Mode-locking in an optical microresonator via soliton formation

T. Herr,1 V. Brasch,1 J.D. Jost,1 C.Y. Wang,1 N.M. Kondratiev,2 M. L. Gorodetsky,2, 3, ∗ and T. J. Kippenberg1, †

1Ecole Polytechnique Federale de Lausanne (EPFL), 1015, Lausanne, Switzerland2Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow 119991, Russia

3Russian Quantum Center, Skolkovo 143025, Russia

The discovery of passive mode-locking of lasers via saturable absorbers[1] has led tooptical femto-second pulses with applications ranging from eye surgery to the analysisof chemical reactions on ultra-short timescales[2, 3]. In the optical frequency domaina train of such pulses corresponds to a frequency comb (equidistant optical laser linesspaced by the pulse repetition rate)[4, 5], which find use in precision spectroscopy andoptical frequency metrology[6, 7]. In an alternative approach, not relying on mode-locking, frequency combs can also be generated in continuously driven, Kerr-nonlinearoptical microresonators via cascaded parametric four-wave mixing[8–14]. Applyinga pulse-shaping mode-locking mechanism could enable compact and robust femto-second pulse generators. However, saturable absorbers are challenging to apply tomicroresonators as they affect the high quality factor essential to nonlinear frequencyconversion. Here, we demonstrate passive mode-locking in microresonators via soli-ton formation. In contrast to soliton mode-locked lasers [15] a stabilizing saturableabsorber is not required due to the parametric gain mechanism. We observe the gen-eration of pulses with 200 fs duration and low noise frequency comb spectra with lowline-to-line power variation. Numerical modeling of the nonlinear coupled mode equa-tions is used to physically understand the mode-locking and in combination with ananalytical description allows the identification of mode-locked regimes. The presentedresults open the route towards compact, high repetition-rate femto-second sources,where the operating wavelength is not bound to the availability of laser gain mediaor saturable absorbers. The smooth optical soliton spectra are essential to frequencydomain applications such as channel generators in advanced telecommunication or infundamental studies such as astrophysical spectrometer calibration[12, 16–18]. More-over, femto-second pulses in conjunction with external broadening provide a viableroute to a microresonator RF-to-optical link[5, 19].

Driven by a wide range of application in telecommunications[18], spectroscopy[20], astronomy[16, 17], and com-pact optical clocks[21] the new class of microresonator[22] based frequency combs has rapidly evolved in variousmaterials and geometries[12]. Recent advances include the demonstration of octave spanning spectra[23, 24], fullphase stabilization[13, 25], low phase noise microwave generation[9, 13, 25, 26], fully planar CMOS compatibledesign[10, 11, 27], extension of spectral coverage to mid-infrared wavelength[28], as well as arbitrary optical waveformgeneration[29]. Microresonator based combs rely on cascaded-four-wave mixing (FWM, cf. Fig 1c) mediated by theparametric Kerr-nonlinearity[8]. A continuous-wave (cw) pump laser is converted into equally spaced optical modes,where the mode spacing corresponds to the free-spectral range (FSR) or, equivalently, the inverse resonator roundtriptime 1/TR. The parametric four-wave mixing process results in coupled phase relations between the optical modesevidenced by a narrow radio-frequency (RF) beat note generated by the beating between neighboring comb lines.This phase-locked state leads to the generation of a periodic time domain output (cf. Fig 1d), which however does notgenerally correspond to pulses (which would occur in a mode-locked state). External line-by-line phase and intensityadjustment have been used after comb generation for pulse-shaping[26, 29], but this is restricted to a small number ofcomb modes. As opposed to phase-locking, mode-locking implies a phase synchronization and intensity adjustmentinside the microresonators, such that stable optical pulses are formed (cf. Fig 1d) circulating inside the resonator witha repetition rate of frep = 1/TR. Beside ultra-short pulses, microresonator mode-locking would result in low-noisefrequency comb spectra, with little line-to-line variation and without spectral gaps. The latter is difficult to reach insystems with mode-spacings below 100 GHz, where multiple, inconsistent subcombs may form [30].

Introducing a saturable absorber or an equivalent optical element into the cavity[1, 3], which is possible in conven-tional laser systems is hindered by the requirement of a high optical quality factor Q. Still, mode-locking mechanismsintrinsic to microresonators could be present. For instance saturable absorption could be induced via self-focusing,affecting both the intermode coupling inside the resonator as well as the coupling to the waveguide. Self-focusing

∗Electronic address: michael.gorodetsky@gmail.com†Electronic address: tobias.kippenberg@epfl.ch

