超快光学-第07章-非线性光学.ppt

NonlinearOptics Whydononlinear opticaleffectsoccur Maxwell sequationsinamediumNonlinear opticalmediaSecond harmonicgenerationSum anddifference frequencygenerationHigher ordernonlinearopticsTheSlowlyVaryingEnvelopeApproximationPhase matchingandConservationlawsforphotons Nonlinearopticsisn 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 Nonlineareffectsinatoms Soanelectron smotionwillalsodepartfromasinewave Anotherwaytolookatnonlinearopticsisthatthepotentialoftheelectronisanatomisnotasimplequadraticpotential Nonlinearopticsandanharmonicoscillators Forweakfields motionisharmonic andlinearopticsprevails Forstrongfields i e lasers anharmonicmotionoccurs andhigherharmonicsoccur bothinthemotionandthelightemission Anucleus inamolecule alsodoesnothaveasimplequadraticpotential Soitsvibrationalmotionisalsononlinear MoleculesExcitedbyLaserLight Laserlightcausesmoleculestovibrateinunison Wesaythatlaserlightpolarizesthemedium Acceleratingchargesemitlight Polarizedmatteremitslightatthefrequencyatwhichitisoscillating Ifthemotionisn tsinusoidal themediumwillalsoemittheadditionalfrequencies Maxwell sEquationsinaMedium Thepolarization P containstheeffectofthemedium Sinewavesofallfrequenciesaresolutionstothewaveequation it sthepolarizationthattellswhichfrequencieswilloccur Thepolarizationisthedrivingtermforthesolutiontothisequation Theseequationsreducetothe scalar waveequation InhomogeneousWaveEquation Solvingthewaveequationinthepresenceoflinearinducedpolarization Forlowirradiances thepolarizationisproportionaltotheincidentfield Inthissimple andmostcommon case thewaveequationbecomes Thisequationhasthesolution Theinducedpolarizationonlychangestherefractiveindex Dull Ifonlythepolarizationcontainedotherfrequencies wherew ckandc c0 nandn 1 c 1 2 Usingthefactthat Simplifying Maxwell sEquationsinaNonlinearMedium Nonlinearopticsiswhathappenswhenthepolarizationistheresultofhigher order nonlinear termsinthefield Whataretheeffectsofsuchnonlinearterms Considerthesecond orderterm 2w 2ndharmonic Harmonicgenerationisoneofmanyexoticeffectsthatcanarise Sum anddifference frequencygeneration Supposetherearetwodifferent colorbeamspresent Notealsothat whenwiisnegativeinsidetheexp theEinfronthasa 2nd harmonicgen 2nd harmonicgen Sum freqgen Diff freqgen dcrectification So 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 wsig Inducedpolarizationfornonlinearopticaleffects Arrowspointingupwardcorrespondtoabsorbedphotonsandcontributeafactoroftheirfield Ei arrowspointingdownwardcorrespondtoemittedphotonsandcontributeafactorofthecomplexconjugateoftheirfield Solvingthewaveequationinnonlinearoptics Recalltheinhomogeneouswaveequation Becauseit ssecond orderinbothspaceandtime andPisanonlinearfunctionofE wecan teasilysolvethisequation Indeed nonlineardifferentialequationsarereallyhard We llhavetomakeapproximations Takeintoaccountthelinearpolarizationbyreplacingc0withc Separation of frequenciesapproximation ThetotalE fieldwillcontainseveralnearlydiscretefrequencies w1 w2 etc Sowe llwriteseparatewaveequationsforeachfrequency consideringonlytheinducedpolarizationatthegivenfrequency whereE1andP1aretheE fieldandpolarizationatfrequencyw1 whereE2andP2aretheE fieldandpolarizationatfrequencyw2 etc Thiswillbeareasonableapproximationevenforrelativelybroadbandultrashortpulses Thenon depletionassumption We llalsoassumethatthenonlinear opticaleffectisweak sowecanassumethatthefieldsattheinputfrequencieswon tchangemuch Thisassumptioniscallednon depletion Asaresult weneedonlyconsiderthewaveequationforthefieldandpolarizationoscillatingatthenewsignalfrequency wsig whereEsigandPsigaretheE fieldandpolarizationatfrequencywsig We llwritethepulseE fieldasaproductofanenvelopeandcomplexexponential Esig z t Esig z t exp i wsigt ksigz We llassumethatthenewpulseenvelopewon tchangetoorapidly ThisistheSlowlyVaryingEnvelopeApproximation SVEA Ifdisthelengthscaleforvariationoftheenvelope SVEAsays d lsig TheSlowlyVaryingEnvelopeApproximation ComparingEsiganditsderivatives We lldothesameintime Iftisthetimescaleforvariationoftheenvelope SVEAsays t TsigwhereTsigisoneopticalperiod 2p wsig TheSlowlyVaryingEnvelopeApproximation continued ComparingEsiganditstimederivatives Andwe lldothesameforthepolarization Psig z t Psig z t exp i wsigt ksigz Iftisthetimescaleforvariationoftheenvelope SVEAsays t TsigwhereTsigisoneopticalperiod 2p wsig TheSlowlyVaryingEnvelopeApproximation continued ComparingPsiganditstimederivatives Computingthederivatives SVEA continued Neglectall2ndderivativesofenvelopeswithrespecttozandt Also neglectthe1stderivativeofthepolarizationenvelope it ssmallcomparedtothewsig2Psigterm WemustkeepEsig sfirstderivatives aswe llseeinthenextslide x Esig z t Esig z t exp i wsigt ksigz Similarly Now becauseksig wsig c thelasttwobracketedtermscancel Andwecancancelthecomplexexponentials leaving TheSlowlyVaryingEnvelopeApproximation Substitutingtheremainingderivativesintotheinhomogeneouswaveequationforthesignalfieldatw0 SlowlyVaryingEnvelopeApproximation Dividingby2iksig IncludingdispersionintheSVEA WecanincludedispersionbyFourier transforming expandingksig w tofirstorderinw andtransformingback Thisreplacescwithvg WecanincludeGVDalso byexpandingto2ndorder yielding Wecanunderstandmostnonlinear opticaleffectsbestbyneglectingGVD sowewill butthisextratermcanbecomeimportantforveryveryshort i e verybroadband pulses Transformingtoamovingco ordinatesystem Defineamovingco ordinatesystem zv ztv t z vg Transformingthederivatives TheSVEAbecomes Cancelingterms theSVEAbecomes We lldropthesub script v tosimplifyourequations Thetimederiva tivescancel IntegratingtheSVEA Usually Psig Psig z t andeventhissimpleequationcanbedifficulttosolve integrate Fornow we lljustassumethatPsigisaconstant andtheintegrationbecomestrivial Andthefieldamplitudegrowslinearlywithdistance Theirradiance intensity thengrowsquadraticallywithdistance whenPsigisconstant Wechoosewsigtobethesumoftheinputw s ButthesignalE fieldandpolarizationk vectorsaren tnecessarilyequal Butthek vectorofthepolarizationis Sokpolmaynotbethesameasksig Andwemaynotbeabletocanceltheexp ikz s Thek vectormagoflightatthisfrequencyis Phase matching Thatkpolmaynotbethesameasksigistheall importanteffectofphase matching Itmustbeconsideredinallnonlinear opticalproblems Ifthek sdon tmatch theinducedpolarizationandthegenerate