Optoelectronics Ch1 KASOP

8/13/2019 Optoelectronics Ch1 KASOP

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OptoelectronicsEE/OPE451,OPT444

Fall2009Section1:T/Th 9:3010:55PM

JohnD.Williams,Ph.D.

DepartmentofElectricalandComputerEngineering

406OpticsBuilding UAHuntsville,Huntsville,AL35899Ph.(256)8242898 email:[email protected]

OfficeHours:Tues/Thurs23PM

JDW,ECEFall2009

1
mailto:[email protected]:[email protected]


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CourseTextbookandTopicsCovered

Ch.1:WaveNatureofLight

Ch.2:DielectricWaveguides Sections2.1and2.2only

Ch.3:SemiconductorScienceandLight

EmittingDiodes

Ch.4:StimulatedEmissionDevices

Sections4.94.14only

Ch.5:Photodetectors

Ch.7:PhotovoltaicDevices

Ch.8:PolarizationandModulationofLight

PrenticeHallInc.

2001S.O.Kasap

ISBN:

0

201

61087

6http://photonics.usask.ca/

2

JDW,ECEFall2009
http://photonics.usask.ca/http://photonics.usask.ca/


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AdditionalSupportiveMaterial

Thebreadthofthiscourseislargerthanasingletextbook

Certainsectionswillhaveaddedmaterialpresentedinclassfromthefollowingtwo

textbooks

Photonicsreadsmorelikeanencyclopediathanatextbookbuthassomeniceapplicationsanddiagrams

Optoelectronicscoverscurrentdeviceconceptsbutlacksontheory

StudentswillbetestedonmaterialfromKasap

FundamentalsofPhotonics

ISBN13: 9780471358329

JohnWiley&Sons

Optoelectroncics:infraredVisibleUV

DevicesandApplications2nded.

ISBN: 9781420067804CRCPress

3

JDW ECEFall2009


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CourseProgrammatics

CoursewillbetaughtonslidespostedonAngelaftereachclass Studentsareexpectedtotaketheirownnotesbasedonclasspresentation

Figuresandkeypointswillbeprovidedafteronslides

AdditionalmaterialfromeachsubjecttopicwillbepostedonAngelasoptionalreadingforanyoneinterested

Coursewillfocusonkeyconceptsandequationsthatdescribethem FirstprinciplederivationswillNOTberequiredunlesstheyarecriticalforstudent

development

Mosthomeworkandtestassignmentswillcanbeansweredbyunderstandingthequestionandapplyingaformula

Classproject(25%ofthetotalgrade)

Studentswillbeaskedtoworkinteamstoresearchaparticulartopicinoptoelectronics

Teamswillturninatermpapernolessthan10pages(1space)

Teamswillpresentthetopicin8minpresentations

HomeworkwillbedueatthebeginningofclasseveryTuesday (15%)

Twoinclassexamsand1comprehensivefinal(60%)

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DefinitionofOptoelectronics

Subfieldofphotonicsinwhich

voltagedrivendevicesareused

tocreate,detect,ormodulate

opticalsignalsusingquantum

mechanicaleffectsoflighton

semiconductorsmaterials

Examplesofoptoelectronic

devices

Photodiode LED

DFBLASER

VSCEL

SemiconductorPhotomultipliers

IntegratedOpticalCircuit

IntroductiontoOptoelectronics

240Opticalcomponentsonachip

InfineraWavelengthMultiplexer2007

JDW,ECEFall2009

siliconphotmultiplier.com

Opticis.com

http://www.fi.isc.cnr.it/users/giovanni.giacomelli/Semic/Samples/samples.html

http://spie.org/x27589.xml?ArticleID=x27589

www.udt.com

http://www.led.scale

train.com/blue0603led.php

5
http://www.fi.isc.cnr.it/users/giovanni.giacomelli/Semic/Samples/samples.htmlhttp://spie.org/x27589.xml?ArticleID=x27589http://www.udt.com/http://www.led.scale-train.com/blue0603led.phphttp://www.led.scale-train.com/blue0603led.phphttp://www.led.scale-train.com/blue0603led.phphttp://www.led.scale-train.com/blue0603led.phphttp://www.led.scale-train.com/blue0603led.phphttp://www.udt.com/http://spie.org/x27589.xml?ArticleID=x27589http://www.fi.isc.cnr.it/users/giovanni.giacomelli/Semic/Samples/samples.html


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WhatisPhotonics? BroadertopicthanOptoelectronicsalone

Studyofwave/particledualitydevicesin

optics.

