微纳制备与新器件集成学习chapter 1 of

Chapter1WaveNatureofLight

Dr.DaoliZhang

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Email:

zhang_d

aoli@

zhang-daoli@163.com

TextbookandReferences

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Outline

LightWavesinaHomogeneousMedium

RefractiveIndexandDispersion

GroupVelocityandGroupIndex

MagneticField,Irradiance,andPoyntingVector

Snell’sLawandTotalInternalReflection(TIR)

Fresnel’sEquations

AntireflectionCoatingsandDielectricMirrors

AbsorptionofLightandComplexRefractiveIndex

TemporalandSpatialCoherence

SuperpositionandInterferenceofWaves

MultipleInterferenceandOpticalResonators

DiffractionPrinciples

AdditionalTopics

Interferometers

ThinFilmOptics:MultipleReflectionsinThinFilms

MultipleReflectionsinPlatesandIncoherentWaves

ScatteringofLight

PhotonicCrystals

1.Lightwavesinahomogeneousmedium

A.PlaneElectromagneticWave

Thewavenatureoflight,quiteasidefromitsphotonicbehavioriswellrecognizedbysuchphenomenaasinterferenceanddiffraction.

Wecantreatlightasanelectromagneticwavewithtimevaryingelectricandmagneticfields,thatisExandByrespectively,whicharepropagatingthroughspaceinsuchawaythattheyare

alwaysperpendiculartoeachotherandthedirectionofpropagatingzdepictedinFigure1.

Anelectromagneticwaveisatravelingwavethathastime-varyingelectricandmagneticfieldsthatareperpendiculartoeachotherandthedirectionofpropagationz.

Thesimplesttravelingwaveissinusoidalwavethat,forpropagationalongz,hasthegeneralmathematicform:

Ex=Eocos(tkz+)

Ex=Electricfieldalongxatpositionzattimetk=Propagationconstant=2/

=Wavelength

=Angularfrequency=2frequency)

Eo=Amplitudeofthewave

=Phaseconstant;att=0andz=0,Exmayormaynotnecessarilybezerodependingonthechoiceoforigin.

(tkz+)==Phaseofthewave

Thisisamonochromaticplanewaveofinfiniteextenttravelinginthepositivezdirection.

Wavefront

Asurfaceoverwhichthephaseofawaveisconstantisreferredtoasawavefront.

Awavefrontofaplanewaveisaplaneperpendiculartothedirectionofpropagation.

Theinteractionofalightwavewithanonconductingmedium(conductivity=0)usestheelectricfieldcomponentExratherthanBy.

Why?

OpticalfieldreferstotheelectricfieldEx.

ItistheelectronicfieldExthatdisplacestheelectronicsinmoleculesorionsandgiverisetopolarizationofmatter

AplaneEMwavetravelingalongz,hasthesameEx(orBy)atanypointinagivenxyplane.Allelectricfieldvectorsinagivenxyplanearethereforeinphase.Thexyplanesareofinfiniteextentinthexandydirections.

PhaseVelocity

Thetimeandspaceevolutionofagivenphase,forexamplethatcorrespondingtoamaximumfieldisdescribedby

=tkz+=constant

Duringatimeintervalt,thisconstantphase(andhencethemaximumfield)movesadistancez.Thephasevelocityofthiswaveisthereforez/t.Thephasevelocityvis

vz

t

k

PhasechangeoveradistanceDz

=tkz+

=

kz

Thephasedifferencebetweentwopointsseparatedbyzissimplykz

sincetisthesameforeachpoint

Ifthisphasedifferenceis0ormultiplesof2thenthetwopointsareinphase.Thus,thephasedifferencecanbeexpressedaskzor2z/

ExponentialNotation

Recallthat

cos=Re[exp(j)]

whereRereferstotherealpart.Wethenneedtotaketherealpartofanycomplexresultattheendofcalculations.Thus,

Ex(z,t)=Re[Eoexp(j)expj(tkz)]

or

Ex(z,t)=Re[Ecexpj(tkz)]

whereEc=Eoexp(jo)isacomplexnumberthatrepresentstheamplitudeofthewaveandincludestheconstantphaseinformation

o.

WaveVectororPropagationVector

Directionofpropagationisindicatedwithavectork,calledthewavevector,whosemagnitudeisthepropagationconstant,k=2/.kisperpendiculartoconstantphaseplanes.

