and frequency, and observe lifting of the degeneracy between the repetition rates of the counter-propagating solitons. The observation of symmetry breaking despite symmetric pumping conditions not only shines new light on the impact of dispersive waves on dissipative Kerr soliton dynamics, but also introduces a novel approach to develop coherent dual-comb spectrometers based on microcombs.

Equations

$${\omega }_{\mu }={\omega }_0+\sum _{j=1}\frac{D_j{\mu }^j}{j!},$$

$$\begin{gathered} \frac{\partial A}{\partial t}+i\sum _{j=2}\frac{D_j}{j!}(\frac{\partial }{i\partial \varphi }{)}^{j}A-ig|A{|}^{2}A= \\ \left(-\frac{\kappa }{2}+i({\omega }_0-{\omega }_p)\right)A+\sqrt{{\kappa }_{\mathrm{ex}}}\cdot s_{\mathrm{in}} \end{gathered}$$

$$A(\varphi )=A_{\mathrm{sol}}+A_{\mathrm{CW}}+A_{\mathrm{CR}}$$

$$\delta f_{\text{rep}}^{\text{CP}}=-\sum _{i\in \text{DW}}{\mu }_i\frac{D_2{\kappa }_i}{D_1\kappa }\left(\frac{P_i^{\text{cw}}}{E^{\text{cw}}}-\frac{P_i^{\text{ccw}}}{E^{\text{ccw}}}\right)$$

Full text

Introduction—Dispersive waves are of critical importance in the study of wave propagation in a wide range of fields including hydraulics Whitham (1965), solid mechanics Sachse and Pao (1978), magnetic systems Rogers et al. (2001) and ecology Frantzen and Van den Bosch (2000). In the optical domain, dispersive wave formation (also known as Soliton Cherenkov radiation) Akhmediev and Karlsson (1995) belongs to one of the most important nonlinear optical processes, and has played a decisive role in the development of self-referenced optical frequency combs Udem et al. (2002); Hänsch (2006) by enabling coherent octave spanning spectra to be attained via supercontinuum generation Ranka et al. (2000); Dudley et al. (2006); Skryabin and Gorbach (2010); Markos et al. (2017). These dispersive waves are emitted when the soliton phase matches a spectrally separated wave, and lead to oscillatory tails at either the trailing or leading edge of the soliton. In a similar vein, the observation of soliton Cherenkov radiation Brasch et al. (2017); Jang et al. (2014) in microresonator-based optical frequency combs (microcombs) Del’Haye et al. (2007); Kippenberg et al. (2011), has provided a method to generate broadband and coherent microcomb spectra via the formation of dissipative Kerr solitons (DKS) Kippenberg et al. (2018). In this context, Cherenkov radiation has been utilized for achieving self-referencing Brasch et al. (2017), octave spanning dual dispersive waves Spencer et al. (2018), self-locking of soliton group velocity (i.e. soliton repetition rate) Skryabin and Kartashov (2017), and deterministic single-soliton generation Bao et al. (2017). In addition, various works have shown that Cherenkov radiation may be used as a means of tuning the soliton group velocity and consequently the repetition rate of microcombs Lucas et al. (2017a); Yi et al. (2017); Cherenkov et al. (2017), which on the one hand may facilitate the agile control and the full stabilization of microcombs, but on the other hand constitutes a noise transduction mechanism for low noise microwave generation Yi et al. (2017); Liu et al. (2019).

Here we observe and explain how soliton Cherenkov radiations can lead to the symmetry breaking of the repetition rates of counterpropagating dissipative Kerr solitons in an optical microresonator. In contrast to earlier work that deliberately imposed an asymmetry in either power Joshi et al. (2018), or laser frequency Yang et al. (2017) to lift the repetition rate degeneracy, we observe that, counter intuitively, even for degenerate pump field parameters (i.e. power, polarization and frequency), the degeneracy in soliton group velocity can be lifted. To explain this phenomenon, we focus our attention on the interference between multiple dispersive waves that are emitted by the individual solitons forming a multi-soliton bound states. Our analysis reveals that the inter-soliton separation variation results in varied dispersive wave intensities through polychromatic Cherenkov radiation interferences. As a consequence the soliton recoil Akhmediev and Karlsson (1995); Milián and Skryabin (2014); Matsko et al. (2016) and the soliton group velocity differ, causing symmetry breaking of the repetition rates in microcombs with a different optical dissipative structure. We observe qualitative agreement between the experimentally observed symmetry breaking and our numerical simulations. The observation of symmetry breaking in the repetition rate, beyond highlighting novel nonlinear dynamics in DKS, constitutes a new method to generate frequency comb spectra for dual comb spectroscopy using a single degenerate pump field, and within a single whispering gallery mode family.

