Imaging soliton dynamics in optical microcavities
IImaging soliton dynamics in optical microcavities
Xu Yi ∗ , Qi-Fan Yang ∗ , Ki Youl Yang, and Kerry Vahala † T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA. ∗ These authors contributed equally to this work. † Corresponding author: [email protected] (Dated: May 22, 2018)
Solitons are self-sustained wavepackets that occurin many physical systems. Their recent demon-stration in optical microresonators has provideda new platform for study of nonlinear opticalphysics with practical implications for miniatur-ization of time standards, spectroscopy tools andfrequency metrology systems. However, despiteits importance to understanding of soliton physicsas well as development of new applications, imag-ing the rich dynamical behaviour of solitons inmicrocavities has not been possible. These phe-nomena require a difficult combination of high-temporal-resolution and long-record-length in or-der to capture the evolving trajectories of closely-spaced microcavity solitons. Here, an imagingmethod is demonstrated that visualizes solitonmotion with sub-picosecond resolution over arbi-trary time spans. A wide range of complex solitontransient behaviors are characterized in the tem-poral or spectral domain, including soliton for-mation, collisions, spectral breathing and solitondecay. This method can serve as a universal visu-alization tool for understanding complex solitonphysics in microcavities.
Temporal solitons are indispensable in optical fibersystems and exhibit remarkable nonlinear phenomena .The potential application of solitons to buffers andmemories as well as interest in soliton physics hasstimulated approaches for experimental visualization ofmulti-soliton trajectories. Along these lines, the displayof solitons trajectories in a co-moving frame allows anobserver to move with the solitons and is being used tomonitor soliton control and interactions of all types infiber systems . However, this useful data visualizationmethod relies upon soliton pulse measurements that areeither limited in bandwidth (pulse resolution) or recordlength. It is therefore challenging to temporally resolvesolitons over the periods often required to observe theircomplete evolution. For example, the time-lens method can provide the required femtosecond-resolution, but hasa limited record length set by the pump pulse. Also, whilethe relative position of closely-spaced soliton complexes can be inferred over time from their composite DFTspectra , Fourier inversion requires the constituent soli-tons to have similar waveforms which restricts the gen-erality of the technique.These limitations are placed in sharp focus byrecent demonstrations of soliton generation in microcavities . This new type of dissipativesoliton was long considered a theoretical possibility and was first observed in optical fiber resonators .Their microcavity embodiment poses severe challengesfor imaging of dynamical phenomena by conventionalmethods, because multi-soliton states feature inherentlyclosely spaced solitons. Nonetheless, the compactness ofthese systems has tremendous practical importance forminiaturization of frequency comb technology throughchip-based microcombs . Indeed, spectroscopysystems , coherent communication , ranging , andfrequency synthesis demonstrations using the newminiature platform have already been reported. More-over, the unique physics of the new soliton microcavitysystem has lead to observation of many unforeseenphysical phenomena involving compound soliton states,such as Stokes solitons , soliton number switching and soliton crystals .In this work, we report imaging of a wide range ofsoliton phenomena in microcavities. Soliton formation,collisions , breathing , Raman shifting as wellas soliton decay are observed. Significantly, femtosecond-time-scale resolution over arbitrary time spans (dis-tances) is demonstrated (and required) in these mea-surements. Also, real-time spectrograms are producedalong-side high-resolution soliton trajectories. Thesefeatures are derived by adapting coherent linear opti-cal sampling to the problem of microcavity solitionimaging. To image the soliton trajectories, a separate op-tical probe pulse stream is generated at a pulse rate thatis close to the rate of the solitons to be imaged in themicrocavity. The small difference in these rates causesa pulse-to-pulse temporal shift of the probe pulses rela-tive to the microcavity signal pulses as illustrated in fig.1a. By heterodyne detection of the combined streams,the probe pulses coherently sample the microcavity signalproducing a temporal interferogram shown in fig. 1a.Ultimately, the time shift per pulse accumulates so thatthe sampling repeats in the interferogram at the “framerate” which is described below, and is close in value tothe difference of sampling and signal rates. Probe pulseshave a sub-picosecond temporal resolution that enablesprecise monitoring of the temporal