Whentheelectromagnetic(EM)waveispropagatingalongsomearbitrarydirectionk,thentheelectricfieldE(r,t)atapointronaplaneperpendiculartokis

E(r,t)=Eocos(tkr+)

Ifpropagationisalongz,krbecomeskz.Ingeneral,ifkhascomponentskx,kyandkzalongx,yandz,thenfromthedefinitionofthedotproduct,kr=kxx+kyy+kzz.

WaveVectork

E(r,t)=Eocos(tkr+)

AtravelingplaneEMwavealongadirectionk

Maxwell’sWaveEquation

(a)Acutof

aplanewaveparalleltothez-axis,theparalleldashed

linesatrightrightanglestothez-directionarewavefronts.Itisanidealizationthatisusefulinanalyzingmanywavephenomena.

Inpractice,therearemanytypesofpossibleEMwaves.Emustobey

Maxwell’sEMwaveequation:

AplanewaveisasolutionofMaxwell’swaveequation

Ex=Eocos(tkz+)

SubstituteintoMaxwell’sEquationtoshowthatthisisasolution.

2E2E2E2E

x2 y2 z2 or ot2 0

AsphericalwaveisdescribedbyatravelingfieldthatemergesfromapointEMsourceandwhoseamplitudedecaywithdistancerfromthesource.Atanypointrfromthesource,thefieldisgivenby

E=Acos(tkr)

Figure1.4(b)illustratedacutofasphericalwavewhereitcanbeseenthatwavefrontsarespherescenteredatthepointsourceO.

Amorepracticalexampleinwhichlightbeamexhibitssome

inevitabledivergencewhilepropagating;thewavefrontsareslowlybentawaytherebyspreadingthewave.

SphericalWave

EAcos(tkr)r

ExamplesofpossibleEMwaves

Opticaldivergencereferstotheangularseparationofwavevectorsonagivenwavefront.

GaussianBeam

TheradiationemittedfromalasercanbeapproximatedbyaGaussianbeam.Gaussianbeamapproximationsarewidelyusedinphotonics.

WavefrontsofaGaussianlightbeam

TheintensityacrossthebeamfollowsaGaussiandistribution

Beamaxis

waistofthebeam

Intensity=I(r,z)=[2P/(w2)]exp(2r2/w2)

2=Farfielddivergence

=w/z=/(wo)

Gaussbeams:Guassianbeamstartsfromafinitewidth2wo(waist,waistradiuswo)wherethewavefrontsareparallelandthenslowlydivergesasthewavefrontscurveoutduringpropagation.Guassianbeamstillhasanexpj(ωt-kz)propagationcharacteristic,buttheamplitude(intensity)variesspatiallyawayfromthebeamaxis,thatisintensityhasaGaussiandistribution.Beamdiameter2w:85%ofthebeampower.

Theincreaseinbeamdiameter2wwithzmakesangle2θ

atO,

whichiscalledthebeamdivergence.

2 4

(20)

TheGaussianIntensityDistributionisNotUnusual

TheGaussianintensitydistributionisalsousedinfiberoptics

ThefundamentalmodeinsinglemodefiberscanbeapproximatedwithaGaussianintensitydistributionacrossthefibercore

I(r)=I(0)exp(2r2/w2)

GaussianBeam

2=Farfielddivergence

zo=wo2/

GaussianBeam

Rayleighrange

1/2

2

z

2w2wo1

2

w

o

1/2

z2

2w2wo1

zo

w2

zo o

RealandIdealGaussianBeams

DefinitionofM2

worr

worr

1/2

2

zM2

M2

2wr2wor1

(/)

wo

2

w

or

RealGaussianBeam

Realbeam

Correctionnote:Page10intextbook,Equation(1.11.1),wshouldbewrasaboveand

wor

shouldbesquaredintheparantheses.

zM221/2

2wr2wor1

w2

or

GaussianBeaminanOpticalCavity

Twosphericalmirrorsreflectwavestoandfromeachother.

Theopticalcavity

containsaGaussianbeam.Thisparticularopticalcavityissymmetricandconfocal;thetwofocalpointscoincideatF.