Influence of Cherenkov Radiation (CR) on the soliton group velocity—A well-known phenomenon of DKS in optical resonators with higher-order dispersion is that resonant CRs are emitted when phase-matched to the dissipative soliton Coen et al. (2013); Brasch et al. (2016). For an optical microresonator with resonance frequencies:

[TABLE]

where $\mu$ is the mode index number and $D_{j}$ is the j-th order dispersion, the approximate condition is met at a mode of index $\mu_{\mathrm{\footnotesize DW}}$ when the integrated dispersion, $D_{\mathrm{int}}$, obeys $D_{\mathrm{int}}(\mu_{\mathrm{DW}})\buildrel\text{d{}ef}\over{=}\omega_{\mu_{\mathrm{DW}}}-(\omega_{0}+D_{1}\,\mu_{\mathrm{DW}})=0$. As a consequence, the dispersive waves can originate from higher order dispersion terms (i.e. $D_{i\,>3}$) of the mode family which supports the DKS Brasch et al. (2016), or can occur due to local mode crossings between different spatial mode families Yang et al. (2016), which can drastically modify the local dispersion profile. Typically the CR manifests as a decaying modulation of the intracavity field background that is bound to the soliton Akhmediev and Karlsson (1995) and which provides a strong trapping potential for multi-soliton states Taheri et al. (2017). As a consequence, for a multi-soliton state, different stable binding sites (and therefore inter-soliton separations) exist, leading both to interference patterns in the spectral envelope Brasch et al. (2016); Herr et al. (2014) and to crystallized states Cole et al. (2017). Moreover, the CR constitutes a loss mechanism where energy is either radiated from the trail or tail of the solitons and induces a positive or negative shift of the soliton group velocity in a deterministic way (the “recoil” effect) Milián and Skryabin (2014); Bao et al. (2017); Yi et al. (2017). In prior works, CRs have been typically treated as single-period oscillations on the intracavity CW background which always constructively interfere when multiple solitons coexist Bao et al. (2017); Parra-Rivas et al. (2017); Vladimirov et al. (2018) thereby creating a periodic potential trap for the solitons with equidistant binding sites. In this picture, every binding site for the multi-soliton state is equivalent and will lead to constructive interference of the CR spectral component, producing a given group velocity shift. However, CRs are fundamentally polychromatic objects which contain multiple spectral components that interfere with each other and which jointly create aperiodic potential traps Wang et al. (2017) with irregularly spaced binding sites. In turn, the interferences, which are not always constructive due to the aperiodicity of the potential, result in the selective enhancement or suppression of specific dispersive waves and the subsequent recoils. Alltogether, these recoils lead to a global shift in the group velocity which is function of the inter-soliton distance. The concept is shown in Fig. 1 on the example of a two-soliton state, where two different temporal arrangements (i.e. separation) lead to different group velocity depending on the CR interference. We note that a similar aperiodic trapping effect was recently reported in the case of 2D soliton bound states Milián et al. (2018). Model— The numerical model describing the intracavity pulse dynamics is the Lugiato-Lefever equation (LLE) Lugiato and Lefever (1987):

[TABLE]

where ${{A}}$ is the envelope of the intracavity field, $\phi$ is the angular coordinate in the co-rotating frame, $g$ is the single photon Kerr induced nonlinear frequency shift, ${\kappa}$ is the cavity decay rate, ${\kappa_{\rm ex}}$ is the external fiber coupling rate, and ${|s_{\rm in}|^{2}}=\frac{P_{\rm in}}{\hbar\omega_{0}}$ is the driving photon flux, where $P_{\rm in}$ is the power of the main pump. In the absence of higher-order dispersion, the solution to the LLE is well approximated by the superposition of stationary solitons, $A_{\rm sol}$, and a CW background, $A_{\rm CW}$ Herr et al. (2014). However, when CR (originating from both higher-order dispersion and mode crossings) are taken into account, additional terms (which correspond to dispersive waves) are included in the solution. This can be expressed as Skryabin and Kartashov (2017):