Studyofopticaldevicesthatutilizephotonsinstead

oftheclassicalelectromagneticwavesolution.

emision,detection,modulation,signalprocessing,

transmissionandamplificationoflightbasedon

QMandSolidStateprinciples

Stateoftheartisthedevelopmentoflight

modulationthroughperiodicstructure

JDW,ECEFall2009

L.H.Gabrielli,NaturePhotonics3,461463(2009).

A.D.Dinsmore,UmassAmherst2009

J.Obrien,USCPhotonicsGroup2009

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Ch.1:WaveNatureofLight

1.1LightWavesinaHomogeneousMedium A.PlaneElectromagneticWave

B.Maxwell'sWaveEquationandDivergingWaves

1.2RefractiveIndex

1.3GroupVelocityandGroupIndex

1.4MagneticField,IrradianceandPoynting Vector

1.5Snell'sLawandTotalInternalReflection(TIR)

1.6Fresnel'sEquations A.AmplitudeReflectionandTransmissionCoefficients

B.Intensity,ReflectanceandTransmittance

1.7MultipleInterferenceandOpticalResonators

1.8GoosHnchen ShiftandOpticalTunneling

1.9TemporalandSpatialCoherence

1.10DiffractionPrinciples A.Fraunhofer Diffraction

B.Diffractiongrating

Chapter1HomeworkProblems:1,2,417

PrenticeHallInc.

2001S.O.Kasap

ISBN:

0

201

61087

6http://photonics.usask.ca/

7

JDW,ECEFall2009
http://photonics.usask.ca/http://photonics.usask.ca/


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WaveNatureofLight PlaneElectromagneticWave

TreatedastimevaryingelectricExandmagnetic,By,fields

EandBarealwaysperpendiculartoeachother

Propagatethroughspaceinthezdirection

Simplestrepresentationisasinusoidalwave(oraMonochromaticplanewave)

WhereEx=electricfieldatpositionzattimet,

Eo=amplitudeoftheelectricfield

k=wavenumber(k=2/)

=wavelength

=angularfrequency

o=phaseconstant

= =phaseofthewave

Ex

z

Direction of Propagation

By

z

x

y

k

An electromagnetic wave is a travelling wave which has timevarying electric and magnetic fields which are perpendicular to eacother and the direction of propagation,z.

1999 S.O. Kasap, Optoelectronics (Prentice Hall)

)cos( oox kztEE +=

)( okzt +

Aplanersurfaceoverwhichthephase

ofthewaveisconstantiscalleda

wavefront

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WaveFronts

)cos( oox kztEE +=

Aplanersurfaceoverwhichthephaseofthewaveisconstantiscalledawavefront

z

Ex=E

osin(tkz)

Ex

z

Propagation

E

B

k

E andB have constant phasein thisxyplane; a wavefront

E

A plane EM wave travelling alongz, has the same Ex(orBy) at any point in agivenxyplane. All electric field vectors in a givenxyplane are therefore in phase.

Thexyplanes are of infinite extent in thexandydirections.


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OpticalField

UseofEfieldstodescribelight

WeknowfromElectrodynamicsthatatimevaryingBfield

resultsintimevaryingEfieldsandviseversa ThusalloscillatingEfieldshaveamutuallyoscillatingBfield

perpendiculartoboththeEfieldandthedirectionofpropagation

However,oneusestheEfieldratherthantheBfieldtodescribethesystem

ItistheEfieldthatdisplaceselectronsinmoleculesandionsinthecrystalsatopticalfrequenciesandtherebygivesrisetothepolarizationofmatter

Notethatthefieldsareindeedsymmetricallylinked,butitistheEfieldthatismostoftenusedtocharacterizethesystem

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OptionalPlaneWaveRepresentations