1/2

z2

z

25m

1

2w2w

2wo (1mm)1.24m20mm

o

z

z

o

o

2.RefractiveIndexandDispersion

WhenanEMwaveistravelinginadielectricmedium,theoscillatingelectricfieldpolarizesthemoleculesofthemediumatthefrequencyofthewave

Thestrongeristheinteractionbetweenthefieldandthedipoles,thesloweristhepropagationofthewave

ForanEMwavetravelinginanonmagneticdielectricmediumof

thephasevelocityvisgivenby

relativepermittivityεr

1

r00

v

εr=1

vvacuum=1/√[ε0μ0=c=3×108ms-1

Theratioofthespeedoflightinfreespacetoitsspeedinamedium

iscalledtherefractiveindexnofthemedium

nc

v

k

=nk

medium

r

λmedium=λ/n

Maxwell’sWaveEquationinanisotropicmedium

E

E

E

2E0

2

2

2

o r o

x2

y2

z2

t2

AplanewaveisasolutionofMaxwell’swaveequation

E=Ecos(tkz+)

x

o

Thephasevelocityofthisplanewaveinthemediumisgivenby

Thephasevelocityinvacuumis

c 1

ko oo

v 1

k oro

PhaseVelocityandr

permittivity r

The

relative

measures

the

ease

with

which

the

of

medium

becomes

polarized

and

hence

itindicates

theextent

interaction

between

the

field

and

the

induced

dipoles.

ForanEMwavetravelinginanonmagneticdielectricmediumof

v

relativepermittivityr,thephasevelocity

isgivenby

ν 1

roo

RefractiveIndexn

PhaseVelocityandr

Refractiveindexn

definition

nc

v r

ν 1

roo

Opticalfrequencies

Typicalfrequenciesthatareinvolvedinoptoelectronicdevicesareintheinfrared(includingfarinfrared),visible,andUV,andwegenericallyrefertothesefrequenciesasopticalfrequenciesSomewhatarbitraryrange:Roughly1012Hzto1016Hz

Lowfrequency(LF)relativepermittivityr(LF)andrefractiveindexn.

RefractiveIndexandPropagationConstant

koko

o

k

Free-spacepropagationconstant(wavevector)2π/

Free-spacewavelength

Propagationconstant(vavevector)inthemediumWavelengthinthemedium

Innoncrystallinematerialssuchasglassesandliquids,thematerialstructureisthesameinalldirectionsandndoesnotdependonthedirection.Therefractiveindexisthenisotropic

nk

ko

RefractiveIndexandWavelength

Itiscustomarytodropthesubscriptoonkand

Infreespace

medium=/n

kmedium=nk

RefractiveIndexandIsotropy

Crystals,ingeneral,havenonisotropic,oranisotropic,properties

Typicallynoncrystallinesolidssuchasglassesandliquids,andcubiccrystalsareopticallyisotropic;theypossessonlyonerefractiveindexforalldirections

ndependsonthewavelength

Dispersionrelation:n=n()

Thesimplestelectronicpolarizationgives

Nat

=Numberofatomsper

unitvolume

Z=Numberofelectronsintheatom(atomicnumber)

o=A“resonantfrequency”

SellmeierEquation

n 1 2 A2 A2

2 2A2 2 2 2 2

1 2 3

1 2 3

NZe2 2 2

n21 at o

m 2c22

o e o

Cauchydispersionrelation:n=n()

n=n-2(h)-2+n0+n2(h)2+n4(h)4

3.GroupVelocityandGroupIndex

Therearenoperfectmonochromaticwaves

Wehavetoconsiderthewayinwhichagroupofwavesdifferingslightlyinwavelengthtravelalongthez-direction

Whentwoperfectlyharmonicwavesoffrequenciesand+

andwavevectorskkandk+kinterfere,theygenerateawavepacketwhichcontainsanoscillatingfieldatthemeanfrequencythatisamplitudemodulatedbyaslowlyvaryingfieldoffrequency

.Themaximumamplitudemoveswithawavevectorkandthus

withagroupvelocitythatisgivenby

d

v

g

dk

GroupVelocity

Twoslightlydifferentwavelengthwavestravelinginthesamedirectionresultinawavepacketthathasanamplitudevariationthattravelsatthegroupvelocity.

d

dk

v

g

GroupVelocity

Considertwosinusoidalwavesthatarecloseinfrequency,thatis,theyhavefrequenciesand+.Theirwavevectorswillbekkandk+k.Theresultantwaveis

Ex(z,t)=Eocos[()t(kk)z]

+Eocos[(+)t(k+k)z]

Byusingthetrigonometricidentity

cosA+cosB=2cos[1/2(AB)]cos[1/2(A+B)]wearriveat

Ex(z,t)=2Eocos[()t(k)z][cos(tkz)]

Ex(z,t)=2Eocos[()t(k)z][cos(tkz)]

Thisrepresentsasinusoidalwaveoffrequency.Thisisamplitudemodulatedbyaveryslowlyvaryingsinusoidaloffrequency.