[TABLE]

Here $A_{\rm CR}=\sum A_{\mu}\exp(i(|\mu|\phi-\theta_{\mu}))$, $\mu$ is the resonance mode number of the positive (higher frequency relative to the pump) and negative Cherenkov radiation peaks (for the pumped resonance $\mu=0$), and $\theta_{\mu}$ is the relative phase of each radiation mode. When multiple CR waves are present, they interfere with each other, and depending on the phase distribution, generate a particular modulation of the intracavity field. This function, which is complicated to derive analytically due to the complex dispersion profile, imposes aperiodic binding sites for the solitons in the multi-soliton states and, in turn, modifies the phase distribution until a bound state is formed when such mutual interaction reaches equilibrium. As depicted schematically in Fig. 1, for each of these possible temporal arrangements, the CRs interfere differently, causing different spectral recoils in the two directions, and breaking the perfect symmetry of the system.

Experiments–We choose to demonstrate the symmetry breaking effect in a configuration of two counter-propagating solitons in a single mode due to the perfect symmetry condition it offers. We generate DKS states in a crystalline MgF2 microcavity Hofer et al. (2010) ($n_{0}$=1.377, $n_{2}\sim$1.10*-20* m2/W) that has been used in other works Lucas et al. (2017a); Guo et al. (2017a). The microresonator has a free spectral range (FSR) of $D_{1}/2\pi$=14.09 GHz, the linewidth of the resonance that is pumped (at 1550 nm) is approximately $\kappa/2\pi\sim$ 100 kHz. The dispersion near 1550 nm is anomalous with $D_{2}/2\pi=$ 2 kHz and $D_{3}/2\pi=\mathcal{O}(\text{1\leavevmode\nobreak\ Hz})$. As shown in Fig. 2a, we pump a single spatial mode with clockwise (CW) and counter-clockwise (CCW) fields derived from the same laser source. By down-sweeping the laser frequency over the mode resonance we are able to generate soliton states in both directions, which can be analysed by observing the comb spectra and detecting the repetition rates as we tune the pump-resonance detuning. When both directions are single-soliton states, the repetition rates of both microcombs remain degenerate as the detuning is tuned over 16 MHz, despite several jumps in repetition rates, which are attributed to the bistability induced by single-mode dispersive waves Yi et al. (2017); Weng et al. (2019) (cf. Fig. 2). This is not surprising, as it has been reported that the back-scattering in microresonators can induce coupling between counter-propagating solitons and thus lead to a repetition rate locking effect Yang et al. (2017). Typically, to lift the degeneracy, one needs to introduce a differential repetition rate shift to overcome the locking effect, which can be done by pumping the two directions with different intensities or frequencies via the Kerr or Raman effects Yang et al. (2017); Joshi et al. (2018). In an auxiliary experiment, we verified that this locking effect was present when we introduce differential repetition rate shifts through CR-induced recoil effects by pumping the two directions with distinct frequencies (see SI). In the following experiments however, the power and frequency of the two driving fields are kept equal to avoid any parasitic influence of the pumps asymmetry in the symmetry breaking mechanism.

As a second experiment, a single-soliton state is generated in the clockwise direction and a double-soliton state in the counter-clockwise direction (see Fig. 3a) by state-switching to the desired DKS number Guo et al. (2017a). We then scan the detuning within the accessible soliton existence range, from 23 MHz to 13 MHz. As shown in Fig. 3d, now that one of the two counter-rotating solitons is in a multi-soliton state, we are able to observe particular detuning regions for which the two solitons exhibit non-degenerate group velocities. This alternation between degenerate and non-degenerate repetition rates depends on the differential strength of the dispersive wave-induced spectral recoils in the CW and CCW direction. The recoils are due to the joint effects of higher-order dispersion (mainly $D_{3}$) and multiple spatial mode-crossings. A thorough evaluation of the total effective repetition rate splitting between the two counter-rotating directions is then non-trivial to infer experimentally as it would require precise knowledge of all dispersive wave effects in both spectra. A simplified study is therefore conducted by considering only the differential recoil induced by the strongest dispersive wave at a particular detuning (see SI), which yields a reasonable quantitative estimation of the repetition rate splitting.

