超快光学-第07章-非线性光学

NonlinearOpticsWhydononlinear-opticaleffectsoccur?Maxwell'sequationsinamediumNonlinear-opticalmediaSecond-harmonicgenerationSum-anddifference-frequencygenerationHigher-ordernonlinearopticsTheSlowlyVaryingEnvelopeApproximationPhase-matchingandConservationlawsforphotonsNonlinearopticsisn’tsomethingyouseeeveryday.Sendinginfraredlightintoacrystalyieldedthisdisplayofgreenlight(second-harmonicgeneration,SHG):Nonlinearopticsallowsustochangethecolorofalightbeam,tochangeitsshapeinspaceandtime,andtocreateultrashortlaserpulses.Whydon'tweseenonlinearopticaleffectsinourdailylife?1.Intensitiesofdailylifearetooweak.2.Normallightsourcesareincoherent.3.Theoccasionalcrystalweseehasthewrongsymmetry(forSHG).4.Phase-matchingisrequired,anditdoesn'tusuallyhappenonitsown.Whydononlinear-opticaleffectsoccur?Recallthat,innormallinearoptics,alightwaveactsonamolecule,whichvibratesandthenemitsitsownlightwave,whichinterfereswiththeoriginallightwave.Wecanalsoimaginethisprocessintermsofthemolecularenergylevels,usingarrowsforthephotonenergies:Whydononlinear-opticaleffectsoccur?(continued)Now,supposetheirradianceishighenoughthatmanymoleculesareexcitedtothehigher-energystate.Thisstatecanthenactasthelowerlevelforadditionalexcitation.Thisyieldsvibrationsatallfrequenciescorrespondingtoallenergydifferencesbetweenpopulatedstates.Nonlinearopticsisanalogoustononlinearelectronics,whichwecanobserveeasily.Sendingahigh-volumesine-wave(purefrequency)signalintoacheapamplifierorcheapspeakersyieldsatruncatedoutputsignal,moreofasquarewavethanasine.Thissquarewavehashigherfrequencies.Wehearthisasdistortion.SharpedgesrequirehigherfrequenciesintheFourierseries.NonlineareffectsinatomsSoanelectron’smotionwillalsodepartfromasinewave.Thepotentialgetsveryflatoutatinfinity,sotheelectron’smotioncaneasilygononlinear!PotentialduetonucleusNucleusEnergyPositionAnotherwaytolookatnonlinearopticsisthatthepotentialoftheelectronisanatomisnotasimplequadraticpotential.NonlinearopticsandanharmonicoscillatorsForweakfields,motionisharmonic,andlinearopticsprevails.Forstrongfields(i.e.,lasers),anharmonicmotionoccurs,andhigherharmonicsoccur,bothinthemotionandthelightemission.Anucleus(inamolecule)alsodoesnothaveasimplequadraticpotential.Soitsvibrationalmotionisalsononlinear:MoleculesExcitedbyLaserLightLaserlightcausesmoleculestovibrateinunison.Wesaythatlaserlightpolarizesthemedium.Acceleratingchargesemitlight.Polarizedmatteremitslightatthefrequencyatwhichitisoscillating.Ifthemotionisn’tsinusoidal,themediumwillalsoemittheadditionalfrequencies!Maxwell'sEquationsinaMediumThepolarization,P,containstheeffectofthemedium:020010tcttBEEEPBB2220222201zcttEEPSinewavesofallfrequenciesaresolutionstothewaveequation;it’sthepolarizationthattellswhichfrequencieswilloccur.Thepolarizationisthedrivingtermforthesolutiontothisequation.Theseequationsreducetothe(scalar)waveequation:InhomogeneousWaveEquationSolvingthewaveequationinthepresenceoflinearinducedpolarizationForlowirradiances,thepolarizationisproportionaltotheincidentfield:0PEInthissimple(andmostcommon)case,thewaveequationbecomes:222222220011zctctEEE(,)(0,)cos()ztEttkzEThisequationhasthesolution:Theinducedpolarizationonlychangestherefractiveindex.Dull.Ifonlythepolarizationcontainedotherfrequencies…where=ckandc=c0/nandn=(1+)1/2Usingthefactthat:20001/c22222010zctEESimplifying:Maxwell'sEquationsinaNonlinearMediumNonlinearopticsiswhathappenswhenthepolarizationistheresultofhigher-order(nonlinear!)termsinthefield:Whataretheeffectsofsuchnonlinearterms?Considerthesecond-orderterm:2=2ndharmonic!Harmonicgenerationisoneofmanyexoticeffectsthatcanarise!**2222(),()exp()ex2exp(2)exp(p2())EEititEEititttEEESince(1)0PE(1)(2)2(3)30...PEEESum-anddifference-frequencygenerationSupposetherearetwodifferent-colorbeamspresent:*1111*2222exp()eexp()exp()x(())pEitEitEitEitEt**121**121212122*2221221222122221112*122exp(exp(2)exp(2exp(exp(2)exp(2))2exp)2ex2)(p())(2)2EitEEEEitEEititEEitEEitEtEEitEitNotealsothat,wheniisnegativeinsidetheexp,theEinfronthasa*.2nd-harmonicgen2nd-harmonicgenSum-freqgenDiff-freqgendcrectificationSo:Complicatednonlinear-opticaleffectscanoccur.Themorephotons(i.e.,thehighertheorder)theweakertheeffect,however.Very-high-ordereffectscanbeseen,buttheyrequireveryhighirradiance.Also,ifthephotonenergiescoincidewiththemedium’senergylevelsasabove,theeffectwillbestronger.Nonlinear-opticalprocessesareoftenreferredtoas:N-wave-mixingprocesseswhereNisthenumberofphotonsinvolved(includingtheemittedone).Thisisasix-wave-mixingprocess.Emitted-lightfrequencysigInducedpolarizationfornonlinearopticaleffectsArrowspointingupwardcorrespondtoabsorbedphotonsandcontributeafactoroftheirfield,Ei;arrowspointingdownwardcorrespondtoemittedphotonsandcontributeafactorofthecomplexconjugateoftheirfield:(5)*012345EEEEEPsigSolvingthewaveequationinnonlinearoptics222022221zcttEEPRecalltheinhomogeneouswaveequation:Becauseit’ssecond-orderinbothspaceandti