PhysRevA.88.023859

Molecularalignmentallowslow-orderharmonicgenerationbycircularlightinagas

J.Houzet,E.Hertz,F.Billard,B.Lavorel,andO.Faucher

LaboratoireInterdisciplinaireCarnotdeBourgogne,UMR6303CNRS,BP47870,21078DijonCedex,France

(Received12April2013;published30August2013)Weexperimentallyinvestigateodd-orderharmonicgenerationinmoleculargasesproducedbycircularlypolarizedlaserfields.Whileforbiddeninisotropicmedium,thiseffectisallowedbysymmetrybreakingresultingfromnonadiabaticlaser-inducedmolecularalignment.ThedemonstrationisprovidedbygeneratingthethirdharmonicinCO2molecules.Attractivepropertiesandchallengingpotentialapplicationsarediscussedinthecontextofhigher-orderharmonicgeneration.DOI:10.1103/PhysRevA.88.023859

I.INTRODUCTION

PACSnumber(s):42.65.Ky,42.50.Md,33.80.−b,42.65.Re

Nonlinearfrequencyup-conversionfromprealignedmolec-ularsampleshasbeenasubjectofgrowinginterestinthelastdecade.Ithasbeenshownforinstancethathigh-orderharmonicgeneration(HHG)fromalignedmoleculescanprovideaccesstotomographicimagingofmolecularorbitals[1,2].Molecularalignmenthasalsobeenexploitedfordeterminingthephaseofthehigh-orderharmonics[3,4]orforcontrollingtheirpolarizationstatewithavarietyofattractiveapplications[5,6].Thecaseoflower-orderharmonicshasalsobeeninvestigated.Bartelsandcoworkers[7]havereportedtheinfluenceofmolecularalignmentonthirdharmonicgeneration(THG)drivenbylinearlypolarizedlaserfields.TheyhaveidentifieddifferenttypesofTHGprocessesandanalyzedthealignmentdependenceofthephasematching.

ThispaperreportsontheexperimentalstudyofTHGpro-ducedbycircularlypolarizedlaserfields.Theeffectisdemon-stratedinCO2gasatroomtemperatureandrelativelylowpressure.Weshouldrecallthatcircularlightdoesnotgenerateharmonicsinisotropicmediabecauseofsymmetryreasons[8].Wethereforetakeadvantageofmolecularalignmenttobreakthesymmetryofthegasmedium.AshortnonresonantlaserpumppulseexcitesarotationalwavepacketbydrivingaseriesofimpulsivestimulatedRamantransitionswithinthevibronicgroundstateofthemolecule.Thefreeevolutionofthesystemafterthepulseturnoffleadstoperiodicrevivalsofmolecularalignment[9,10].Ashorttime-delayedprobepulse,circularlypolarized,generatesthird-orderharmonicradiationinthesampleofalignedmolecules.AnextendedcontroloftheTHGefficiencyuponmolecularalignmentisobserved.Itisnoticedthatthesignalevolveswiththepump-probedelayasacombinationoftherotationalexpectationvalues󰀆cos2θ󰀇(t)and󰀆cos4θ󰀇(t).Thepresentmethodprovidesthereforeadirectmeasurementof󰀆cos4θ󰀇(t),atermgenerallyinvolvedinthedescriptionofhigh-orderharmonicsignalsproducedbyalignedmolecules[11–14].Theobservationofthisexpectationvalueprovidesalsoricherinformationabouttheevolutionofrotationalwavepackets,whichcouldbeexploitedforprobingcollisionsindissipativemediaassuggestedinRef.[15].Themethodfeaturesadditionalattractiveproperties.WeshowinparticularthatthepolarizationoftheTHGradiationvarieswiththedegreeofalignmentandthatcircularlypolarizedfieldcanbeproducedfromprealignedmolecules.OnecanthereforeenvisionthattheextensionofthemethodtoHHGcouldallowproductionofVUVorXUVradiationwithellipticorcircular

1050-2947/2013/88(2)/023859(4)

polarizationthatwouldbehighlydesirableforinstanceinchiralrecognition[16–20].Finally,thestandardrescatteringmodel[21]isconsideredtobeinadequatetoexplainthegenerationofhigh-orderharmonicsbycircularlightsothattheunderlyingmechanismcouldshedlightonappealingnewphysicsaspredictedinRefs.[22,23].