2

could also lead to an intensity dependent surface loss. Moreover, a small saturable component of material absorptioncould facilitate mode-locking. Interestingly, pump laser detuning dependent transitions from high to low phase noisecomb states have been observed[14, 24, 30] and interpreted as a signature of potential phase-locking [14, 24, 31], butan understanding is still lacking. Theoretical analysis of the damped, driven nonlinear Schrodinger equation (NLS)suggests that mode-locked regimes could exist in microresonators[32]. It has also been proposed that temporal, dis-sipative Kerr-cavity solitons may form in microresonators [33, 34]. This soliton formation, in essence a pulse shapingmechanism, could in principle lead to mode-locking similar to soliton mode-locked lasers, where pulse duration muchshorter then the recovery time of the saturable absorber are achievable[15]. In contrast to soliton mode-locked lasers,which still require a saturable absorber to stabilize the soliton against a growing cw background field [15] a saturableabsorber is not neccessary in the present case. This fact is a consequence of the different gain mechanism. As opposedto a laser gain medium the phase-coherent and frequency-selective parametric gain does not support the growth ofa destabilizing cw background (cf. SI for details). Indeed, stable solitons have been induced experimentally in acontinuously driven fiber-loop cavity via control pulses[33]. To date, however, no definitive proof nor mechanism ofpulse-forming mode-locking has been found in microresonators.

To reveal potentially mode-locked states we scan the pump laser (wavelength 1553 nm) with decreasing opticalfrequency ωp over a high-Q resonance of a crystalline MgF2 resonator (Q = 4 × 108, free-spectral range FSR=35.2GHz; cf. Fig. 1a and Methods) [28, 30, 35]. The RF signal (electronically down-mixed to 20 MHz), serves as anindicator of mode-locked states. We sample and subsequently Fourier-transform the RF signal, while the laser scanis performed. In mode-locked states the resulting spectral beat signal is expected to be a single low-noise peakcorresponding to the pulse repetition rate. This criterion is necessary but not sufficient and may also be observed inphase-locked states.

Fig. 2b shows the evolution of the optical spectrum during the laser scan. Reducing the laser-cavity detuning leadsto a build-up of intra-cavity power and once a critical power threshold is reached widely spaced primary comb modesare generated, followed by secondary modes filling in the spectral gaps as frequently observed[29, 30, 35]. The increaseof intra-cavity power also results in a shift of the resonance frequency towards lower frequencies via a combined effectof thermal expansion, thermal refractive index change(induced by absorptive heating) and Kerr-effect. This leadsto a non-Lorentzian, triangular resonance shape (thermal triangle) when the pump laser is scanned with decreasingoptical frequency over the resonance (cf. Fig 2a, inset)[36, 37].

The Fourier-transformed, sampled RF signal is contained in Fig. 2c. We observe a transition from a broad, noisyRF signal to a single, low-noise RF beat note. Strikingly, this transition coincides with the beginning of a seriesof discrete steps in the transmission, which deviates markedly from the expected thermal triangle (the RF beatnoteremains narrow throughout all steps). This observations is highly indicative of mode-locking. Importantly, the stepsare only readily observed transiently when scanning the pump laser quickly (within a few milliseconds) over theresonance such that thermal resonance shifts are small. We note that the observations of the step signature is nota spectrally local particularity but is observed for all resonances of the same mode family (verified for 34 adjacentresonances).

To understand the intriguing observations of discrete steps in the transmission, and the possible connection tomode-locking we carry out numerical simulations based on the coupled mode-equations approach[38]. The simulatedsystem corresponds to a MgF2 microresonator with resonance frequencies ωµ = ω0 + D1µ + 1

2D2µ2 + 1

6D3µ3, where

D1 = 2π ·FSR, D2 and D3 correspond to the FSR, second and third order dispersion around the resonance frequencyω0 of the pumped mode[30] with mode index µ = 0. The remaining resonator parameters are refractive and nonlinearindices as well as the effective mode-volume. We neglect effects of non-unity mode-overlap, interactions with othermode families, any particularities of the resonator geometry and thermal effects. The resulting set of coupled modeequations is propagated in time using an adaptive step-size Runge-Kutta integrator (cf. methods and SI). Results ofa numerical simulation including 101 comb modes are shown in Fig. 3. The blue curve in Fig. 3a shows the simulatedintracavity power as function of the normalized detuning ζo = 2(ω0 − ωp)/κ of the pump laser ωp from the coldresonance frequency, where κ = ω0/Q denotes the cavity decay rate. Note that due to the nonlinear Kerr-shift of theresonance frequency, the effective detuning between pump laser and resonance is smaller than ζ0. The blue curve ismirrored horizontally for convenience when comparing to the equivalent experimental transmission trace in Fig. 2a(an increased transmission corresponds to a drop in intracavity power). Remarkably, the step features are very wellreproduced, implying that the simulation includes all relevant physical mechanism. In agreement with the experimentthe number and height of steps fluctuate in repeated numerical scans. This last part of comb evolution is sensitive tofluctuations in the pump power and short pump power drops can be used to deliberately induce a transition to a newcomb state. Numerically tracing out all possible comb evolutions yields the orange curves in Fig. 3a. The evolutionof the optical spectrum corresponding to the blue curve in Fig. 3a is shown in Fig. 3b. This spectral evolutionfollows the typical path[30]. With each step in the transmission the optical spectrum becomes less modulated until iteventually reaches a perfectly smooth envelope state (frame XI).

