超快光学-第04章-脉冲(2)

UltrashortLaserPulsesIIMoresecond-orderphaseHigher-orderspectralphasedistortionsRelativeimportanceofspectrumandspectralphasePulseandspectralwidthsTime-bandwidthproductProf.RickTrebinoGeorgiaTech200012()...1!2!RecalltheTaylorseriesfor():Asinthetimedomain,onlythefirstfewtermsaretypicallyrequiredtodescribewell-behavedpulses.Ofcourse,we’llconsiderbadlybehavedpulses,whichhavehigher-ordertermsin().01ddwhereisthegroupdelay.0222ddiscalledthe“group-delaydispersion.”TheFouriertransformofachirpedpulseWritingalinearlychirpedGaussianpulseas:or:Fourier-Transformingyields:Rationalizingthedenominatorandseparatingtherealandimagparts:AGaussianwithacomplexwidth!AchirpedGaussianpulseFourier-Transformstoitself!!!2200()expexp.tEtittccE200()expexp..tEititccE2001/4()expEEi220002222/4/4()expexpEEiwhere21/tneglectingthenegative-frequencytermduetothec.c.Butwhenthepulseislong(0):whichistheinverseoftheinstantaneousfrequencyvs.time.Thegroupdelayvs.forachirpedpulseThegroupdelayofawaveisthederivativeofthespectralphase:()/grtdd2022/4()012grt022/2grtSo:ForalinearlychirpedGaussianpulse,thespectralphaseis:0()2instttAndthedelayvs.frequencyislinear.Thisisnottheinverseoftheinstantaneousfrequency,whichis:2nd-orderphase:positivelinearchirpNumericalexample:Gaussian-intensitypulsew/positivelinearchirp,2=14.5radfs2.Herethequadraticphasehasstretchedwhatwouldhavebeena3-fspulse(giventhespectrum)toa13.9-fsone.2nd-orderphase:negativelinearchirpNumericalexample:Gaussian-intensitypulsew/negativelinearchirp,2=–14.5radfs2.Aswithpositivechirp,thequadraticphasehasstretchedwhatwouldhavebeena3-fspulse(giventhespectrum)toa13.9-fsone.Thefrequencyofalightwavecanalsovarynonlinearlywithtime.ThisistheelectricfieldofaGaussianpulsewhosefre-quencyvariesquadraticallywithtime:Thislightwavehastheexpression:Arbitrarilycomplexfrequency-vs.-timebehaviorispossible.Butweusuallydescribephasedistortionsinthefrequencydomain.Nonlinearlychirpedpulses2300()Reexp/expGEtEtitt20()3instttE-fieldvs.timeE(t)3rd-orderspectralphase:quadraticchirpLongerandshorterwavelengthscoincideintimeandinterfere(beat).Trailingsatellitepulsesintimeindicatepositivespectralcubicphase,andleadingonesindicatenegativespectralcubicphase.S()tg()()Spectrumandspectralphase400500600700Becausewe’replottingvs.wavelength(notfrequency),there’saminussigninthegroupdelay,sotheplotiscorrect.3rd-orderspectralphase:quadraticchirpNumericalexample:Gaussianspectrumandpositivecubicspectralphase,with3=750radfs3Trailingsatellitepulsesintimeindicatepositivespectralcubicphase.Negative3rd-orderspectralphaseAnothernumericalexample:Gaussianspectrumandnegativecubicspectralphase,with3=–750radfs3Leadingsatellitepulsesintimeindicatenegativespectralcubicphase.4th-orderspectralphaseNumericalexample:Gaussianspectrumandpositivequarticspectralphase,4=5000radfs4.Leadingandtrailingwingsintimeindicatequarticphase.Higher-frequenciesinthetrailingwingmeanpositivequarticphase.Negative4th-orderspectralphaseNumericalexample:Gaussianspectrumandnegativequarticspectralphase,4=–5000radfs4.Leadingandtrailingwingsintimeindicatequarticphase.Higher-frequenciesintheleadingwingmeannegativequarticphase.5th-orderspectralphaseNumericalexample:Gaussianspectrumandpositivequinticspectralphase,5=4.4×104radfs5.Anoscillatorytrailingwingintimeindicatespositivequinticphase.Negative5th-orderspectralphaseNumericalexample:Gaussianspectrumandnegativequinticspectralphase,5=–4.4×104radfs5.Anoscillatoryleadingwingintimeindicatesnegativequinticphase.TherelativeimportanceofintensityandphasePhotographsofmywifeLindaandme:CompositephotographmadeusingthespectralintensityofLinda’sphotoandthespectralphaseofmine(andinverse-Fourier-transforming)CompositephotographmadeusingthespectralintensityofmyphotoandthespectralphaseofLinda’s(andinverse-Fourier-transforming)Thespectralphaseismoreimportantfordeterminingtheintensity!PulsepropagationWhathappenstoapulseasitpropagatesthroughamedium?Alwaysmodel(linear)propagationinthefrequencydomain.Also,youmustknowtheentirefield(i.e.,theintensityandphase)todoso.()()exp[()/2]exp[()]outinEELinkLInthetimedomain,propagationisaconvolution—muchharder.()inE()outE()()n()()exp[()]outinSSL()()()outinnLcPulsepropagation(continued)()()exp[()/2]exp[()]outinEELinkL()exp[()/2]exp[()]inELinLcusingk=/c:Separatingoutthespectrumandspectralphase:Rewritingthisexpression:()inE()outEThepulselengthTherearemanydefinitionsofthewidthorlengthofawaveorpulse.Theeffectivewidthisthewidthofarectanglewhoseheightandareaarethesameasthoseofthepulse.Effectivewidth≡Area/height:Advantage:It’seasytounderstand.Disadvantages:TheAbsvalueisinconvenient.Wemustintegrateto±∞.1()(0)efftftdtftf(0)0tefftt(Absvalueisunnecessaryforintensity.)ThermspulsewidthTheroot-mean-squaredwidthorrmswidth:Advantages:Integralsare