Thissystemofwaves,i.e.themodulation,travelsalongzataspeeddeterminedbythemodulatingterm,cos[()t(k)z].Themaximuminthefieldoccurswhen[()t(k)z]=2m=constant(misaninteger),whichtravelswithavelocity

dz

or

k

dt

Thisisthegroupvelocityofthewavesbecauseitdeterminesthespeedofpropagationofthemaximumelectricfieldalongz.

v d

g dk

Thegroupvelocitythereforedefinesthespeedwithwhich

energyorinformationispropagated.

=2c/oandk=2n/o,oisthefreespacewavelength.

d=2c/o)do

2

Differentiatetheabove

dn

n(1/)d

/)

do

dn

dk2

(2

2

(2

/)

n

d

2

dk

o

o

o

d

d

o

o

o

o

o

d

(2

c/)d

2

vg

o o

dk

dn

d

(2/)n

2

d

o

o

o

wheren=n()isafunctionofthewavelength.

cdnn

o od

o

v d

g dk

inamediumisgivenby,

Thegroupvelocityvg

Thiscanbewrittenas

v(medium)c

g N

g

v(medium)d c

g dk ndnd

GroupIndex

isdefinedasthegroupindexofthemedium

Ingeneral,formanymaterialstherefractiveindexnandhencethegroupindexNgdependonthewavelengthoflight.Suchmaterialsarecalleddispersive

N ndn

g d

RefractiveindexnandthegroupindexNgofpureSiO2(silica)glassasafunctionofwavelength.

4.MagneticField,Irradianceand

PoyntingVector

Themagneticfield(magneticinduction)componentByalwaysaccompaniesExinanEMwavepropagation.

IfvisthephasevelocityofanEMwaveinanisotropicdielectric

mediumandnistherefractiveindex,then

wherev=(oro)1/2andn=1/2

E vB cB

x y n y

EMwavecarriesenergyalongthedirectionofpropagationk.Whatistheradiationpowerflowperunitarea?

AplaneEMwavetravelingalongkcrossesanareaAatrightanglestothedirectionofpropagation.Intimet,theenergyinthecylindricalvolumeAt(showndashed)flowsthroughA.

EnergyDensityinanEMWave

AstheEMwavepropagatesinthedirectionofthewavevectork,thereisanenergyflowinthisdirection.Thewavebringswithitelectromagneticenergy.

TheenergydensitiesintheExandByfieldsarethesame,

ThetotalenergydensityinthewaveisthereforeorEx2.

1E2 1 B2

2 o r x 2 y

o

PoyntingVectorandEMPowerFlow

IfSistheEMpowerflowperunitarea,

S=Energyflowperunittimeperunitarea

(Avt)(E2)

2

2

vorExvorExBy

o r x

S

At

Inanisotropicmedium,theenergyflowisinthedirectionofwavepropagation.IfweusethevectorsEandBtorepresenttheelectricandmagneticfieldsintheEMwave,thentheEMpowerflowperunitareacanbewrittenas

whereS,calledthePoyntingvector

S=v2orEB

PoyntingVectorandIntensity

SrepresentstheenergyflowperunittimeperunitareainadirectiondeterminedbyEB(directionofpropagation).Itsmagnitude,powerflowperunitarea,iscalledtheirradiance(instantaneousirradiance,orintensity).

Theaverageirradianceis

ISaverage1vorE2

2 o

AverageIrradianceorIntensity

Sincev=c/nandr=n2wecanwrite

cnE

ISaverage

10 )nE

2

3

2

o

(1.33

1

2

o

o

Theinstantaneousirradiancecanonlybemeasuredifthepowermetercanrespondmorequicklythantheoscillationsoftheelectricfield.Sincethisisintheopticalfrequenciesrange,allpracticalmeasurementsyieldtheaverageirradiancebecausealldetectorshavearesponseratemuchslowerthanthefrequencyofthewave.

