DOI: 10.1038/NPHOTON.015.137 Mode-locked dark pulse Kerr combs in normal-dispersion microresonators Xiaoxiao Xue 1, Yi Xuan 1,, Yang Liu 1, Pei-Hsun Wang 1, Steven Chen 1, Jian Wang 1,, Dan E. Leaird 1, Minghao Qi 1,, and Andrew M. Weiner 1,* 1 School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue, West Lafayette, Indiana 47907-035, USA Birck Nanotechnology Center, Purdue University, 105 West State Street, West Lafayette, Indiana 47907, USA *amw@purdue.edu 1. Heterodyne beat note measurement The mode-locking transitions shown in Fig. 1d, Fig. 3d and Fig. 3g in the main text were also verified by measuring the beat note of a selected comb line with a narrow-linewidth reference laser. Figure S1 shows the results at different stages of the transition which coincide with the intensity noise measurements. The mode-locked state is indicated by the narrow linewidth of the beat note. a I b I c I Relative power (0 db/div) II III Relative power (0 db/div) II III Relative power (0 db/div) II III -1 0 1 Frequency (GHz) -1 0 1 Frequency (GHz) -1 0 1 Frequency (GHz) Fig. S1. Heterodyne beat note of the comb line with a narrow-linewidth reference laser. The short-term linewidth of the laser and the reference laser is <100 khz and <00 khz, respectively. The comb line tested is the first line to the red of the. Subplot a corresponds to Figs. 1d,e in the main text; subplot b corresponds to Figs. 3d,e; subplot c corresponds Figs. 3g,h. The resolution bandwidth of the electrical spectrum analyzer is 1 MHz.. Lugiato-Lefever (L-L) equation The mean-field L-L equation is given by [S1, S] Et, t i il ilet Et E t R 0,, in (S1) where Et, is the intra field; t slow time; fast time; R roundtrip amplitude loss; i t roundtrip time; ( ) roundtrip intensity loss due to absorption and scattering in the ; coupling i NATURE PHOTONICS www.nature.com/naturephotonics 1
DOI: 10.1038/NPHOTON.015.137 intensity loss (see Fig. S); 0 m 0 where 0 is the overall round-trip phase shift and m is the order of the closest resonance; the dispersion induced relative change of propagation constant within the detuning range is generally very small and can be neglected, the phase detuning is thus approximated as 0 ( 0 p) t R where p is the frequency and 0 is the resonance frequency closest to p ; L is the roundtrip length; d d the second-order dispersion coicient; nonlinear Kerr coicient; E 0 in external driving field (i.e., field). The normalized L-L equation is given by [S1] F t', ' t ' The normalization is performed as follows 1 ii i F t', ' Ft', ' S ' R (S) t t ' (S3) t ' (S4) L L F E (S5) where t ' is the slow time scaled with respect to the photon lifetime; L S Ein (S6) 3 0 (S7) sign( ) (S8) ' normalized fast time; F normalized intra field; S normalized field; phase detuning scaled with respect to the loss; sign of the second-order dispersion coicient. Note that Eq. (S) is normalized such that loss, Kerr coicient, and dispersion are all of unit strength. α i θ Fig. S. Microring resonator. 3. Modulational instability in normal-dispersion microresonators The CW steady-state solutions of Eq. (S) satisfy the well-known cubic equation [S1] where X S and Y F. For 3 3 X Y Y 1 Y (S9), Y X is single-valued; whereas for 3, Y X has the characteristic S-shape of the bistable hysteresis cycle (Fig. S3). The negative slope branch of the response curve is unstable with respect to CW perturbations, i.e., for intensities that lie in between the two values NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION 1/ Y [ ( 3) ] 3. It has been demonstrated that modulational instability may occur in normal-dispersion nonlinear cavities by optimizing the phase detuning and the intra intensity in the region [S1, S3, S4] 1Y, with. (S10) It refers to a fraction of the lower branch near the limit point (Fig. S3). In experiments, the laser frequency usually starts to the blue of the cold- resonance (Δ < 0) and is then continuously tuned toward the red (i.e., from higher to lower frequency) to overcome resonance red-shifting caused by thermal ect and Kerr ect [S5]. This process corresponds to continuously increasing the cold- phase detuning (under the sign convention in Eqs. (S1) and (S)) between the frequency and the cold-state while the power is fixed, and prevents the intra power getting to the modulational instability region on the lower branch (Fig. S4). Due to the Kerr ect induced resonance shift, the ective detuning may have sign different than the cold- detuning. The upper branch is ectively blue detuned while the lower