1Dsolution

]Re[),(

]Re[),(

)cos(),()cos(

)(

)(

kztj

c

kztjj

o

oox

oox

eEtzE

eeEtzE

kztEtzEE

kztEE

o

=

=

+==+=

]Re[)cos( ie=Since

Generalsolution

]Re[),(

]Re[),()cos(),(

)(

)(

rktj

c

rktjj

o

oo

eEtrE

eeEtrE

kztEtrEEo

r

r

r

r

r

r

=

=+==

zkykxkrk zyx ++=r

r

Where

y

z

k

Direction of propagation

r

O

E(r,t)r

A travelling plane EM wave along a direction 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

k=wavevectorwhosemagnitudeis2/

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PhaseVelocity

Foraplanewave,therelationshipbetweentimeandspaceforanygive

phase,,isconstant

= =constant)( okzt + Duringanytimeinterval,t,thisconstantphase(andhencemaximum

valueofE)movesadistancer.

Thephasevelocityofthewaveis

Thephase

difference,

at

any

given

time

between

two

points

on

awave

thatareseparatedbyadistancezis

Thefieldsaresaidtobeinphaseifhephasedifferenceiszeroif=0or2multiplesofkzwithregardstotheinitialvalue.

v

v

kdt

dr

t

r=====

/2

2v

zzk

==

2Sincetisthesameforeachpoint

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MaxwellsWaveEqn.andDivergingWaves

Planewavesemanatefromsurfaceofrelativelyinfinitesize

Wavefrontsareplanes AsphericalwaveemanatesfromaEMpointsourcewhoseamplitude

decayswithdistance

Wavefrontsarespherescenteredatthepointsource

Adivergentbeamemanatesfromadefinedsurface Theopticaldivergencereferstotheangularseparationofthewavevectorsonagiven

wavefront

k

Wave fronts

r

E

k

Wave fronts(constant phase surfaces)

z

Wave fronts

PO

P

A perfect spherical waveA perfect plane wave A divergent beam

(a) (b) (c)

Examples of possible EM waves


2

2

2

22

t

E

t

EE oor

=

= )cos( krt

r

AE =

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BeamDivergence ConsideraGaussianlaseremittingfromaslaboffiniteradius(orwaistradius)2wo Wedefinetheinitialwaistofthebeamaswo Asthebeammovesfarenoughfromthesurfacesuchthatsourcenolongerlookslike

aninfiniteplane,thenthewavefrontsbegintodivergeataconstantangle Thehalfangleofthedivergenceis

Thebeamdiameter,2w,atanydistancezfromtheoriginisdefinedsuchthatthecrosssectionalareaofthebeam(w2)contains85%ofthetotalbeampower.

Thebeam

divergence

isthe

angle

2which

iscalculated

from

the

waist

radiusy

x

Wave fronts

z Beam axis

r

Intensity

(a)

(b)

(c)

2wo

O

Gaussian

2w

(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam crosssection. (c) Light irradiance (intensity) vs. radial distance rfrom beam axis (z).


Beamdivergence

)2(

42

ow

=

Inradians!!!!!!

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Example1

ConsideraHeNelaserbeamat633nmwithaspotsizeof10

mm. AssumingaGaussianbeam,whatisthedivergenceof

thebeam?

Atwhatdistanceisthespotsizeofthedivergedbeamequal

to

1

m?

Beamdivergence

)2(

42

ow

= orad

m

m

0046.01006.8)1010(

)10633(4 53

9

==

=

0.5m0.0046o

z

mmz o 62272

1)0046.0(tan 1 =

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RefractiveIndex

AssumeanEMwavetravelinginadielectricmediumwithpermittivity=ro

EMpropagationisequaltothepropagationofthepolarizationinthemedium

Duringpropagation,theinducedmoleculardipolesbecomecoupledandthepolarizationdecaysthepropagationoftheEMwave

ie THEWAVESLOWSDOWNandthevelocityofthewavedependsdirectlyonthe

permittivityand permeabilityofthematerialitistravelingthrough

ForanEMwavetravelingthroughanonmagneticdielectric,thephasevelocityofthewaveis:

Inavacuum,v=c=speedoflight=3x108 m/s wherec

Therefractiveindex,n, istherelativeratioofc/v

2

2

2

2

2

2

2

2 1

1

t

E

t

E

t

E

vE oor

oor

=

=

=

=

Fromthewaveeqn.v

oo

1=

r=

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OpticalConstantsinamedium

Indexofrefraction,n=c/v

Wavevector,kmedium=nk

Wavelength,medium=n Innoncrystallinematerialssuchasglassesandliquids,thematerialstructureis

statisticallythesameinaldirections,andthusndoesnotdependondirection.