To reveal mode-locking we investigate the time dependent waveform in Fig. 3c by summing the simulated individual

3

optical modes. Indeed, the first step (frame V) corresponds to a transition to a mode-locked state where multiplepulses inside the cavity exist. Further steps can be associated with a stepwise reduction of the number of pulsespropagating in the resonator, where the system remains mode-locked. The separation between multiple pulses in theresonator is random.

The presence of second order anomalous dispersion (cf. Fig. 1b) and Kerr-nonlinearity in the cavity suggests thatthe simulated pulses may actually be bright, temporal, dissipative Kerr-cavity solitons[33]. Moreover, observationssimilar to the discrete transmission steps have been observed in a χ(2) nonlinear microwave resonator, and connectedto soliton formation[39].

To confirm the soliton nature of these pulses we perform a simulation of 501 modes (cf. Fig. 3d,e,f) and an-alyze a state of five pulses. We compare the numerical simulation with an approximate analytical solution forthe damped, driven nonlinear Schrodinger (NLS) equation(Lugiato-Lefever equation[40]), which describes the Kerr-nonlinear microresonator system (equivalent to the coupled mode-equations when third and higher order dispersioncan be neglected[32], which is the case in the present microresonator, cf. Fig. 1b). A stable solution to this equationare dissipative, temporal Kerr-cavity solitons[41, 42]. These solitons exist on top of the cw pump light field, whichprovides power to the solitons. Essentially, the stable solitons rely on the double-balance between anomalous cavitydispersion and Kerr-nonlinearity on the one hand, and cavity loss and parametric frequency conversion of the pumplight on the other hand [33, 41, 42]. For multiple solitons the analytical solution [41] has the form

Ψ(φ) ' C1 + C2 ·N∑j=1

sech(

√2(ω0 − ωp)

D2(φ− φj)) (1)

where Ψ denotes the complex field amplitude, φ the angular coordinate inside the resonator, φj the angular coordinateof the jth soliton, ωp the pump frequency, and where N is the number of solitons. The complex numbers C1 andC2 are fully determined by the resonator parameters and the pump conditions and the ratio of soliton peak power tocw background can typically exceed several hundreds (cf. SI for details). Indeed, the close to perfect match betweenanalytic solution and numerical result shows that the pulses forming in the microresonator are stable temporal Kerr-cavity solitons.

Eq. 1 allows for the estimation of the minimal temporal soliton width (FWHM)

∆tFWHMmin ≈ 2

√−β2

γFPin, (2)

where F is the cavity’s finesse, Pin the coupled power, β2 = −n0

c D2/D21 the group velocity dispersion and γ = ω

cn2

Aeff

the effective nonlinearity with the nonlinear mode area Aeff and the nonlinear refractive index n2 (cf. SI).Having shown the soliton nature of the pulses in numerical simulation, we can interpret the blue curve in Fig. 3a

based on general limits applying to solitons as solutions of the driven, damped NLS[43]. Adopting these criteria forthe present case we identify three main regions in Fig. 3a colored red, yellow and green (cf. SI). Solitons with aconstant temporal envelope can only exist in the green area. While the yellow area still allows for solitons with atime varying envelope (breather solitons [44]), solitons can not exist in the red area. Note that in the red area on theleft, the system may undergo chaotic Hopf-bifurcations[43]. The correspondence of these analytical limits with thenumerical results is remarkable.

For different number of solitons we can derive the total power inside the resonator by averaging the respectiveanalytic soliton solution (eq. 1) over one cavity roundtrip time. The result is shown as dark gray dashed lines in Fig.3a, and is in excellent agreement with the numerically observed steps (to account for the limitation due to the lowmode number an additional correction factor of order unity is applied). The intracavity power changes discontinuouslywith the number of solitons present in the cavity (cf. SI).

The simulations, in combination with the analytic description, show that microresonators can intrinsically bemode-locked via the formation of temporal Kerr-cavity solitons. Notably, a saturable absorber is not included in thesimulation and thus not required to achieve mode-locking in a microresonator.

Next, we investigate if stable operation of the transiently observed mode locked states is possible. The obstacle inreaching stable mode-locked states experimentally in our MgF2 resonator lies in the absorption induced thermal shiftof the resonance frequency[36, 37]. This implies that a stable effective laser cavity detuning (as required for stableoperation) is only realized in thermal equilibrium, where the laser and cavity undergo thermal and Kerr-nonlinear self-locking[37] (for details cf. SI). As long as the transmission during comb evolution is continuously and monotonicallydecreasing, this can be achieved by slowly (adiabatically) tuning the laser frequency transitioning between differentcomb states. However, adiabatic laser tuning does not generally allow reaching comb states preceded by discretetransmission steps. In this case, the transmission (and therefore the intracavity power) undergoes sudden changes,

4

inducing a rapid change of resonance frequency, which destabilizes the respective comb state. To avoid this instability,we apply an adaptive laser tuning scheme, whereby the laser is scanned quickly to the targeted mode-locked state.The laser scan speed is chosen such that the resulting temperature at the destination is equal to the correspondingequilibrium temperature (cf. SI for more details). We note that soliton states may also be reached by stabilizing therelative laser to cavity detuning via thermal (such as laser power variation) or mechanical actuation on the cavityresonance frequency or via injection locking.

