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Second-order nonlinear-optical effectsSymmetryissuesPhase-matchinginSHGPhase-matchingbandwidthGroup-velocitymismatchNonlinear-opticalcrystalsPracticalnumbersforSHGElectro-opticsDifference-frequencygenerationandopticalparametricgenerationSecond-ordernonlinear-opticalFirst demonstration of second-harmonic generationP.A.Franken,etal,PhysicalReviewLetters7,p.118(1961)Thesecond-harmonicbeamwasveryweakbecausetheprocesswasnotphase-matched.Firstdemonstrationofsecond-First demonstration of SHG:the dataTheactualpublishedresultsInputbeamThesecondharmonicNotethattheveryweakspotduetothesecondharmonicismissing.ItwasremovedbyanoverzealousPhysicalReviewLetterseditor,whothoughtitwasaspeckofdirtanddidntasktheauthorsfirst.FirstdemonstrationofSHG:tSymmetry in second-harmonic generationForthistohold,(2)mustbezeroformediawithinversionsymmetry.Mostmaterialshaveinversionsymmetry,soyoujustdontseeSHGoranyothereven-ordernonlinear-opticaleffecteveryday.E(t)E 2(t)Esig(x,t)(2)E 2(x,t)Ifweimagineinvertingspace:Esig(x,t)-Esig(x,t)E(x,t)-E(x,t)Now,ifthemediumissymmetrical,(2)remainsunchanged.So:-Esig(x,t)(2)-E(x,t)2 =(2)E(x,t)2 Esig(x,t)Symmetryinsecond-harmonicgePhase-matching in second-harmonic generationHowdoesphase-matchingaffectSHG?Itsamajoreffect,anotherimportantreasonyoujustdontseeSHGoranyothernonlinear-opticaleffectseveryday.Phase-matchinginsecond-harmoSinusoidal dependence of SHG intensity on lengthLargeDkSmallDkTheSHGintensityissharplymaximizedifDk=0.SinusoidaldependenceofSHGiwhichwillonlybesatisfiedwhen:Unfortunately,dispersionpreventsthisfromeverhappening!Phase-matching second-harmonic generationSowerecreatinglightatwsig=2w.FrequencyRefractiveindexAndthek-vectorofthepolarizationis:Thephase-matchingconditionis:Thek-vectorofthesecond-harmonicis:whichwillonlybesatisfiedwWecannowsatisfythephase-matchingcondition.Usetheextraordinarypolarizationforwandtheordinaryfor2w.Phase-matching second-harmonic generation using birefringenceBirefringentmaterialshavedifferentrefractiveindicesfordifferentpolarizations.Ordinaryandextraordinaryrefractiveindicescanbedifferentbyupto0.1forSHGcrystals.FrequencyRefractiveindexnedependsonthepropagationangle,sowecantuneforagivenw.Somecrystalshavene no,sotheoppositepolarizationswork.WecannowsatisfythePhase-mNoncollinear phase-matchingBut:Sothephase-matchingconditionbecomes:zxqqDroppingthe“o”and“e”subscriptsforgenerality.Noncollinearphase-matchingButPhase-matching bandwidthPhase-matchingonlyworksexactlyforonewavelength,say0.Sinceultrashortpulseshavelotsofbandwidth,achievingapproximatephase-matchingforallfrequenciesisabigissue.Therangeofwavelengths(orfrequencies)thatachieveapproximatephase-matchingisthephase-matchingbandwidth.WavelengthRefractiveindexRecallthattheintensityoutofanSHGcrystaloflengthLis:where:IsigDkPhase-matchingbandwidthPhase-Calculation of phase-matching bandwidthWhentheinputwavelengthchangesby,thesecond-harmonicwavelengthchangesbyonly/2.Thephase-mismatchis:Assumingtheprocessisphase-matchedat0,letsseewhatthephase-mismatchwillbeat=0+.xxButtheprocessisphase-matchedat0tofirstorderinFirst:Nowthistermyieldsonlysecond-ordertermsandsocanbeneglected.Calculationofphase-matchingThesinc2curvewilldecreasebyafactorof2whenk L/2=1.39.Sosolvingforthewavelengthrangethatyields|k|widlerDifference-Frequency Generation:Optical Parametric Generation,Amplification,Oscillationw1w3w2OpticalParametricAmplification(OPA)w1w1w3w2OpticalParametricGeneration(OPG)Difference-frequencygenerationtakesmanyusefulforms.mirrormirrorw1w1w3w2=w3-w1ParametricDOptical Parametric GenerationEquationsarejustaboutidenticaltothoseforSHG:where:ki=wavevectorofithwave Dk=k1+k2-k3vgi=groupvelocityofithwaveThesolutionsforE1andE2involveexponentialgain!OPAsetc.areidealusesofultrashortpulses,whoseintensitiesarehigh.OpticalParametricGenerationEPhase-matching applies.Wecanvarythecrystalangleintheusualmanner,orwecanvarythecrystaltemperature(sincendependsonT).Phase-matchingapplies.WecanFree code to perform OPO,OPA,and OPG calculationsPublicdomainsoftwaremaintainedbyArleeSmithatSandiaNationalLabs.Justweb-searchSNLO.Youcanuseittoselectthebestnonlinearcrystalforyourparticularapplicationorperformdetailedsimulationsofnonlinearmixingprocessesincrystals.FunctionsinSNLO:1.Crystalproperties2.Modelingofnonlinearcrystalsinvariousapplications.3.Designingofstablecavities,computingGaussian focusparametersanddisplayingthehelpfile.FreecodetoperformOPO,OPA,Optical Parametric GenerationResultsusingthenonlinearmedium,periodicallypoledRbTiOAsO4Sibbett,etal.,Opt.Lett.,22,1397(1997).signal:idler:OpticalParametricGenerationRAn ultrafast noncol-linear OPA(NOPA)Continuumgeneratesanarbitrary-colorseedpulse.Anultrafastnoncol-linearOPANOPA specsNOPAspecsCrystals for far-IR generationWithunusualcrystals,suchasAgGaS2,AgGaSe2orGaSe,onecanobtainradiationtowavelengthsaslongas20m.Theselongwavelengthsareusefulforvibrationalspectroscopy.GavinD.Reid,UniversityofLeeds,andKlaasWynne,UniversityofStrathclyde10m1mWavelengthCrystalsforfar-IRgenerationElsaesser,etal.,Opt.Lett.,23,861(1998)Difference-frequency generation in GaSeAngle-tunedwavelengthElsaesser,etal.,Opt.Lett.,Another 2nd-order process:Electro-opticsApplyingavoltagetoacrystalchangesitsrefractiveindicesandintroducesbirefringence.Inasense,thisissum-frequencygenerationwithabeamofzerofrequency(butnotzerofield!).AfewkVcanturnacrystalintoahalf-orquarter-waveplate.VIfV=0,thepulsepolarizationdoesntchange.IfV=Vp,thepulsepolarizationswitchestoitsorthogonalstate.AbruptlyswitchingaPockelscellallowsustoswitchapulseintooroutofalaser.“Pockelscell”(voltagemaybetransverseorlongitudinal)PolarizerAnother2nd-orderprocess:Ele
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