纳米光子学-第一讲.pdf

纳米光子学-第一讲.pdf

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1Lecture1:

LightInteractionwithMatter5nm2LightInteractionwithMatterMaxwellsEquations0=B=BEt=+DHJtDivergenceequationsCurlequationsf=DD=ElectricfluxdensityB=MagneticfluxdensityE=ElectricfieldvectorH=MagneticfieldvectorJ=currentdensity=chargedensity3ConstitutiveRelationsConstitutiverelationsrelatefluxdensitytopolarizationofamedium（）0=+=DEPEEElectricpolarizationvectorMaterialdependent!

WhenPisproportionaltoEElectricE-+-Totalelectricfluxdensity=FluxfromexternalE-field+fluxduetomaterialpolarization=Materialdependentdielectricconstant0=Dielectricconstantofvacuum=8.8510-12C2N-1m-2F/m（）00=+BHMHMagneticfluxdensityMagneticfieldvectorMagneticMagneticpolarizationvector0=permeabilityoffreespace=4x10-7H/mNote:

Fornow,wewillfocusonmaterialsforwhich0=M0=BH4DivergenceEquationsHowdidpeoplecomeupwith:

?

=DCoulombChargesofsamesignrepeleachother（+and+orand-）Chargesofoppositesignattracteachother（+and-）Heexplainedthisusingtheconceptofanelectricfield:

F=qEEverychargehassomefieldlinesassociatedwithit+-Hefound:

LargerchargesgiverisetostrongerforcesbetweenchargesCoulombexplainedthiswithastrongerfield（morefieldlines）5DivergenceEquationsEdS+GausssLaw（Gauss1777-1855）AAVdEddv=DSSE-fieldrelatedtoenclosedchargeGausssTheorem（verygeneral）AVddv=FSFCombiningthe2GausssAVVddvdv=DSD=D0Ad=BSTheotherdivergenceeq.0=Bisderivedinasimilarwayfrom6CurlEquationsHHowdidpeoplecomeupwith:

=+DHJtJHDDincreasingwhenchargingthecapacitorCA?

JAmpere（1775-1836）CAddt=+?

DHlJSChangesinel.fluxMagneticfieldinducedby:

Electricalcurrents7CurlEquationsCAddt=+?

DHlJS（）CAdd=?

FlFSAmpere:

（）CAAdddt=+?

DHlHSJSStokestheorem:

=+DHJtBECA=BEtOthercurleq.DerivedinasimilarwayfromCAddt=BElS?

（）CAAdddt=?

BElESSStokes8SummaryMaxwellsEquationsDivergenceequationsCurlequations=BEt=D=+DHJt0=BFluxlinesstartandendonchargesorpolesChangesinfluxesgiverisetofieldsCurrentsgiverisetoH-fieldsNote:

Noconstantssuchas00,c,.appearwhenEqsarewrittenthisway.9TheWaveEquationPlausibilityargumentforexistenceofEMwavesEHHEE.Curlequations:

ChangingE-fieldresultsinchangingH-fieldresultsinchangingE-field.Therealthing（）（）2222,1,rtrtv=UUtGoal:

Deriveawaveequation:

forEandH（）（）（）0,Reexptit=UrUrSolution:

Wavespropagatingwitha（phase）velocityvPositionTimeStartingpoint:

Thecurlequations10TheWaveEquationfortheE-field（）（）2222,1,rtrtv=EEtGoal:

0t=BHEtCurlEqs:

a）（MaterialswithM=0only）=+DHJtb）Step1:

TryandobtainpartialdifferentialequationthatjustdependsonEApplycurlonbothsideofa）（）00tt=HHEStep2:

Substituteb）intoa）222000000222ttttt=DJEPJE0=+DEPCool!

.lookslikeawaveequationalready1122000022ttt=EPJE22221v=EEtWith:

!

（）2=EEEUsevectoridentity:

VerifythatE=0when1）f=02）（r）doesnotvarysignificantlywithinadistance222000022ttt=+EPJEResult:

TheWaveEquationfortheE-fieldCompare:

1）FindP（E）2）FindJ（E）somethinglikeOhmslaw:

J（E）=Ewewilllookatthislater.fornowassume:

J（E）=0Inordertosolvethisweneed:

12DielectricMediaLinear,Homogeneous,andIsotropicMedia222000022ttt=+EPJEPlinearlyproportionaltoE:

0=PEisascalarconstantcalledthe“electricsusceptibility”Allthematerialsproperties（）22220000002221ttt=+=+EEEE22002rt=EEResultsfromPDefinerelativedielectricconstantas:

1r=+

（2）2（3）3000.=+PEEENote1:

InanisotropicmediaPandEarenotnecessarilyparallel:

0iijjjPE=Note2:

Innon-linearmedia:

13PropertiesofEMWavesinBulkMaterialsWehavederivedawaveequationforEMwaves!

22002rt=EENowwhat?

Euh.Letslookatsomeoftheirproperties14SpeedofanEMWaveinMatterSpeedoftheEMwave:

22002rt=EE22221v=EEtCompareand2200011rrcv=Wherec02=1/（00）=1/（8.85x10-12C2/m3kg）（4x10-7mkg/C2）=（3.0x108m/s）2Opticalrefractiveindex1rcv=+Refractiveindexisdefinedby:

nNote:

Includingpolarizationresultsinsamewaveequationwithadifferentrcbecomesv15RefractiveIndexVariousMaterials2.03.01.03.40.11.010（m）Refractiveindex:

n16DispersionRelation（）（）（）,Re,expztzikrit=+EE（）（）22222,tntc=ErErtDispersionrelation:

=（k）DerivedfromwaveequationSubstitute:

2222ckn=2222nkc=gvResult:

Checkthis!

kgdvdkGroupvelocity:

1phrcccvkn=+Phasevelocity:

17ElectromagneticWaves（）（）22222,tntc=ErErtSolutionto:

（）（）（）,Re,exptiit=+ErEkkrMonochromaticwaves:

Checkthesearesolutions!

（）（）（）,Re,exptiit=+HrHkkrTEMwaveSymmetryMaxwellsEquationsresultinEHpropagationdirectionOpticalintensity（）（）（）,ttt=rErHrTimeaverageofPoyntingvector:

S18LightPropagationDispersiveMediaRelationbetweenPandEisdynamicTherelation:

assumesaninstantaneousresponse（）（）0,tt=PrEr（）（）（）0,tdtxttt+=PrErInreallife:

PresultsfromresponsetoEoversomecharacteristictime:

Functionx（t）isascalarfunctionlastingacharacteristictime:

x（t-t）t=t-E（t）x（t-t）=0fortt（causality）tt=t19EMwavesinDispersiveMedia（）（）（）,Re,exptiit=+ErEkkr（）（）（）0,tdtxttt+=PrErRelationbetweenPandE