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1、476J.Opt.Soc.Am.B/Vol.12,No.3/March1995Magnietal.AstigmatisminGaussian-beamself-focusingandinresonatorsforKerr-lensmodelockingV.Magni,G.Cerullo,S.DeSilvestri,andA.MonguzziDipartimentodiFisicadelPolitecnico,CentrodiElettronicaQuantisticaeStrumentazioneEle
2、ttronicadelConsiglioNazionaledelleRicerche,PiazzaLeonardodaVinci32,20133Milano,ItalyReceivedAugust8,1994;revisedmanuscriptreceivedOctober20,1994AnanalyticaltreatmentforthepropagationofastigmaticGaussianbeamsinKerrmediaispresented.Thismethod,whichisvalidt
3、othefirstorderinthebeampower,allowsustocalculateinclosedformthenonlinearvariationsofthespotsizeandtheradiusofcurvature.Thisformalismisappliedtothecalculationoftheself-consistentGaussianmodeofanastigmaticresonatorwithaKerrmedium,suchasthoseusedforKerr-len
4、smode-lockedlasers.TheoreticalpredictionsareconfirmedbytheexperimentalresultsobtainedwithafemtosecondTi:sapphirelaser.Inparticular,thedesignofanoptimizedresonatorpermitsself-startingofthemode-lockingmechanism.1.INTRODUCTIONmediumisthereforeessential.Fora
5、stigmaticGauss-ianbeamstheaberrationlessapproximationprovidesInpastyearsself-focusingeffectsinmaterialswithfastdifferentialequationsdescribingthepropagationoftheKerrnonlinearitieshavefoundanimportantapplicationbeamspotsizesalongtransversedirections.22The
6、so-inlasermodelockingforthegenerationofultrashortlightlutionoftheseequationswasfoundanalyticallyinanpulses.IntheKerr-lensmode-locking(KLM)techniqueimplicitform,23,24which,howeverisnothighlyamenabletheintracavityself-focusingeffect,combinedwithanin-toadir
7、ectapplication.Themainfeaturesofastigmaticternalaperture,resultsinnonlinearpower-dependentGaussianbeamsinKerrmediaatvariouspowerlevelsresonatorlossesthatsustainthemode-lockingregime.1,2werealsostudiednumerically.25TypicallytheKerrnonlinearityactsonbeamsw
8、ithaOurpurposeinthispaperistoderiveclosed-formequa-powersignificantlylowerthanthecriticalpowerforself-tionsthatdescribetheself-focusingofastigmaticGauss-trapping,andthetransverseprofileremainsessentiallyianbeamsandtocalcul