Having access to stable operation of these soliton states, we experimentally investigate - in addition to their RFbeatnote and optical spectrum - their temporal characteristics by performing a frequency-resolved optical gating exper-iment (FROG, Fig. 4a). This corresponds to a second-harmonic generation (SHG) autocorrelation experiment, wherethe frequency-doubled light is spectrally resolved (Fig. 4d and cf. methods). In contrast to intensity auto-correlation,the FROG method allows a reliable identification of ultra-short pulses via the associated minimal bandwidth of theSHG spectrum given by the time-bandwidth-product (TBP).

In full consistency with the numerical simulations, we observe single and multiple pulse states as shown in Fig.3c. The one pulse state is characterized by a smooth spectral envelope, without spectral gaps. The power spectralenvelope exhibits a sech2-shape (3 dB bandwidth 1.6 THz) as expected from the Fourier transform of a sech-shapedsoliton pulse. Based on the TBP for solitons of 0.315 the expected pulse duration is 197 fs (cf. SI). The observed lowphase noise RF beatnote is resolution bandwidth limited to 1 kHz and its signal-to-noise ratio exceeds 60 dB. TheFROG trace shows a train of pulses well separated by the cavity roundtrip time of TR =28.4 ps, corresponding to theFSR of 35.2 GHz. The multi-pulse states (here shown for the case of two and five pulses), show a more structuredoptical spectrum. This structure reflects the number and distribution of pulses in the cavity (cf. SI). The RF beatnote generated in the multi-pulse states is of similar quality as in the single pulse state. Importantly, the FROGmeasurement allows for a full reconstruction (neglecting a time direction ambiguity) of intensity and phase of thepulses (cf. Fig. 4c). The reconstructed intensity profile is consistent with the expected sech2-shape for solitons andthe reconstructed temporal width of 200 fs (FWHM) is in agreement with the bandwidth of the optical spectrum andthe expectation based on eq. (2). The FROG traces show that it is the full spectrum that contributes to each pulse.This is in particular true for the multi-soliton case.

To further corroborate the presented results an independent intensity sampling method is applied to a singlesoliton state. The high repetition rate prohibits a direct sampling, which would require hundreds of GHz bandwidthin detection and recording. This limitation can be overcome by stretching the optical waveform in time using anUltrafast Temporal Magnifier[45] (PicoLuz LLC, Fig. 4f). While the time resolution of about 2.5 ps does not allowfor determining the duration of the pulse, this single-shot method, as opposed to auto-correlation-type experiments,does not rely on averaging. The results in Fig. 4e clearly show optical pulses separated by TR, with constant pulseamplitudes, as expected for solitons. We note that in the temporal characterization experiments no spectral filtering,phase or intensity adjustment (except for pump suppression) is applied. The gain window (≥ 4 THz) of the opticalamplifier used before temporal characterization supports more than 100 comb modes.

Combining experimental, numerical and analytical results, we have demonstrated mode-locking in a MgF2 mi-croresonator via soliton formation, which is only based on the generic properties of Kerr-nonlinear microresonatorswith anomalous dispersion. Consequently, our observations are generic and apply equally to other microresonatorcomb platforms. In this context, we note that independent research has very recently reported on femto-secondpulse generation and mode-locking in Si3N4 microresonators[46]. From a frequency domain perspective, mode-lockedmicro-resonators enables based frequency combs with low noise and smooth spectral envelopes. From a time domainperspective it enables ultra-short soliton pulse generation in a microresonator. In combination with chip-scale[10, 11]integration this opens the route towards compact, stable and low cost ultra-short pulse sources[47], which can alsooperate in wavelength regimes (such as the mid-infrared[28]), where broadband laser gain media or saturable absorbersdo not exist. Due to the broadband nature of the parametric gain, which can significantly exceed the spectral widthof conventional laser gain media, the direct synthesis of sub-cycle pulses may be conceivable.