location of the soli-ton pulses. Moreover, the coherent mixing of probe andsoliton pulses allows extraction of each soliton’s spectralevolution by fast Fourier transform of the interferogram.In principle, the probe pulses can be generated by a sec-ond microcavity soliton source operating in steady state.However, in the present measurement, an electro-optical a r X i v : . [ phy s i c s . op ti c s ] M a y Time ( µ s)0 20 30 40 5010 T r an s m i ss i on ( a . u . ) C o -r o t a t i ng t i m e ( p s ) cde MI: parametric oscillation MI: non-periodic Single soliton a Multiple solitons0 20 30 40 5010 I n t en s i t y ( a . u . ) I n t e r f e r og r a m I n t en s i t y ( a . u . ) b InterferogramEnvelope A ngu l a r po s i t i on π P o w e r ( d B / d i v ) Wavelength (nm) W a v e l eng t h ( n m )
46 ps-2 -1 0 1 2 I n t en s i t y ~800 fs FIG. 1:
Coherent sampling of dissipative Kerr soliton dynamics. a,
Conceptual schematic showing micro-cavity signal (red) combined with the probe sampling pulse train (blue) using a bidirectional coupler. The probepulse train repetition rate is offset slightly from the microcavity signal. It temporally samples the signal uponphoto detection to produce an interferogram signal shown in the lower panel. The measured interferogram showsseveral frame periods during which two solitons appear with one of the solitons experiencing decay. b, Left panelis the optical spectrum and right panel is the FROG trace of the probe EO comb (pulse repetition period is shownas 46 ps). An intensity autocorrelation in the inset shows a full-width-half-maximum pulse width of 800 fs. c, Mi-croresonator pump power transmission when the pump laser frequency scans from higher to lower frequency. Mul-tiple “steps” indicate the formation of solitons. d, Imaging of soliton formation corresponding to the scan in panel c . The x-axis is time and the y-axis is time in a frame that rotates with the solitons (full scale is one round-triptime). The right vertical axis is scaled in radians around the microcavity. Four soliton trajectories are labeled andfold-back into the cavity coordinate system. The color bar gives their signal intensity. e, Soliton intensity patternsmeasured at four moments in time are projected onto the microcavity coordinate frame. The patterns correspondto initial parametric oscillation in the modulation instability (MI) regime , non-periodic behavior (MI regime),four soliton and single soliton states .(EO) comb is used . The EO comb pulse rate isconveniently adjusted electronically to match the ratesof various phenomena being probed within the microcav-ity.The soliton signal is produced by a 3 mm diameter sil-ica wedge resonator with FSR of 22 GHz and intrinsicquality factor above 200 million . The device gener-ates femtosecond soliton pulses when pumped at frequen-cies slightly lower than a cavity resonant frequency .To sample the 22 GHz soliton signal the EO comb wasformed by modulation of a tunable continuous wave(CW) laser. The EO comb features ∼ . The typical pump powerand laser scan speed are ∼
70 mW and ∼ − µ s,respectively.As described above, heterodyne-detection of the soli-ton signal and the EO-comb pulse produces the electricalinterferogram. The period of the signals in the intero-gram is adjusted by tuning the EO-comb repetition rate.In the initial measurements, it is set to ∼
10 MHz lowerthan the rate of the microcavity signal so that the nom-inal period in the interferogram is ∼
100 ns. To displaythe interferogram signal a co-rotating frame is applied.First, a frame period T is chosen that is close to the pe-riod of signals of interest in the interferogram. Integersteps (i.e., mT ) are plotted along the x-axis while theinterferogram is plotted along the y-axis, but offset intime by the x-axis time step (i.e., t − mT ). To make con-nection to the physical time scale of the solitons, the y-axis time scale is also compressed by the same bandwidth Time ( µ s)0 63 C o -r o t a t i ng t i m e ( p s ) C o -r o t a t i ng t i m e ( p s ) cba d Intensity (a.u.)0 1 Time ( µ s)0 84 M ode nu m be r -20200 C o -r o t a t i ng t i m e ( p s ) I n t en s i t y ( a . u . ) I n t en s i t y ( d B ) FIG. 2:
Temporal and spectral measurements of non-repetitive soliton events. a,
