洛伦兹模型(LorentzModel)

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1、2.LORENTZMODELOFLIGHTMATTERINTERACTION2.1.FrommicroscopictomacroscopicresponseReviewthemainconceptsinbasicatom-fieldinteractions.InparticulartheLorentzmodel,apre-quantummechanicsmodel,anditsasymptoticcaseformetals,theDrudemodel.TheLorentzmodelexplainsmuchofclassicalopticsviaaphysicalpict

2、ureborrowedfrommechanics.Thestartingpointisthe“massonaspring”descriptionofelectronsconnectedtonuclei.Thus,theincidentelectricfieldinducesdisplacementtotheelectronthatisundertheinfluenceofaspring-likerestoringforceduetothenucleus.1Theequationofmotionfortheelectroncanbeexpressedas2dxt()dxt(

3、)2exE(),t(0.1)20dtdtmisthedampingconstant,0istheresonantfrequency,eistheelectroniccharge,mmassoftheelectron,andEtheincidentfield.and0arecharacteristicsofthematerial,thefirstdescribingtheenergydissipationpropertyofthemediumandthesecondtheabilityofthemediumtostoreenergy.SinceEq.5

4、7isalineardifferentialequation,Fouriertransformingbothsidesoftheequationgivesthefrequency-domainsolution.2UsingtheFourierpropertyofthedifferentialoperator,ndn(),i(0.2)ndtweobtainEq.57inthefrequencydomain22exi()x()x()E().(0.3)0mThus,wefindthesolutionforthechargedisplacemen

5、tinthefrequencydomain,eE()mx().(0.4)22i0Toobtainthetimedomainsolution,x(t),weneedtoFourier-transformEq.60.However,weexplorefurtherthefrequencydomainsolution.Theinduceddipolemomentduetothechargedisplacementx()ispx()e()(0.5)3InEqs.59-60weobtainedmicroscopicquantities,theatom

6、ic-levelresponse.ThemacroscopicbehaviorofthemediumisobtainedfromtheinducedpolarizationP,whichcapturesthecontributionofalldipolemomentswithinacertainvolume,PpN.(0.6)-3Nisthevolumeconcentrationofdipoles(m)andtheangularbracketsdenoteensembleaverage.Assumingthatallinduceddipolesareparallel

7、withinthevolume,weobtain2NeE()P().(0.7)22mi04Generally,eachatomhasmultipleresonancesordipole-activemodes,suchthatEq.63canbegeneralizedto2NeE()P()i,(0.8)22mii0iiThesummationisoverallmodes,characterizedbydifferentresonantfrequenciesanddam