Methods

Microresonator based frequency comb generation:

The pump laser (fiber laser; 1553 nm wavelength; shortterm linewidth < 100 kHz) is amplified by an erbium dopedfiber amplifer (EDFA) and evanescently coupled to the MgF2 resonator (free spectral range 35.2 GHz; refractive indexn0 = 1.37) via a tapered optical fiber. The coupled resonance width of κ/2π = 450 kHz (quality factor Q = 4× 108,Finesse F = 78 × 104)has been measured using modulation sidebands of a scanning laser. The dispersion of theresonator has been determined following ref. [48] to D2/2π = 16 kHz that is β2 = −9.39 ps2km−1. A typical coupledpump power of Pin = 5 − 30 mW leads to circulating powers Pcirc of several hundreds of Watts. The estimated

5

effective mode area Aeff = 90× 10−12m2 (effective mode volume Veff = 5.6× 10−13 m3) and the nonlinear refractiveindex n2 = 0.9× 10−20 m2W−1 yields an estimated nonlinear parameter of γ =

ωp

cn2

Aeff= 4.1× 10−4m−1W−1.

Numerical simulation:

The simulations are based on the coupled mode equations (cf. SI) which are numerically propagated in time. Thecoupled mode-equations are equivalent to the damped, driven NLS equation(cf. main text) describing the systemwhen third order dispersion is neglected [32]. The simulated resonator is defined similar to the experimental resonatorby its resonance frequencies ωµ given by D1/2π = 35.2 GHz, D2/2π = 10 kHz, D3/2π = −130 Hz and a quality factorof Q = 2×108. All other parameters are equal to the values listed for the experimental resonator. The coupled pumppower is set to Pin = 100 mW at a pump frequency of ωp/2π = 193 THz.

FROG Experiment:

Prior to the FROG experiment the optical spectra are sent through a fiber-Bragg grating for pump supression (−30dB) and are subsequently amplified to 50 mW. Dispersion compensating fiber (DCF3,DCF38) is used for approximatedispersion compensation. In the FROG setup (cf. Fig 4d) the generated optical pulses, are interferometrically splitand recombined with a variable delay in a nonlinear BBO crystal. This results in second harmonic generation (SHG)whenever the optical pulses in the two arms of the interferometer overlap temporally in the BBO crystal. Thegenerated SHG light is spectrally resolved and recorded as a function of delay by a CCD-spectrometer, yielding aFROG trace. Each FROG trace consists of nearly 1000 spectra with individual exposure times of 800 ms. TheN-by-N (N=63) FROG trace of the single pulse state in Fig. 4c is analyzed using a principal component generalizedprojection algorithm[49], after noise removal via Fourier-filtering. The FROG reconstruction error is defined as

ε = 1N

√∑Ni,j(M

measij −M reco

ij )2, where Mmeasij and M reco

ij denote the elements of the N ×N matrices representing the

measured and reconstructed FROG traces.

Acknowledgemets

The authors thank R. Salem and A. Gaeta for providing the PicoLuz, LLC Ultrafast Temporal Magnifier and advicewhen evaluating the data. The authors acknowledge valuable advice by K. Hartinger on dispersion compensationas well as helpful discussions with F. Kartner and U. Keller. This work was supported by the DARPA programQuASAR, the Swiss National Science Foundation (SFN) as well as a Marie Curie IAPP program. MLG acknowledgessupport from the ‘Dynasty’ foundation and RFBR grant 11-02-00383-a. The research leading to these results hasreceived funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreementno. 263500.

Author Contribution

TH designed and performed the experiments and analyzed the data. MLG and TH performed the numericalsimulations. MLG developed the analytic description. VB assisted in the experiments. JDJ assisted in the TemporalMagnifier experiment. TH and MLG fabricated the sample. CYW assisted in sample fabrication. NMK assisted indeveloping the analytic description. TH, MLG, and TJK wrote the manuscript. TJK supervised the project.

[1] Haus, H. A. Mode-locking of lasers. IEEE Journal of Selected Topics In Quantum Electronics 6, 1173–1185 (2000).[2] Steinmeyer, G., Sutter, D. H., Gallmann, L., Matuschek, N. & Keller, U. Frontiers in ultrashort pulse generation: Pushing

the limits in linear and nonlinear optics. Science 286, 1507–1512 (1999).[3] Keller, U. Recent developments in compact ultrafast lasers. Nature 424, 831–838 (2003).[4] Holzwarth, R. et al. Optical frequency synthesizer for precision spectroscopy. Physical Review Letters 85, 2264–2267

(2000).[5] Cundiff, S. T. & Ye, J. Colloquium: Femtosecond optical frequency combs. Reviews of Modern Physics 75, 325 (2003).[6] Udem, T., Holzwarth, R. & Hansch, T. W. Optical frequency metrology. Nature 416, 233–237 (2002).