Two solitons collideand annihilate. b, Two solitons survive a collision, but collide again and one soliton is annihilated. c, Motion ofa single soliton state showing peak power breathing along its trajectory. A zoom-in view of the white rectangularregion is shown as the inset. d, Spectral dynamics corresponding to panel c . The y-axis is the relative longitudinalmode number corresponding to specific spectral lines of the soliton. Mode zero is the pumped microcavity mode.The soliton spectral width breaths as the soliton peak power modulates. The spectrum is widest when peak poweris maximum. The frame rate is 50 MHz for all panels.compression factor ( T × FSR) that accompanies the sam-pling process. The y-axis scale is accordingly set to spanone microcavity round-trip time. A typical measurementplotted in this manner is given in fig. 1d. Because thisway of plotting the data creates a co-rotating referenceframe, a hypothetical soliton pulse with an interferogramperiod equal to the frame rate T would appear as a hori-zontal line in fig. 1d. On the other hand, slower (higher)rate solitons would appear as lines tilted upward (down-ward) in the plot. In creating the imaging plot, a Hilberttransformation is applied to the interferogram followedby taking the square of its amplitude to produce a pulseenvelope intensity profile. The vertical co-rotating timeaxis can be readily mapped into an image of the solitonangular position within the circular microcavity as shownin fig. 1d.Imaging of soliton formation and multi-soliton trajec-tories is observable in fig. 1d. For comparison with thetransmitted power, the time-axis scale is identical in fig.1c and fig. 1d. As the pump laser frequency initiallyscans towards the microcavity resonant frequency its cou-pled power increases. At ∼ µ s the resonator enters themodulation instability regime . Initially, a periodictemporal pattern is observable in fig. 1d correspondingto parametric oscillation . Soon after, the cavity en-ters a regime of non-periodic oscillation. At ∼ µ s,this regime suddenly transitions into four soliton pulses.The soliton positions evolve with scan time and disap-pear one-by-one. All solitons have upward curved trajec-tories, showing that the soliton repetition rate decreases as the scan progresses. This soliton rate shift is causedby the combination of the Raman self-frequency shift ef-fect and anomalous dispersion in the silica resonator and a similar effect on soliton trajectory is observed inoptical fiber cavities . The features of soliton formationand evolution observed in fig. 1d compare well with nu-merical simulations presented in Supplementary fig. S2.Moreover, relative soliton positions can be extracted fromthe interferogram measurement (Supplementary fig. S3)and illustrate solitons stabilizing their relative positions.Movies of the corresponding multi-soliton motion aroundthe microcavity are also provided in the SupplementarySection. Finally, the cavity states at four moments intime are plotted within the circular microcavity in fig. 1e.These correspond to parametric oscillation, non-periodicmodulational instability, four soliton and single solitonstates.A variety of non-repetitive multi and single soliton phe-nomena were measured in both temporal and spectral do-mains. To enable more rapid imaging the repetation rateof EO comb was adjusted to produce an interferogram ata rate of approximately 50 MHz. The frame period, T ,was then reduced accordingly to approximately 20 nsec.Fig. 2a-b present observations of two solitons interact-ing. Soliton annihilation is observed in fig. 2a, whereintwo solitons move towards each other, collide, create anintense peak upon collision and then disappear. A newphenomena, a “wave splash”, is observed immediatelyfollowing the collision. In fig. 2b, two solitons collide butquickly recover and then collide again, after which point ab
400 600 800Time (ns)Resolution limit0.00.20.40.6 E n v e l ope ( a . u . ) P u l s e w i d t h τ s ( p s ) τ s A E ( a . u . ) FIG. 3:
Characterization of soliton decay. a,
Inter-ferogram envelope showing a single soliton experiencingdecay. An exponential fitting is given as the dashedblack line. b, The measured pulse width (blue) is plot-ted versus time and its resolution limit is set by the EOcomb pulse width. The product of soliton amplitudeand pulse width is plotted in red.one soliton is annihilated. Significantly, the observationof these complex motions requires measurement of eventsin close temporal proximity over long time spans. Figure2c-d shows measurement of a breathing soliton in boththe temporal and frequency domains. The spectrogram isobtained by applying a Fourier transform to the interfer-ogram signal . The spectrum is widest when the solitonhas maximum peak power. As an observation unrelatedto the breathing action, the soliton spectral envelope infig. 2d is continuously red shifted in frequency by theRaman self-frequency shift as its average power in-creases (increasing time in the plot).Finally, soliton decay is analyzed using the samplingmethod. The measurement results are shown in fig. 3. Inthe experiment, the pump laser frequency is continuouslytuned towards lower frequencies. After soliton formation,at some point the cavity-laser frequency detuning exceedsthe soliton existence range and the soliton decays .Fig. 3a shows the interferogram signal just before andduring the decay. Pulse widths ( τ s ) are extracted duringthe decay process and are plotted in fig. 3b. Also plottedin fig. 3b is the product of pulse width and soliton peakamplitude ( A E ). Curiously, the soliton pulse width andpeak amplitude preserve the same