branch is ectively red detuned [S5]. The modulational instability region on the lower branch is generally thermally unstable for microresonators [S6]. Although it might be possible to get to the modulational instability region by controlling the phase detuning and the power in tandem, experiments and simulations have shown that the intra power may switch to the upper branch due to the instability of Turing patterns in this region [S7]. No dark pulse formation has ever been found in this process. In the case of mode interaction, modulational instability may occur on the upper branch [S8, S9]. In experiments, it provides an easy way to get mode-interaction-aided initial comb lines which may act as a source for exciting dark pulses. 8 6 Upper branch Y Y 4 CW unstable Modulationally unstable Y = 1 Lower branch Y 0 0 5 10 15 0 5 30 Fig. S3. Steady-state intra intensity versus driving intensity when the phase detuning 5. X a 6 4 X = 4.95 = 5 = 3 = 3 1/ b 6 4 Y Y 0 0 10 0 30 X 0-5 0 5 10 Fig. S4. Intra intensity when the cold- phase detuning is continuously increased. a, Intra intensity versus driving intensity for 3, 3, 5. b, Intra intensity versus phase detuning. The driving intensity is 4.95. The intra intensity stays on the upper branch before it drops down to the lower branch. NATURE PHOTONICS www.nature.com/naturephotonics 3
DOI: 10.1038/NPHOTON.015.137 4. Pump line in the Pulse shaping experiments performed on the output comb field allow us to determine the waveform at the output waveguide (usually the through port). In order to reconstruct the comb field internal to the microresonator, we need to relate the internal field to the externally measured field. This takes some ort, as the line at the through port is the coherent summation of the output coupled fraction of the internal field with the input field transmitted directly to the through port. In the following we first outline a procedure whereby the phase of the internal field is obtained from the amplitudes of the internal comb lines; we test the obtained relation by comparing with simulation results (section 4.1). Then in section 4., we show how to use the comb spectrum measured at the through port, together with the results of section 4.1, to determine the internal field. Experimental results validating the procedure are shown in section 4.3. Sections 4.4 and 4.5 described methods for extracting the linear parameters and identifying the ective detuning region, respectively. 4.1. Retrieving phase from the comb spectral amplitude From the perspective of the line, comb generation will introduce an additional loss to the line. According to energy conservation, the ective loss for the line is given by all comb lines P P (S11) where P all comb lines is power of all the comb lines in the including the ; P is power of the line only. Suppose that the ective phase detuning for the line is denoted as, the intra line is then given by the build up E Ein i (S1) It is worth noting that is different from 0 in Eq. (S1) which represents the phase detuning between the frequency and the cold-state resonance. Here represents the phase detuning between the frequency and the ectively shifted resonance seen by the. Equation (S1) is only valid for coherent comb states in which the amplitude and phase of each comb line are stable, for which a time-invariant phase delay per round-trip can be found between the intra and the input. The formalism here is not intended to be applied for incoherent comb regimes such as chaotic states. Simulations (see for example Fig. S5) show that for bright pulse formation, the frequency is generally red detuned with respective to the shifted resonance (i.e., 0, as reported in [S5]); whereas for dark pulse formation, the frequency is generally blue detuned (i.e., 0 ). Assuming knowledge of the amplitude information of the intra comb spectrum and the coupling condition of the cold-state, one can calculate the phase of the line without knowing the phase of the other comb lines or the exact time-domain waveform. Once the stable intra comb spectral amplitude as well as the detuning region are known, can be obtained by solving the following equation E Ein i (S13) The phase of the line can then be calculated by substituting into Eq. (S1). Numerical simulations are performed based on the normalized L-L equation (Eq. (S)). In normalized form, equations (S11) and (S1) are given by 4 NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION all comb lines P P ' (S14) F S ' i (S15) Figure S5 shows the simulation