By choosing an appropriate detuning (e.g. 18 MHz as in Fig. 3a,3b (left panel) and 3c), we can make a dual-comb system from a single microresonator by generating counter propagating solitons with monochromatic pumping and equal power. Two distinct repetition rates are then detected on the ESA, see Fig. 3b (left panel), separated by 8 kHz. The weak peaks on the sides (50 dB below the main signals) are modulation sidebands due to the solitons regularly colliding in the microresonator. The dual-comb configuration is corroborated by the baseband structure observed on the ESA (see Fig. 3c). It consists of a comb of equidistant radio-frequencies starting from DC with a line spacing imposed by the difference between soliton group velocities. The comb structure starts at DC and is an experimental evidence that the two Kerr frequency combs that are generated are pumped by the same laser field. We emphasize that this result is fundamentally distinct from previous attempts in the literature that were either using different whispering gallery mode families Lucas et al. (2018), or counter-propagating modes with asymmetric pumps (unequal powers Joshi et al. (2018) or unequal frequencies Yang et al. (2017)).

This setup is then an ideal candidate for developing a dual-comb spectrometer where two combs with different repetition rates are used to probe the spectral information of an optical sample and map it to the RF domain Keilmann et al. (2004); Coddington et al. (2016); Suh et al. (2016). Here our repetition rate difference is in the 10 kHz range, and adjustable down to $\sim$1 kHz by acting on the pump-to-cavity detuning. It’s important to keep in mind that the RF comb which results from multi-heterodyne interference of the mixed combs has a repetition rate which is equal to the difference in individual comb repetition rates ($f_{\text{rep}}^{RF}=\Delta f_{\text{rep}}=f_{\text{rep}}^{\text{cw}}-f_{\text{rep}}^{\text{ccw}}$). This fact, combined with the exceptionally small $\Delta f_{rep}$ which we can achieve (along with the fact that pump degeneracy leads to the RF comb always beginning at DC), strongly relaxes the bandwidth requirements of the photodetector used, and leads to a very large optical-to-RF mapping factor of approximately $10^{6}$.

Simulations and discussion–We corroborate the soliton group velocity symmetry breaking mechanism with extensive numerical simulations based on the LLE as discussed above. The parameters in the simulation are derived from the experimental setup presented above (see SI for details). The results confirm that, when higher-order dispersion effects are included, a multi-soliton state will adopt a particular (and seemingly random) temporal arrangement that results in a particular dispersive wave intensity which yields a particular soliton group velocity (through spectral recoil). However, if this hypothesis is valid, then any two counter-rotating multi-soliton states with the exact same temporal arrangement must always have degenerate group velocities, due to the preserved symmetry. We experimentally verify this hypothesis by investigating two double-soliton states which exhibit (within our experimental precision) the same temporal distance. Here we scan the detuning in the usual manner, and show that in this case [Fig. 4b (upper panels)], the repetition rate degeneracy is never lifted, irrespective of detuning 111Note that this case is in principle extremely unlikely given the formidable number of possible temporal arrangements. However, over a hundred of instances, we observed this case a couple of times, directly emerging from the chaotic modulation instability state of the intracavity field (see SI).. Conversely, when the temporal separations are distinct (lower panels) we have a symmetry breaking of the soliton group velocities. This confirms our previous hypothesis, and demonstrates that polychromatic CR interference is the underlying mechanism of the observed group velocity symmetry breaking.

In conclusion, we have observed and explained a novel mechanism that leads to symmetry breaking of counterpropagating soliton group velocities, despite identical pump frequency and power, due to polychromatic CR radiations. This counter-intuitive work sheds new light on the role of dispersive waves in multi-soliton state formation and, more practically, is an important step towards achieving compact monolithic spectrometers which require only one pump.