soliton product rela-tionship as prior to decay. This is an indication that thedecaying soliton pulse in the microcavity is constantlyadapting itself to maintain the soliton waveform. A sim-ilar behavior is known to occur for conventional solitonsin optical fiber . To the authors knowledge, this is thefirst time this behavior has been observed in real time.In the Method section the amplitude decay of the soliton in the interferogram trace is analyzed to extract a decaytime and the cavity Q factor.Coherent sampling induces a large bandwidth com-pression of the ultrafast signal that is equal to the sam-pling rate divided by the difference in the signal rate andthe sampling rate. This compression is well known inthe related techniques of dual comb spectroscopy anddual comb ranging , and is also present in sampling ofoptical signals by four-wave mixing in optical fibers .In order to avoid spectral folding, the compressed signalbandwidth must lie within half of the EO comb samplingrate (the Nyquist condition for sampling). As shownin the Method section, this basic condition establishes thefollowing relationship between temporal resolution ( τ ),frame rate ( f ) and the sampling rate (approximately themicrocavity free-spectral-range, FSR): f < τ FSR / used for coherent sampling. It is alsopossible to replace the EO-comb with a microcomb thatis closely matched to the FSR of a microcavity to be sam-pled. Such matching has been recently used to implementdual soliton microcomb spectroscopy measurements . Inthis case, even higher sampling rates would be possiblethat would enable GHz-scale frame rates. The coherentsampling method can serve as a general real-time statevisualization tool to monitor the dynamics of microcav-ity systems. It would provide an ideal way to monitorthe formation and evolution of soliton complexes such asStokes solitons , soliton number switching and soli-ton crystals . It can also be used to monitor the stateof chip-based optical memories based on microresonatorsolitons. Methods
Time constant in soliton decay.
In the soliton decay process,the average intracavity energy decays exponentially and its timeconstant equals the dissipation rate of the cavity ( κ = ω/Q ), where ω is the optical frequency and Q is the loaded cavity Q factor. Forlarge cavity-laser frequency detuning , the average intracavityenergy is approximately the soliton energy, τ s A E , such that τ s ( t ) A E ( t ) = τ s (0) A E (0) e − κt . (1)When the dissipation rate is relatively small compared to solitonKerr nonlinear shift, the dissipation is a perturbation and the pulsemaintains its soliton waveform . The corresponding balance ofdispersion and Kerr nonlinearity requires that the product of soli-ton amplitude and pulse width be constant. This condition wasalso verified experimentally in figure 3b , τ s ( t ) A E ( t ) = τ s (0) A E (0) . (2) Inserting eq. (2) into eq. (1) gives, A E ( t ) = A E (0) e − κt , τ s ( t ) = τ s (0) e κt , A E ( t ) = A E (0) e − κt . (3)In particular, the soliton amplitude decays at the cavity dissipationrate, the pulse width exponentially grows, and the soliton peakpower decays twice as fast as the cavity dissipation rate. In theexperiment, the fitted decay constant of the soliton amplitude is133 ns, which corresponds to κ/ (2 π ) = 1 . Q = 161million. This value is in reasonable agreement with the measuredloaded-Q factor of 140 million. Nyquist condition for sampling.
In the EO comb samplingprocess the optical to electrical conversion is accompanied by alarge bandwidth compression of the sampled signal. In effect, sam-pling stretches the time scale so that, for example, the opticaltemporal resolution ( τ ) is stretched to τ × FSR /f after conver-sion to the electrical signal where f is the frame rate given by f ≈ FSR − f comb . This stretching means that the THz EO combresolution bandwidth is compressed to an electrical bandwidth of f/ ( τ FSR). To avoid nonsensical signals in the electrical spectrum,the compressed bandwidth should lie within the Nyquist frequencyset by the FSR . This gives the condition f/ ( τF SR ) < FSR /
2, or f < τ
FSR /
2. In practice, when the oscilloscope bandwidth ( f osc )is smaller than the Nyquist frequency, the interferogram signal willbe limited by the oscilloscope instead of the Nyquist frequncy, such that f/ ( τ FSR) < f osc , or f < τf osc
FSR. This is, in fact, the casein the present measurement as the oscilloscope bandwidth is 4 GHzwhile the Nyquist frequency is 11 GHz. In addition, the frequencycomponents of the interferogram signal must be positive to avoidfrequency folding near zero frequency. This requires that the car-rier frequency of the interferogram signal is larger than half of theelectrical bandwidth. In the present measurement, the carrier fre-quency is the frequency offset between the EO comb pump laserand the microcavity pump laser (defined as ∆Ω). As a result, thiscondition is expressed as ∆Ω > f/ (2 τ FSR).
