超快光学-第03章-脉冲(1)

UltrashortLaserPulsesIDescriptionofpulsesIntensityandphaseTheinstantaneousfrequencyandgroupdelayZerothandfirst-orderphaseThelinearlychirpedGaussianpulseProf.RickTrebinoGeorgiaTech()exp{[]}().().titcctItENeglectingthespatialdependencefornow,thepulseelectricfieldisgivenby:IntensityPhaseCarrierfrequencyAsharplypeakedfunctionfortheintensityyieldsanultrashortpulse.Thephasetellsusthecolorevolutionofthepulseintime.()ItElectricfieldE(t)Time[fs]Therealandcomplexpulseamplitudes()exp{}()()EIttitRemovingthe1/2,thec.c.,andtheexponentialfactorwiththecarrierfrequencyyieldsthecomplexamplitude,E(t),ofthepulse:Thisremovestherapidlyvaryingpartofthepulseelectricfieldandyieldsacomplexquantity,whichisactuallyeasiertocalculatewith.()Itisoftencalledtherealamplitude,A(t),ofthepulse.()ItElectricfieldE(t)Time[fs]TheGaussianpulsewheretHW1/eisthefieldhalf-width-half-maximum,andtFWHMistheintensityfull-width-half-maximum.Theintensityis:201/2020()exp(/)exp2ln2(/)exp1.38(/)HWeFWHMFWHMEtEtEtEttttForalmostallcalculations,agoodfirstapproximationforanyultrashortpulseistheGaussianpulse(withzerophase).220220()exp4ln2(/)exp2.76(/)FWHMFWHMItEtEtttIntensityvs.amplitudeTheintensityofaGaussianpulseis√2shorterthanitsrealamplitude.Thisfactorvariesfrompulseshapetopulseshape.Thephaseofthispulseisconstant,(t)=0,andisnotplotted.It’seasytogobackandforthbetweentheelectricfieldandtheintensityandphase:Theintensity:CalculatingtheintensityandthephaseIm[()]arcta()nRe[()]EttEt(t)=Im{ln[E(t)]}Thephase:Equivalently,(ti)ReImE(ti)I(t)=|E(t)|2Also,we’llstopwriting“proportionalto”intheseexpressionsandtakeE,E,I,andStobethefield,intensity,andspectrumdimensionlessshapesvs.time.TheFourierTransformTothinkaboutultrashortlaserpulses,theFourierTransformisessential.()()exp()titdtEE1()()exp()2titdEEWealwaysperformFouriertransformsontherealorcomplexpulseelectricfield,andnottheintensity,unlessotherwisespecified.Thefrequency-domainelectricfieldThefrequency-domainequivalentsoftheintensityandphasearethespectrumandspectralphase.Fourier-transformingthepulseelectricfield:102()exp{[]}().().titcctItEyields:12100020()exp{[]}()exp{[()](()})iSiSEThefrequency-domainelectricfieldhaspositive-andnegative-frequencycomponents.Notethatandaredifferent!NotethatthesetwotermsarenotcomplexconjugatesofeachotherbecausetheFTintegralisthesameforeach!Thecomplexfrequency-domainpulsefieldSincethenegative-frequencycomponentcontainsthesameinfor-mationasthepositive-frequencycomponent,weusuallyneglectit.Wealsocenterthepulseonitsactualfrequency,notzero.Sothemostcommonlyusedcomplexfrequency-domainpulsefieldis:Thus,thefrequency-domainelectricfieldalsohasanintensityandphase.Sisthespectrum,andisthespectralphase.()exp{()()}SiEThespectrumwithandwithoutthecarrierfrequencyFouriertransformingE(t)andE(t)yieldsdifferentfunctions.Weusuallyusejustthiscomponent.()E()EThespectrumandspectralphaseThespectrumandspectralphaseareobtainedfromthefrequency-domainfieldthesamewaytheintensityandphasearefromthetime-domainelectricfield.Im[()]arctanRe[)()](EEImln[()()]Eor2(())SEIntensityandphaseofaGaussianTheGaussianisreal,soitsphaseiszero.Timedomain:Frequencydomain:Sothespectralphaseiszero,too.AGaussiantransformstoaGaussianIntensityandPhaseSpectrumandSpectralPhaseThespectralphaseofatime-shiftedpulse()exp()()ftaiaFFRecalltheShiftTheorem:Soatime-shiftsimplyaddssomelinearspectralphasetothepulse!Time-shiftedGaussianpulse(withaflatphase):IntensityandPhaseSpectrumandSpectralPhaseWhatisthespectralphase?Thespectralphaseisthephaseofeachfrequencyinthewave-form.t0Allofthesefrequencieshavezerophase.Sothispulsehas:()=0Notethatthiswave-formseesconstructiveinterference,andhencepeaks,att=0.Andithascancellationeverywhereelse.123456Nowtryalinearspectralphase:()=a.BytheShiftTheorem,alinearspectralphaseisjustadelayintime.Andthisiswhatoccurs!t(1)=0(2)=0.2(3)=0.4(4)=0.6(5)=0.8(6)=Totransformthespectrum,notethattheenergyisthesame,whetherweintegratethespectrumoverfrequencyorwavelength:TransformingbetweenwavelengthandfrequencyThespectrumandspectralphasevs.frequencydifferfromthespectrumandspectralphasevs.wavelength.()(2c/)()()SdSdChangingvariables:S(2c/)2c2dS()S(2c/)2c2Thespectralphaseiseasilytransformed:22(2/)cScd22dcd2cThespectrumandspectralphasevs.wavelengthandfrequencyExample:AGaussianspectrumwithalinearspectralphasevs.frequencyvs.Frequencyvs.WavelengthNotethedifferentshapesofthespectrumandspectralphasewhenplottedvs.wavelengthandfrequency.Bandwidthinvariousunits(1/)cInfrequency,bytheUncertaintyPrinciple,a1-pspulsehasbandwidth:=~1/2THzc(1/)/cSo(1/)=(0.5×1012/s)/(3×1010cm/s)or:(1/)=17cm-1Inwavelength:41(800nm)(.810cm)(17cm)Assumingan800-nmwavelength:usingt~½21(1/)2(1/)or:=1nmInwavenumbers(cm-1),wecanwrite:Thetemporalphase,(t