I Supplementary material for: “Cherenkov radiation induced symmetry breaking in counter propagating dissipative Kerr solitons”

II Experimental setup

The light from a continuously tunable laser (Toptica CTL) is split in two paths, amplified in two indepedent EDFAs, frequency-shifted in two acousto-optic modulators (AOMs) and coupled evenescently to a MgF2 microresonator in counter-propagating directions via two fibered circulators connected at each end of a tapered fiber. The pump powers at the taper inputs are adjusted to the same power ($\sim 450$ mW). The third output ports of the circulators containing the counter-propagating lights are partially sent to two optical spectrum analyzers (OSA) while the remaining parts are recombined in a coupler and sent to an electrical spectrum analyzer (ESA) and an oscilloscope. The countinuous wave (CW) laser is offset sideband locked from one mode of the microresonator using a standard Pound-Drever-Hall technique Drever et al. (1983); Thorpe et al. (2008) in order to stabilize the effective detuning of the pump to the cavity resonance Lucas et al. (2017b); Weng et al. (2019). Dissipative Kerr soliton states are simultaneously generated in the two counter-propagating directions using the forward detuning method Herr et al. (2014). In order to stably access a detuning for which the two counter-propagating states are supported, we look simultaneously at the transmission from both rotating directions by detecting the mixed light on a photodetector, and adapt the detuning to land on the desired step after scanning the pump laser frequency.

We performed two main experiments: in the first one the AOMs drive frequencies are adjusted in order to induce a small frequency shift between the two pumps (typically several kHz), while in the second one (described in the main text) the AOMs are not present and a single drive is used. The microresonator used in our experiments is a crystalline MgF2 microcavity ($n_{0}$=1.377, $n_{2}\sim$ 1.10*-20* m2/W) that has been used in other works Lucas et al. (2017a, c); Guo et al. (2017a, b). The microresonator has a free spectral range (FSR) of $D_{1}/2\pi$=14.09 GHz, the linewidth of the resonance that is pumped is approximately $\kappa/2\pi\sim$ 100 kHz. The dispersion near 1550 nm is anomalous with $D_{2}/2\pi=$ 2 kHz and $D_{3}/2\pi=\mathcal{O}(\text{1\leavevmode\nobreak\ Hz})$. The effective mode area is $A_{\text{eff}}\sim$150 $\mu$m2.