results. The phase retrieved according to Eqs. (S11) (S13) agrees well with the actual phase. a Pump phase (rad) -1.0-1.6-1.3 Retrieved phase Actual phase ': 1.6518; : 4.9355; ': 1.3075; : 4.8433; 1 3 Soliton number ':.0313; : 4.9901; d Pump phase (rad) 0.39 0.36 0.33 0.30 ': 1.0701; : -0.431; ': 1.1474; : -0.4150; 1 3 Soliton number Retrieved phase Actual phase ': 1.330; : -0.3888; b 1 soliton c 1 soliton e 1 soliton f 1 soliton Amplitude (a.u.) solitons 3 solitons Relative power (0 db/div) solitons 3 solitons Amplitude (a.u.) solitons 3 solitons Relative power (0 db/div) solitons 3 solitons -40-0 0 0 40-100 -50 0 50 100 Fast time (a.u.) Mode number -40-0 0 0 40-100 -50 0 50 100 Fast time (a.u.) Mode number Fig. S5. Numerical simulations of retrieving the phase from the comb spectral amplitude. a c, Bright solitons. Three cases corresponding to 1,, 3 solitons are shown. The time-domain waveforms and frequency-domain comb spectra are shown in (b) and (c) respectively. d f, Dark solitons. Three cases corresponding to 1,, 3 solitons are shown. The time-domain waveforms and frequency-domain comb spectra are shown in (e) and (f) respectively. The simulation parameters are as follows: S.55, FSR 0.01, 5, 1 for subplots a c, 1 for subplots d f. The distance between the initial bright or dark pulses is randomly selected. 4.. Correction of the line measured at through port When the comb spectrum is measured at the through port, the line consists of two components: one from coupling out of the, the other from the bus waveguide. To reconstruct the time-domain waveform, the measured line at the through port should be corrected to represent the component from the. Suppose that the correction factor is denoted as E C from from E total from from bus E E E (S16) where from E and from bus E represent the component from the and the bus waveguide respectively. The NATURE PHOTONICS www.nature.com/naturephotonics 5
DOI: 10.1038/NPHOTON.015.137 ective loss for the line is then total total total total C P C P Pother comb lines (S17) where P E is the power at the frequency measured at the through port and P other comb lines is the combined power of the other comb lines, again measured at the through port. The component from coupling out of the is given by E from Ein i (S18) And the component from the bus waveguide E from bus 1 E (S19) in The amplitude drop of the line in comb generation compared to when the frequency is out of resonance can be measured in experiments and is given by L 1 E E i from from bus from bus E 1 (S0) The correction factor is given by i C 1 i (S1) Equations (S17), (S0) and (S1) are three independent equations for, and C, and can be numerically solved. For the microring shown in Fig. 1a of the main paper, the parameters are as follows 3 3.3978 10 ; 3 5.3613 10 ; total Pother comb lines P 0.8937 ; L 0.5957 ; The retrieved ective phase detuning, ective loss, and correction factor are 3 3.81 10 ; 3 4.563 10 ; C 0.3946 1.5659i (corresponding to amplitude 4. db and phase 1.3 rad). In reconstructing the time-domain waveform in the, the amplitude measured at the through port is increased by 4. db and the phase is shifted by 1.3 rad (which is inverse of the phase of C because of the different sign conventions we use in simulations and experiments). 6 NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION 4.3. Experimental results Here we show experimental results which validate the through-port correction method outlined above. The same microring shown in Fig. 3a of the main paper was ed with ~0.8 W. Since the ed power is reduced compared to that in Fig. 3 of the main paper, the comb has a narrower spectrum and different features. The comb spectrum also becomes more asymmetric with more comb power shifted to the longer wavelength range which falls in the passband of our pulse shaper. The comb was characterized both at the through port and the drop port. With the through port correcting method, the estimated complex amplitude gets very close to the drop port value. For the 1-FSR comb, the intensity agreement is improved from 8.14 db (without correction procedure) to 0.73 db; the phase agreement is improved from.19 rad to 0.01 rad. For the -FSR comb, the intensity agreement is improved from 4.5 db to 0.58 db; the phase agreement is improved from.6 rad to 0.14 rad. The reconstructed intra time-domain waveforms from the through-port (Figs. S6b,f) and the drop port (Figs. S6d,h) are very close to each other. The agreement validates our procedure to correct for the strong superimposed field in through port measurements (e.g., Fig. d of the main paper). The parameters used in correction are as follows. 3 3.0981 10, 3 1.937 10. For the 1-FSR comb, total Pother comb lines P 0.0879 ; L 0.7713 ; 3 3.6784 10 ; 3 4.5981 10 ; C0.498 0.3450i. For the -FSR comb, total Pother comb lines P 0.1439 ; L 0.6446 ; 3 1.9669 10 ; 4.1363 10 3 ; C0.4887 0.4365i. NATURE PHOTONICS www.nature.com/naturephotonics 7