Data availability.
The data that support the plotswithin this paper and other findings of this study areavailable from the corresponding author upon reasonablerequest.
Acknowledgement
The authors thank Stephane Coen and Yun-Feng Xiaofor helpful comments during the preparation of thismanuscript and gratefully acknowledge the Air Force Of-fice of Scientific Research (AFOSR), NASA and the KavliNanoscience Institute. Kivshar, Y. S. & Agrawal, G.
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Supplemental Information: Imaging soliton dynamics in optical microcavities
Xu Yi ∗ , Qi-Fan Yang ∗ , Ki Youl Yang, and Kerry Vahala † T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA. ∗ These authors contributed equally to this work. † Corresponding author: [email protected]
SUPPLEMENTARY NOTE 1: EXPERIMENTAL SETUP
Microresonator signal setup Data acquisition
EDFA AOM µ DiskCW laser PC Oscilloscope10% % EO comb probe setup
EDFA WS EDFAPM1Amp PS ATT DCAmpPS AmpPSPM2 PM3 IMCW laser PC PC PDPDPD C o m b po w e r T r an s m i ss i on I n t e r f e r og r a m FIG. S1:
Experimental setup.
CW laser: continuous-wave laser; EDFA: erbium-doped-fiber-amplifier; AOM:acousto-optic modulator; BPF: bandpass filter; PC: polarization controller; PM: phase modulator; IM: intensitymodulator; PS: phase shifter; ATT: attenuator; Amp: RF amplifier; DC: DC voltage source; WS: opticalwaveshaper; FBG: fiber-Bragg-gating; PD: photodetector.Fig. S1 divides the experimental setup into three sections. In the microresonator section, a tunable, continuous-wave (cw) laser is used to pump the microcavity for production of solitons. An erbium-doped fiber amplifier (EDFA)amplifies its power to 500 mW and an acousto-optic modulator (AOM) is used for rapid control of power to themicrocavity. A tunable bandpass filter (BPF) is used to block the spontaneous emission noise from the EDFA. Thepump is coupled into the microcavity through a tapered-fiber S1 . The emitted power from the microcavity (alongwith transmitted pump power) is split by a 90/10 fiber coupler. 10 percent of the power is sent to a fiber-Bragggrating (FBG) filter to separate the pump power and the microcomb power. The drop port output is the pump powertransmission, while the through-port output is the comb power. Both the pump transmission and the microcombpower are detected with photodetectors (125 MHz bandwidth). The other 90 percent of the power is combined withthe electro-optic (EO) modulation comb sampling pulse using a second fiber coupler.In the EO comb setup, a pump laser is amplified by an EDFA to 200 mW and then phase modulated by threetandem lithium niobate modulators. The EO comb and microcavity setup can share the same pump laser when theacousto-optic modulator can provide a frequency offset higher than half of the electrical bandwidth of the interferogramsignal (to avoid frequency folding). This is the case in figure 1 of the main text. However, they can also use separatepump lasers, which is demonstrated in the main text from figure 2 to figure 3. The modulators are driven by amplifiedelectrical signals at frequency close to 22 GHz that are synchronized by electrical phase shifters. The output power ofthe electrical amplifiers is 33 dBm. The phase modulated pump is then coupled to an intensity modulator to select onlyportions of the waveform with a uniform chirp. The intensity modulator is driven by the recycled microwave signalfrom the external termination port of the first phase modulator. The modulation intensity and phase are controlledby an electrical attenuator and phase shifter. A programmable line-by-line waveshaper is used to flatten the EOcomb optical spectrum and to nullify the linear chirping so as to form a transform-limited sinc-shaped temporal pulse.The average power from the waveshaper output is around 100 µ W. The EO pulses are amplified by an EDFA beforecombining with the microresonator signal.In the interferogram measurement, the microcavity signal and the EO pulses are combined in a 90/10 coupler andare then detected by a fast photodetector with 50 GHz bandwidth. An FBG filter is used to block the pump laserof the microcavity to avoid saturation in the photodetector. All photodetected signals are recorded using a 4 GHzbandwidth, 20 GSa/s sampling rate oscilloscope.
