9.MeasuringUltrashortLaserPulsesII:FROGTheMusicalScoreandtheSpectrogramFrequency-ResolvedOpticalGating(FROG)1Dvs.2DPhaseRetrievalFROGasa2DPhase-retrievalProblemSecond-harmonic-generation(SHG)FROG(andothergeometries)MeasuringtheshortesteventevercreatedSingle-shotFROG,XFROG,andTREEFROGPerhapsit’stimetoaskhowresearchersinotherfieldsdealwiththeirwaveforms…Consider,forexample,acousticwaveforms.Autocorrelationandrelatedtechniquesyieldlittleinformationaboutthepulse.It’saplotoffrequencyvs.time,withinformationontopabouttheintensity.Themusicalscorelivesinthe“time-frequencydomain.”Mostpeoplethinkofacousticwavesintermsofamusicalscore.IfE(t)isthewaveformofinterest,itsspectrogramis:whereg(t-t)isavariable-delaygatefunctionandtisthedelay.Withoutg(t-t),SpE(w,)wouldsimplybethespectrum.SpectrogramSSpectrogramFrequencyFrequencyTimeDelaySpectrogramsforLinearlyChirpedPulsesPropertiesoftheSpectrogramThespectrogramresolvesthedilemma!Itdoesn’tneedtheshorterevent!AlgorithmsexisttoretrieveE(t)fromitsspectrogram.Thespectrogramessentiallyuniquelydeterminesthewaveformintensity,I(t),andphase,(t).

Thereareafewambiguities,buttheyare“trivial.”Thegateneednotbe—andshouldnotbe—significantlyshorterthanE(t). Supposeweuseadelta-functiongatepulse:=TheIntensity.Nophaseinformation!“Polarization-Gate”GeometrySpectro-meterFrequency-ResolvedOpticalGating(FROG)Trebino,etal.,Rev.Sci.Instr.,68,3277(1997).KaneandTrebino,Opt.Lett.,18,823(1993).FROGFrequencyFrequencyTimeDelayFROGTracesforLinearlyChirpedPulsesFrequencyFrequencyIntensityTimeDelayFROGTracesforMoreComplexPulsesFrequencyDelay

Unfortunately,spectrograminversionalgorithmsrequirethatweknowthegatefunction.SubstitutingforEsig(t,)intheexpressionfortheFROGtrace:yields:Esig(t,)E(t)|E(t–)|2where:g(t–)

|E(t–)|2TheFROGtraceisaspectrogramofE(t).IfEsig(t,),isthe1DFouriertransformwithrespecttodelaytofsomenewsignalfield,Esig(t,W),then:SowemustinvertthisintegralequationandsolveforEsig(t,W).Thisintegral-inversionproblemisthe2Dphase-retrievalproblem,forwhichthesolutionexistsandisunique.Andsimplealgorithmsexistforfindingit.andTheinputpulse,E(t),iseasilyobtainedfromEsig(t,W):E(t)

Esig(t,)Instead,considerFROGasatwo-

dimensionalphase-retrievalproblem.Stark,ImageRecovery,AcademicPress,1987.1DPhaseRetrieval:SupposewemeasureS(w)anddesireE(t),where:GivenS(kx,ky),thereisessentiallyonesolutionforE(x,y)!!!Itturnsoutthatit’spossibletoretrievethe2Dspectralphase!.GivenS(w),thereareinfinitelymanysolutions

forE(t).Welackthespectralphase.2DPhaseRetrieval:SupposewemeasureS(kx,ky)anddesireE(x,y):TheseresultsarerelatedtotheFundamentalTheoremofAlgebra.WeassumethatE(t)andE(x,y)areoffiniteextent.1Dvs.2DPhaseRetrievalStark,ImageRecovery,AcademicPress,1987.TheFundamentalTheoremofAlgebrastatesthatallpolynomialscanbefactored:fN-1zN-1+fN-2zN-2+…+f1z+f0=fN-1(z–z1)

(z–z2)…(z–zN–1)TheFundamentalTheoremofAlgebrafailsforpolynomialsoftwovariables.Onlyasetofmeasurezerocanbefactored.fN-1,M-1yN-1zM-1+fN-1,M-2yN-1

zM-2+…+f0,0=?Whydoesthismatter?Theexistenceofthe1DFundamentalTheoremofAlgebraimpliesthat1Dphaseretrievalisimpossible.Thenon-existenceofthe2DFundamentalTheoremofAlgebraimpliesthat2Dphaseretrievalispossible.PhaseRetrievalandtheFundamentalTheoremofAlgebra1DPhaseRetrievalandtheFundamentalTheoremofAlgebraTheFouriertransform{F0,…,FN-1}ofadiscrete1Ddataset,{f0,…,fN-1},is:wherez=e–ikTheFundamentalTheoremofAlgebrastatesthatanypolynomial,fN-1zN-1+…+f0,canbefactoredtoyield:fN-1(z–z1)

(z–z2)…(z–zN–1)SothemagnitudeoftheFouriertransformofourdatacanbewritten:|Fk|=|fN-1(z–z1)