IrradianceofaSphericalWave

Perfectsphericalwave

I Po

4r2

Sphericalwavefront

Source

O

Po

A

4A

9A

r

2r

3r

I Po

4r2

AGaussianBeam

I(r,z)=[2P/(w2)]exp(2r2/w2)

o=w/z=/(wo)

2o=Farfielddivergence

PowerinaGaussianBeam

I(r)2I(0)2exp[2(r/w)2]

and

Areaofacircularthinstrip(annulus)withradiusris2rdr.PowerpassingthroughthisstripisproportionaltoI(r)(2r)dr

w

I(r)2rdr

0

Fractionofopticalpowerwithin2w

0.865

=

I(r)2rdr

0

5.Snell’sLaworDescartes’sLawand

TotalInternalReflection

Snell'sLaw

sinisint

n2

n1

DerivationofSnell’sLaw

Alightwavetravelinginamediumwithagreaterrefractiveindex(n1>n2)suffersreflectionandrefractionattheboundary.(Noticethattisslightlylongerthan)

Wecanuseconstructiveinterferencetoshowthattherecanonlybeonereflectedwavewhichoccursatanangleequaltotheincidenceangle.ThetwowavesalongAiandBiareinphase.

WhenthesewavesarereflectedtobecomewavesArandBrthentheymuststillbeinphase,otherwisetheywillinterferedestructivelyanddestroyeachother.Theonlywaythetwowavescanstayinphaseisifr=i.AllotheranglesleadtothewavesArandBrbeingoutofphaseandinterferingdestructively.

UnlessthetwowavesatAandBstillhavethesamephase,therewillbenotransmittedwave.AandBpointsonthefrontareonlyinphaseforoneparticulartransmittedangle,t.

IttakestimetforthephaseatBonwaveBitoreachBBB=v1t=ct/n1

Duringthistimet,thephaseAhasprogressedtoA

AA=v2t=ct/n2

AandBbelongtothesamefrontjustlikeAandBsothatABis

perpendiculartokiinmedium1andABisperpendiculartoktinmedium2.Fromgeometricalconsiderations,

AB=BB/siniandAB=AA/sintsothat

v1t

sini

v2t

sint

AB

or

sini v1 n2

sint v2 n1

ThisisSnell'slawwhichrelatestheanglesofincidenceandrefractiontotherefractiveindicesofthemedia.

Whenn1>n2thenobviouslythetransmittedangleisgreaterthantheincidenceangleasapparentinthefigure.Whentherefractionangletreaches90°,theincidenceangleiscalledthecriticalangle

cwhichisgivenby

n2

sin

c

n

1

n1 i n2 t

nsinconstant

sin

sin

Whentheincidenceangleiexceedscthenthereisnotransmittedwavebutonlyareflectedwave.Thelatterphenomenoniscalledtotalinternalreflection(TIR).TIRphenomenonthatleadstothepropagationofwavesinadielectricmediumsurroundedbyamediumofsmallerrefractiveindexasinopticalwaveguides,e.g.opticalfibers.

AlthoughSnell'slawfori>cshowsthatsint>1andhencetisan"imaginary"angleofrefraction,thereishoweveranattenuated

wavecalledtheevanescentwave.

TotalInternalReflection

Lightwavetravelinginamoredensemediumstrikesalessdensemedium.Dependingontheincidenceanglewithrespecttoc,whichisdeterminedbytheratiooftherefractiveindices,thewavemaybetransmitted(refracted)orreflected.

(a)i<c(b)i=c(c)i>candtotalinternalreflection(TIR).

Prisms

LateralDisplacement

dsin cosi

L i1 2 2

(n/n) sini

Lighttravelsbytotalinternalreflectioninopticalfibers

Anopticalfiberlinkfortransmittingdigitalinformationincommunications.Thefibercorehasahigherrefractiveindexsothatthelighttravelsalongthefiberinsidethefibercore

bytotalinternalreflectionatthecore-claddinginterface.

Asmallholeismadeinaplasticbottlefullofwatertogenerateawaterjet.Whentheholeisilluminatedwithalaserbeam(fromagreenlaserpointer),thelightisguidedbytotalinternalreflectionsalongthejettothetray.ThelightguidingbyawaterjetwasfirstdemonstratedbyJean-DanielColladan,aSwissscientist(Waterwithairbubbleswasusedtoincreasethevisibilityoflight.Airbubblesscatterlight.)[Left:Copyright:S.O.Kasap,2005][Right:ComptesRendes,15,800–802,October24,1842;Cnum,ConservatoireNumériquedesArtsetMétiers,France

6.Fresnel'sEquations

Lightwavetravelinginamoredensemediumstrikesalessdensemedium.Theplaneofincidenceistheplaneofthepaperandisperpendiculartotheflatinterfacebetweenthetwomedia.Theelectricfieldisnormaltothedirectionofpropagation.Itcanberesolvedintoperpendicularandparallelcomponents.

