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Tampere University of Technology

Real-time measurements of dissipative solitons in a mode-locked fiber laser

CitationRyczkowski, P., Närhi, M., Billet, C., Merolla, J. -M., Genty, G., & Dudley, J. M. (2017). Real-time measurementsof dissipative solitons in a mode-locked fiber laser. arXiv eprint.

Year2017

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Published inarXiv eprint

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1

Real‐timefullfieldmeasurementsrevealtransientdissipative

solitondynamicsinamode‐lockedlaserP.Ryczkowski1†,M.Närhi1†,C.Billet2†,J.‐M.Merolla2,G.Genty1,J.M.Dudley2*

1 LaboratoryofPhotonics,TampereUniversityofTechnology,Tampere,Finland.

2 InstitutFEMTO‐ST,UMR6174CNRS‐UniversitéBourgogneFranche‐Comté,Besançon,France.

† Equal contributions

* Corresponding author john.dudley@univ-fcomte.fr

Dissipativesolitonsareremarkablelocalizedstatesofaphysicalsystemthatarisefromthe

dynamicalbalancebetweennonlinearity,dispersionandenvironmentalenergyexchange.

Theyarethemostuniversalformofsolitonthatcanexistinnature,andareseeninfar‐

from‐equilibriumsystemsinmanyfieldsincludingchemistry,biology,andphysics.There

hasbeenparticularinterestinstudyingtheirpropertiesinmode‐lockedlasersproducing

ultrashort light pulses, but experiments have been limited by the lack of convenient

measurement techniques able to track the soliton evolution in real‐time.Here,we use

dispersiveFouriertransformandtimelensmeasurementstosimultaneouslymeasurereal‐

timespectralandtemporalevolutionofdissipativesolitonsinafiberlaserastheturn‐on

dynamicspassthroughatransientunstableregimewithcomplexbreak‐upandcollision

dynamics before stabilizing to a regularmode‐locked pulse train. Ourmeasurements

enable reconstruction of the soliton amplitude and phase and calculation of the

correspondingcomplex‐valuedeigenvalue spectrum toprovide furtherphysical insight.

These findingsaresignificant inshowinghowreal‐timemeasurementscanprovidenew

perspectivesintotheultrafasttransientdynamicsofcomplexsystems.

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INTRODUCTION

Whenoperatingintheirsteady‐stateregime,mode‐lockedlasersproducehighlyregular

stable pulse trains which have found widespread application in many fields of science (1‐3).

However,themode‐lockedlaserisalsowell‐knowntoexhibitarichvarietyofunstabledynamical

phenomenawhendetunedfromitssteady‐state,orasstablemode‐lockingbuildsupfromnoise

(4‐6).Thesetransient instabilitiesareasubjectofparticular interest inpassivelymode‐locked

fiberlasers,sincetheinterplayofthefibernonlinearityanddispersionwithcavitygainandloss

yieldsarichlandscapeofdissipativesolitondynamics(7‐10).Dissipativesolitonsareinfactthe

mostuniversalclassoflocalizedsolitonstateinphysics,existingforanextendedperiodoftimein

thepresenceofnonlinearityanddispersion,evenwhilepartsofthestructurecanexperiencegain

andloss.Incontrasttosolitonsinenergy‐conservingsystems,dissipativesolitonsexistinsystems

farfromequilibrium,andaredynamicalobjectsthatdisplayhighlynon‐trivialbehaviour(7).

Althoughthereisanextensivetheoreticalliteratureondissipativesolitonsinfiberlasers,

experimentalstudieshavebeenmorelimited,andoftenrestrictedtomeasurementsusingonly

fastphotodetectors(6,9,11‐13).Theseexperimentshavecertainlyprovidedinsightintounstable

modesoflaseroperation,yetthelimitedtemporalresolutionofthedetectorsusedhasgenerally

preventedanydetailedstudyoftheunderlyingsolitondynamics.