II.EXPERIMENTALSETUP

ThemainfeaturesoftheexperimentalsetuparedepictedinFig.1.ATi:sapphirechirpedpulseamplifierdelivers5mJpulsesof90fsdurationaround800nmat100Hzrepetitionrate.Theoutputbeamissplitintwopartstoproducethepumpandprobebeams.Therelativedelayτbetweenthemisadjustedviaamotorizedtranslationstage.Thetwobeamsarefocusedandcrossedatsmallangle(≈4◦)inastaticcellfilledwithCO2atlowpressure(lessthan0.1bar).TheTHGradiation,filteredfromthefundamentalradiationbymeansofabandpassfilterandtwodielectricmirrorsat266nm,isdetectedbyaphotomultiplier.Theprobefieldiscircularlypolarizedbymeansofazero-orderquarter-waveplate.Specialattentionispaidtoavoidanydetrimentalharmonicsignalassociatedtoresidualellipticitybycarefullytiltingthequarter-waveplatearounditsneutralaxessuchastominimizetheTHGwhenthepumpfieldisoff.ThepumpfieldislinearlypolarizedalongtheverticalzaxisandtheTHGisexpectedtohavefieldcomponentsalongtheyandzdirections(Fig.1).Inordertodetecteachcomponentindependently,weuseahalf-waveplateat266nm(producedbyaBerekcompensator)inconjunctionwithaglassplatesetatBrewsterangle.Thelatterselectstheverticalpolarizationforthedetectionwhilethehalf-waveplateallowsanoverallrotationofthethirdharmonicfieldattheexitofthecell.Thisschemeallowsmeasurementoftheharmonic

3ω3ω

signalsSyandSzproducedperpendicularlyandalongthealigningpulsepolarization,respectively.WehavecheckedthatthesignalproducedbyrotatingthepolarizationoftheTHGthroughthisprocedureisequivalenttotheoneobtainedbyrotatingthepumppolarizationaxis.

III.RESULTSANDDISCUSSION

Typicalsignalsrecordedat0.07bararedepictedinFigs.2(a)and2(b).Asshown,thesignaltonoiseratioisverygoodinspiteoftherelativelylowpressureindicatingthatthepresenttechniquecanbeseenasavaluableversatilemethodformeasuringalignment.BesidestheusualCO2transient

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BSPHYSICALREVIEWA88,023859(2013)

PumpLSCLzyx/2 (266)Probe/4−→

(y,z)planeandatemporalfieldenvelopeε(t)sothatEpr=0,ε(t−τ),±iε(t−τ)].Foralinearmolecule,itcanbeshownthatthecomponentsα=y,zoftheinduceddipolemomentoperatorresponsibleforTHGwritesas

ω23μ3α(t)=c2α(󰀆cosθ󰀇(t)−1/3)ε(t−τ)

(1)

+c4α(󰀆cos4θ󰀇(t)−1/5)ε(t−τ)3,

M(266)M(266)PMwith

c2y=−9/4γZZZZ−9/4γXXYY+9γXXZZ,c4y=15/8γZZZZ+45/8γXXYY−45/4γXXZZ,c2z=±i(3/2γZZZZ+9γXXYY−27/2γXXZZ),c4z=±i(−5/2γZZZZ−15/2γXXYY+15γXXZZ),andwhereγIJKLdenotesthecomponentsofthesecond

hyperpolarizabilitytensorinthemolecularframe(X,Y,Z)withvaluesgiveninRef.[24].Theexpectationvalues󰀆cos2θ󰀇(t)and󰀆cos4θ󰀇(t)arecalculatedbysolvingthetime-dependentSchr¨odingerequation[10].