III Non-degenerate pumps

In this experiment, the two counter-propagating soliton states are generated using different pump frequencies ($\delta\omega_{\text{\tiny Pump}}=\omega_{\text{\tiny CW}}\leavevmode\nobreak\ -\leavevmode\nobreak\ \omega_{\text{\tiny CCW}}\neq\leavevmode\nobreak\ 0$) hereby introducing an asymmetry in the system. The configuration is similar to what was reported by Yang et al. in the SiO2 microresonator platform Yang et al. (2017). The typical results obtained are displayed in Fig. 6. We observe that when the pumps are detuned from each other by more than 140 kHz, the repetition rates of the two counter-propagating solitons split and two repetition rate peaks are observable after detecting the mixed light from the two directions on the same photodiode. The lift of degeneracy is induced because of the different effective detunings of the pumps respective to the cold cavity resonance. However, here unlike previous works in SiO2 platforms, the tunability of the repetition rates with the detuning is not due to the Raman self-frequency shift Karpov et al. (2016); Yi et al. (2016) because the Raman gain in MgF2 material is spectrally narrow (the strongest phonon mode close to room temperature is typically a lorentzian having a linewidth of 250 GHz Porto et al. (1967); Grudinin et al. (2013)). In our case the tunability originates from the higher-order dispersion effects in the microresonator that leads to a modification of the group velocity of the soliton (and then the repetition rate) with the detuning 11footnotetext: A rapid estimation of the effect of the third order dispersion for a detuning of 1 MHz yields a repetition rate difference of $\delta f_{r}=\delta\omega D_{3}/(3\times 2\pi D_{2})\sim$100 Hz, the factor ten difference between the experimental and theoretical values can be explained in part by the error on the measurement of $D_{3}$, by the effect of mode-crossings and by the contibution from the Kerr shock bao2017soliton.Yang et al. (2016); Cherenkov et al. (2017). When the pump frequency difference is smaller than 140 kHz, the repetition rate splitting becomes small enough so that it cannot counteract the trapping effect of the backscattered light and the two solitons will lock to each other Yang et al. (2017), as is confirmed by the unique RF beat at 14.09 GHz in the RF spectrum (blue curve in the Fig. 6 (a)). This backscattering-induced soliton interlocking can be seen as a very efficient injection locking where the whole comb spectra participate to the locking. Besides, in this experiment we carefully choose the reference detuning $\delta\omega$ (the laser detuning when the two pumps are degenerate in frequency) so that no strong mode-crossing is disturbing the repetition rate splitting through abrupt changes in the repetition rate Matsko et al. (2016); Yang et al. (2016). We note that we find experimentally a locking range close to what was observed in a SiO2 platform Yang et al. (2017). This can appear surprising as the scattering effects should be much less pronounced in MgF2 materials, however we have a significantly higher quality factor (109 instead of 200$\times$106) that can compensate for this parameter and yield the same locking range. The asymmetric cross-phase modulation of the CW and CCW pump light can also yield a lift of degeneracy of the two counter-propagating DKS repetition rates when the pump power ratio is not equal to unity. In this case the differential pumps powers causes differential non-linear phase shifts in the two directions, which subsequently results in a difference in the repetition rates Del Bino et al. (2017). This phenomenon has been used to generate dual-comb from a single Si3N4 microresonator with a single pump frequency and repetition rate splitting at the MHz level were obtained Joshi et al. (2018). However in that case, the locking range due to backscattering was more than several MHz so that a power ratio was required to be typically superior to 10% to obtain a repetition rate splitting. In our case, given the material property of the MgF2 microresonator compared to Si3N4 chips ($n_{2}$ 25 times weaker, FSR 14 times lower and the effective mode area 150 times larger) this effect is extremely minute and a two-fold difference between the CW and CCW pumps would only induce a repetition rate splitting of a few tens of Hz that would not counteract the solitons interlocking due to backscattering. In our main experiment with a single pump frequency, we nevertheless keep the power difference within $\pm$5% to suppress the aforementioned effect.

IV Degenerate pumps: simplified evaluation of the differential repetition rate shift

As described in the main text, when we obtain a lift of degeneracy of the counter-rotating soliton group velocities due to the presence of a multi-soliton state with CR interference, a thorough evaluation of the repetition rates splitting would require taking into account the effect of every dispersive wave in each comb (see Fig. 7) and calculate the spectral recoils and the subsequent repetition rate shifts using a reformulation of the analytical expressions derived in Yi et al. (2017):

[TABLE]

where $\mu_{i}$ is the number of the mode in which the dispersive wave is generated (counted from the pump), $\kappa$ is the linewidth of the pumped mode generating the soliton, $\kappa_{i}$ is the linewidth of the crossing mode which is inducing the dispersive wave, $E^{\text{\tiny cw (ccw)}}$ is the circulating soliton energy and $P^{\text{\tiny cw (ccw)}}_{i}$ is the power of a given dispersive wave for the CW (CCW) solitons. The analysis can be singificantly simplified by noting that an intense single-mode dispersive wave is dominating in the comb spectra. The formula can then be evaluated for a single dispersive wave and yield an estimation of the repetition rate splitting. We look at the detuning region close to 18 MHz where we know from the experiment that the dispersive wave in the mode $\mu=$ -246 undergoes a sheer change in power in the CW comb but not in the CCW comb that is causing a repetition rate splitting of $\delta f_{\text{rep}}\sim$ 8 kHz (see main text). Evaluating (4) with the parameters of our microresonators and calculating $E$ and $P_{-246}$ from the experimental optical spectrum of the two combs, we find: $\delta f_{\text{rep}}\sim\frac{\kappa_{-246}}{\kappa}\times 2.5\,[\text{kHz}]$222The absolute power of the soliton combs and of the dispersive waves need not to be known as only the power ratio intervenes in the equation.. The power loss rate of the crossing mode causing the dispersive wave is unknown but the value required to match the experimental data is: $\kappa_{-246}\,\sim 3.2\,\kappa$, which is totally acceptable given that the crossing mode linewidth is slightly broader than the intrinsic linewidth of the pumped mode (we deliberately choose to pump the mode family with the highest quality factor). The important result behind this is that the right order of magnitude is found to explain the repetition rate splitting between the counter-propagating solitons as induced by a differential spectral recoil due to the strongest dispersive wave.