Recentyears,however,haveseendramaticadvancesintechniquescapableofreal‐time

measurementsofnon‐repetitiveopticalsignals(14‐25).Thefirstmethodtoseewidespreaduse

wasthedispersiveFouriertransform(DFT)techniqueforreal‐timespectralmeasurementsatthe

MHzrepetitionrate typicalofmode‐locked lasers (14). TheDFTwas firstused tostudy fiber

instabilitiessuchasroguewavedynamics,modulationinstabilityandsupercontinuumgeneration

(15‐18), andmore recently has been applied to study the spectral properties ofmode‐locked

lasers.Theabilitytoresolvethespectrumaftereachcavityroundtripinalaserhasrevealednovel

decoherenceandsolitonexplosiondynamicsina fiber laser(19,20),andwavelengthevolution

dynamics in a Kerr‐lens Ti:Sapphire laser (21). In fact, soliton dynamics in the Kerr‐lens

3

Ti:SapphirelaserhavealsobeenstudiedusingamodifiedversionoftheDFTadaptedforreal‐time

spectralinterferometry(22).Inparallelwiththeseadvancesinreal‐timespectralmeasurements,

thedevelopmentofatemporalanalogofaspatialthinlenshasresultedinreal‐timepulseintensity

measurementsinwithsub‐picosecondresolution(23).Thetime‐lenshasbeenusedinstudiesof

incoherentsolitonpropagationinopticalturbulence(24),andstochasticbreatheremergencein

modulationinstability(25).

In this work, we combine, for the first time, real‐time spectral DFT and time lens

techniques to perform simultaneous measurements of the spectral and temporal profiles of

dissipative solitons evolving during the turn‐on phase of a mode‐locked fiber laser. Our

measurementsaremadewithsub‐nmandsub‐psresolution,providingaunifiedsnapshotofhow

thesolitonsevolvefromroundtriptoroundtrip,anddirectlyrevealingarangeofhighlycomplex

dynamicstypicalofdissipativesolitons.Inaddition,phaseretrievalallowsthereconstructionof

the full field (intensity and phase) of the measured pulses, allowing us to determine the

correspondingcomplexeigenvaluespectrumofthepulsesindifferentdynamicalregimes.These

results are significant in providing a unique picture of the internal evolution of fiber laser

dissipativesolitons,andweanticipatetheirapplicationintheoptimizationanddesignoflasers

withimprovedstabilitycharacteristics.Moregenerally,webelievethatourresultswillstimulate

the widespread use of simultaneous temporal and spectral characterization as a standard

techniqueforthestudyofultrafastcomplexopticalsystems.

RESULTS

Ourexperimentsstudiedthetransient turn‐ondynamicsofasaturableabsorbermode‐locked

fiberlaserconfigured(instableoperation)togeneratepulsesof~4.5psdurationat1545nmwith

20MHzrepetitionrate(i.e.cavityroundtriptimeof50ns).Thelasercavityhasnetanomalous

dispersion such that in steady‐state operation the circulating pulses are close to fundamental

solitons,andindeedfrequency‐resolvedopticalgatingmeasurementsconfirmedthatthestable

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outputpulseswerewell‐fittedbyahyperbolicsecantprofileandweretransformlimitedwitha

uniformtemporalphase(26).

Wecharacterizedtheturn‐ondynamicsusingbothdirectphotodiodedetectionaswellas

simultaneousDFTandtime‐lensmeasurements(seeMaterialsandMethodsforfulldetails).Fig.1

showsresultsusingdirectphotodiodedetection.Specifically,Fig.1(a)plotstheenvelopeofthe

pulses measured with reduced detection bandwidth to capture a time‐window of 130ms,

correspondingto2.6 million cavityroundtrips.The figureclearlyrevealsa~100ms transient

regime of Q‐switched mode‐locked operation before the appearance of a stable pulse train.

Figure1(b)showsaseparatemeasurementofaportionofthetransientregimeasindicated,but

hereusing30GHzdetectionbandwidthsothatitispossibletoresolveaseriesof~10μsbursts

separatedbya~70 μsperiod.TheexpandedviewinFig.1(b)showshowundereachburst,an

irregulartrainofmode‐lockedpulses(at50nsperiod)ispresent.Incontrastwiththisunstable

regime,Fig.1(c)showsmeasurementsinthestablemode‐lockedregimeusing30GHzdetection,

toillustratearegularpulsetrainofconstantintensity.Ofcourse,suchcomplextransientdynamics

havebeenseeninbothnumericalandexperimentalstudiesofarangeofotherpassively‐mode‐

lockedlasers(27‐34),butweincludethesemeasurementshereforcompleteness.Thekeypoint

isthat,evenwithadetectionbandwidthof30GHz,thetemporalwidthofthepulsesseeninthe

photodiode intensity record is~30ps,precludinganydetailedstudyof theunderlyingsoliton

dynamicsinthetransientregime.It isthis limitationthatweovercomewithoursimultaneous

real‐timeDFTandtime‐lensmeasurements.

