纳米光子学-第二讲.pdf
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纳米光子学-第二讲.pdf
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1Lecture2:
DispersioninMaterials5nm2CourseInfoCoursewebpageIsnowupandrunninghttp:
/shay.ecn.purdue.edu/ece695sLetmeknowwhatyouthink!
Topics?
Format?
Askquestionsanytime.DirectquestionsBigcommentsonthenanocoursearemostwelcome!
3WhatHappenedinthePreviousLecture?
MaxwellsEquationsf=D0=B=BEt=+DHJtCurlEquationsleadto22200022tt=+EPE0=PEWaveEquationLinear,Homogeneous,andIsotropicMedia(undercertainconditions)()()22222,tntc=ErErtInreallife:
Responseofmatter(P)isnotinstantaneous()()()0,tdtxttt+=PrErSolutions:
EMwaves()(),Re,exp2ztzizzit=+EE0kn=02kn=whereandPhasepropagationabsorption()=()nn=()=()nn=Boldfacelettersarevectors!
4Today:
MicroscopicOrigin-ResponseofMatterOriginfrequencydependenceofinrealmaterialsLorentzmodel(harmonicoscillatormodel)RealandimaginarypartofarelinkedKramers-KronigInsulators(Latticeabsorption,colorcenters)Semiconductors(Energybands,Urbachtail,excitons)Metals(ACconductivity,Plasmaoscillations,interbandtransitions)Butfirst.WhenshouldIworkwith,orn?
Theyallseemtodescribetheopticalpropertiesofmaterials!
5nandnvsandvsandAllpairs(nandn,and,and)describethesamephysicsForsomeproblemsonesetispreferableforothersanothernandnusedwhendiscussingwavepropagation()(),Re,exp2ztzizzit=+EE0kn=02kn=whereandPhasepropagationabsorptionandandusedwhendiscussingmicroscopicoriginofopticaleffectsAswewillseetodayInterrelationships()()22rnn=2rnn=()()222rrrn+=()()222rrrn+=Example:
nandrn=Fromandrrnini+=+6BehaviorofboundelectronsinanelectromagneticfieldChargesinamaterialaretreatedasharmonicoscillatorsOpticalpropertiesofinsulatorsaredeterminedbyboundelectrons,elELocalDampingSpringm=+aFFF()22expLddmmCeitdtdt+=rrrE()0expit=ppGuessasolutionoftheform:
22000LmimCe+=pppEe()0;expdiitdt=ppTheelectricdipolemomentofthissystemis:
e=pr()222expLddmmCeitdtdt+=pppE()2202;expditdt=ppSolveforp0(EL)LinearDielectricResponseofMatterLorentzmodel(oneoscillator)Nucleuse-,me=pr+C,rLE7AtomicPolarizabilityDeterminationofatomicpolarizability22000LmimCe+=pppELastslide:
202201Lemi=pE22000LCeimm+=pppE(Dividebym)Defineas02(turnsouttobetheresonance)Atomicpolarizability(inSIunits)Resonancefrequency()2022001ELpemi=0Defineatomicpolarizability:
Dampingterm8CharacteristicsoftheAtomicPolarizability()()2022001expELpeAimi=0Atomicpolarizability:
()21/222222001eAm=+AmplitudeResponseofmatter(P)isnotinstantaneousPhaselagofwithE:
1220tan=AmplitudePhaselag0018090smallersmaller-dependentresponse9RelationAtomicPolarizability()and:
2casesCase1:
Rarifiedmedia(.gasses)()0jjL=pEDipolemomentofoneatom,j:
Polarizationvector:
001jjLjLjjNVV=PpEEsumoverallatoms()222001jemi=()202201LLNemi=PEEE-fieldphoton()222001Nemi=Microscopicoriginsusceptibility:
Plasmafrequencydefinedas:
220pNem=()2220pi=DensityOccursinMaxwellsequation.10Remember:
andnfollowdirectlyfromRelationofto:
222011pi=+=+Frequencydependence()()()222022222011p=+=+()()2222220p=+222011piii+=+=+0()2201p+10=220p=p11PropagationofEM-waves:
Neednandn0()2201p+1()nnn1n=0Relationbetweennandn=()()22rnn=2rnn=0:
Highnlowvph=c/nn=1vph=c1222201kkkNemi=Resonancesoccurduetomotionoftheatoms(low)andelectrons(high)WhereNkisthedensityoftheelectrons/atomswitharesonanceatkRealisticatomshavemanyresonancesRealisticRarefiedMedia=2k0nn213Log()MoleculerotationAtomicvibrationElectronexcitationn1indicatespresencehighoscillatorsExampleofarealisticdependenceofnandn13BacktoRelationAtomicPolarizability()and:
Case2:
SolidsFieldwithoutmatterLocalfield:
SolidE-field?
piAtom“feels”fieldfrom:
1)Incidentlightbeam2)Induceddipolarfieldfromotheratoms,piLocalfieldInduceddipolarfieldfromalltheotheratoms0LI=+EEE14ElectricSusceptibilityofaSolidLocalfieldFieldwithoutmatterAlltheotheratomsLocalfield:
LocalfieldInduceddipolarfieldExample:
Forcubicsymmetry:
03I=PE(SolidstatePhys.Books,e.g.Kittel,Ashcroft.)0LI=+EEE003L=+PEEPolarizationofasolid00000033jjLjjjjjjjjjjNNNN=+=+PPPEEE0113jjjjjjNPEN=(Similarrelationscanbederivedforanysolid)00113jjjjjjNN=PEjpSolidconsistsofatomtypejataconcentrationNj150113jjjjjjNPEN=Clausius-MossottiRelationPolarizationofasolidSusceptibility:
Limitoflowatomicconcentration:
jjjNBydefinition:
Clausius-Mossotti1=+01123jjjN=+IIIIIIRearrangingIgivesConclusion:
DielectricpropertiesofsolidsrelatedtoatomicpolarizabilityThisisverygeneral!
.orweakpolarizability:
prettygoodforgassesandglasses
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