超快光学-第08章-非线性二阶效应

Second-ordernonlinear-opticaleffects,SymmetryissuesPhase-matchinginSHGPhase-matchingbandwidthGroup-velocitymismatchNonlinear-opticalcrystalsPracticalnumbersforSHGElectro-opticsDifference-frequencygenerationandopticalparametricgeneration,Firstdemonstrationofsecond-harmonicgeneration,P.A.Franken,etal,PhysicalReviewLetters7,p.118(1961),Thesecond-harmonicbeamwasveryweakbecausetheprocesswasnotphase-matched.,FirstdemonstrationofSHG:thedata,Theactualpublishedresults,Inputbeam,Thesecondharmonic,Notethattheveryweakspotduetothesecondharmonicismissing.ItwasremovedbyanoverzealousPhysicalReviewLetterseditor,whothoughtitwasaspeckofdirtanddidntasktheauthorsfirst.,Symmetryinsecond-harmonicgeneration,Forthistohold,c(2)mustbezeroformediawithinversionsymmetry.Mostmaterialshaveinversionsymmetry,soyoujustdontseeSHGoranyothereven-ordernonlinear-opticaleffecteveryday.,E(t),E2(t),Esig(x,t)c(2)E2(x,t),Ifweimagineinvertingspace:,Esig(x,t)-Esig(x,t),E(x,t)-E(x,t),Now,ifthemediumissymmetrical,c(2)remainsunchanged.So:,-Esig(x,t)c(2)-E(x,t)2=c(2)E(x,t)2Esig(x,t),Phase-matchinginsecond-harmonicgeneration,Howdoesphase-matchingaffectSHG?Itsamajoreffect,anotherimportantreasonyoujustdontseeSHGoranyothernonlinear-opticaleffectseveryday.,SinusoidaldependenceofSHGintensityonlength,LargeDk,SmallDk,TheSHGintensityissharplymaximizedifDk=0.,whichwillonlybesatisfiedwhen:Unfortunately,dispersionpreventsthisfromeverhappening!,Phase-matchingsecond-harmonicgeneration,Sowerecreatinglightatwsig=2w.,Frequency,Refractiveindex,Andthek-vectorofthepolarizationis:,Thephase-matchingconditionis:,Thek-vectorofthesecond-harmonicis:,Wecannowsatisfythephase-matchingcondition.Usetheextraordinarypolarizationforwandtheordinaryfor2w.,Phase-matchingsecond-harmonicgenerationusingbirefringence,Birefringentmaterialshavedifferentrefractiveindicesfordifferentpolarizations.Ordinaryandextraordinaryrefractiveindicescanbedifferentbyupto0.1forSHGcrystals.,nedependsonthepropagationangle,sowecantuneforagivenw.Somecrystalshaveneno,sotheoppositepolarizationswork.,Noncollinearphase-matching,But:,Sothephase-matchingconditionbecomes:,Droppingthe“o”and“e”subscriptsforgenerality.,Phase-matchingbandwidth,Phase-matchingonlyworksexactlyforonewavelength,sayl0.Sinceultrashortpulseshavelotsofbandwidth,achievingapproximatephase-matchingforallfrequenciesisabigissue.Therangeofwavelengths(orfrequencies)thatachieveapproximatephase-matchingisthephase-matchingbandwidth.,Wavelength,Refractiveindex,RecallthattheintensityoutofanSHGcrystaloflengthLis:,where:,Calculationofphase-matchingbandwidth,Whentheinputwavelengthchangesby,thesecond-harmonicwavelengthchangesbyonly/2.,Thephase-mismatchis:,Assumingtheprocessisphase-matchedat0,letsseewhatthephase-mismatchwillbeat=0+.,x,x,Buttheprocessisphase-matchedat0,tofirstorderind,First:,Nowthistermyieldsonlysecond-ordertermsandsocanbeneglected.,Thesinc2curvewilldecreasebyafactorof2whenkL/2=1.39.Sosolvingforthewavelengthrangethatyields|k|widler,Difference-FrequencyGeneration:OpticalParametricGeneration,Amplification,Oscillation,w1,w3,w2,OpticalParametricAmplification(OPA),w1,w1,w3,w2,OpticalParametricGeneration(OPG),Difference-frequencygenerationtakesmanyusefulforms.,mirror,mirror,OpticalParametricGeneration,EquationsarejustaboutidenticaltothoseforSHG:,where:ki=wavevectorofithwaveDk=k1+k2-k3vgi=groupvelocityofithwave,ThesolutionsforE1andE2involveexponentialgain!,OPAsetc.areidealusesofultrashortpulses,whoseintensitiesarehigh.,Phase-matchingapplies.,Wecanvarythecrystalangleintheusualmanner,orwecanvarythecrystaltemperature(sincendependsonT).,FreecodetoperformOPO,OPA,andOPGcalculations,PublicdomainsoftwaremaintainedbyArleeSmithatSandiaNationalLabs.Justweb-searchSNLO.Youcanuseittoselectthebestnonlinearcrystalforyourparticularapplicationorperformdetailedsimulationsofnonlinearmixingprocessesincrystals.FunctionsinSNLO:1.Crystalproperties2.Modelingofnonlinearcrystalsinvariousapplications.3.Designingofstablecavities,computingGaussianfocusparametersanddisplayingthehelpfile.,OpticalParametricGeneration,Resultsusingthenonlinearmedium,periodicallypoledRbTiOAsO4,Sibbett,etal.,Opt.Lett.,22,1397(1997).,signal:,idler:,Anultrafastnoncol-linearOPA(NOPA),Continuumgeneratesanarbitrary-colorseedpulse.,NOPAspecs,Crystalsforfar-IRgeneration,Withunusualcrystals,suchasAgGaS2,AgGaSe2orGaSe,onecanobtainradiationtowavelengthsaslongas20mm.Theselongwavelengthsareusefulforvibrationalspectroscopy.,GavinD.Reid,UniversityofLeeds,andKlaasWynne,UniversityofStrathclyde,10mm,1mm,Wavelength,Elsaesser,etal.,Opt.Lett.,23,861(1998),Difference-frequencygenerationinGaSe,Angle-tunedwavelength,Another2nd-orderprocess:Electro-optics,Applyingavoltagetoacrystalchangesitsrefractiveindicesandintroducesbirefringence.Inasense,thisissum-frequencygenerationwithabeamofzerofrequency(butnotzerofield!).AfewkVcanturnacrystalintoahalf-