Thesignaldetectedbythephotomultiplierasafunctionofthepump-probedelayτisdirectlyproportionaltothecomplexmodulusofthedipolemoment:

󰀁∞

󰀃3ω󰀃2

3ω󰀃μ(t)󰀃dt.(3)Sα(τ)∝α

−∞

GPPB(2)

FIG.1.(Coloronline)Experimentalsetup.BS:beamsplitter,

L:lenses(f=150mm),SC:staticcell,λ/4:quarter-waveplateat800nm,λ/2(266):half-waveplateat266nm,M(266):mirrorscoatedformaximizingthereflectionat266nm,GP:glassplateatBrewsterangle,PB:pass-bandfilter(250-280nm),PM:photomultiplier.

revivalsproducedaround10.7and21.4ps(i.e.,aroundnTr/4,withTr=42.7pstheclassicalrotationalperiodofthemolecule[10]),thesignalfeaturesadditionaltinypeaksaround(2n+1)Tr/8(i.e.,5.3and16.0ps).Inordertointerpretthesignal,thedependencyoftheTHGuponmolecularalignmentisdescribedwithperturbationtheory.Weconsiderforthecalculationapumppulsepolarizedalongthezaxis.Theprobefieldisassumedtohaveapurelycircularpolarizationinthe

τFIG.2.(Coloronline)CO2harmonicsignals(blacklineswithsymbols)measuredwithapolarizationalongtheyaxis(a)andzaxis(b)togetherwiththesimulation(redlines).Experimentalconditions:pumpenergy150μJ,probeenergy50μJandpressure0.07bar.

AsdiscussedinRef.[7],apartoftheproducedTHGresultsfromamixofyandzfieldcomponentsasinatype-IIconversionprocess.Thephasemismatchdependsinthiscaseonthedegreeofalignmentandvarieswiththepump-probedelay,whichsignificantlycomplicatestheanalysisoftheTHGsignal.However,asimpleestimationshowsthatthiseffectcanbeneglectedforthepressureinvestigatedinthepresentwork.ThisisconfirmedbythesimulationsshowninredlinesinFigs.2(a)and2(b)thatreproducefairlywelltheexperimentaldata.Weemphasizethat,besidesthe

3ω3ω

andSztemporalshape,therelativeamplitudesbetweenSy

arealsowellreproduced.Theexpectationvalues󰀆cos2θ󰀇(τ)and󰀆cos4θ󰀇(τ)aredepictedinFig.3.Asshown,bothtermscontributetothelargeTHGsignalatthenTr/4revivals.How-ever,thesimilitudebetweentheirtemporalshapespreventsanycleardisentanglementoftheirrespectivecontributions.Thesmalltransientsaround(2n+1)Tr/8,resultingfromcoherencesexcitedbetweenJandJ+4rotationalstates,areinturnsolelyrelatedto󰀆cos4θ󰀇(τ).Theirobservationprovidesthereforeaclearexperimentalevidenceofthisterm[11].TheaccuracyofthesimulationsdisplayedinFig.2,including

3ω3ω

backgroundandrelativeamplitudeofSyandSz,providesastringenttestforthedeterminationof󰀆cos4θ󰀇.Thisisrelevantconsideringtheroleplayedbythistermintheproductionofhigh-orderharmonicsinseveralmolecularsystems[11–14].MeasurementsconductedatvariouspumpintensitiesandintheO2moleculeallowedustodrawsimilarconclusions.It

3ω3ω

shouldbepointedoutthatthesignalsSyandSzproducedat(2n+1)Tr/8displayalmostoppositevariationswithrespecttothebaselinesothatthesesmallrevivalswouldbehardlydetectablewithoutapolarizationanalysis.Inthiscase,thesumofthetwoTHGfieldcomponentswouldindeedleadtoasignificantcancellationofthesignalat(2n+1)Tr/8

ωω023859-2

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(c)45-45045-90(a)1.51.00.50.000.5190β (deg)-9000.5190θS3ω10.010.511.0pump-probe delay τ (ps) 11.53ω3ω

FIG.4.(Coloronline)(a)THGsignalsSyandSzrecordedalongtheyaxis(blueline)andthezaxis(reddashedline),respectively.Upperpanels(b)and(c)depictthepolarizationanalysisforthetwospecificdelaysindicatedbyarrows,βbeingtheanglebetweenthedetectedTHGpolarizationandz(seetext).