Describetheincident,reflectedandrefractedwavesbytheexponentialrepresentationofatravelingplanewave,i.e.

Ei=Eioexpj(tkir)Er=Eroexpj(tkrr)Et=Etoexpj(tktr)

IncidentwaveReflectedwave

Transmittedwave

whereristhepositionvector,thewavevectorski,krandktdescribethedirectionsoftheincident,reflectedandtransmittedwavesandEio,EroandEtoaretherespectiveamplitudes.

Thesearetravelingplanewaves

Anyphasechangessuchasrandtinthereflectedandtransmittedwaveswithrespecttothephaseoftheincidentwaveareincorporatedintothecomplexamplitudes,EroandEto.OurobjectiveistofindEroandEtowithrespecttoEio.

Theelectricandmagneticfieldsanywhereonthewavemustbeperpendiculartoeachotherasarequirementofelectromagneticwavetheory.ThismeansthatwithE//intheEMwavewehaveamagneticfieldBassociatedwithitsuchthat,B(n/c)E//.SimilarlyEwillhaveamagneticfieldB//associatedwithitsuchthatB//(n/c)E.

Weuseboundaryconditions

Etangential(1)=Etangential(2)

Non-magneticmedia(relativepermeability,r=1),

Btangential(1)=Btangential(2)

Usingtheaboveboundaryconditionsforthefieldsaty=0,andtherelationshipbetweentheelectricandmagneticfields,wecanfindthereflectedandtransmittedwavesintermsoftheincidentwave.

Theboundaryconditionscanonlybesatisfiedifthereflectionandincidenceanglesareequal,r=iandtheanglesforthetransmittedandincidentwaveobeySnell'slaw,n1sin1=n2sin2

Ei=Eioexpj(tkir)Er=Eroexpj(tkrr)Et=Etoexpj(tktr)

IncidentwaveReflectedwaveTransmittedwave

Applyingmedium

theboundary

conditionsto

the

EM

wavegoing

from

1

to

2,

the

amplitudes

of

the

reflected

and

transmitted

wavescanbereadilyobtainedintermsofn1,n2andtheincidenceangleialone.TheserelationshipsarecalledFresnel'sequations.Ifwedefinen=n2/n1,astherelativerefractiveindexofmedium2tothatof1,thenthereflectionandtransmissioncoefficientsforEare,

E cosn2sin21/2

r r0, i i1/2

E cos n2 sin2

i0, i i

TherearecorrespondingcoefficientsfortheE//fieldswithcorrespondingreflectionandtransmissioncoefficients,r//andt//,

t Et0,// 2ncosi 1/2

// E n2cos n2sin2

i0,// i i

E n2sin21/2n2cos

r//E n2sin21/2n2cos

r0,// i i

i0,// i i

t Et0, 2cosi 1/2

E cos n2sin2

i0, i i

Further,theabovecoefficientsarerelatedby

r//

=

1

and

+

1

+nt//

r

=t

ForconveniencewetakeEiotobearealnumbersothatphaseanglesofr andt correspondtothephasechangesmeasuredwithrespecttotheincidentwave.

Fornormalincidence(i=0)intoFresnel'sequationswefind,

rr n1n2

// nn

1 2

Internalreflection

Magnitudeofthereflectioncoefficientsr//andrvs.angleofincidenceiforn1

=1.44andn2=1.00.Thecriticalangleis44.

Thecorrespondingchanges//andvs.incidenceangle.

ReflectionandPolarizationAngle

asp,

We

find

a

special

incidence

angle,

labeled

by

solving

the

r//

Fresnelequationfor

=0.Thefieldinthereflectedwaveisthen

alwaysperpendicular

to

the

plane

of

incidence

and

hence

well-

defined.

This

special

angle

is

called

the

polarization

angle

or

Brewster'sangle,

n2

tan

Forbothn1>n2

orn1<n2.

p

n

1

E n2sin21/2n2cos

r//E n2sin21/2n2cos

r0,// i i 0

i0,// i i

PolarizedLight

y

Planeofpolarization

E

x

z

Alinearlypolarizedwavehasitselectricfieldoscillationsdefinedalongalineperpendiculartothedirectionofpropagation,z.ThefieldvectorEandzdefineaplaneofpolarization.