DOI: 10.1038/NPHOTON.015.137 Fig. S6. Experimental results of correcting the through-port complex. a d, 1-FSR comb. e h, -FSR comb. a & e, Comb spectrum measured at through port. c & g, Comb spectrum measured at drop port. b & f, Reconstructed time-domain waveform based on through-port data. d & h, Reconstructed time-domain waveform based on drop-port data. Inst. freq.: instantaneous frequency. 4.4. Extracting parameters The loss and the bus waveguide coupling coicient can be extracted from the measured loaded quality factor ( Q loaded ) and through-port extinction ratio (EXR), i.e. solving the following equations Q t 0R loaded, (S) EXR, (S3) where 0 is the resonance frequency and t R is the round-trip time. The EXR is defined as the complex amplitude when the is out of resonance over that when the is in the resonance center. EXR is positive (negative) when the is under (over) coupled. The coupling condition can be characterized by measuring the phase response of the microresonator. Figure S7a shows the experimental setup of sideband sweeping method. The tunable laser is tuned close to the resonant frequency, and is modulated by a radiofrequency signal through single-sideband modulation. By sweeping the radiofrequency, the 8 NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION sideband sweeps across the resonance of the microresonator. The response of the microresonator is then transferred to the electrical domain through beating of the sideband with the carrier. Figure S7b shows the measured response of the microring shown in Fig. 1a of the main paper. The phase curve indicates that the resonator is over-coupled. Figure S7c shows the measured result of the microring in Fig. 3a of the main paper. Since this microring has a drop port which introduces an additional coupling loss comparable to the through port, the phase curve indicates an under-coupled condition. Fig. S7. Measuring the amplitude and phase response of the microresonator. microring in Fig. 1a and the microring in Fig. 3a of the main paper, respectively. a, Experimental setup. b & c, Results for the 4.5. Identifying the ective detuning region In Fig. S6, the estimated intra phase based on the blue detuning condition agrees well with the actual value measured at drop port. The good agreement confirms that the ective detuning is indeed in the blue region. If ective red detuning is assumed, poor agreement is obtained. For the microring with no drop port shown in Fig. 1a of the main paper, the ective detuning can be identified by using the method shown in Fig. S8a which is in principle similar to the method of detecting the Pound Drever Hall (PDH) signal employed in [S5]. A dithering voltage is applied on the microheater. The line power after the microring is modulated because of the slight shift of the resonance. The detuning region can be distinguished by comparing the phases of the dithering signal ( V d ) and the line variation (converted to a voltage V o through a photodetector). Example curves of V d and V o for the cold (microring shown in Fig. 1a of the main paper) are shown in Fig. S8b. When the laser wavelength is shorter than the resonant wavelength (i.e., blue detuned), V o is in phase with V d. In the other case of red detuned region, V o is out of phase with V d. The measured result for the ed under dark pulse action (example comb spectrum shown in Fig. 1d III of the main paper) is shown in Fig. S8c. The curve of V o is in phase with V d suggesting an ectively blue detuned region. NATURE PHOTONICS www.nature.com/naturephotonics 9