Figure2showsthesetupused.Lightfromthefiberlaseroutputwassplitintotwopaths

andsenttothereal‐timeDFTandtime‐lensacquisitionarms.Realtimespectralmeasurements

usedastandardDFTsetup(18)withspectralresolutionof~0.3nm.Thetimelensset‐upwas

similar to that described inRef. (25) andwas capable of real‐timemeasurements of intensity

profilesupto60psdurationwithaneffectivetemporalresolutionof400fs.BothDFTandtime

lenssignalswererecordedusingahighstoragecapacitydigitaloscilloscope,withdataacquisition

triggeredbythetime‐lenssignaldetectedafterthelaserwasturnedon.Wecouldsimultaneously

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measurespectralandtemporalintensityprofilesoveramaximumof400cavityroundtrips(see

MaterialsandMethods).

We recorded multiple data sequences to examine the spectral and temporal soliton

dynamicsbothintheQ‐switchedmode‐lockedandthestable‐modelockingregime.Webeginin

Fig.3(a)byshowingresultsforthestablemodelockingregimewhereasexpectedthespectral(i)

and temporal (ii) intensity profiles are constant with roundtrip. The extracted spectral and

temporalwidths and the integratedpulse energy are shownas the right subfigures; the small

(noise)variationsinthesefiguresisoftheorderoftheexperimentalresolutionofthespectraland

temporalmeasurements.

Althoughtheabilitytomeasurebothtemporalandspectralintensityinrealtimeisitselfa

highly significant advance, we extend this technique further using Gerchberg‐Saxton phase‐

retrieval(35,36)torecoverthecorrespondingcomplexelectricfieldofeachpulse(seeMaterials

andMethods).TheseresultsareshowninFig.3(a‐iii)whereweplottheretrievedintensity(left

axis)andphase(rightaxis)revealingthesolitoncharacteristicsofnear‐uniformphase.Accessto

thefullcomplexelectricfieldallowsustocalculatetheassociatedwavelength‐timespectrogram,

and these results are shown in Fig.3(a‐iv), confirming the localized soliton‐like nature of the

pulses.Fig4showssimilar results,but fromadatasequence justprior to theregimeof stable

modelocking.Here,overarelativelysmallnumberofroundtrips,weseesignificantmodulationin

boththemeasuredspectralandtemporalamplitudes,aswellasenergyvariationof~30%.This

“breathing” of the intracavitypulses justprior to theonsetof stability is aknownpropertyof

dissipativesolitondynamicsinmode‐lockedlasers(21),butourresultsarethefirsttobeableto

experimentallycharacterizeitcompletelyinboththetimeandfrequencydomains.

Dissipative solitons candisplayamuch richer rangeof interactiondynamics (7‐9), and

Fig.4showsexamplesof thisbehavior in theevolutionofpulsesundera transientQ‐switched

burst. To capture different bursts at different timeswe used a variable hold off time for our

measurementwindowtoscanvariouspointsofthetransientregime.Infact,wefoundthatwhilst

thegeneraldynamicsweresimilarforallburstdurations(i.e.showingtheemergenceofmultiple

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pulsesfromaninitialnoisyfieldfollowedbysubsequentdecay)thedetailedevolutiondynamics

varied significantly for different bursts. Indeed, ourmeasurement set‐up allowed us to reveal

previously unobserved regimes of dissipative soliton propagation and interaction under the

transientenvelope.

Remarkingfirstlythatthespectralandtemporalevolutionwithroundtripisplottedfrom

toptobottom,twotypicalresultsareshowninFig.4(a)and4(b)toillustratethedifferenttypes

ofbehaviorobserved.Alsonotethatenergyisnotconstantinthetransientturn‐onregime(12).

