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 llalsoas