RESEARCH ARTICLE◥

NONLINEAR OPTICS

Real-time spectral interferometryprobes the internal dynamicsof femtosecond soliton moleculesG. Herink,1,2* F. Kurtz,1 B. Jalali,2 D. R. Solli,1,2 C. Ropers1,3*

Solitons, particle-like excitations ubiquitous in many fields of physics, have been shownto exhibit bound states akin to molecules. The formation of such temporal soliton boundstates and their internal dynamics have escaped direct experimental observation. Bymeans of an emerging time-stretch technique, we resolve the evolution of femtosecondsoliton molecules in the cavity of a few-cycle mode-locked laser. We track two- andthree-soliton bound states over hundreds of thousands of consecutive cavity roundtrips,identifying fixed points and periodic and aperiodic molecular orbits. A class of trajectoriesacquires a path-dependent geometrical phase, implying that its dynamics may betopologically protected. These findings highlight the importance of real-time detectionin resolving interactions in complex nonlinear systems, including the dynamics of solitonbound states, breathers, and rogue waves.

Despite their overwhelming complexity, non-linear systems exhibit universal featuresthat facilitate an understanding of theirdynamics, including periodic attractors andchaos. The prototypical excitations of many

nonlinear systems are solitons, localized struc-tures balanced by nonlinearity and dispersion.Soliton dynamics attract considerable attentionin numerous contexts, including fluids, Bose-Einstein condensates, plasmas, polymers, andoptical systems (1–5). The soliton’s stability againstperturbations endows it with particle-like charac-teristics, which may follow from topological pro-tection, asmanifested in skyrmions or edge states(e.g., in topological insulators).Interactions between individual solitons create

the possibility of bound states, which were theo-retically predicted and successfully demonstratedin various physical forms and degrees of freedom.The self-trapping of multiple co-propagatingmodes was discovered, for example, in opticalfiber for temporal solitons of different polariza-tion (6) or wavelength (7) and for multicomponentspatial solitons in photorefractive media (8, 9).Stable anddynamically evolvingbound stateswerefound to arise from various couplingmechanisms(3,6–12), resulting indistinct relationships betweenmutual amplitudes, phases, and separations. Inone-dimensional or single-modal propagation,attractive and repulsive interactions betweentemporal optical solitons result in bound states,

which are frequently referred to as soliton mole-cules (13–18).In dissipative nonlinear systems, a twofold ba-

lance of energy loss with gain and dispersionwithnonlinearity (19) allows for large families of boundstates between multiple solitons. Beyond station-ary solutions, these systems also support solitonmolecules with time-varying properties. Access tosuch dynamics has been gained mostly by nu-merical simulations. In particular, in various typesof lasers, numerical studies predict stationary,periodic, or chaotic bound-state evolutions (19–25).Experimentally, time-averagedmeasurements haveresolved static solitonmolecules, and internal mo-tions have been inferred from partial coherencelosses (26–30).Whereas the femtosecond or even attosecond

time scales typically associated with the forma-tion and dissociation of atomic bonds can onlybe traced via temporal reconstruction (pump-probe techniques), the dynamics of femtosecondsoliton molecules often span the nanosecond tomicrosecond range. However, bound states format unpredictable times, and their subsequent evo-lution may be nonrepetitive. Thus, observation ofthese dynamics requires real-time detection of thetiming and relative phase within femtosecondmolecules over long recording intervals.In our experimental setup, we resolve the for-

mation and internal dynamics of a diverse setof femtosecond soliton molecules in a broad-band Kerr lens mode-locked Ti:sapphire laserwith chirped-mirror dispersion compensation. Al-though single-pulse operation is commonly pre-ferred, it is well known that such systems supportdouble ormultisoliton complexes (26–29, 31).Weprepare dynamic soliton molecules (Fig. 1A), andresolve their evolution using a relatively simplebut powerful technique, known as the time-

stretch dispersive Fourier transform (TS-DFT)(32): Spectral information is mapped into thetime domain—using chromatic dispersion in along optical fiber—and is detected via a high-speed photodetector and real-time digitization.This method is increasingly used for measure-ments of rapid signals (33–35) and, in recent ex-periments with mode-locked sources, has beenapplied to record the build-up of femtosecondmode locking (36) and soliton instabilities in afiber oscillator (37).Typically, the TS-DFT is used to obtain spec-