DOI: 10.1038/NPHOTON.015.137 b (V) V ~ d 0.1 0.0-0.1 a Pump laser Polarization controller Blue detuned Vc Red detuned ~ Vd - + ~ Microresonator -0-10 0 10-9 0 Time (ms) 9 6 3 0-3 -6 (mv) Resonance Blue detuned Red detuned λ Microheater Pump laser V ~ o Fig. S8. Identifying the detuning region. a, Experimental setup. The tunable filter after microresonator is used to select the line in comb generation. By changing the dc voltage ( V ) applied on the microheater, the detuning can be changed to c either blue detuned ( laser wavelength shorter than resonant wavelength) or red detuned ( laser wavelength longer than resonant wavelength). b, Example curves of V d and V o for the cold (microring shown in Fig. 1a of the main paper) measured under low power, showing that V o is in phase (out of phase) with V d in the blue (red) detuned region. The frequency of the dithering voltage ( V d ) is 100 Hz. c, Measured result for the ed in comb generation. V o is in phase with V d suggesting an ectively blue detuned region. The wavelength is 1549.3 nm. The measurement is done after the comb transitions to a low-noise mode-locked state related to dark pulse formation (example comb spectrum shown in Fig. 1d III of the main paper). c (V) V ~ d Tunable filter 0. 0.1 0.0-0.1 Photodetector ~ V o 0.4 0. 0.0-0. -0. -0.4-0 -10 0 10 0 Time (ms) (mv) V ~ o 5. Complex structure of dark solitons Figure S9 shows the retrieved comb phase and reconstructed time-domain waveform measured over a larger spectral range for the dark pulse in Fig. e of the main paper. Here a Finisar WaveShaper 4000S which can operate in both the lightwave C and L bands was used to shape the comb, allowing access to more comb lines. Compared to Fig. e, here the extra comb bandwidth used in the reconstruction yields sharper edges and stronger frequency modulation. The chirped ripples at the bottom of the dark pulse also become more pronounced. Unlike bright soliton pulses in the anomalous dispersion region, dark pulses can have complex and quite distinct features. Numerical simulations are performed to model the comb generation behavior for this microring. The power is 0.3 W. The initial intra field is the steady-state continuous-wave solution on the upper branch of the bistability curve plus weak noise (~ 1 pw/mode). The phase detuning at the beginning is 3 10 rad ; and an additional phase shift of 0.815 rad per roundtrip is applied to modes 54 and 55. Figure S10a shows the evolution of the comb spectrum versus the slow time; figures S10b and S10c show the transient comb spectrum and time-domain waveform at different time. The comb grows up and shows some random variations with the slow time, corresponding to a high intensity noise which is similar to the experimental observation (see Fig. 1e I of the main paper). The phase detuning is increased to 3.75 10 rad after 60 ns; and the additional phase shift applied to modes 54 and 55 is set to 0. The field then evolves to a stable dark pulse. The width of the dark pulse is ~1.1 ps which agrees well with the experimental result. 10 NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION Fig. S9. Characterization of the dark pulse shown in Fig. e of the main paper with a larger pulse shaping range. The wavelength is 1549.3 nm. a, Comb spectrum and phase. The red circles are the comb phases retrieved experimentally. The green triangles correspond to additional comb lines that fall outside of the pulse shaper operating band, for which we assume phases based on symmetry about the line. b, Reconstructed time-domain waveform. The comb lines used for reconstruction contain 99.9% of the total comb power excluding the (comb lines with green triangle phases based on a symmetry assumption account for 7% of the power excluding ). Inst. freq.: instantaneous frequency. Fig. S10. Numerical simulation for the microring in Fig. 1a of the main paper. a, Evolution of the comb spectrum versus the slow time. b & c, Transient comb spectra and waveforms at different slow times. Inst. freq.: instantaneous frequency. Since the general L-L equation uses an exp( it) convention [S1, S] while our experimental measurements use the exp( it) convention common in ultrafast optics [S10], the comb phase from the simulation is adjusted to be consistent with the exp( it) convention in order to facilitate comparison of subplots b with the experiments. 6. Fronts and dark pulses In simulations, we found that the formation of dark pulses and breathers is related to interactions of fronts which connect the two stable steady-state solutions in the [S11-S13]. Fronts can be formed from weak perturbations in an externally driven Kerr resonator subject to normal dispersion [S14]. To show the physics more clearly, we did some simulations using the normalized L-L equation. We found that dark pulses with different features can be excited by different initial intra fields even when the driving amplitude and the phase detuning are the same. Figure S11 shows the results of one simulation example. The phase detuning 5 3 so that bistable steady-state intra solutions exist (Fig. S11a). The initial field is a square dark pulse which contains two edges connecting the upper and lower branches (Fig. S11b). The amplitude and phase of the dark pulse top are equal to the amplitude and phase of the upper branch, while the amplitude and phase of the bottom are equal to those of the lower branch. The distance between the two edges (i.e., the width of the initial dark square pulse) is 6. The stable structure evolved from the initial NATURE PHOTONICS www.nature.com/naturephotonics 11