Therefractiveindexisthensaidtobeisotropic

Incrystalstheatomicarrangementsbetweenatomsoftendemonstratedifferentpermittivitiesindifferentdirections. Suchmaterialsaresaidtobeanisotropic

IngeneralthepropagationofanEMfieldinasolidwilldependonthepermittivity

ofthesolidalongthekdirection.

Anisotropicpermittivitiesthatintroducearelativephaseshiftalongthedirectionofpropagationhavecomplextermsintheoffdiagonalstermsofthepermittivity

matrixandwillbethediscussionofvariousdeviceconceptsdescribedinCh.7

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FrequencyDependentPermittivity Materialsdonotoftendemonstrateasingledegreeofpolarizationalonganyonedirection

acrosstheentirefrequencyrange.

Infactthefrequencydependenceofpermittivityiswhatgivesrisetopropertiessuchas

absorptionwithinasolidandallowsonetoseeobjectsincolor.

Mostmaterialsofopticalinteresthaveabsorptionbandsinwhichthepermittivity,andthus

therefractiveindex,changesdrastically. Shiftingoftheseconstantsbydopingthematerial,

(oraddinglargemagneticfields)hasallowedforthedevelopmentofbandgap

semiconductorswithspecificopticalpropertiesforopticalgenerationanddetection.

Considerthe

simplest

expression

used

tocalculate

the

permittivity

WhereNisthenumberofpolarizablemoleculesperunitvolume, and isthe

polarizabilitypermolecule.

IfIcaninjectorremovetherelativeNvalueinasolid,thenonecanchangethepermittivity

ofthatsolidandthereforeitselectronicandopticalproperties.

IfthesolidisastackofsemiconductormaterialswithdifferentNvaluesthatrespond

opticallywhenbiased,thenonecancreateanoptoelectronicdevice!!!

Ifthepolarizability,,isfrequencydependent(anditis),thenouroptoelectronicdevicewillworkoveraparticularfrequencyrangewhichcanbeengineeredforthespectralbandof

interest!!!!

or N /1 +=

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GroupVelocity Firstandforemost: THEREARENOPERFECT

MONOCHROMATICWAVESinpractice

Therearealwaysbundlesofwaveswithslightlydifferentfrequenciesandwavevectors

Assumethewavestravelwithslightlydifferentfrequencies, +and

ThewavevectorsarethereforerepresentedbyK+ kandk k

Thecombinedtransformgeneratesawave

packet

oscillating

at

a

mean

beat

frequency

thatisamplitudemodulatedbyaslowlytimevaryingfieldat

Themaximumamplitudemoveswithawavevectork

Thevelocityofthepacketiscalledthegroup

velocityandisdefinedas

Thegroupvelocitydefinesthespeedatwhich

theenergyispropagatedsinceitdefinesthespeedoftheenvelopeoftheamplitudevariation

d

dvg =

+

kEmax Emax

Wave packet

Two slightly different wavelength waves travelling in the samdirection result in a wave packet that has an amplitude variatiowhich travels at the group velocity.

1999 S.O. Kasap,Optoelectronics (Prentice Hall)

http://newton.ex.ac.uk/tea

ching/resources/au/phy11

06/animationpages/

For

video

of

wave

packetswithandwithoutv=vg:

19

JDW,ECEFall2009
http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/http://newton.ex.ac.uk/teaching/resources/au/phy1106/animationpages/


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Example:GroupVelocity

Resultingwaveis:

UsingtheTrigidentity:

Weget:

Maximumfieldoccurswhen:

Yieldsvelocity:

+

kEmax Emax

Wave packet

Two slightly different wavelength waves travelling in the samdirection result in a wave packet that has an amplitude variatiowhich travels at the group velocity.

1999 S.O. Kasap,Optoelectronics (Prentice Hall)

( ) ( )[ ] ( ) ( )[ ]zkktEzkktEtzE oox +++= coscos),(

( )[ ] ( )[ ]BABABA +=+ 2/1cos2/1cos2coscos

( ) ( )[ ] [ ]kztzktEtzE ox = coscos2),(

( ) ( )[ ] mzkt 2=

gvd

d

dt

dz==

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GroupIndex

SupposevdependsontheorK

Bydefinition,thegroupvelocityisthen

WedefineNgasthegroupindexofthemedium.