6

[7] Newbury, N. R. Searching for applications with a fine-tooth comb. Nature Photonics 5, 186–188 (2011).[8] Del’Haye, P. et al. Optical frequency comb generation from a monolithic microresonator. Nature 450, 1214–1217 (2007).[9] Savchenkov, A. A. et al. Tunable optical frequency comb with a crystalline whispering gallery mode resonator. Physical

Review Letters 101, 093902 (2008).[10] Levy, J. S. et al. CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects. Nature Photonics 4,

37–40 (2010).[11] Razzari, L. et al. CMOS-compatible integrated optical hyper-parametric oscillator. Nature Photonics 4, 41–45 (2010).[12] Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonator-based optical frequency combs. Science 332, 555

(2011).[13] Papp, S. B., Del’Haye, P. & Diddams, S. A. Mechanical control of a microrod-resonator optical frequency comb. ArXiv

e-prints (2012). 1205.4272.[14] Li, J., Lee, H., Chen, T. & Vahala, K. J. Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-wave Repetition

Rate Operation in Microcombs. ArXiv e-prints (2012). 1208.5256.[15] Kartner, F., Jung, I. & Keller, U. Soliton mode-locking with saturable absorbers. IEEE Journal of Selected Topics in

Quantum Electronics 2, 540 –556 (1996).[16] Steinmetz, T. et al. Laser frequency combs for astronomical observations. Science 321, 1335–1337 (2008).[17] Li, C.-H. et al. A laser frequency comb that enables radial velocity measurements with a precision of 1 cm s−1. Nature

452, 610–612 (2008).[18] Pfeifle, J. et al. Microresonator-based optical frequency combs for high-bitrate WDM data transmission. In Optical Fiber

Communication Conference, OW1C.4 (Optical Society of America, 2012).[19] Telle, H. R. et al. Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and

ultrashort pulse generation. Applied Physics B-Lasers And Optics 69, 327–332 (1999).[20] Diddams, S. A., Hollberg, L. & Mbele, V. Molecular fingerprinting with the resolved modes of a femtosecond laser frequency

comb. Nature 445, 627–630 (2007).[21] Savchenkov, A. A. et al. Miniature Optical Atomic Clock: Stabilization of a Kerr Comb Oscillator. ArXiv e-prints (2013).

1301.3198.[22] Vahala, K. J. Optical microcavities. Nature 424, 839–846 (2003).[23] Del’Haye, P. et al. Octave spanning tunable frequency comb from a microresonator. Phys. Rev. Lett. 107, 063901 (2011).[24] Okawachi, Y. et al. Octave-spanning frequency comb generation in a silicon nitride chip. Opt. Lett. 36, 3398–3400 (2011).

URL http://ol.osa.org/abstract.cfm?URI=ol-36-17-3398.[25] Del’Haye, P., Arcizet, O., Schliesser, A., Holzwarth, R. & Kippenberg, T. J. Full stabilization of a microresonator-based

optical frequency comb. Physical Review Letters 101, 053903 (2008).[26] Papp, S. B. & Diddams, S. A. Spectral and temporal characterization of a fused-quartz-microresonator optical frequency

comb. Phys. Rev. A 84, 053833 (2011).[27] Foster, M. A. et al. Silicon-based monolithic optical frequency comb source. Opt. Express 19, 14233–14239 (2011).[28] Wang, C. Y. et al. Mid-infrared optical frequency combs at 2.5 µm based on crystalline microresonators. Nature Commu-

nications 4 (2013).[29] Ferdous, F. et al. Spectral line-by-line pulse shaping of on-chip microresonator frequency combs. Nature Photonics 5,

770–776 (2011). 1103.2330.[30] Herr, T. et al. Universal formation dynamics and noise of Kerr-frequency combs in microresonators. Nature Photonics 6,

480–487 (2012).[31] Johnson, A. R. et al. Chip-based frequency combs with sub-100 GHz repetition rates. Opt. Lett. 37, 875–877 (2012).[32] Matsko, A. B. et al. Mode-locked kerr frequency combs. Optics Letters 36, 2845–2847 (2011).[33] Leo, F. et al. Temporal cavity solitons in one-dimensional kerr media as bits in an all-optical buffer. Nature Photonics 4,

471–476 (2010).[34] 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 (2013).[35] Grudinin, I. S., Baumgartel, L. & Yu, N. Frequency comb from a microresonator with engineered spectrum. Opt. Express

20, 6604–6609 (2012).[36] Gorodetsky, M. L. & Ilchenko, V. S. Thermal nonlinear effects in optical whispering-gallery microresonators. Laser Physics

2, 1004–1009 (1992).[37] Carmon, T., Yang, L. & Vahala, K. J. Dynamical thermal behavior and thermal selfstability of microcavities. Optics

Express 12, 4742–4750 (2004).[38] Chembo, Y. K. & Yu, N. Modal expansion approach to optical-frequency-comb generation with monolithic whispering-

gallery-mode resonators. Physical Reveiw A 82, 033801 (2010).[39] Gasch, A., Wedding, B. & Jager, D. Multistability and soliton modes in nonlinear microwave resonators. Applied Physics

Letters 44, 1105–1107 (1984).[40] Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Physical Review Letters 21, 2209–2211

(1987).[41] Wabnitz, S. Suppression of interactions in a phase-locked soliton optical memory. Opt. Lett. 18, 601–603 (1993).[42] Firth, W. J. & Weiss, C. O. Cavity and feedback solitons. Opt. Photon. News 13, 54–58 (2002).[43] Barashenkov, I. V. & Smirnov, Y. S. Existence and stability chart for the ac-driven, damped nonlinear schrodinger solitons.