We first discuss Fig.4(a)which shows an initial noisy field splitting into 3 distinct pulses (of

duration~5ps)thatpropagatecoherentlytogetherover~100roundtrips.Eachpulsedisplaysa

well‐localizedtemporalintensitypeak,andthethreepulsesaremutuallycoherentasisseenby

thefactthatthereisdistinctmodulationinthecorrespondingspectrum.Significantly,thepulse

separationdoesnotvaryover~100roundtripsasthepulsesevolvewithoutseemingtointeract

beforedecaying.Figure4(b)showsaqualitativelydifferentcase.Here, twopulsesemerge,but

rather than propagating without interacting, they undergo attraction and eventually collide,

displayingamorecomplexnonlinearphaseprofileatthispoint.

Althoughadetailedstudyofthesedynamicsisbeyondthescopeofthispaper,weremark

thatanalysisoftheseresults(andamoreextensivedatasetofover100similarcases)revealthat

whetherornottheemergentsolitonsevolvewithoutinteractingorcollidedependsonwhether

ornotthepulseshavethesamecentralfrequency.Thecentralfrequenciesofthepulsescanbe

clearly seenon thespectrogramplots, and inFig.4(a)weseehowthe three (non‐interacting)

pulseshaveslightlydifferentcentralfrequencieswhereasinFig.4(b)thetwopulsesthatcollide

have the same central frequency and similar amplitudes. These results are consistent with

previousstudiesofboundsolitondynamics(37‐39).Figure4(c)ontheotherhandshowsanother

exampleofburstevolutionwhereacombinationof theabovedynamicsareobservedwiththe

emergenceof3solitons.Here,thetwosolitonswiththesamecentralfrequencyattracteachother

and collideafter~100 roundtrips,whereas the remaining solitonwith loweramplitudeand a

slightlydifferentfrequencypropagateswithoutinteractionbeforeeventuallydecaying(40,41).

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Theabilitytocompletelyreconstructthecomplexelectricfieldfromthemeasuredpulse

intensityandphaseishighlysignificant,andindeedallowscalculationofderivedquantitiesthat

yieldadditionalandimportantphysical insights. Inparticular, it ispossibletoapplynumerical

techniquesfromscatteringtheory(42‐45)tocalculateanonlinearFouriertransformthatyields

spectraleigenvalueportraitsofthepulsestructureswhichcanthenbecomparedwiththeknown

signaturesof ideal solitons (42).Although theuseof suchscattering theory in fiberopticshas

generallybeenfromatheoreticalperspectivetoobtainanalyticsolutionstothegoverningpartial

differentialequations,therehasalsobeenrecentinterestinnumericalapproachestoyieldinsight

intoroguewavedynamics(46)andasanovelapproachtoovercomingchannellimitsinoptical

communications(47).Significantly,although the theoreticalanalysis isonlystrictlyvalid inan

integrablesystem,numericalcomputationoftheeigenvaluespectrumforthedissipativesoliton

systemconsideredhere(seeMaterialsandMethods)nonethelessyieldsresultsthathaveaclear

physicalinterpretation.Thisisbecauseweconsiderevolutionregionswherethepulseproperties

areprimarilydeterminedbynonlinearanddispersiveeffects(i.e.wecanneglectresonantgain

shaping) so the systemmaybe considered tobeonlyweaklynon‐integrable. In this case the

evolutionofpulsesthroughthedifferentdiscreteelementsofthecavitycanbeapproximatedby

anequivalentnonlinearSchrödingerequation(NLSE)withuniformdistributednonlinearityand

dispersion,andthenonlinearspectrumcanbeconsideredasyieldinglocaleigenvaluesthatcan

revealapproximatesolitoncontent(48).

TheresultsofthisanalysisareshowninFig.5wherethereddotsineachsubfigureshow

thecalculateddiscreteeigenspectrumforthenormalizedpulseintensityprofilesshownasinsets.

In this case,we associate the stable pulse regime show in Fig. 3(a)with solitons havingwith

eigenvaluesat=±0.5iinthecomplexplane.BasedonthemeasurementsinFig.3(a),wecanthen

normalizetheintensitiesofmeasuredpulsesinthetransientregimewithrespecttotheseideal

solitonsandperformdirect scattering analysis todetermine the correspondingmore complex

eigenvaluespectrum(seeMaterialsandMethods).

