UltrashortLaserPulsesI DescriptionofpulsesIntensityandphaseTheinstantaneousfrequencyandgroupdelayZerothandfirst orderphaseThelinearlychirpedGaussianpulse Prof RickTrebinoGeorgiaTechwww frog gatech edu Anultrashortlaserpulsehasanintensityandphasevs time Neglectingthespatialdependencefornow thepulseelectricfieldisgivenby Intensity Phase Carrierfrequency Asharplypeakedfunctionfortheintensityyieldsanultrashortpulse Thephasetellsusthecolorevolutionofthepulseintime Therealandcomplexpulseamplitudes Removingthe1 2 thec c andtheexponentialfactorwiththecarrierfrequencyyieldsthecomplexamplitude E t ofthepulse Thisremovestherapidlyvaryingpartofthepulseelectricfieldandyieldsacomplexquantity whichisactuallyeasiertocalculatewith isoftencalledtherealamplitude A t ofthepulse ElectricfieldE t Time fs TheGaussianpulse wheretHW1 eisthefieldhalf width half maximum andtFWHMistheintensityfull width half maximum Theintensityis Foralmostallcalculations agoodfirstapproximationforanyultrashortpulseistheGaussianpulse withzerophase Intensityvs amplitude TheintensityofaGaussianpulseis 2shorterthanitsrealamplitude Thisfactorvariesfrompulseshapetopulseshape It seasytogobackandforthbetweentheelectricfieldandtheintensityandphase Theintensity Calculatingtheintensityandthephase f t Im ln E t Thephase Equivalently I t E t 2 Also we llstopwriting proportionalto intheseexpressionsandtakeE E I andStobethefield intensity andspectrumdimensionlessshapesvs time TheFourierTransform Tothinkaboutultrashortlaserpulses theFourierTransformisessential WealwaysperformFouriertransformsontherealorcomplexpulseelectricfield andnottheintensity unlessotherwisespecified Thefrequency domainelectricfield Thefrequency domainequivalentsoftheintensityandphasearethespectrumandspectralphase Fourier transformingthepulseelectricfield yields Thefrequency domainelectricfieldhaspositive andnegative frequencycomponents Notethatfandjaredifferent NotethatthesetwotermsarenotcomplexconjugatesofeachotherbecausetheFTintegralisthesameforeach Thecomplexfrequency domainpulsefield Sincethenegative frequencycomponentcontainsthesameinfor mationasthepositive frequencycomponent weusuallyneglectit Wealsocenterthepulseonitsactualfrequency notzero Sothemostcommonlyusedcomplexfrequency domainpulsefieldis Thus thefrequency domainelectricfieldalsohasanintensityandphase Sisthespectrum andjisthespectralphase Thespectrumwithandwithoutthecarrierfrequency FouriertransformingE t andE t yieldsdifferentfunctions Thespectrumandspectralphase Thespectrumandspectralphaseareobtainedfromthefrequency domainfieldthesamewaytheintensityandphasearefromthetime domainelectricfield or IntensityandphaseofaGaussian TheGaussianisreal soitsphaseiszero Timedomain Frequencydomain Sothespectralphaseiszero too AGaussiantransformstoaGaussian IntensityandPhase SpectrumandSpectralPhase Thespectralphaseofatime shiftedpulse RecalltheShiftTheorem Soatime shiftsimplyaddssomelinearspectralphasetothepulse Time shiftedGaussianpulse withaflatphase Whatisthespectralphase Thespectralphaseisthephaseofeachfrequencyinthewave form 0 Allofthesefrequencieshavezerophase Sothispulsehas j w 0Notethatthiswave formseesconstructiveinterference andhencepeaks att 0 Andithascancellationeverywhereelse w1w2w3w4w5w6 Nowtryalinearspectralphase j w aw BytheShiftTheorem alinearspectralphaseisjustadelayintime Andthisiswhatoccurs t j w1 0 j w2 0 2p j w3 0 4p j w4 0 6p j w5 0 8p j w6 p Totransformthespectrum notethattheenergyisthesame whetherweintegratethespectrumoverfrequencyorwavelength Transformingbetweenwavelengthandfrequency Thespectrumandspectralphasevs frequencydifferfromthespectrumandspectralphasevs wavelength Changingvariables Thespectralphaseiseasilytransformed Thespectrumandspectralphasevs wavelengthandfrequency Example AGaussianspectrumwithalinearspectralphasevs frequency Notethedifferentshapesofthespectrumandspectralphasewhenplottedvs wavelengthandfrequency Bandwidthinvariousunits Infrequency bytheUncertaintyPrinciple a1 pspulsehasbandwidth dn 1 2THz Sod 1 l 0 5 1012 s 3 1010cm s or d 1 l 17cm 1 Inwavelength Assumingan800 nmwavelength usingdndt or dl 1nm Inwavenumbers cm 1 wecanwrite Thetemporalphase t containsfrequency vs timeinformation Thepulseinstantaneousangularfrequency inst t isdefinedas TheInstantaneousfrequency Thisiseasytosee Atsometime t considerthetotalphaseofthewave Callthisquantity 0 Exactlyoneperiod T later thetotalphasewill bydefinition increaseto 0 2p where t T istheslowlyvaryingphaseatthetime t T Subtractingthesetwoequations DividingbyTandrecognizingthat2 Tisafrequency callit inst t inst t 2 T 0 t T t TButTissmall so t T t Tisthederivative d dt Sowe redone Usually however we llthinkintermsoftheinstantaneousfrequency inst t sowe llneedtodivideby2 inst t 0 d dt 2 Whiletheinstantaneousfrequencyisn talwaysarigorousquantity it sfineforultrashortpulses whichhavebroadbandwidths Instantaneousfrequency cont d Groupdelay Whilethetemporalphasecontainsfrequency vs timeinformation thespectralphasecontainstime vs frequencyinformation Sowecandefin