第八讲-超短脉冲强度测量说课讲解

Thegoal:measuringtheintensityandphasevs.time(orfrequency)Theways:TheSpectrometerandMichelsonInterferometerScanningAutocorrelationSingle-shotautocorrelationTheAutocorrelationandSpectrumThird-orderAutocorrelationInterferometricAutocorrelationMeasuringUltrashortLaserPulsesI:AutocorrelationE(t)E(t–t)Inordertomeasureaneventintime,youneedashorterone.Tostudythisevent,youneedastrobelightpulsethat’sshorter.Butthen,tomeasurethestrobelightpulse,youneedadetectorwhoseresponsetimeisevenshorter.Andsoon…So,now,howdoyoumeasuretheshortestevent?PhotographtakenbyHaroldEdgerton,MITTheDilemmaTodeterminethetemporalresolutionofanexperimentusingit.Todeterminewhetheritcanbemadeevenshorter.Tobetterunderstandthelasersthatemitthemandtoverifymodelsofultrashortpulsegeneration.Tobetterstudymedia:thebetterweknowthelightinandlightout,thebetterweknowthemediumwestudywiththem.Tousepulsesofspecificintensityandphasevs.timetocontrolchemicalreactions:“Coherentcontrol.”Tounderstandpulse-shapingeffortsfortelecommunications,etc.Becauseit’sthere.Whymeasureanultrashortlaserpulse?Asamoleculedissociates,

itsemissionchangescolor(i.e.,thephasechanges),revealingmuchaboutthemoleculardynamics,notavail-ablefromthemerespectrum,oreventheintensityvs.time.ExcitationtoexcitedstateEmissionGroundstateExcitedstateExp’tTheoryLinearLinearornonlinearmediumMeasuringtheintensityandphaseofthepulsesintoandoutofamediumtellsusasmuchaspossibleaboutthelinearandnonlineareffectsinthemedium.StudyingMediabyMeasuringtheIntensityandPhaseofLightPulsesWithalinearmedium,welearnthemedium’sabsorptioncoefficientandrefractiveindexvs.

Withanonlinear-opticalmedium,wecanlearnaboutself-phasemodulation,forexample,forwhichthetheoryismuchmorecomplex.Indeed,theoreticalmodelscanbetested.Time(fs)IntensityPhaseNonlinearEaton,etal.,JQE35,451(1999).Time(fs)IntensityPhaseAlaserpulsehasthetime-domainelectricfield:EI(t)1/

2exp[iw0t–i

(t)]}IntensityPhase(t)=Re{Equivalently,vs.frequency:exp[-ij(w–w0)]}SpectralPhase(neglectingthenegative-frequencycomponent)E(w)=Re{~S(ww0)1/

2Wemustmeasureanultrashortlaserpulse’s

intensityandphasevs.timeorfrequency.SpectrumKnowledgeoftheintensityandphaseorthespectrumandspectralphaseissufficienttodeterminethepulse.

t

d

dtTheinstantaneousfrequency:Example:“Linearchirp”Phase,

(t)timetimeFrequency,w(t)timeWe’dliketobeabletomeasure,notonlylinearlychirpedpulses,butalsopulseswitharbitrarilycomplexphasesandfrequenciesvs.time.Thephasedeterminesthepulse’sfrequency

(i.e.,color)vs.time.Thespectrometermeasuresthespectrum,ofcourse.Wavelengthvariesacrossthecamera,andthespectrumcanbemeasuredforasinglepulse.PulseMeasurementintheFrequencyDomain:

TheSpectrometerCollimatingMirror“Czerny-Turner”arrangementEntranceSlitCameraorLinearDetectorArrayFocusingMirrorGrating“Imagingspectrometers”allowmanyspectratobemeasuredsimultaneously.Broad-bandpulsePulseMeasurementintheTimeDomain:DetectorsExamples:Photo-diodes,Photo-multipliersDetectorsaredevicesthatemitelectronsinresponsetophotons.Detectorshaveveryslowriseandfalltimes:~1nanosecond.Asfaraswe’reconcerned,detectorshaveinfinitelyslowresponses.Theymeasurethetimeintegralofthepulseintensityfrom–

to+

:Thedetectoroutputvoltageisproportionaltothepulseenergy.Bythemselves,detectorstelluslittleaboutapulse.Anothersymbolforadetector:DetectorDetectorPulseMeasurementintheTimeDomain:

VaryingthepulsedelaySincedetectorsareessentiallyinfinitelyslow,howdowemaketime-domainmeasurementsonorusingultrashortlaserpulses????We’lldelayapulseintime.Andhowwillwedothat?Bysimplymovingamirror!Sincelighttravels300µmperps,300µmofmirrordisplacementyieldsadelayof2ps.Thisisveryconvenient.MovingamirrorbackwardbyadistanceLyieldsadelayof:Donotforgetthefactorof2!Lightmusttraveltheextradistancetothemirror—andback!TranslationstageInputpulseE(t)E(t–t)MirrorOutputpulseWecanalsovarythedelayusing

amirrorpairorcornercube.Mirrorpairsinvolvetworeflectionsanddisplacethereturnbeaminspace:Butout-of-planetiltyieldsanonparallelreturnbeam.Cornercubesinvolvethreereflectionsandalsodisplacethereturnbeaminspace.Evenbetter,theyalwaysyieldaparallelreturnbeam:“Hollowcornercubes”avoidpropagationthroughglass.TranslationstageInputpulseE(t)E(t–t)MirrorsOutputpulse[EdmundScientific]Measuringtheinterferogramisequivalenttomeasuringthespectrum.PulseMeasurementintheTimeDomain:

TheMichelsonInterferometerPulseenergy(boring)Fieldautocorrelation(maybeinteresting,but…){TheFTofthefieldautocorrelationisjustthespectrum!Beam-splitterInputpulseDelaySlowdetectorMirrorMirrorE(t)E(t–t)Okay,sohowdowemeasureapulse?V.Wong&I.A.Walmsley,Opt.Lett.19,287-289(1994)I.A.Walmsley&V.Wong,J.Opt.Soc.AmB,13,2453-2463(1996)Result:Usingonlytime-independent,linearfilters,completecharacterizationofapulseisNOTpossiblewithaslowdetector.Translation:Ifyoudon'thaveadetectorormodulatorthatisfastcomparedtothepulsewidth,youCANNOTmeasurethepulseintensityandphase.withonlylinearmeasurements,suchasadetector,interferometer,oraspectrometer.Weneedashorterevent,andwedon’thaveone.Butwedohavethepulseitself,whichisastart.Andwecandevisemethodsforthepulsetogateitselfusingopticalnonlinearities.PulseMeasurementintheTimeDomain:

TheIntensityAutocorrelatorCrossingbeamsinanSHGcrystal,varyingthedelaybetweenthem,andmeasuringthesecond-harmonic(SH)pulseenergyvs.delayyieldstheIntensityAutocorrelation:TheIntensityAutocorrelation:DelayBeam-splitterInputpulseApertureeliminatesinputpulsesandalsoanySHcreatedbytheindividualinputbeams.SlowdetectorMirrorE(t)E(t–t)MirrorsSHGcrystalLensSingle-ShotAutocorrelationWhilethiseffectintroducesarangeofdelaysonanygivenpulseandcouldcauseabroadeningofthetraceinmulti-shotmeasurements,itallowsustomeasureapulseonasinglelasershotifweusealargebeamandalargebeamangletoachievethedesiredrangeofdelays.Single-ShotAutocorrelationc2Single-ShotAutocorrelationInputpulse(expandedinspaceto~1cm)Beam-splitterSHGcrystalCameraE(t)E(t–t)Cylindricallensfocusesthebeamintheverticaldirection(forhighintensity),whilethedelayvarieshorizontally.Nomirrormoves!Crossingthebeamsatalargeangle,focusingwithacylindricallens,anddetectingvs.transversepositionyieldstheautocorrelationforasinglepulse.LensimagescrystalontocameraandhencedelayontopositionatcameraThebeammusthaveconstantintensityvs.horizontalpositiontoavoidbiases.ApertureSingle-ShotAutocorrelationofLongerPulsesIfalongerpulseistobemeasured,alargerrangeofdelaysisrequired.Alongerrangecanbeachievedusingadispersiveelement,suchasaprismorgrating,whichtiltsthepulsefronts.Angulardispersionisundesired,however.Fortunately,ifweneedtousethistrick,it’sbecausethepulseislong.Asaresult,itsbandwidthisusuallysmall,soangulardispersionislessofaproblem(forpulses>10ps).PracticalIssuesinAutocorrelationGroup-velocitymismatchmustbenegligible,orthemeasurementwillbedistorted.Equivalently,thephase-matchingbandwidthmustbesufficient.Soverythincrystals(<100µm!)mustbeused.Thisreducestheefficiencyandhencethesensitivityofthedevice.Conversionefficiencymustbekeptlow,ordistortionsdueto“depletion”ofinputlightfieldswilloccur.Insingle-shotmeasurements,thebeammusthaveaconstantintensityvs.position.Inmulti-shotmeasurements,thebeamoverlapinspacemustbemaintainedasthedelayisscanned.MinimalamountsofglassmustbeusedinthebeambeforethecrystaltominimizetheGVDintroducedintothepulsebytheautocorrelator.It’seasytointroducesystematicerror.Theonlyfeedbackonthe