Brewster'sangle

E

Reflectedlightati=phasonlyE

forbothn1>n2

orn1<n2.

TotalInternalReflection

Inlinearlypolarizedlight,however,thefieldoscillationsarecontainedwithinawelldefinedplane.LightemittedfrommanylightsourcessuchasatungstenlightbulboranLEDdiodeisunpolarizedandthefieldisrandomlyorientedinadirectionthatisperpendiculartothedirectionofpropagation.

Atthecriticalangleandbeyond(past44°inthefigure),i.e.when

ic,themagnitudesofboth

r//

and

rgotounitysothatthe

reflectedwavehasthesameamplitudeastheincidentwave.Theincidentwavehassufferedtotalinternalreflection,TIR.

Phasechangeupontotalinternalreflection

Wheni>c,inthepresenceofTIR,thereflectioncoefficientsbecomecomplexquantitiesofthetype

r=1exp(j)andr//=1exp(j)

withthephaseanglesand//beingotherthanzeroor180°.Thereflectedwavethereforesuffersphasechanges,and//,inthe

componentsEandE//.Thesephasechangesdependonthe

incidenceangle,andonnandn.

1

2

Thephasechange

isgivenby

FortheE//component,thephasechange//isgivenby

1/2

sin

n

2

2

tan

1

2

1

2

i

n2cos

//

i

1 sin2n212

tan i

2 cosi

ExternalReflection

Thereflectioncoefficientsr//andrversusangleofincidenceiforn1=1.00andn2=1.44.

EvanescentWave

Ininternalreflection(n1>n2),theamplitudeofthereflectedwavefromTIRisequaltotheamplitudeoftheincidentwavebutitsphasehasshifted.

Whathappenstothetransmittedwavewheni>c?

Accordingtotheboundaryconditions,theremuststillbeanelectricfieldinmedium2,otherwise,theboundaryconditionscannotbesatisfied.Wheni>c,thefieldinmedium2isattenuated(decreaseswithy,andiscalledtheevanescentwave.

Wheni>c,foraplanewavethatisreflected,thereisanevanescentwaveattheboundarypropagatingalongz.

Evanescentwavewhenplanewavesareincidentandreflected

wherekiz=kisiniisthewavevectoroftheincidentwavealongthez-axis,and2isanattenuationcoefficientfortheelectricfieldpenetratingintomedium2

2nn2 1/2

2 2 1sini1

2

n2

y

Et,(y,z,t)e 2 expj(tkizz)

Penetrationdepthofevanescentwave

2=

Attenuationcoefficientfortheelectricfieldpenetratinginto

medium2

Thefieldoftheevanescentwaveise1inmedium2when

y=1/2==Penetrationdepth

2nn2

2 1sin21

2 n i

2

Goos-HänchenShift

Virtualreflectingplane

n2

y

z

A

n1>n2

i

r

z

Reflectedlight

Incidentlight

z=2tani

B d

OpticalTunneling

y

d

z

A

n

>n

i

r

1

2

Reflectedlight

issmall),thefieldpenetratesfromtheABinterface

Incidentlight

Bisthin(thicknessd

Whenmedium

intomediumBandreachesBCinterface,andgivesrisetoatransmittedwaveinmediumC.TheeffectisthetunnelingoftheincidentbeaminAthroughBtoC.Themaximumfield

Emax

oftheevanescentwaveinBdecaysinBalongyandbutisfiniteattheBCboundary

andexcitesthetransmittedwave.

C n1

B n2

BeamSplitters

FrustratedTotalInternalReflection(FTIR)

(b)Twoprismsseparatedbyathinlowrefractiveindexfilmformingabeam-splittercube.TheincidentbeamissplitintotwobeamsbyFTIR.

(a)AlightincidentatthelongfaceofaglassprismsuffersTIR;theprismdeflectsthelight.

Beamsplittercubes(CourtesyofCVIMellesGriot)

Twoprismsseparatedbyathinlowrefractiveindexfilmformingabeam-splittercube.TheincidentbeamissplitintotwobeamsbyFTIR.

OpticalTunneling

Lightpropagationalonganopticalguidebytotalinternalreflections

Couplingoflaserlightintoathinlayer

-opticalguide-usingaprism.Thelight