tral dynamics rather than temporal informationon ultrashort time scales. Yet an ensemble ofclosely spaced pulses exhibits spectral interference,which encodes both precise timing and phase in-formation (Fig. 1B). The extraction of timing in-formation of bound states in nonlinear cavitiesand fiber oscillators via TS-DFT was recently dis-cussed (38, 39). Generally, both timing and phasecan be obtained from the interferogram by con-sidering a bound state as a superposition of tem-porally separated individual solitons. For example,the bound state field of a doublet with the solitonenvelopes E1(t), E2(t) at a common carrier fre-quency w0 can be expressed as

EðtÞ ¼ Ref½E1ðtÞ þ E2ðtÞ�expðiw0tÞg ð1Þ

(16, 20). In the case of identical envelopes, E2(t)can be replaced by E1(t + t) exp(iDϕ) with solitonseparation t and a possible relative phase Dϕ.In the frequency domain, the temporal separa-tion translates to a phase factor exp(iwt), whichmodulates the spectrum E(w) with a fringe pe-riod of 1/t. Thus, the pulse separation is mappedinto a modulation observed as an interferogramS(w) = |E(w)|2, and the phase of the fringe patternat w0 encodes the relative phase of the pulses inthe doublet:

SðwÞºjE1ðw − w0Þj2½1þ cosðwt − w0tþ DϕÞ�(2)

We implement the TS-DFT with an optical fiberof group velocity dispersion (GVD) parameter b =4.7 × 10–2 ps2/m and length L = 400 m (Fig. 1C).The spectral fringe period 1/t encoding the bound-state separation is mapped into the time domainand stretched to Dt = 2pbL/t. The signal is de-tectedwith a fast photodiode (bandwidth > 8GHz)and a high-speed real-time oscilloscope (band-width 16 GHz). Thus, we can record spectral inter-ferograms as a continuous real-time series at therepetition rate of the mode-locked laser (78 MHz)over an interval of typically T = 4 ms, spanning~300,000 consecutive roundtrips.

Formation of soliton molecules

In the real-time series of interferograms trackingthe formation of a stable soliton molecule fromtwo individual solitons (Fig. 2, A and B), the stateis prepared by adjusting the cavity alignmentand applying a slight mechanical perturbation(see supplementarymaterials). The real-time dataexhibit interferenceswith increasing fringe periodand a complex temporal evolution. The Fouriertransform of each single-shot spectrum (Fig. 2C)

RESEARCH

Herink et al., Science 356, 50–54 (2017) 7 April 2017 1 of 4

1IV. Physical Institute—Solids and Nanostructures, Universityof Göttingen, Göttingen, Germany. 2Department of ElectricalEngineering, University of California, Los Angeles, CA, USA.3International Center for Advanced Studies of EnergyConversion (ICASEC), University of Göttingen, Göttingen, Germany.*Corresponding author. Email: gherink@gwdg.de (G.H.); claus.ropers@uni-goettingen.de (C.R.)

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corresponds to a field autocorrelation, which di-rectly represents the separation between the twosolitons. During the transition, solitons approacheach other via steps toward a stable binding se-paration of 180 fs, accompaniedby a characteristicsequence in the relative phase evolution and alocked final phase relation (compare to interfer-ence fringes in Fig. 2A).The natural representation of the bound-state

configuration space is the polar diagram or “in-teraction plane,” assigning the pulse separationto the radius and the relative phase to the angle(20). Constructed over 6000 roundtrips prior tothe establishment of the bound state in Fig. 2D,the trajectory illustrates the evolution throughperiodic metastable soliton separations. Final-ly, the system reaches a fixed point with lockedrelative phase, settling at a constant binding sep-aration. The continuous tracing of successive in-terferograms yields the direction of the phasechange: We observe a predominant shift of thefringe pattern toward shorter wavelengths, cor-responding to a decreasing relative phase and aclockwise rotation in the interaction plane. Thisfinding can be explained by a slight intensitydifference of the two constituents of the boundstate, which have different phase velocities in thegain medium due to the intensity-dependent re-fractive index (Kerr effect). A persistent intensitydifference of the two constituents during the ap-proach results in different carrier envelope phaseshifts after each roundtrip and, in turn, leads to along-term evolution of the relative phase, as theo-retically described (21, 30).The present interferometric detection over