t

measurementqualityisthatitshouldbemaximalatt=0andsymmetricalindelay:

tbecauseSquarePulseandItsAutocorrelationttPulseAutocorrelationGaussianPulseandItsAutocorrelationPulseAutocorrelationttPulseAutocorrelationttSech2PulseandItsAutocorrelationSincetheoreticalmodelsforidealultrafastlasersusuallypredictsech2pulseshapes,peopleusuallysimplydividetheautocorrelationwidthby1.54andcallitthepulsewidth.EvenwhentheautocorrelationisGaussian…LorentzianPulseandItsAutocorrelationPulseAutocorrelationttAutocorrelationsofmorecomplexintensitiesAutocorrelationsnearlyalwayshaveconsiderablylessstructurethanthecorrespondingintensity.Anautocorrelationtypicallycorrespondstomorethanoneintensity.Thustheautocorrelationdoesnotuniquelydeterminetheintensity.Autocorr2Autocorrelationsofcomplexpulses:firstconsideradoublepulsePulseAutocorrelationtwhere:tAutocorrelationofVeryComplexPulsesIntensityAutocorrelationAstheintensityincreasesincomplexity,itsautocorrelationapproachesabroaddiffusebackgroundwithacoherencespike.PulseMeasurementinBothDomains:

CombiningtheSpectrumandAutocorrelationPerhapsthecombinedinformationoftheautocorrelationandthespectrumcoulddeterminethepulseintensityandphase.Thisideahasbeencalled:“TemporalInformationViaIntensity(TIVI)”J.PeatrossandA.Rundquist,J.Opt.Soc.AmB15,216-222(1998)Itinvolvesaniterativealgorithmtofindanintensityconsistentwiththeautocorrelation.Thenitinvolvesanotheriterativealgorithmtofindthetemporalandspectralphasesconsistentwiththeintensityandspectrum.Neitherstephasauniquesolution,sothisdoesn’twork.AmbiguitiesinTIVI:PulseswiththeSameAutocorrelationandSpectrumPulse#1Pulse#2SpectraandspectralphasesforPulses#1and#2AutocorrelationsforPulses#1and#2IntensityIntensityPhasePhasetFWHM=24fstFWHM=21fs#1#2ChungandWeiner,IEEEJSTQE,2001.SpectraThesepulses—especiallythephases—areverydifferent.AmbiguitiesinTIVI:MorePulseswiththeSameAutocorrelationandSpectrumPulse#3Pulse#4SpectraandspectralphasesforPulses#3and#4AutocorrelationsforPulses#3and#4IntensityIntensityPhasePhasetFWHM=37fstFWHM=28fs#4#3ChungandWeiner,IEEEJSTQE,2001.Despitehavingverydifferentlengths,thesepulseshavethesameauto-correlationandspectrum!There’snowaytoknowallthepulseshavingagivenautocorrelationandspectrum.SpectraThird-OrderAutocorrelationPolarizationGating(PG)wSelf-diffrac-tion(SD)wThird-har-monicgen-eration(THG)

wThird-ordernonlinear-opticaleffectspro-videthe3rd-orderintensityautocorrelation:Notethe2Thethird-orderautocorrelationisnotsymmetrical,soityieldsslightlymoreinformation,butnotthefullpulse.Third-ordereffectsareweaker,soit’slesssensitiveandisusedonlyforamplifiedpulses(>1µJ).Whenashorterreferencepulseisavailable:TheIntensityCross-CorrelationTheIntensityCross-correlation:DelayUnknownpulseSlowdetectorE(t)Eg(t–t)SFGcrystalLensReferencepulseIfashorterreferencepulseisavailable(itneednotbeknown),thenitcanbeusedtomeasuretheunknownpulse.Inthiscase,weperformsum-frequencygeneration,andmeasuretheenergyvs.delay.Ifthereferencepulseismuchshorterthantheunknownpulse,thentheintensitycross-correlationfullydeterminestheunknownpulseintensity.PulseMeasurementintheTimeDomain:

