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1、NonlinearOptics Whydononlinear opticaleffectsoccur Maxwell sequationsinamediumNonlinear opticalmediaSecond harmonicgenerationSum anddifference frequencygenerationHigher ordernonlinearopticsTheSlowlyVaryingEnvelopeApproximationPhase matchingandConservationlawsforphotons Nonlinearopticsisn tsomethingy
2、ouseeeveryday Sendinginfraredlightintoacrystalyieldedthisdisplayofgreenlight second harmonicgeneration SHG Nonlinearopticsallowsustochangethecolorofalightbeam tochangeitsshapeinspaceandtime andtocreateultrashortlaserpulses Whydon tweseenonlinearopticaleffectsinourdailylife 1 Intensitiesofdailylifear
3、etooweak 2 Normallightsourcesareincoherent 3 Theoccasionalcrystalweseehasthewrongsymmetry forSHG 4 Phase matchingisrequired anditdoesn tusuallyhappenonitsown Whydononlinear opticaleffectsoccur Recallthat innormallinearoptics alightwaveactsonamolecule whichvibratesandthenemitsitsownlightwave whichint
4、erfereswiththeoriginallightwave Wecanalsoimaginethisprocessintermsofthemolecularenergylevels usingarrowsforthephotonenergies Whydononlinear opticaleffectsoccur continued Now supposetheirradianceishighenoughthatmanymoleculesareexcitedtothehigher energystate Thisstatecanthenactasthelowerlevelforadditi
5、onalexcitation Thisyieldsvibrationsatallfrequenciescorrespondingtoallenergydifferencesbetweenpopulatedstates Nonlinearopticsisanalogoustononlinearelectronics whichwecanobserveeasily Sendingahigh volumesine wave purefrequency signalintoacheapamplifierorcheapspeakersyieldsatruncatedoutputsignal moreof
6、asquarewavethanasine Thissquarewavehashigherfrequencies Wehearthisasdistortion SharpedgesrequirehigherfrequenciesintheFourierseries Nonlineareffectsinatoms Soanelectron smotionwillalsodepartfromasinewave Anotherwaytolookatnonlinearopticsisthatthepotentialoftheelectronisanatomisnotasimplequadraticpot
7、ential Nonlinearopticsandanharmonicoscillators Forweakfields motionisharmonic andlinearopticsprevails Forstrongfields i e lasers anharmonicmotionoccurs andhigherharmonicsoccur bothinthemotionandthelightemission Anucleus inamolecule alsodoesnothaveasimplequadraticpotential Soitsvibrationalmotionisals
8、ononlinear MoleculesExcitedbyLaserLight Laserlightcausesmoleculestovibrateinunison Wesaythatlaserlightpolarizesthemedium Acceleratingchargesemitlight Polarizedmatteremitslightatthefrequencyatwhichitisoscillating Ifthemotionisn tsinusoidal themediumwillalsoemittheadditionalfrequencies Maxwell sEquati
9、onsinaMedium Thepolarization P containstheeffectofthemedium Sinewavesofallfrequenciesaresolutionstothewaveequation it sthepolarizationthattellswhichfrequencieswilloccur Thepolarizationisthedrivingtermforthesolutiontothisequation Theseequationsreducetothe scalar waveequation InhomogeneousWaveEquation
10、 Solvingthewaveequationinthepresenceoflinearinducedpolarization Forlowirradiances thepolarizationisproportionaltotheincidentfield Inthissimple andmostcommon case thewaveequationbecomes Thisequationhasthesolution Theinducedpolarizationonlychangestherefractiveindex Dull Ifonlythepolarizationcontainedo
11、therfrequencies wherew ckandc c0 nandn 1 c 1 2 Usingthefactthat Simplifying Maxwell sEquationsinaNonlinearMedium Nonlinearopticsiswhathappenswhenthepolarizationistheresultofhigher order nonlinear termsinthefield Whataretheeffectsofsuchnonlinearterms Considerthesecond orderterm 2w 2ndharmonic Harmoni
12、cgenerationisoneofmanyexoticeffectsthatcanarise Sum anddifference frequencygeneration Supposetherearetwodifferent colorbeamspresent Notealsothat whenwiisnegativeinsidetheexp theEinfronthasa 2nd harmonicgen 2nd harmonicgen Sum freqgen Diff freqgen dcrectification So Complicatednonlinear opticaleffect
13、scanoccur Themorephotons i e thehighertheorder theweakertheeffect however Very high ordereffectscanbeseen buttheyrequireveryhighirradiance Also ifthephotonenergiescoincidewiththemedium senergylevelsasabove theeffectwillbestronger Nonlinear opticalprocessesareoftenreferredtoas N wave mixingprocessesw
14、hereNisthenumberofphotonsinvolved includingtheemittedone Thisisasix wave mixingprocess Emitted lightfrequency wsig Inducedpolarizationfornonlinearopticaleffects Arrowspointingupwardcorrespondtoabsorbedphotonsandcontributeafactoroftheirfield Ei arrowspointingdownwardcorrespondtoemittedphotonsandcontr
15、ibuteafactorofthecomplexconjugateoftheirfield Solvingthewaveequationinnonlinearoptics Recalltheinhomogeneouswaveequation Becauseit ssecond orderinbothspaceandtime andPisanonlinearfunctionofE wecan teasilysolvethisequation Indeed nonlineardifferentialequationsarereallyhard We llhavetomakeapproximatio
16、ns Takeintoaccountthelinearpolarizationbyreplacingc0withc Separation of frequenciesapproximation ThetotalE fieldwillcontainseveralnearlydiscretefrequencies w1 w2 etc Sowe llwriteseparatewaveequationsforeachfrequency consideringonlytheinducedpolarizationatthegivenfrequency whereE1andP1aretheE fieldandpolarizationatfrequencyw1 whereE2andP2aretheE fieldandpolarizationatfrequencyw2 etc Thiswillbeareasonableapproximationevenforrelativelybroadbandultrashortpulses Thenon depletionassumption We llalsoas