θτFIG.3.(Coloronline)Expectationvalues󰀆cos2θ󰀇(τ)and󰀆cos4θ󰀇(τ)forapumpintensityof55TW/cm2correspondingtoFig.2.

thatwouldpreventthedeterminationof󰀆cos4θ󰀇(τ).Anothercriterionfortheobservationof󰀆cos4θ󰀇(τ)referstothemoleculardensity.Athighmoleculardensity,theprobebeamsuffersfromdepolarizationasitpropagatesthroughasampleofalignedmolecules.Theinducedbirefringenceresultingfrommolecularalignmentisproportionalto󰀆cos2θ󰀇(t)[10]sothatdepolarizationmainlyoccursatnTr/4.TheweakdepolarizationoftheprobecontributestoalargeenhancementoftheTHGsincefrequencyconversionbyellipticpolarizationisnotforbiddeninisotropicmedium.Thecorrespondingenhancementcanreachafactorof100at1baraccordingtoourcalculation.Intheseconditions,thetinypeaksaround(2n+1)Tr/8thatdonotbenefitfromthisincreasearenotdetectableanymore.Thisfeaturehasbeenexperimentallyverifiedandisasubjectforfuturework.Weemphasizethatanincreaseoftheanglebetweenpumpandprobebeamscanpreventtheeffectofdepolarization.Inthiscase,theprobepulsewillinteractwiththesampleofalignedmoleculesonlyatthelaserfocuswhereTHGisproduced.

Itisworthmentioningthatthepresentmeasurementsdonotinvolveanysignificantcontributionofplasma-enhancedTHG.Recently,enhancementofTHGbypreionizingagasmediumwithapumplaserpulsehasbeenreported.Twodifferentinterpretationshavebeenproposed,bothrelatedtotheplasmaleftoutbythepumppulse.Thefirstone[25]arguesforaneffectiveplasma-enhancedthird-orderopticalsusceptibility,whilethesecondone[26]invokesareductionoftheinteractionlengththroughapropagationeffect,whichlimitsthepartialTHGcancellationduetotheGouyphase.Inbothcases,theplasma-enhancedTHGwouldresultinasubstantialbackgroundatpositivepump-probedelaysthatwouldobscuretheanalysisofTHGsignal.ThishasbeenconfirmedbyproducingtheTHGwithaprobelinearlypolarized.However,itcanbeshownthatcircularpolarizationdoesnotallowplasma-enhancedTHGinacentrosymmetricmediumsothatthiseffectisassumedtobemarginalfor

themoderatemolecularalignmentachievedinthepresentstudy.

AnotherinterestingprospectisrelatedtothecontroloftheTHGellipticity,withthepossibilityofproducingaTHGfieldwithcircularpolarizationasultimategoal.AccordingtoEqs.(1)–(3),theharmonicfieldsalongtheyandzaxisareproducedinphasequadrature,resultinginanellipticfieldinmostcases.However,dependingonthetimedelayτ,theTHGfieldamplitudesalongyandzaredifferentlymodulatedbythe󰀆cos2θ󰀇(τ)and󰀆cos4θ󰀇(τ)termsasshownbythemeasurementspresentedinFig.4(a).Forcertainvaluesofτ,equalamplitudecomponentscanbeobservedmeetingthereforetherequirementsforthegenerationofcircularlypolarizedTHG.