TheInterferometricAutocorrelatorWhatifweuseacollinearbeamgeometry,andallowtheautocorrelatorsignallighttointerferewiththeSHGfromeachindividualbeam?DevelopedbyJ-CDielsUsualAutocor-relationtermNewtermsAlsocalledthe“Fringe-ResolvedAutocorrelation”FilterSlowdetectorSHGcrystalLensBeam-splitterInputpulseDelayMirrorMirrorE(t)E(t–t)MichelsonInterferometerDielsandRudolph,UltrashortLaserPulsePhenomena,AcademicPress,1996.InterferometricAutocorrelationMathThemeasuredintensityvs.delayis:Multiplyingthisout:whereTheInterferometricAutocorrelationisthe

sumoffourdifferentquantities.Constant(uninteresting)Sum-of-intensities-weightedw

“interferogram”ofE(t)

w(oscillatesatwindelay)IntensityautocorrelationInterferogramofthesecondharmonic;equivalenttothespectrumoftheSHw(oscillatesat2windelay)Theinterferometricautocorrelationsimplycombinesseveralmeasuresofthepulseintoone(admittedlycomplex)trace.Conveniently,however,theyoccurwithdifferentoscillationfrequencies:0,w,and2w.InterferometricAutocorrelationandStabilizationInterferometricAutocorrelationTracesforaFlat-phaseGaussianpulse:PulselengthFortunately,it’snotalwaysnecessarytoresolvethefringes.WithstabilizationWithoutstabilizationToresolvethewand2wfringes,whicharespacedbyonlylandl/2,wemustactivelystabilizetheapparatustocanceloutvibrations,whichwperturbthedelaybymanyl.C.Rulliere,FemtosecondLaserPulses,Springer,1998.InterferometricAutocorrelation:ExamplesTheextentofthefringes(atwand2w)indicatestheapproximatewidthoftheinterferogram,whichisthecoherencetime.Ifit’sthesameasthew

widthofthethelow-frequencycomponent,whichistheintensityw

autocorrelation,then

thepulseisnear-Fourier-transformlimited.w

Unchirpedpulse(short)~Coherencetime~PulselengthChirpedpulse(long)~Coherencetime~PulselengthTheinterferometricautocorrelationnicelyrevealstheapproximatepulselengthandcoherencetime,and,inparticular,theirrelativevalues.Solidblacklineshavebeenadded.Theytracetheintensityautocorrelationcomponent(forreference).C.Rulliere,FemtosecondLaserPulses,Springer,1998.Doestheinterferometricautocorrelationyieldthepulseintensityandphase?No.TheclaimhasbeenmadethattheInterferometricAutocorrelation,combinedwiththepulseinterferogram(i.e.,thespectrum),coulddosoNaganuma,IEEEJ.Quant.Electron.25,1225-1233(1989).Buttherequirediterativealgorithmrarelyconverges.Thefactisthattheinterferometricautocorrelationyieldslittlemoreinformationthantheautocorrelationandspectrum.Weshouldn’texpectittoyieldthefullpulseintensityandphase.Indeed,verydifferentpulseshaveverysimilarinterferometricautocorrelations.PulseswithVerySimilarInterferometricAutocorrelationsPulse#1IntensityPhasetFWHM=24fsPulse#2IntensityPhasetFWHM=21fsWithouttryingtofindambiguities,wecanjusttryPulses#1and#2:Despitetheverydifferentpulses,theseIAtracesarenearlyidentical!ChungandWeiner,IEEEJSTQE,2001.InterferometricAutocorrelationsforPulses#1and#2Difference:#1and#2PulseswithVerySimilarInterferometricAutocorrelationsChungandWeiner,IEEEJSTQE,2001.It’sevenhardertodistinguishthetraceswhenthepulsesareshorter,andtherearefewerfringes.ConsiderPulses#1and#2,but1/5aslong:InterferometricAutocorrelationsforShorterPulses#1and#2#1and#2Pulse#1IntensityPhasetFWHM=4.8fs-20-1001020Pulse#2IntensityPhasetFWHM=4.2fs-20-1001020Inpractice,itwouldbevirtuallyimpossibletodistinguishthesetracesalso.Difference:MorePulseswithSimilarInterferometricAutocorrelationsChungandWeiner,IEEEJSTQE,2001.Withouttryingtofindambiguities,wecantryPulses#3and#4:IntensityPhasetF