纳米光子学-第二讲.pdf

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