DOI: 10.1038/NPHOTON.015.137 field is shown in Figs. S11c,d. Figures S11e,f show another possible stable solution which is evolved from a slightly narrower square dark pulse (of which the initial width is 4). Breathers which are oscillating dark pulses in slow time are also possible. Figures S11g,h show a breather which is excited by a square dark pulse with a width of, and then transitions to a stationary dark pulse when the phase detuning is slightly increased (Fig. S11g) or the driving amplitude is slightly reduced (Fig. S11h). This transition is similar to that observed in our experiments (see Fig. 1d and Fig. 3d of the main paper). Fig. S11. Numerical simulation of fronts and dark pulses in normal dispersion region. a, Steady-state intra amplitude versus the external driving amplitude. b, Initial intra field which is a square dark pulse. The width is 6. The phase detuning 5. The driving amplitude S.55. The top amplitude and phase are equal to the upper-branch steady-state values, while the bottom amplitude and phase are equal to the lower-branch steady-state values. c d, Stable dark pulse evolved from the initial field shown in subplot b. c, Time-domain amplitude and phase. The color shows frequency chirp. Inst. freq.: instantaneous frequency. d, Frequency-domain comb amplitude and phase. e f, Another possible solution of dark pulse with different features compared to subplots c & d. The pulse is excited by an initial square dark pulse with a width of 4. e, Time-domain amplitude and phase. f, Frequency-domain comb amplitude and phase. g h, Breather which is excited by an initial square dark pulse with a width of. The breather transitions to a stable dark pulse after (g) the phase detuning is increased to 5.5 or (h) the driving amplitude is reduced to.45 at slow time 33. As in the previous figure, the phase plotted in subplots b, d, f is adjusted to conform to the exp( i t) convention used in ultrafast optics, thereby facilitating comparison with our experimental plots. 1 NATURE PHOTONICS www.nature.com/naturephotonics
DOI: 10.1038/NPHOTON.015.137 SUPPLEMENTARY INFORMATION References for supplementary information [S1] Haelterman, M., Trillo, S. & Wabnitz, S. Dissipative modulation instability in a nonlinear dispersive ring. Opt. Commun. 91, 401 407 (199). [S] Coen, S., Randle, H. G., Sylvestre, T. & Erkintalo, M. Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato Lefever model. Opt. Lett. 38, 37 39 (013). [S3] Coen, S. & Haelterman, M. Modulational instability induced by boundary conditions in a normally dispersive optical fiber. Phys. Rev. Lett. 79, 4139 414 (1997). [S4] Hansson, T., Modotto, D. & Wabnitz, S. Dynamics of the modulational instability in microresonator frequency combs. Phys. Rev. A 88, 03819 (013). [S5] Herr, T., et al. Temporal solitons in optical microresonators. Nature Photon. 8, 145 15 (014). [S6] Carmon, T., Yang, L. & Vahala, K. J. Dynamical thermal behavior and thermal self-stability of microcavities. Opt. Express 1, 474 4750 (004). [S7] Coen, S., et al. Bistable switching induced by modulational instability in a normally dispersive all-fibre ring. J. Opt. B: Quantum Semiclass. Opt. 1, 36 4 (1999). [S8] Savchenkov, A. A., et al. Kerr frequency comb generation in overmoded resonators. Opt. Express 0, 790 798 (01). [S9] Liu, Y., et al. Investigation of mode coupling in normal-dispersion silicon nitride microresonators for Kerr frequency comb feneration. Optica 1, 137 144 (014). [S10] Weiner, A. M. Ultrafast Optics (John Wiley & Sons, 009). [S11] Rosanov, N. N. Spatial Hysteresis and Optical Patterns (Springer, 00). [S1] Rosanov, N. N. Transverse patterns in wide-aperture nonlinear optical systems. Progress in Optics XXXV, 1 60 (Elsevier, 1996). [S13] Boardman, A. D. & Sukhorukov, A. P. Soliton-driven Photonics (Springer, 001). [S14] Malaguti, S., Bellanca, G. & Trillo, S. Dispersive wave-breaking in coherently driven passive cavities. Opt. Lett. 39, 475 478 (014). NATURE PHOTONICS www.nature.com/naturephotonics 13