Phys. Rev. E 54, 5707–5725 (1996).[44] Matsko, A. B., Savchenkov, A. A. & Maleki, L. On excitation of breather solitons in an optical microresonator. Opt. Lett.

7

37, 4856–4858 (2012).[45] Salem, R. et al. High-speed optical sampling using a silicon-chiptemporal magnifier. Opt. Express 17, 4324–4329 (2009).[46] Saha, K. et al. Modelocking and Femtosecond Pulse Generation in Chip-Based Frequency Combs. ArXiv e-prints (2012).

1211.1096.[47] Grelu, P. & Akhmediev, N. Dissipative solitons for mode-locked lasers. Nature Photonics 6, 84–92 (2012).[48] Del’Haye, P., Arciezt, O., Gorodetsky, M. L., Holzwarth, R. & Kippenberg, T. J. Frequency comb assisted diode laser

spectroscopy for measurement of microcavity dispersion. Nature Photonics 529–533 (2009).[49] Kane, D. J. Principal components generalized projections: a review. J. Opt. Soc. Am. B 25, A120–A132 (2008).

8

2mm

a

randomphases

equalphases

mode

locking

periodic intensity pulsed intensity

I

Pump

ω

d

TR = neffL/c

pulse

x6.5

x140

b

rel. mode number μ0-100-200 100

0

300 Deviation from uniform FSR(ωμ - ω0 - μD1)/2π

25 nm

MH

z

FSR

c

FIG. 1: Mode-locking and frequency comb generation in microresonators. a. MgF2 crystal carrying two whispering-gallery-mode microresonators of different size (the smaller one with an FSR of 35.2 GHz is used). A optical whispering-gallery-mode propagates along the circumference L of the resonator within the roundtrip time TR. The smaller panels show amagnified view of the resonator and the simulated optical mode profile. b. Second order anomalous dispersion (FSR increaseswith mode number) of the microresonator with 35.2 GHz FSR shown as the deviation of the measured resonance frequencies(blue dots) ωµ = ω0+D1µ+ 1

2D2µ

2+ ... from an equidistant frequency grid ω0+µD1 (horizontal grey line), where µ denotes therelative mode number and D1 corresponds to the FSR at the frequency ω0. The resonator’s anomalous dispersion is accuratelydescribed by D2/2π = 16 kHz and higher order terms neglected (red dashed line). The grey vertical lines mark spectral intervalsof 25 nm width (µ = 0 corresponds to 1553 nm). c. Possible pathway to frequency comb generation in a Kerr-nonlinearmicroresonator via four-wave-mixing, where the newly generated optical modes are supported by the cavity’s resonances (gray).Widely spaced primary modes emerge first, followed by secondary comb modes filling the gaps. d. Optical whispering gallerycomb modes (blue) and resulting intensity (red). If the relative phases (indicated by black arrows) are constant but random(i.e. phase-locked), the resulting intensity is periodic with TR; mode-locking implies a synchronization of phases such that theresulting intensity is pulsed with a pulse repetition rate of 1/TR.

9

a.u.

3.4 3.6 3.8 4

0.9

1

scan time, ms

trans

mis

sion

, a.u

.

0 1 2 3 4

0.7

1

scan time, ms

dow

n-m

ixed

RF

sign

al, M

Hz

0 1 2 3 4

10

20

30

nm

30dB

/div

1540 1570

Resonator

Oscillo-scope

FPC

OSA

ESA

Scan

LO

FBG

PD

PD

CW Pump Laser + EDFA

3.2scan time, ms

0

10

20

dB

laser scan

expected transmission:

wavelength

trans

mis

sion

III III

I

II

III

a

b

c dMain Setup

I II III

high noise (multiple and broad

RF beatnote)low noise

(narrow RF beatnote)

FIG. 2: Transmission and beatnote a. Transmission observed when scanning a laser over a high-Q Kerr-nonlinear resonancein a MgF2 resonator (coupled pump power 5 mW). The transmission signal follows the expected thermal triangle (cf. inset)with deviations in the form of discrete steps (green shading). b. Evolution of the optical power spectrum for three differentpositions in the scan; the spectrum (II) and in particular the mesa shaped one (III) exhibit a high-noise RF beat signal. c.Down-mixed RF beat signal. d. Experimental main setup composed of pump laser and resonator followed by an opticalspectrum analyzer (OSA), an oscilloscope to record the transmission and to sample the down-mixed beatnote (via the thirdharmonic of a local oscillator LO at 11.7 GHz), and an electrical spectrum analyzer (ESA) to monitor the beatnote. Beforebeatnote detection the pump is filtered out by a narrow fiber-Bragg grating (FBG) in transmission; FPC: Fiber polarizationcontroller, EDFA: erbium-doped fiber amplifier, PD: photodetector.

