非线性光学英文版课件：Lecture 3- Descriptions of nonlinear optical processes(Continued)

2011-8-29DepartmentofPhysicsSoutheastUniversityLecturethree:

DescriptionsofNonlinearOpticalProcesses

(Third-OrderOpticalSusceptibility)1、Third-OrderNonlinearOpticalProcessesWenowconsiderthethird-ordercontributiontothenonlinearpolarization(1)

(2)

(3)1.1.Third-HarmonicGenerationThefirstterminEq.(3)describesaresponseatfrequency3ωthatiscreatedbyanappliedfieldatfrequencyω.Thistermleadstotheprocessofthird-harmonicgeneration,seeFig..2.Intensity-DependentRefractiveIndexThesecondterminEq.(3)describesanonlinearcontributiontothepolarizationatthefrequencyoftheincidentfield;thistermhenceleadstoanonlinearcontributiontotherefractiveindexexperiencedbyawaveatfrequencyω.Weshallseelaterthattherefractiveindexinthepresenceofthistypeofnonlinearitycanberepresentedas

Self-Focusing:Oneoftheprocessesthatcanoccurasaresultoftheintensity-dependentrefractiveindexisself-focusing,whichisillustratedinFig.1.2.6.Thisprocesscanoccurwhenabeamoflighthavinganonuniformtransverseintensitydistributionpropagatesthroughamaterialforwhichn2ispositive.Undertheseconditions,thematerialeffectivelyactsasapositivelens,whichcausestheraystocurvetowardeachother.Thisprocessisofgreatpracticalimportancebecausetheintensityatthefocalspotoftheself-focusedbeamisusuallysufficientlylargetoleadtoopticaldamageofthematerial.1.3.Third-OrderInteractions(GeneralCase)Letusnextexaminetheformofthethird-ordernonlinearpolarization(1)inducedbyanappliedfieldthatconsistsofthreefrequencycomponents:(4)

Againrepresentingthenonlinearpolarizationas(5)wecanwritethecomplexamplitudesofthenonlinearpolarizationforeachofthepositivefrequenciesas(6)Wehavedisplayedtheseexpressionsincompletedetailbecauseitisveryinstructivetostudytheirform.Also,thenumericalfactor(1,3,or6)thatappearsineachtermontheright-handsideofeachequationisequaltothenumberofdistinctpermutationsofthefieldfrequenciesthatcontributetothatterm.

SomeofthenonlinearopticalmixingprocessesdescribedbyEq.(6)areillustratedinFig.、ParametricvsNonparametricProcessesParametricprocessdenotesaprocessinwhichtheinitialandfinalquantum-mechanicalstatesofthesystemareidentical.Consequently,inaparametricprocesspopulationcanberemovedfromthegroundstateonlyforthosebriefintervalsoftimewhenitresidesinavirtuallevel.Nonparametricprocessinvolvesthetransferofpopulationfromonerealleveltoanother.Forexample,saturableabsorption,two-photonabsorptionstimulatedRamanscatteringDifferences:Parametricprocessescanalwaysbedescribedbyarealsusceptibility,whilenonparametricprocessesaredescribedbyacomplexsusceptibility.Inaparametricprocess,photonenergyisalwaysconserved,whileinanonparametricprocessphotonenergyneednotbeconserved.Particularly,forthecaseoftheusual(linear)indexofrefraction,therealpartoftherefractiveindexdescribesaresponsethatoccursasaconsequenceofparametricprocesses,whereastheimaginarypartoccursasaconsequenceofnonparametricprocesses.2.1SaturableAbsorption

(7)2.2Two-PhotonAbsorption

2.3StimulatedRamanScattering

Theefficiencyofthisprocesscanbequitelarge,withoften10%ormoreofthepoweroftheincidentlightbeingconvertedtotheStokesfrequency.Incontrast,theefficiencyofnormalorspontaneousRamanscatteringistypicallymanyordersofmagnitudesmaller.3、FormalDefinitionoftheNonlinearSusceptibilityNonlinearopticalinteractionscanbedescribedintermsofanonlinearpolarizationgivenbyEq.(1.1.2)(seetextbook)onlyforamaterialsystemthatislosslessanddispersionless.Weconsiderthemoregeneralcaseofamaterialwithdispersionand/orloss.Inthismoregeneralcasethenonlinearsusceptibilitybecomesacomplexquantityrelatingthecomplexamplitudesoftheelectricfieldandpolarization.TheprimeonthesummationsignofEq.(8)indicatesthatthesummationistobetakenoverpositivefrequenciesonly.Weassumethatwecanrepresenttheelectricfieldvectoroftheopticalwaveasthediscretesumofanumberoffrequencycomponentsas(8)(12)Sometimes,weshallexpressthesefieldamplitudesusingthealternativenotationwhere(11)sothat

(9)(10)wheretheunprimedsummationsymboldenotesasummationoverallfrequencies,bothpositiveandnegative.Usingthisnewnotation,wecanwritethetotalfieldinthemorecompactform(13)Similarly,wecanexpressthenonlinearpolarizationas(14)

(15)

LetusexaminesomeoftheconsequencesofthedefinitionofthenonlinearsusceptibilityasgivenbyEq.(15)byconsideringtwosimpleexamples.(16)3.1Sum-frequencygeneration

Accordingtotheintrinsicpermutationsymmetry,wehave(17)Throughuseofthisrelation,theexpressionforthenonlinearpolarizationbecomes.(18)andforthespecialcaseinwhichbothinputfieldsarepolarizedinthexdirectionthepolarizationbecomes(19)3.2Second-harmonicgeneration

(20)Againassumingthespecialcaseofaninputfieldpolarizationalongthexdirectio