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WefirstconsidertheresultsinFig.5(a)and(b)whichcorrespondtosinglepulsesattwo

pointsofthebreathercycleshowninFig.3(b‐iii).Ineachcasethescatteringanalysisyieldsone

distincteigenvalue,andweseethatasthepulseintensityvariesbelowandabovetheunityvalue

ofanormalizedfundamentalsoliton(dashedblackline),theeigenvalueIm()alsovariesbelow

andabovetheidealsolitonvalueof±0.5i.Inotherwords,thevariationintheeigenvaluereflects

thebreathingofthepulsepropertiesinthisregime.

Figure5(c) and (d) considermore complex evolution. Figure5(c)plots the eigenvalue

spectrumforthedouble‐solitonpulseshowninFig.4(b‐iii)wherethedirectscatteringprocedure

yieldstwodiscreteeigenvalues,bothwithIm()~±0.5i.Figure5(d)showsresultsforthethree‐

solitoncaseinFig.4(a)andhereweseetwodiscreteeigenvalueswithIm(~±0.5iandathird

eigenvalueataslightlylowervalue.Inallcases,theclearseparationoftheeigenvaluesfromthe

realaxisindicatesthatthisparticulardynamicalregimewithinthelaserburstspriortosteady‐

statemodelockingcanbeconsideredascreatingdiscretepulsecomplexes,similartothoseseen

withmultipleboundsolitons(7‐9).Thisisaveryimportantphysicalinsightthatthenonlinear

Fouriertransformrevealsdirectly(48).

DISCUSSION

Dissipativenonlinearsystemscanbehighlycomplex,butitisclearthatrecentyearshave

seentremendousadvancesinbeingabletounderstandtheirbehaviorthroughnoveltheoretical

approachesandnumericalmodelling.Ofcourse,afullunderstandingofsuchcomplexdynamical

phenomena requires that theoretical and numerical results are carefully compared with

experiment,andtothisendthemeasurementtechniqueandphaseretrievalanalysisreportedin

thispapermakeasignificantcontribution inenablingcompletecharacterizationofpicosecond

dissipativesolitonsystems.

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Theresultsobtainedshowawiderangeofultrafasttemporalandspectraldynamicsneverbefore

seendirectly,andweanticipatethatourresultswillstimulatemanyfuturetheoreticalstudiesand

numericalmodellingthatcannowbecompareddirectlywithexperimentinawaythatwasnot

previouslypossible.

A further area of particular significance concerns our use of the nonlinear Fourier

transformtocalculatealocaleigenspectrumofthedissipativesolitonsinthelaser.Thecalculated

spectraclearlyshowthepresenceoflocalsolitoncontentinthecomplexpulseprofilesmeasured,

whichprovidesanewwindowintothephysicsoftheunderlyinglaserdynamics.Wealsobelieve

ourresultswillmotivateinterestinmuchbroaderapplicationsofthenonlinearFouriertransform

in all dissipative systems where field evolution through discrete elements can be modelled

approximately by uniformly distributed nonlinearity and dispersion (49,50). In optics, we

anticipateparticularinterestinreal‐timespectralandtemporalstudiesofnonlinearsingle‐pass

propagationdynamicsinopticalfiberssuchasroguewaveandmodulationinstabilitythatalso

displaycomplextransientnoisespikes.

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SUPPLEMENTARYMATERIAL

MATERIALSANDMETHODS

ExperimentalSetup

ThefiberlaserusedinourexperimentswasacommercialPritelFFL‐500modelusinga

linearFabry‐Perotcavityconfiguration,similartothatdescribedin(51)butwitha978nmpump

laserandEr:dopedfibergainmedium.Modelockingissustainedbya2mthickbulksaturable

absorber(InGaAsonInPsubstrate)contactbondedtooneofthecavitymirrors.Thesteady‐state

modelockingdynamicsaredominatedbysolitonpropagationeffectsbecauseofthenetanomalous

dispersioninthecavity,withthegenerationofhyperbolic‐secantlikepulsesofflatphaseinstable

operation.

