超快光学-第08章-非线性二阶效应

Second-ordernonlinear-opticaleffectsSymmetryissuesPhase-matchinginSHGPhase-matchingbandwidthGroup-velocitymismatchNonlinear-opticalcrystalsPracticalnumbersforSHGElectro-opticsDifference-frequencygenerationandopticalparametricgenerationFirstdemonstrationofsecond-harmonicgenerationP.A.Franken,etal,PhysicalReviewLetters7,p.118(1961)Thesecond-harmonicbeamwasveryweakbecausetheprocesswasnotphase-matched.FirstdemonstrationofSHG:thedataTheactualpublishedresults…InputbeamThesecondharmonicNotethattheveryweakspotduetothesecondharmonicismissing.ItwasremovedbyanoverzealousPhysicalReviewLetterseditor,whothoughtitwasaspeckofdirtanddidn’tasktheauthorsfirst.Symmetryinsecond-harmonicgenerationForthistohold,c(2)mustbezeroformediawithinversionsymmetry.Mostmaterialshaveinversionsymmetry,soyoujustdon’tseeSHG—oranyothereven-ordernonlinear-opticaleffect—everyday.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)2Esig(x,t)Phase-matchinginsecond-harmonicgenerationHowdoesphase-matchingaffectSHG?It’samajoreffect,anotherimportantreasonyoujustdon’tseeSHG—oranyothernonlinear-opticaleffects—everyday.SinusoidaldependenceofSHGintensityonlengthLargeDkSmallDkTheSHGintensityissharplymaximizedifDk=0.whichwillonlybesatisfiedwhen:Unfortunately,dispersionpreventsthisfromeverhappening!Phase-matchingsecond-harmonicgenerationSowe’recreatinglightatwsig=2w.022()polkkncww00(2)()(2)sigsigsigknnccsigpolkk(2)()nnwww2wFrequencyRefractiveindexAndthek-vectorofthepolarizationis:Thephase-matchingconditionis:Thek-vectorofthesecond-harmonicis:Wecannowsatisfythephase-matchingcondition.Usetheextraordinarypolarizationforwandtheordinaryfor2w.Phase-matchingsecond-harmonicgenerationusingbirefringenceBirefringentmaterialshavedifferentrefractiveindicesfordifferentpolarizations.Ordinaryandextraordinaryrefractiveindicescanbedifferentbyupto~0.1forSHGcrystals.(2)()oennwww2wFrequencyRefractiveindexneonnedependsonthepropagationangle,sowecantuneforagivenw.Somecrystalshaveneno,sotheoppositepolarizationswork.Noncollinearphase-matching0ˆ2cos2()cospolpolkkkkzkncww02(2)sigkncww(2)()cosnnwwˆˆcossinkkzkx-ˆˆcossinkkzkxBut:Sothephase-matchingconditionbecomes:zxDroppingthe“o”and“e”subscriptsforgenerality.Phase-matchingbandwidthPhase-matchingonlyworksexactlyforonewavelength,sayl0.Sinceultrashortpulseshavelotsofbandwidth,achievingapproximatephase-matchingforallfrequenciesisabigissue.Therangeofwavelengths(orfrequencies)thatachieveapproximatephase-matchingisthephase-matchingbandwidth.4()()(/2)knnllllD-0l02lWavelengthRefractiveindexneno22()(/)sinc(/2)sigILLkLlDRecallthattheintensityoutofanSHGcrystaloflengthLis:where:())/2(nnll2llIsigDkCalculationofphase-matchingbandwidth4()()(/2)knnllllD-00000041()1()()(/2)(/2)2knnnnllllllllllD---Whentheinputwavelengthchangesbyl,thesecond-harmonicwavelengthchangesbyonlyl/2.Thephase-mismatchis:Assumingtheprocessisphase-matchedatl0,let’sseewhatthephase-mismatchwillbeatl=l0+l.xxButtheprocessisphase-matchedatl000041()()(/2)2knnlllllD-tofirstorderinlFirst:0000441411/lllllll-Nowthistermyieldsonlysecond-ordertermsandsocanbeneglected.Thesinc2curvewilldecreasebyafactorof2whenDkL/2=±1.39.Sosolvingforthewavelengthrangethatyields|Dk|2.78/Lyieldsthephase-matchinghalf-bandwidthl/2.010020.44/()(/2)FWHMLnnllll-0004(/2)1()(/2)2.78/2nnLllll-Calculationofphase-matchingbandwidth(cont’d)FWHM2.78/L-2.78/Lsinc2(DkL/2)IsigDkPhase-matchingefficiencyvs.wavelengthforBBOThesecurvesalsotakeintoaccountthe(L/l)2factor.Whilethecurvesarescaledinarbitraryunits,therelativemagnitudescanbecomparedamongthethreeplots.(Thesecurvesdon’t,however,includethenonlinearsusceptibility,c(2)).Phase-matchingefficiencyvs.wavelengthforthenonlinear-opticalcrystal,beta-bariumborate(BBO),fordifferentcrystalthicknesses:10m100m1000mNotethehugedifferencesinphase-matchingbandwidthandefficiencywithcrystalthickness.Phase-matchingefficiencyvs.wavelengthforKDPThecurvesforthethincrystalsdon’tfalltozeroatlongwavelengthsbecauseKDPsimultaneouslyphase-matchesfortwowavelengths,thatshownandalonger(IR)wavelength,whosephase-matchingrangesbegintooverlapwhenthecrystalisthin.Phase-matchingefficiencyvs.wavelengthforthenonlinear-opticalcrystal,potassiumdihydrogenphosphate(KDP),fordifferentcrystalthicknesses:10m100m1000mThehugedifferencesinphase-matchingbandwidthandefficiencywithcrystalthicknessoccurforallcrystals.AthincrystalgeneratesSHforallfrequenciesoftheinputpulseintheforwardandotherdirections.ThinSHGcrystalThemorethepropagationdirectiondiffersfromtheprecisephase-matchingangle,thelesstheefficiency.Thisfall-offisfasterforthickercrystals.Phase-matchingbandwidth(inSHG)AthickcrystalgeneratesSHforonlysomefrequenciesintheinputpulseineachdirection.ThickSHGcrystalPolarplotsofSHoutputintensityvs.angleforagivenfrequencyGroup-VelocityMismatchisanotherwayofdescribingthephase-matchingbandwidth.Inthecr