long record intervals enables sensitivity to verysmall phase shifts at the single-shot level. The ob-servation of discrete separations during the bondformation may be connected to the periodic spa-tial dependence of the interaction potential, aspreviously reported theoretically and experimen-tally for stable bound states (15, 28). However,the detailed origin of this strong transient sta-bilization remains a subject for further study.

Doublet and triplet states withdynamical phase evolution

To induce dynamic bound states that continu-ously evolve, we initially generate a soliton mol-ecule with a separation of ~170 fs and lockedrelative phase, similar to the previous final state(Fig. 3A). Then, reducing the pump power belowa critical threshold, we observe a highly stablestepwise evolution of the relative phase, evidentin the dynamic fringe pattern in Fig. 3B. Each stepis composed of a steady decrease in relative phaseby a small fraction of p, followed by a rapid com-pletion of a (–2p)-cycle. This evolution of the rela-tive phase is plotted on a normalized time axis fortwo periods in Fig. 3F. Such steplike progressionsof the phase in mode-locked lasers were attrib-uted to gain saturation dynamics in numericalstudies (25). From the observed phase slip perroundtrip, we can readily infer that the intensitydifference of the soliton molecule’s constituentsis ~0.5%, which would be extremely difficult tomeasure in any other way.

Herink et al., Science 356, 50–54 (2017) 7 April 2017 2 of 4

Fig. 1. Spectral interferometry of soliton molecule dynamics. (A) Sketch of the temporal dynamicsduring solitonmolecule formation in a femtosecond oscillator: Starting from two individual solitons, a boundstate with fixed separation and locked relative phase evolves. (B) In the frequency domain, the separationand relative phase of the solitons are encoded in the spectral interferogram. (C) Experimentally, such multi-soliton states are detected via real-time spectroscopy: Using the time-stretch dispersive Fourier transform(TS-DFT) in an optical fiber, spectral interferograms are mapped into the time domain and recorded at thelaser repetition rate, thereby tracing rapid evolutions of soliton molecules.

Fig. 2. Formation of a solitonmolecule. (A) Experimental real-time interferograms during the formationof a soliton bound state with locked relative phases and a separation of 180 fs. The decrease in pulseseparation is inversely proportional to the fringe period. Shifts in the fringe pattern denote changes of therelative phase. (B) Exemplary single-shot spectrum, corresponding to the last frame in (A). (C) The Fouriertransforms (magnitude) of each single-shot spectrum represent the field autocorrelations of the mo-mentary bound state, tracing the separation between both solitons and thus the merging into a stablebound state. (D) Dynamics of the soliton molecule formation mapped in the interaction plane over 6000roundtrips. The radius and angle represent the momentary pulse separation and relative phase, respec-tively. The system ultimately settles on the fixed point (black), a stable soliton bound state with lockedrelative phase.

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A further reduction in pump power (Fig. 3, Cand D) not only accelerates the phase dynamicsbut also results in the transition from a stepwiseevolution toward a linear phase shift over time(Fig. 3F). The decreasing number of roundtripsper period is depicted in the inset of Fig. 3F. Theinteraction planes for all pump powers in Fig. 3Dreveal that below a threshold pump value of about4.5 W, the system evolves on limit cycles of fixedintersoliton separation. The bound state tracesorbits around the singularity at t = 0 with therelative phase constantly accumulating Dϕ = –2pin each revolution. This finite phase may be re-garded as a geometric or Pancharatnam-Berryphase. The topology of these dynamics in the con-figuration space can be characterized by a wind-ing numberm= ∮Cðdϕ=2pÞ = –1, whereC denotesa closed integration path.These findings directly relate to numerical re-