WeimplementedtheDFTtechniqueusing850mofdispersioncompensatingfiber(DCF)

withgroupvelocitydispersioncoefficientof100ps/nm.kmanddispersionslope0.33ps/nm2.km

at 1550nm.We attenuated the input to the DCF in order to ensure linear propagation, and

confirmedthefidelityofthetime‐stretchingtechniquewhenthefiberlaserwasoperatedinthe

stablemode‐lockedregimebycomparingtheDFTspectrumwiththatmeasuredusinganoptical

spectrum analyser (Anritsu MS9710B). The real‐time DFT signal was measured by a 35GHz

photodiode (New Focus 1474A) connected to a 30GHz channel of a real‐time oscilloscope

(LeCroy845Zi‐A,80GS/s),resultinginaspectralresolutionof0.3nm.

The time lens measurements used a commercial Picoluz UTM-1500 system described

previously in (24) withatemporalmagnificationfactorof76.4.Totalaccumulateddispersionfor

theinputandoutputpropagationstepswas:D1=4.16ps/nmandD2=318ps/nmrespectively,

withmagnification|M|=D2/D1.Thetemporalquadraticphase(toreproducetheeffectofathin

lens)wasimposedthroughfourwavemixingfromapumppulse(100MHzMenloC‐FiberSync

andP100‐EDFA)withlinearchirpaccumulatedfrompropagationinapre‐chirpingfiberDp. The

imaging condition for magnification is 2/Dp = 1/D1 +1/D2 so that the dispersion for the pump is

around twice that of the signal input step. Thesignalatthetimelensoutputisrecordedbya

11

13GHzphotodiode(Miteq135GE)connectedtothe30GHzchannelofthereal‐timeoscilloscope

at a sampling rate of 80Gs/s, resulting in an effective 400 fs temporal resolution over a

(demagnified) timewindowof 60ps.The temporalwindow isdeterminedby thepumppulse

durationusedinthetimelenstoimposetherequiredquadraticchirpviafour‐wavemixingina

highlynonlinearSiwaveguide(52).Instablemode‐lockedoperationitispossibletosynchronize

the20MHzlaserunderstudywiththe100MHzpumplaser,butthisisnotthecasewhenstudying

the transient dynamics as there is significant amplitude and phase noise that precludes the

detectionofawell‐defined20MHzharmonicforrepetition‐ratelocking.Thetimelensistherefore

operatedinasynchronousmodewithfree‐runningacquisitiontriggeredbythearrivalofthetime

lenssignal,althoughthislimitsthenumberofroundtripsthatcanbesimultaneouslymeasuredto

~400(asthereisawalk‐offbetweentheQ‐switchedmode‐lockedpulsesrelativetothetimelens

gate).ToobtainrepresentativedatasetsatdifferentpointsintheQ‐switchedmode‐lockedburst

(whichtypicallydevelopsover200‐300roundtrips)weperformedmultiplemeasurementswith

differentdelaysbetweentheswitch‐ontimeofthefiberlaserandthetimelenstrigger.Inthisway,

wewere able to characterize different phases of the pulse evolution in the transient regime.

Finally,wenotethatbecausetheopticalpathlengthsoftheDFTandtimelensstepsweredifferent,

thedelaybetween time‐lens andDFT recordswas calibrated (in a separatemeasurement)by

comparing thearrival timeson theoscilloscopeofa characteristic intensitypattern imprinted

ontoaCWlaser.Thisallowedustomatchwithoutambiguitythereal‐timetemporalandspectral

intensityprofilesofthedissipativesolitonsduringtheirtransientevolutionphase.

PhaseRetrieval

PhaseretrievalwasperformedwiththeGerchberg‐Saxtonalgorithm(35).Asidefromafewtrivial

ambiguities,thisalgorithmisknowntobeaccurateforpulseretrievalwhenintensityenvelopes

aremeasuredbothinthespectralandtimedomains.Thealgorithmconstructsaninitialguessfor

theprofileoftheelectricfieldusingthemeasuredtemporalintensityIM(t)andaninitialrandom

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phase rand(t): which is then Fourier transformed to the spectral

domaintoyieldacorrespondinginitialguessforthespectralamplitudeandphaseFT[ ]=

. The next step involves retaining the calculated spectral phase but replacing the

calculated spectral amplitude with the measured spectral amplitude i.e. . This

updated spectral profile is then transformed back into the time domainwherewe retain the

calculated temporal phase but again replace the calculated amplitude with that from

experiment . This procedure is iteratively repeated until the root mean square error

betweenthemeasurementsandretrievedintensityprofilesbecomessmallerthanachosenvalue