10

−50 0 50

40

dB/d

iv

0 1

a.u.

−10 0 10 20 30

a.u.

−250 0 250

10 d

B/d

iv

0 10

1a.

u.

0

a.u.

a b c

d e f

III

III

IV

V

VI

VII

VIII

IX

X

XI

XI

X

IX

VII

VII

VI

V

IV

III

II

I

(20dB/div)

−0.02 0.02

1

XI

X

VII

V

IV

III

II

I

VI

VII

IX

ζ0 (normalized laser detuning from ω0) mode number

mode number

time/TR

time/TR time/TR

FIG. 3: Numerical Simulations a. Horizontally mirrored intracavity power (blue, corresponding to the transmission signalin Fig. 2a) during a simulated laser scan (101 simulated modes) over a resonance in a MgF2 resonator. The step featuresare well reproduced. The orange lines trace out all possible evolutions of the system during the scan. The dashed lines showan analytical description of the steps. The green area corresponds to the area where solitons exist, the yellow area allows forsolitons with a time variable envelope; no solitons can exist in the red area. b/c. Optical spectrum and intracavity intensity fordifferent positions I-XI in the laser scan. d. Optical spectrum obtained when simulating 501 modes and stopping the simulatedlaser scan in the soliton-regime. e. Intracavity intensity for the comb state in (d) showing 5 solitons. f. Zoom into one of thesoliton states showing the numerical results for the field real (red) and imaginary part (dark blue). The respective analyticalsoliton solutions are shown in light blue and orange.

11

−20 0 20 40 60 delay, ps

−2

0

2

20 d

B/d

iv

1520 1540 1560 1580 nm

~sech2, FWHM = 1.6 THz

kHz + 35.2 GHz

20 d

B/d

iv

−100 0 100

−20 0 20 40 60 delay, ps

−2

0

2

20 d

B/d

iv

1520 1540 1560 1580 nm

kHz + 35.2 GHz

10 d

B/d

iv

−100 0 100

−20 0 20 40 60 delay, ps

−2

0

2

20 d

B/d

iv

1520 1540 1560 1580 nm

kHz + 35.2 GHz

20 d

B/d

iv

−100 0 100

Main SetupFBG DCF

EDFA

DelaySHG

CCD

Grating

LensLens HWPQWP

measured reconstructed

RBW1 kHz

RBW1 kHz

RBW1 kHz

28.4 ps

1 Soliton

2 Solitons

5 Solitons

a

c d

−1 0 1, ps

−3

0

3

−1 0 1, psωS

HG−

386

THz

, TH

z

0

2π

a.u. rad

−1 0 1, ps0

1reconst.intensityphase

trace size: 63 x 63

measured reconstructed

error: 1.7%

~sech2

FWHM = 200 fs

1

0

−10 0 10

0

a.u.

time, ps −10 0 10 −10 0 10

frame 1 frame 2 frame 3

28.4 ps

Main SetupFBG DCF

EDFA

Ultrafast TemporalMagnifier x 105.5

t

FLC

Oscillo-scope

PD

FPC

e f

b

ωS

HG

/2π

− 3

86 T

Hz

, TH

zω

SH

G /2

π −

386

TH

z , T

Hz

ωS

HG

/2π

− 3

86 T

Hz

, TH

z

FIG. 4: Experimental demonstration of soliton states a. Optical spectra of three select comb states with one, two andfive solitons, respectively. The insets show the RF beat note, which is resolution bandwidth limited to 1 kHz width in all cases.The red line in the optical spectrum of the one pulse state shows the spectral hyperbolic-secant envelope expected for solitonpulses with a of 3 dB bandwidth of 1.6 THz. b. FROG traces of the comb states in (a) displaying the signal of the single andmultiple pulses. c. Higher resolution experimental FROG trace of a one soliton pulse (left). The reconstruction converges toa FROG error of ε = 1.7% in good agreement with the experimental trace (middle). The reconstruction (right) of intensityand phase yields an estimated pulse duration of 200 fs (FWHM). d. Setup of the FROG experiment. e. Sampled opticalintensity of the microresonator output over a duration of 40 ps. The three measurements (frame 1,2,3) are separated from oneanother by a duration of 4 ns corresponding to approx. 140 round-trip times TR = 28.4 ps f. Setup of the intensity samplingexperiment including the PicoLuz LLC Temporal Magnifier and a 4 GHz sampling oscilloscope. FBG: fiber-Bragg grating (intransmission), DCF: dispersion compensating fiber, EDFA: erbium-doped fiber amplifier, SHG: second harmonic generation,FLC: fiber laser comb (250MHz rep. rate), PD: photodetector, FPC: fiber polarization controller, HWP: half-wave-plate, QWP:quarter-wave-plate.