sults predicting periodic bound-state dynamics,which, although not discussed in terms of a wind-ing number, would imply solutions ofm = –1 (25)orm = –2 (21). Onemay speculate to what degreethese different dynamic bound statesmay be topo-logically protected against perturbation, addinganother topological aspect to the field of pho-tonics and soliton interactions (11, 40). In thespectral domain, the continuous phase revolu-tion by m2p within k roundtrips (k need not beinteger) implies an overall geometrical shift ofthe corresponding frequency comb (41) by fgeo =frep/k, adding to the carrier envelope offset fre-quency fCEO and the fundamental repetition rate

frep, such that the frequency of the nth comb linewill be f(n) = nfrep + fCEO + fgeo. In our observa-tions, this contribution accounts for a downwardshift by up to 0.4 MHz. Further investigationsmay address possible couplings between theseperiodic changes in the relative phase and ab-solute carrier envelope phase of such multisoli-ton complexes (i.e., relating fCEO and fgeo).In addition, it is also possible to induce bound

triplet states for identical laser adjustment atenhanced pump power. At 4.8 W, we observetriplets with fixed separations and dynamicallyevolving relative phases. The recorded interfero-grams in Fig. 3G exhibit three superimposedfringe periodicities arising from the three tem-poral delays present in the state (the correspond-ing autocorrelation is shown in Fig. 3H). Eachrelative phase continuously evolves in time (Fig.3I). The rate of phase slip is larger between pulseswith greater separation, indicating that thethree intensities of the bound solitons representamonotonic sequence. Via reduction of the pumppower, the triplet decays into the doublet statesshown in Fig. 3, A to E.

Soliton molecule vibrations

Finally, we present another set of qualitativelydifferent dynamic solitonmolecules with shorterbinding separations. Shown in the left panels ofFig. 4, A to D, is a series of experimental real-timeinterferograms of soliton doublets, prepared viareducing the pump power in each case. Here,both the pulse separation and their relative phase

evolve rapidly on a time scale of a few hundredroundtrips. The extracted binding separationsand relative phases are shown in the center panels,with corresponding interaction planes in the rightpanels. In this set of measurements, all trajecto-ries evolve between separations of 95 fs and 115 fsin configuration space (dashed radii in Fig. 3,right). For the highest pump power (Fig. 4A), thebound state periodically traces a closed orbit inthe interaction plane, given by two partial circlesover phases of 3p/2. In contrast to previous mea-surements with larger, invariant separation (Fig. 3),here the relative phase does not monotonically in-crease over time but oscillates between two turningpoints, reminiscent of theoretical predictions (24, 25).In those studies, the regular internal motion ofthe bound state was attributed to a flipping be-tween two unstable solutions, facilitated by gaindynamics and soliton interaction.By reducing the pump power, the trajectories

extend over phases above 4p (Fig. 4D). Specifi-cally, in this state, we can readily observe that theclosed orbits comprise interleaving spiral seg-ments. The outer spiral corresponds to the stageof growing soliton separation with increasingrelative phase (counterclockwise rotation). At theturning point, the trajectory passes into the innerspiral, both solitons approach, and the relativephase decreases (clockwise rotation). These obser-vations are consistent with a Kerr-mediated inter-action: Throughout the inner spiral, a dominantleading pulse experiences nonlinear phase retar-dation and propagates with reduced velocity [com-pare to the “heavy pulse” of (27)]; as a result, therelative phase and separation decrease. At theturning points, the maximum intensity passesfrom one pulse to the other, effectively reversingthe motion and yielding the complementary spiralin the interaction plane.Upon reduction of the pump level, the initial

periodic orbit in Fig. 4A exhibits increased sen-sitivity to fluctuations, resulting in aperiodic dy-namics (Fig. 4B). In the interaction plane, thebound state features several turning points con-sisting of partially periodic segments that twistbecause of aperiodic shifts in the phase. As a con-sequence, the configuration space progressivelyfills over time. Further reduction of the pump level,however, leads to a stabilization on closed or-bits (21) (Fig. 4C, right). Contrary to the closedorbits in Fig. 4, A and D (m = 0), the relativephase continuously advances over time, yielding awinding number of m = –1. In this regard, thedynamics are related to the step-like phaseevolution of the bound states in Fig. 3 with equalwinding number. In contrast, the present state(Fig. 4C) exhibits extensive internal motion,demonstrated by six turning points in the configu-ration space. This multifaceted closed orbitdemonstrates complicated, periodic trajectoriesas a result of intricate multisoliton interactionsemerging from the interplay of gain dynamicsand Kerr nonlinearity.

