超快光学-第09章-非线性三阶效应

Third-ordernonlinearities:Four-wavemixingThird-harmonicgenerationInducedgratingsPhaseconjugationNonlinearrefractiveindexSelf-focusingSelf-phasemodulationContinuumgenerationThird-harmonicgeneration333exp[(33)]*(,)otheertexp[(33)m]rsEitkzEitkrztE*exp([()]exp[(,))]EitkzrtEitkzE3THGMediumWemustnowcubetheinputfield:Third-harmonicgenerationisweakerthansecond-harmonicandsum-frequencygeneration,sothethirdharmonicisusuallygeneratedusingSHGfollowedbySFG,ratherthanbydirectTHG.Noncollinearthird-harmonicgenerationWecanalsoallowtwodifferentinputbeams,whosefrequenciescanbedifferent.Soinadditiontogeneratingthethirdharmonicofeachinputbeam,themediumwillgenerateinterestingsumfrequencies.221221212112213exp{[(2)(2))]}exp{[(2)(2))(,)otherter}ms]EEitEEitkrkkrtkrETHGmediumSignal#1Signal#2Third-orderdifference-frequencygeneration:Self-diffraction22*212121*3121212exp{[(2)(2))]}exp{[(2)(2))(,)otherter}ms]EEitEEitkrkkrtkrEConsidersomeofthedifference-frequencyterms:NonlinearmediumSignal#1Signal#2122kk212kkSamplemediumSignalTheexcite-probegeometry113122exp[(,)otherter(]s)mEitrtEkrEOnefieldcancontributetwofactors,oneEandtheotherE*.Thiswillinvolvebothaddingandsubtractingthefrequencyanditsk-vector.Thiseffectisautomaticallyphase-matched!11122221kkkkTheexcite-probebeamgeometryhasmanyapplications,especiallytoultrafastspectroscopy.Thesignalbeamcanbedifficulttoseparatefromtheinputbeam,E1,however.NonlinearmediumWaveplateyielding45°polarizationSignalPolarizationgating*11122v3ˆexp[()]v(,)+.).(.hhrtEEkEtirEHerefield#2contributestwofactors,oneEandtheotherE*.Butoneisverticallypolarized,whiletheotherishorizontallypolarized.Thisyieldsasignalbeamthat’sorthogonallypolarizedtotheinputbeamE1.IfE1ishorizontallypolarized,thesignalwillbeverticallypolarized:Thisarrangementisalsoautomaticallyphase-matched.It’salsoreferredtoaspolarizationspectroscopyduetoitsmanyusesinbothultrafastandfrequency-domainspectroscopy.Theinputbeaminthesignalbeamdirectionisrejectedbypolarizer!x1k2kTheirradianceoftwocrossedbeamsissinusoidal,inducingasinusoidalabsorptionorrefractiveindexinthemedium––adiffractiongrating!Aninducedgratingresultsfromthecrosstermintheirradiance:*12EE*12Reexp(cossinexp(cossinReexp2sincos(2sin)EitkzkxEitkzkxikxkxManynonlinear-opticaleffectscanbeconsideredasinducedgratings.xxxx2ˆcosˆsinkkzkx1ˆcosˆsinkkzkxTime-independentfringesDiffractionoffaninducedgrating*123sigsigEPEEEAthirdbeamwillthendiffractintoadifferentdirection.Thisresultsinabeamthat’stheproductofE1,E2*,andE3:Thisisjustagenericfour-wave-mixingeffect.x1k2kInducedgratings2123exp{[(,).(}.)].prsigpexrxeexEitrkkkrtEEPhase-matchingcondition:1212cossin2sˆinˆˆexexesigprprsigprpolprprprpxexexerprxkkkkkkxkzkxkAssume:12exexexThediffractedbeamhasthesamefrequencyandk-vectormagnitudeastheprobebeam,butitsdirectionwillbedifferent.12exexexEEE12exexkkbutDiffractedsignalex1ex2sigprprexzprsigxPhase-matchinginducedgratingsPhase-matching:垐coscosprsigprprkzkz垐sin[2sinsin]prprexexprprkxkkx2sin2sinprprexexkkz-component:x-component:exexprexzprsigx22sinsinprexprexsigprTheminussignisjusttheexcite-probeeffect.sinsinprprexexThe“BraggCondition”ˆˆsincossigprsigprsigkkxkz垐ˆ2sincossinpolexexprprprprkkxkzkxsigpolkkInducedgratingswithdifferentfrequenciesNonlinearmediumDiffractedsignal33*1231231233exp{[()()...()]}EEEitkkkrtEPhase-matchingcondition:123123sigpolkkkkThiseffectiscallednon-degeneratefour-wavemixing.Inthiscase,theintensityfringessweepthroughthemedium:amovinggrating.Thesetofpossiblebeamgeometriesiscomplex.Iwroteathesisonthis!Acousto-opticsinvolvesdiffractinglightoffagratinginducedbyanacousticwave.Acousto-opticsworksbecauseacousticwaveshaveaboutthesamewavelengthsas(visible)lightwaves.Suchdiffractioncanbequitestrong:~70%.Acousto-opticsisthebasisofusefuldevices.QuartzDiffractedbeamacousticAcoustictransducerPressure,density,andrefractive-indexvariationsduetoacousticwaveInputbeamAnacousticwaveinducessinusoidaldensity,andhencesinusoidalrefractive-index,variationsinamedium.Inducedgratingswithplanewavesandmorecomplexbeams(ofthesamefrequency)Allsuchinducedgratingswilldiffractaplanewave,reproducingthedistortedwave:AplanewaveandaverydistortedwaveAplanewaveandaslightlydistortedwaveTwoplanewavesAplanewaveandaslightlydistortedwave3*123123()[(,)]exp{()}...tExyEEikkkrEE2andE3areplanewaves.Holographyisaninduced-gratingprocess.Oneofthewritebeamshasacomplexspatialpattern—theimage.Differentincidenceanglescorrespondtodifferentfringespacings.Differentobjectviewsarestoredasdifferentfringespacings.Athirdbeam(aplanewave)diffractsoffthegrating,acquiringtheimageinfor-mation.Differentfringespacingsyielddifferentdiffractionangles––hence3D!Thelightphasestorestheangularinfo.PhaseconjugationWhenanonlinear-opticaleffectproducesalightwaveproportionaltoE*,theprocessis