(z–z2)…(z–zN–1)|wherez=e–ikComplexconjugationofanyfactor(s)leavesthemagnitudeunchanged,butchangesthephase,yieldinganambiguity!So1Dphaseretrievalisimpossible!PhaseRetrieval&theFundThmofAlgebra2polynomial!2DPhaseRetrievalandtheFundamentalTheoremofAlgebraTheFouriertransform{F0,0,…,FN-1,N-1}ofadiscrete2Ddataset,{f0.0,…,fN-1,N-1},is:wherey=e–ikandz=e–iqButwecannotfactorpolynomialsoftwovariables.Sowecanonlycomplexconjugatetheentireexpression(yieldingatrivialambiguity).Onlyasetofpolynomialsofmeasurezerocanbefactored.So2Dphaseretrievalispossible!Andtheambiguitiesareverysparse.PhaseRetrieval&theFundThmofAlgebra2Polynomialof2variables!Esig(t,)E(t)|E(t–)|2Esig(t,)GeneralizedProjectionsEsig(t,)E(t)AlgorithmFindsuchthatandisascloseaspossibletoEsig(t,)DeLongandTrebino,Opt.Lett.,19,2152(1994)CodeisavailablecommerciallyfromFemtosoftTechnologies.WemustfindsuchthatandisascloseaspossibletoEsig(t,).ApplyingtheSignalFieldConstraintThewaytodothisistofindthefield,E(t),thatminimizes:OncewefindtheE(t)thatminimizesZ,wewritethenewsignalfieldas:Thisisthenewsignalfieldintheiteration.DeLongandTrebino,Opt.Lett.,19,2152(1994)KohlertracesSecond-harmonicgeneration(SHG)isthestrongestNLOeffect.Second-Harmonic-GenerationFROGKaneandTrebino,JQE,29,571(1993).DeLongandTrebino,JOSAB,11,2206(1994).FrequencyFrequencyTimeDelaySHGFROGtracesaresymmetricalwithrespecttodelay.FrequencyFrequencyIntensityTimeDelaySHGFROGtracesforcomplexpulsesFrequencyDelaySHGFROGMeasurementsofaFree-ElectronLaserSHGFROGworksverywell,eveninthemid-IRandfordifficultsources.Richman,etal.,Opt.Lett.,22,721(1997).Time(ps)Wavelength(nm)507651125148-4-2024IntensitySpectralIntensityPhase(rad)SpectralPhase(rad)043215043215ShortestpulselengthYearShortestpulsevs.yearPlotpreparedin1994(byErichIppen,MIT)reflectingthestateofaffairsatthattime.Themeasuredpulsespectrumhadtwohumps,andthemeasuredautocorrelationhadwings.10-fsspectraandautocorrsDatacourtesyofKapteynandMurnane,WSUDespitedifferentpredictionsforthepulseshape,boththeorieswereconsistentwiththedata.Twodifferenttheoriesemerged,andbothagreedwiththedata.FromHarveyet.al,Opt.Lett.,v.19,p.972(1994)FromChristovet.al,Opt.Lett.,v.19,p.1465(1994)FROGdistinguishesbetweenthetheories.Taft,etal.,J.SpecialTopicsinQuant.Electron.,3,575(1996).SHGFROGMeasurementsofa4.5-fsPulse!Baltuska,Pshenichnikov,andWeirsma,J.Quant.Electron.,35,459(1999).Pulseretrievalremainsequivalenttothe2Dphase-retrievalproblem.Manyinteractionshavebeenused,e.g.,polarizationrotationinafiber.GeneralizingFROGtoarbitrarynonlinear-opticalinteractionsFROGtracesofthesamepulsefordifferentgeometriesFROGgeometries:ProsandConsSecond-harmonicgenerationThird-harmonicgenerationTransient-gratingPolarization-gateSelf-diffractionmostsensitive;mostaccuratetightlyfocusedbeamsusefulforUV&transient-gratingexperimentssimple,intuitive,bestschemeforamplifiedpulsesusefulforUVGeneralizedProjectionswithanArbitraryMediumResponseDeLong,etal.,Opt.Lett.20,486(1995).Single-shotFROGSingle-ShotPolarization-GateFROGKaneandTrebino,Opt.Lett.,18,823(1993).Thiseliminatestheneedforabulkyexpensivespectrometeraswellastheneedtoalignthebeamthroughatinyentranceslit(whichwouldinvolvethreesensitivealignmentparameters)!Wecanusethefocusofthebeaminthenonlinearmediumastheentranceslitforahome-madeimagingspectrometer(inmulti-shotandsingle-shotFROGmeasurements).FROGallowsaverysimpleimagingspectrometer.PulsesfromfirsthalfofFROGCollimatingLensCameraorLinearDetectorArrayGratingImagingLensNonlinearMediumSignalpulseWhenaknownreferencepulseisavailable:

Cross-correlationFROG(XFROG)TheXFROGtrace(aspectrogram):UnknownpulseCameraE(t)Eg(t–t)SFGcrystalLensKnownpulseIfaknownpulseisavailable(itneednotbeshorter),thenitcanbeusedtofullymeasuretheunknownpulse.Inthiscase,weperformsum-frequencygeneration,andmeasurethespectrumvs.delay.Spectro-meterXFROGcompletelydetermin