Conclusions

Real-time access to multipulse interactions in afemtosecond laser oscillator allows us to track

Herink et al., Science 356, 50–54 (2017) 7 April 2017 3 of 4

Fig. 3. Solitonmolecules with stepping phase. (A) Experimental real-time interferograms of the initialsoliton bound state with a separation of 170 fs and fixed relative phase. (B, C, and E) Upon reduction ofpump power, the relative phase of the soliton molecule evolves in a highly regular, stepping fashion,which accelerates for lower pump levels. (D) The common interaction plane representation illustratesthe fixed point (P = 4.8 W) and the periodic orbits. (F) Upon reduction of the pump power, the stepwisephase evolution approaches a linear shift in time, as illustrated by plotting the phase for various pumplevels on a common normalized time axis. Inset: The number of roundtrips per 2p-cycle decreases uponreduction of pump power. (G) Triplet bound states with fixed separations and dynamically evolvingrelative phases at enhanced pump power. (H) The autocorrelation (averaged) displays three temporaldelays present in the state. (I) Each relative phase continuously evolves in time; their rates of evolutionincrease with the intrapulse separation.

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the formation of stable soliton molecules anduncover rapid internal motions for a diverse setof bound states. Highly complex excitations, bothperiodic and aperiodic, are resolved via the directmapping of kinematic orbits in configuration space.In particular, we find oscillatory and progressivelyevolving bound states whose topologies are cha-racterized by different winding numbers, pointingtoward a possible topological protection of thedynamic excitations. The approach is expectedto provide real-time insight into a wider class ofphenomena in nonlinear systems, including thedynamics of breathers, the scattering of solitonsor rogue waves, and, generally, transient inter-actions in the emergence of rare and nonrepeti-tive events (4, 42).

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SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/356/6333/50/suppl/DC1Supplementary TextFig. S1

5 December 2016; accepted 15 March 201710.1126/science.aal5326

Herink et al., Science 356, 50–54 (2017) 7 April 2017 4 of 4

Fig. 4. Soliton molecules withoscillating separation andphase. (A to D) Left: Experimen-tal real-time interferograms ofdynamic doublet bound states,accessed via the tuning of pumppower. Center: Soliton doubletseparation and relative phase asa function of round trips along thesections marked by the whitelines (guide to the eye) in the leftcolumn. Right: Correspondinginteraction planes over multiplecycles.The radius represents thebound-state separation; theangle corresponds to the relativephase. All trajectories evolvebetween separations of 95 fs and115 fs (dashed radii). In addition,the line color encodes the evolu-tion of the accumulated relativephase. Continuous color gra-dients in (A) and (D) visualizeperiodic bound states with oscil-lating accumulated relativephases. In contrast, the periodicorbit with finite geometric phasefeatures a red/blue discontinuityin (C) (arrow), resulting from theaccumulated phase shift of –2pafter a completion of one orbit.For the aperiodic bound state in(B), the interaction plane illus-trates theprogressive fillingof theconfiguration space.

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moleculesReal-time spectral interferometry probes the internal dynamics of femtosecond soliton

G. Herink, F. Kurtz, B. Jalali, D. R. Solli and C. Ropers

DOI: 10.1126/science.aal5326 (6333), 50-54.356Science 

, this issue p. 50Sciencecomplex interaction dynamics could help in modeling other nonlinear systems.formation of soliton complexes as they propagated in a laser cavity. Real-time access to the formation processes and

used spectral interferometry to image and track theet al.localized structures known as solitons or optical bullets. Herink diffusing. However, under certain circumstances, the dispersion processes can be balanced by nonlinearities to produce

As a pulse of light propagates through a medium, scattering and dispersion processes usually result in the pulseProbing the interaction of solitons

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