V Generation of the soliton states

In order to generate the different soliton states in the two directions we first sweep the diode laser frequency and use the forward tuning method Herr et al. (2014). Once the two counter-propagating solitons are generated, they are usually multi-soliton states. To obtain fewer soliton numbers in one of the direction, we scan the pump-to-cavity detuning by sweeping the offset frequency of the Pound-Drever-Hall phase lock loop Weng et al. (2019). The solitons number can then iteratively be decreased by using the backward tuning method Guo et al. (2017a). When generating dual two-soliton states with this method they usually exhibit different temporal arrangements. However, in the striking case where we obtained totally symmetric two-solitons states (with the exact same temporal arrangement) it was always obtained directly with the forward tuning method, emerging from the chaotic modulation instability state. It can be related to some synchronization mechanism occuring during the chaotic state, which would represent an interesting feature to investigate in future works.

VI Dispersive wave intensity versus inter-soliton separation: simulation and experiment

We experimentally generated two-soliton states in a repetitive manner and each time measured the intensity of a strong single-mode dispersive wave (SMDW) on the long-wavelength wing of the comb spectrum. The SMDW is indicated by a red arrow in the spectra presented in Fig. 7. We also derived the inter-soliton separations between the two solitons from the interference patterns of the comb spectra. After having generated 37 such soliton states while keeping the pump power and the effective detuning constant we plotted the measured SMDW intensity versus the inter-soliton separation in Fig. 8. The results show that the intensity of the SMDW is erratic with respect to the inter-soliton separation. We then measured the SMDW intensity for 6 different single-soliton states and we found that the measured intensity is constant in all 6 events, as indicated by the red line in Fig. 8. The comparison shows that in two-soliton states, the SMDW intensity is randomly distributed between the levels of total destructive interference (which corresponds to 0 in normalized intensity) and total constructive interference (which corresponds to 4 in normalized intensity).

To verify our postulation that the randomly distributed SMDW intensities are caused by the interference of polychomatic dispersive waves, we carried out numerical simulations based on the Lugiato-Lefever equation (LLE). In the simulation the second-order dispersion coefficient is set as $\frac{D_{2}}{2\pi}=2$ kHz. We introduce a SMDW by changing the resonance frequency of the corresponding mode (deviation of $\sim 48\kappa$ of the resonance of mode 400). We also use the approach detailed in other works Yi et al. (2017); Guo et al. (2017b), to introduce a local dispersion disruption due to the effect of mode coupling. The integrated dispersions of the pumped and coupled mode families are displayed in Fig. 9 (right panel). We set the coupling factor $\frac{g}{2\pi}=600$ kHz and the loss rate of the coupled mode family $\frac{\kappa_{\rm{c}}}{2\pi}=800$ kHz. Then we initiate the split-step simulation of two-soliton states by seeding the solitons with random inter-soliton separation. After a period of at least ten photon-decay times when the group velocities of the solitons and the inter-soliton separations settle, the SMDW intensity is derived from the simulated optical spectrum. These observations show that different inter-soliton separations in a multi-soliton state lead to different SMDW intensities, that we attribute to interference between polychromatic dispersive waves emitted by individual solitons.

From the same set of simulated data, we also calculated the DKS repetition rate shift by numerically fitting the intracavity motion of solitons. Fig. 10 (a) shows the relation between the repetition rate shift and the intensity of the SMDW. We observe that the repetition rate shift increases monotonically as the SMDW intensity rises, which is due to the increased soliton frequency recoil with stronger dispersive wave. Fig. 10 (b) shows the evolution of the intracavity field for one randomly seeded state. The inset shows a shot of the intracavity field, in which we also observe the dispersive wave as field oscillations of the CW background. This simulation confirms that different SMDW intensities will induce different shifts in the group velocity of a multi-soliton state. This can cause a symmetry breaking between the group velocities of counter-propagating solitons sharing the exact same pump, provided that the soliton states are not identical in each direction (i.e. different number of solitons or different inter-soliton separations).