(3×10‐5inourcase).Convergencewasimprovedbyapplyingthemeasuredtemporalandspectral

constraintsonlywherethemeasuredintensitieswerewellabovethenoisefloor(‐20dBfromthe

maximum).Elsewheretheamplitudesweremultipliedwithasmallconstantvalueof0.001forcing

thembelowthenoise.Ifthealgorithmwasdetectedtostagnate(nochangeintheretrievalerror

beforereachingthedesiredvalue),asmalladditionalrandomphasecontributionwasadded.The

reliabilityofthealgorithmwastestedwithsimulatedpulseswithcomplexpropertiessimilarto

thoseseeninexperiments,andbyperformingmultipleretrievalsonthesameexperimentaldata,

whichallconvergedtothesameresults(withintheretrievalerror).

NonlinearFourierTransform

The nonlinear Fourier transform (also known as the direct scattering transform) is a

mathematicalprocedurethatidentifiesandquantifiessolitoncontentinagivenpulsestructure.

Significantly,whilstitsuseintheoreticalanalysistoobtainclosed‐formanalyticsolitonsolutions

is only strictly valid in an integrable system, computation of the eigenvalue spectrum can be

performednumericallyforanyarbitraryopticalfield.Ofcourse,thequestioninthiscaseishow

such results should be interpreted, but for a laser where the pulse properties are primarily

determined by nonlinear and dispersive effects, the nonlinear spectrum can be considered as

yieldinglocaleigenvaluesthatcanrevealapproximatesolitoncontent(48).Thisisbecauseifwe

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canneglectstrongpulsefilteringeffects(e.g.fromtheresonanttransitionorsaturableabsorber)

then the evolution through the different discrete elements of the cavity can be considered as

yielding an average pulse that is a solution to an equivalent NLSE with uniform distributed

nonlinearityanddispersion.Infactthisapproachisthebasisofthewell‐knownuseofthecomplex

GinzbergLandau(orHausMasterEquation)modelthathasprovenhighlysuccessfulinmodelling

mode‐lockedlasers(8,9,53).

UndertheseconditionsconsideringasystemdescribedbytheNLSEinnormalisedform:

(1)

Theassociatedscatteringproblemyieldsthefollowingsystem(42):

(2)

where v1 and v2 are the amplitudes of the waves scattered by the potential, and is the

correspondingcomplexeigenvalue.Forourresults,stablemodelockedoperationwasassumedto

correspondto(intracavity)solitonswith||=1,andthefieldprofilescorrespondingtotheother

pulsesanalysedwerenormalisedrelativetothisvalue.Standardnumericaltechniques(matrix

methods)wereusedtodeterminetheeigenvaluespectrum(48).

22

2

10

2i

14

ACKNOWLEDGEMENTS

Funding

ThisworkwassupportedbytheAgenceNationaledelaRechercheprojectLABEXACTIONANR11‐

LABX‐0001‐01,theRegionofFranche‐ComtéProjectCORPSandtheAcademyofFinland(Grants

267576and298463).

Contributions

Allauthorsparticipatedintheexperimentalworkanddataanalysisreported,andtothewriting

and reviewof the finalmanuscript.G.G. and J.M.D.planned the researchproject andprovided

overallsupervision.TheauthorsalsothankK.V.Reddyforprovidingtechnicaldetailsconcerning

thesolitonoperatingregimeofthePritellaserusedintheseexperiments.

Competingfinancialinterests

Theauthorsdeclarenocompetingfinancialinterests.

Correspondingauthors

Correspondenceto:JohnM.Dudley

15

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Figure1.Directphotodetectormeasurementof transient laserdynamics. (a)Results recorded

with reduced bandwidth detection of 20MHz illustrating the~100ms transient regime ofQ‐

switched mode‐locked operation before stable mode‐locking. The total time record shown

corresponds to2.6millionround trips. (b)Results fromseparatemeasurementsusing30GHz

bandwidthdetection,showinghowthelaseroutputduringtheQ‐switchedmode‐lockedregime

consistsoftransientburstsoftemporalwidth~10sseparatedby~70speriod.Theexpanded

viewshowshowunstablemode‐lockedpulsesat50nsperiodaregeneratedundereachburst.