SUPPLEMENTARY NOTE 2: SIMULATION OF SOLITON FORMATION
FIG. S2:
Simulation and measurement of microcavity soliton formation. a.
Simulated intracavity powerplotted versus time as the pumping laser is tuned across a cavity resonance from higher to lower frequencies. Thestep features correspond to the formation of solitons. b. Simulation results corresponding to panel a and showingthe formation of multiple solitons. The slow (horizontal) and co-rotating (vertical) time axes are defined in the maintext. In the simulation, the Raman effect and avoided mode crossing are included. c, d.
Measured solitontrajectories. The frame rate is 10 MHz for these measurements and the resonator is the same one described in themain text.The soliton formation process is governed by the Lugiato-Lefever equation (LLE) S2 augmented by Raman S3,S4 andavoided mode crossing S5 effects. The formation process can be simulated numerically using the split-step method S6 .The simulated intracavity power and temporal profile are presented in figure S2 (panels a and b, respectively). Thetime domain result is plotted in the slow and co-rotating time frame. In the simulation, the laser frequency is linearlyscanned from higher to lower frequency. For comparison, two measurement results showing soliton formation arepresented in figure S2 (c) and S2 (d). Concerning the vertical axis scale, it is noted that because the periodicity ofthe soliton interferogram signals varies by less than 1 %, the vertical co-rotating time axis can be readily mapped intosoliton angular position axis within the circular microcavity as shown in fig. 1d-e in the main text. SUPPLEMENTARY NOTE 3: MEASUREMENT OF RELATIVE SOLITON POSITION
The soliton positions can be extracted from the measurement by a peak-finding algorithm. One soliton is selectedto be the reference and is always positioned at the zero point of the angular position so as to eliminate the change insoliton repetition rate. The angular position is defined from − π to π . Four representative results are shown in figureS3. In the measurement, the laser frequency is scanned from high to low frequency. In panel (a) and (b), the solitonsstabilize relative to each other within a few µ s after formation. In panel (c), the relative soliton positions stabilizeimmediately after soliton formation. In panel (d), the relative soliton positions stabilize from 9 to 22 µ s and are thenobserved to destabilize. Note that at some point in time the solitons in all panels are annihilated when the laser tunesbeyond the existence detuning range. π - π/2 π/2π A ngu l a r po s i t i on (r ad ) - π - π/2 π/2π A ngu l a r po s i t i on (r ad ) - π - π/2 π/2π A ngu l a r po s i t i on (r ad ) - π - π/2 π/2π A ngu l a r po s i t i on (r ad ) Time ( µ s)Time ( µ s)Time ( µ s) Time ( µ s) ac db FIG. S3:
Measurement of relative soliton positions in multiple soliton states.
The positions of each solitonare measured relative to a reference soliton (located at angular position zero) and are plotted versus scan time. Thesampling rate for panel a and b is 10 MHz, while the rate is 50 MHz for panel c and d . The laser frequency isscanned from high to low frequency for all panels. SUPPLEMENTARY NOTE 4: NUMERICAL SIMULATION OF SOLITON COLLISION
Numerical simulation is used to reproduce soliton collisions using the method described in Note 2 above. Foursimulation results are shown in figure S4. Note the appearance of the soliton “splash” at points of annihilation. Thisphenomena is noted for observations presented in the main text.
SUPPLEMENTARY NOTE 5: NUMERICAL SIMULATION OF SOLITON ANNIHILATION
Numerical simulation (described in Note 2 above) is used to examine soliton properties during annihilation. Thelaser frequency scans from higher to lower frequency and when the cavity-laser detuning frequency exceeds the solitonexistence range, the soliton begins to decay. The calculated soliton amplitude ( A E ) and pulse width ( τ s ) from thesimulation are plotted in figure S5 (a). Their product τ s A E is shown to be approximately constant in figure S5 (b).An oscillation of the parameters is seen when the soliton amplitude decays to a small value. [S1] H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photon. 6, 369 (2012).[S2] L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).[S3] M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev.Lett. 116, 103902 (2016). Numerical simulation showing transient soliton scattering events. a,
Two solitons collide andannihilate. A soliton “splash” appears after annihilation. b, Two solitons survive a collision. c, Two solitons collideand merge into one soliton. d, One soliton hops in location when another soliton is annihilated. Parameters are setsimilar to experimental condition.FIG. S5:
Numerical simulation of soliton annihilation.