WehaveanalyzedtheTHGfieldpolarizationforthedelayτ=10.8pssatisfyingthisconditionandindicatedbytheleftarrowinFig.4(a).Figure4(b)depictsthesignalobtainedbyinducinganoverallrotationβoftheTHGfieldattheexitofthecellwiththehalf-waveplate.Asobserved,themeasuredsignaldoesnotexhibitanysignificantvariationwithβbearingoutthecircularpolarizationoftheproducedTHGfield.Incontrast,thepolarizationanalysisperformedatadelayτ=11.2ps,where

3ω3ω

pronounceddifferencebetweenSyandSzareobserved,displayslargevariations[Fig.4(c)]revealingthesignificantellipticityoftheTHGfield.Productionofacircularlypolarizedharmonicfieldisanimportantissue,especiallyinthecontextofHHGwithapplicationsinultrafastspindynamics,nano-lithography,andmagneticcirculardichromism.ExtensionofthepresenttechniquetoHHGprovidesanalternativeapproachtootherexistingmethods[16–20].

IV.SUMMARYANDCONCLUSION

Inconclusion,opticalharmonicgenerationhasbeenpro-ducedinamoleculargaswithacircularlypolarizedlaserfield.Theconversionprocessismadefeasiblebybreakingthesym-metryofthegasmediumthroughnonadiabaticlaser-inducedmolecularalignment.Althoughtheeffectisdemonstratedusingthird-harmonicgeneration,itcanbestraightforwardlyappliedtohigher-orderharmonics.ExperimentalconditionsallowingTHGobservationarediscussed.Besidesasimple

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waytomeasurethealignment,themethodfeaturesseveralattractiveproperties.First,itprovidesadirectmeasurementoftheexpectationvalue󰀆cos4θ󰀇(t).ThistermplaysakeyroleinHHGsignalsproducedbyalignedmoleculesbutitsmeasure-mentalsofeaturesabroadrangeofadditionalapplications.Forinstance,theexpectationvalue󰀆cos2θ󰀇(t)isproportionaltothesecondmomentoftherotationaldistributionwhilethecompleterotationaldistributionimpliesallmoments.Inthisregard,measurementofhigher-orderrotationalexpectationvaluesofferspotentialforrotationalwave-packetimaging[27]orforprobingdissipativemedia[15]byprovidinginformationthatisnotrevealedby󰀆cos2θ󰀇(t).Second,thesimulationsofFig.2havebeenperformedusingthecomponentsofthesecondhyperpolarizabilitytensorgiveninRef.[24].Ouranalysishasrevealedanoticeablesensitivityofthemodelwithrespecttothissetofelements.Othervaluesfoundintheliterature

3ω

didnotallowustoreachasatisfactoryagreementforSy

3ω

togetherwithSz,includingtherelativeamplitudebetweenthem.Measurementofthehyperpolarizabilitycomponentsandinparticularoftheoff-diagonalones(i.e.,γXXZZ)isknown

tobeatrickytask.Webelievethatthepresentmethodcouldbeveryhelpfulinthisissueandshouldbeconsideredasafutureprospectofourwork.Third,thetechniqueenablestheproductionofcircularlypolarizedharmonics.ExtensionofthemethodtoHHGcouldallowproductionofellipticallyorcircularlypolarizedfields,whichrepresentsasubjectofactiveresearch[16–20].Finally,HHGinalignedmoleculesbycircularlypolarizedpulsescouldsignificantlydifferfromthewell-knownrecollisionmodel[21]andthusrevealsparticularphysicalmechanisms[23].Unforeseendynamicsarepredictedwith,forinstance,harmonicspectrarevealingtwodistinguishableplateaus[22].Thepresentworkprovidestheguidelinesforsuchforthcomingexperiments.

ACKNOWLEDGMENTS

ThisworkwassupportedbytheConseilR´egionaldeBourgogne,theCNRS,andtheLABEX“ACTION”.AuthorssincerelyacknowledgeP.B´ejotandG.Karrasforfruitfuldiscussions.

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