(c)Resultsusing30GHzdetectionbutinthestablemode‐lockedregime,showingaregulartrain

ofpulseswithconstantamplitude.

21

Figure2.Setupusedforreal‐timecharacterization.Pulsesfroma20MHzpassivelymode‐locked

fiberlaseraresplitintotwopathsinafibercoupler(FC)andsenttoaDFTrealtimespectralsetup

andatime‐lens.TheDFTused850mofdispersioncompensatingfiber(DCF)toperformtimeto

wavelengthconversion,withdetectionusinga35GHzbandwidthphotodiode(PD2).Anoptical

spectrumanalyzerwasalsousedforcontrolmeasurementsofthespectrum.Thetimelensuses

twodispersivepropagationsteps(D1andD2),oneoneachsideofasiliconwaveguidestagethat

appliesaquadratictemporalphasethroughfourwavemixing(FWM)withlinearly‐chirpedpump

pulsesgeneratedfroma100MHzfemtosecondpulsefiberlaser(TLPump)afterstretchingina

dispersivestretching fiberDp.Thetime lenssignal isextracted fromtheFWMspectrumbyan

opticalfilterandwasmeasuredusinga13GHzbandwidthphotodiode(PD1).BothPD1andPD2

wereinputtoa30GHzchannelofthereal‐timeoscilloscope.ValuesfordispersionmodulesD1,D2

andDpinthetimelensaregiveninMaterialsandMethods.TheinputstoboththeDFTandtime

lenswereattenuated(Att.)toavoidnonlineareffectsandpolarizationcontrol(PC)wasneededto

ensureoptimalsignalfromthetime‐lens.

22

Figure3.Resultsover200roundtripsshowingreal‐timespectralandtemporalcharacterization

for(a)stableand(b)breathingmode‐lockingregimes.(i)Measuredspectralintensitywiththe

extractedspectralwidthshownintherightsubfigure.(ii)Measuredtemporalintensitywiththe

extractedtemporalwidthandenergyshownintherightsubfigures.(iii)Correspondingtemporal

intensityandphaseextractedusingtheGerchberg‐Saxtonalgorithmforpulsesatpointsindicated

by arrows. For each of the extracted pulses in (iii), the plots in (iv) show the calculated

wavelength‐timespectrogram.

23

Figure4.Resultsover200roundtripsshowingreal‐timespectralandtemporalcharacterization

for (a) a non‐interacting triplet of three solitons, (b) more complex break‐up and collision

dynamicsofasolitondoublepulseand(c)acombinationofasinglenon‐interactingsolitonanda

twopulsecollision.(i)Measuredspectral intensity.(ii)Measuredtemporal intensity.Theright

subfigure shows the integrated energy. (iii) Corresponding temporal intensity and phase

extractedusingtheGerchberg‐Saxtonalgorithmforpulsesatpointsindicatedbyarrows.Foreach

oftheextractedpulsesin(iii),theplotsin(iv)showthecalculatedwavelength‐timespectrogram.

24

Figure5.Resultsapplying thescattering transformto intensityandphasemeasurementsofa

selectionofmeasureddissipativesolitons.Ineachcasethereddotsshowthecalculateddiscrete

eigenspectrum corresponding to the normalized pulse intensity profiles shown as insets. The

resultsinFig5(a)and(b)correspondtothetwopointsinthedissipativesolitonbreathercycle

showninFig.3(b).Asthepulseintensityvariesbelowandabovetheunityvalueofanormalized

fundamentalsoliton,theretrievedeigenvaluesIm()variesbelowandabove±0.5i.(c)Thedouble

solitoncaseinFig.4(b)whereweseetwodiscreteeigenvalueswithIm()~±0.5i.(d)Thethree

solitoncaseinFig.4(a)whereweseethreediscreteeigenvalueswithIm()around±0.5i.(Note

that the intensity profiles plotted correspond to results shown in Fig. 3 and 4 but have been

smoothedforclaritywhenshownasinsets.)Thedashedlineintheinsetsshowstheintensityof

anidealsoliton.