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1、UltrashortLaserPulsesII Moresecond orderphaseHigher orderspectralphasedistortionsRelativeimportanceofspectrumandspectralphasePulseandspectralwidthsTime bandwidthproduct Prof RickTrebinoGeorgiaTechwww frog gatech edu Frequency domainphaseexpansion RecalltheTaylorseriesfor Asinthetimedomain onlythefir
2、stfewtermsaretypicallyrequiredtodescribewell behavedpulses Ofcourse we llconsiderbadlybehavedpulses whichhavehigher ordertermsin where isthegroupdelay iscalledthe group delaydispersion Butwhenthepulseislong a0 whichistheinverseoftheinstantaneousfrequencyvs time Thegroupdelayvs wforachirpedpulse Theg
3、roupdelayofawaveisthederivativeofthespectralphase So ForalinearlychirpedGaussianpulse thespectralphaseis Andthedelayvs frequencyislinear Thisisnottheinverseoftheinstantaneousfrequency whichis 2nd orderphase positivelinearchirp Numericalexample Gaussian intensitypulsew positivelinearchirp 2 14 5radfs
4、2 Herethequadraticphasehasstretchedwhatwouldhavebeena3 fspulse giventhespectrum toa13 9 fsone 2nd orderphase negativelinearchirp Numericalexample Gaussian intensitypulsew negativelinearchirp 2 14 5radfs2 Aswithpositivechirp thequadraticphasehasstretchedwhatwouldhavebeena3 fspulse giventhespectrum to
5、a13 9 fsone Thefrequencyofalightwavecanalsovarynonlinearlywithtime ThisistheelectricfieldofaGaussianpulsewhosefre quencyvariesquadraticallywithtime Thislightwavehastheexpression Arbitrarilycomplexfrequency vs timebehaviorispossible Butweusuallydescribephasedistortionsinthefrequencydomain Nonlinearly
6、chirpedpulses 3rd orderspectralphase quadraticchirp Longerandshorterwavelengthscoincideintimeandinterfere beat Trailingsatellitepulsesintimeindicatepositivespectralcubicphase andleadingonesindicatenegativespectralcubicphase S w tg w j w Spectrumandspectralphase 400500600700 Becausewe replottingvs wa
7、velength notfrequency there saminussigninthegroupdelay sotheplotiscorrect 3rd orderspectralphase quadraticchirp Numericalexample Gaussianspectrumandpositivecubicspectralphase with 3 750radfs3 Trailingsatellitepulsesintimeindicatepositivespectralcubicphase Negative3rd orderspectralphase Anothernumeri
8、calexample Gaussianspectrumandnegativecubicspectralphase with 3 750radfs3 Leadingsatellitepulsesintimeindicatenegativespectralcubicphase 4th orderspectralphase Numericalexample Gaussianspectrumandpositivequarticspectralphase 4 5000radfs4 Leadingandtrailingwingsintimeindicatequarticphase Higher frequ
9、enciesinthetrailingwingmeanpositivequarticphase Negative4th orderspectralphase Numericalexample Gaussianspectrumandnegativequarticspectralphase 4 5000radfs4 Leadingandtrailingwingsintimeindicatequarticphase Higher frequenciesintheleadingwingmeannegativequarticphase 5th orderspectralphase Numericalex
10、ample Gaussianspectrumandpositivequinticspectralphase 5 4 4 104radfs5 Anoscillatorytrailingwingintimeindicatespositivequinticphase Negative5th orderspectralphase Numericalexample Gaussianspectrumandnegativequinticspectralphase 5 4 4 104radfs5 Anoscillatoryleadingwingintimeindicatesnegativequinticpha
11、se Therelativeimportanceofintensityandphase PhotographsofmywifeLindaandme CompositephotographmadeusingthespectralintensityofLinda sphotoandthespectralphaseofmine andinverse Fourier transforming CompositephotographmadeusingthespectralintensityofmyphotoandthespectralphaseofLinda s andinverse Fourier t
12、ransforming Thespectralphaseismoreimportantfordeterminingtheintensity Pulsepropagation Whathappenstoapulseasitpropagatesthroughamedium Alwaysmodel linear propagationinthefrequencydomain Also youmustknowtheentirefield i e theintensityandphase todoso Inthetimedomain propagationisaconvolution muchharde
13、r Pulsepropagation continued usingk w c Separatingoutthespectrumandspectralphase Rewritingthisexpression Thepulselength Therearemanydefinitionsofthewidthorlengthofawaveorpulse Theeffectivewidthisthewidthofarectanglewhoseheightandareaarethesameasthoseofthepulse Effectivewidth Area height Advantage It
14、 seasytounderstand Disadvantages TheAbsvalueisinconvenient Wemustintegrateto Absvalueisunnecessaryforintensity Thermspulsewidth Theroot mean squaredwidthorrmswidth Advantages Integralsareofteneasytodoanalytically Disadvantages Itweightswingsevenmoreheavily soit sdifficulttouseforexperiments whichcan
15、 tscanto Thermswidthisthenormalizedsecond ordermoment TheFull Width Half Maximum Full width half maximumisthedistancebetweenthehalf maximumpoints Advantages Experimentallyeasy Disadvantages Itignoressatellitepulseswithheights 49 99 ofthepeak Also wecandefinethesewidthsintermsoff t orofitsintensity f
16、 t 2 Definespectralwidths Dw similarlyinthefrequencydomain t w TheUncertaintyPrinciple TheUncertaintyPrinciplesaysthattheproductofafunction swidthsinthetimedomain Dt andthefrequencydomain Dw hasaminimum Combiningresults or Useeffectivewidthsassumingf t andF w peakat0 1 Otherwidthdefinitionsyieldslightlydifferentnumbers Foragivenwave theproductofthetime domainwidth Dt andthefrequency domainwidth Dn istheTime BandwidthProduct TBP DnDt TBPApulse sTBPwillalwaysbegreaterthanthetheoreticalminimumgiven