Wenowhaveawaytodeterminetheeffectofthemediumonthegroupvelocityatdifferentwavelengths(frequency

dependence!!!) Therefractiveindex,n,andgroupindex,Ng,

dependonthepermittivityofthematerial,r Wedefineadispersivemediumisamedium

inwhichboththegroupandphasevelocities

dependon

the

wavelength.

Allmaterialsaresaidtobedispersiveoverparticularfrequencyranges

==

2

n

cvk wheren=n()

g

gN

c

d

dnn

c

dk

dv =

==

Refractive index nand the group indexNgof pureSiO2(silica) glass as a function of wavelength.

Ng

n

500 700 900 1100 1300 1500 1700 1900

1.44

1.45

1.46

1.47

1.48

1.49

Wavelength (nm)


Ngandnarefrequency(wavelength)dependent

NoticetheminimaforNgat1300nm.

Ngiswavelengthindependentnear1300

nm Lightat1300nmtravelsthroughSiO2atthesamegroupvelocitywithoutdispersion

Dispersivemediumexample:SiO2

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Example:EffectsofaDispersiveMedium

Consider1umwavelengthlightpropagatingthroughSiO2 Atthiswavelength,Ngandnarebothfrequencydependentwithno

localminima

Thusthemediumisdispersive Nowwemustaskthequestion,arethegroupandphasevelocitiesof

thepropagatingwavepacketthesame?

PhaseVelocity

GroupVelocity

Answer:NO!!!!

Thegroupvelocityis0.9%slowerthanthephasevelocity Refractive index nand the group indexNgof pure

SiO2(silica) glass as a function of wavelength.

Ng

n

500 700 900 1100 1300 1500 1700 1900

1.44

1.45

1.46

1.47

1.48

1.49

Wavelength (nm)


s

mx

s

mx

n

c

dt

dzv

88 10069.2450.1/103 =

===

s

mx

s

mx

N

c

dk

dv

g

g

88 10051.2463.1/103 =

===

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EnergyFlowinEMWaves

LetusrecallthatthereisindeedaBfieldintheEMwave.

Recallfromelectrostaticsthat

where

AstheEMwavepropagatesalongthedirectionk,thereisanenergyflowinthatdirection

Electrostaticenergydensity

Magnetostatic energydensity

TheEnergyflowperunittimeperunitarea,S,isdefinedasthePoynting Vector

( )

r

oor

yyx

n

v

BncvBE

=

=

==1

2

2

2

1

2

1

yo

xor

B

E

Whereboththesevaluesareequal

( )( ) ( )

BEvS

BEvEv

tA

EtAvS

or

yxorxorxor

vrv

=

==

=

2

222

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Irradiance

MagnitudeofthePointingVectoriscalledtheirradiance

Notethatbecausewearediscussingsinusoidalwaveforms,thattheinstantaneousirradiance

oflightpropagatinginphaseistakenfromtheinstantaneousamplitudeofEandB

respectively

Instantaneousirradiancecanonlybemeasuredifthepowermeterrespondsmorequickly

thantheelectricfieldoscillations.

Asonemightimagine,atopticalfrequencies,allpracticemeasurementsaremadeusingthe

averageirradiance.

Theaverageirradianceis

yxor BEvS 2=

2822

2

1033.12

1

2

1

21

ooooor

ooravg

nEs

mEcnE

n

c

EvSI

===

==

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Example: ElectricandMagneticFieldsinLight

Theintensity(irradiance)oftheredlaserfromaHeNelaseratacertainlocationwas

measuredtoabout1mW/cm2.

Whatarethemagnitudesoftheelectricandmagneticfields?

Whatarethemagnitudesifthisbeamisinaglassmediumwithrefractiveindex1.45?

( )

TcEB

m

V

n

s

m

Wm

cn

IE

oo

o

o

29.0/

87

11033.1

)10101(22

8

243

==

=

=

==

( )

TcnEB

m

V

s

m

Wm

cn

IE

oo

o

o

35.0/

72

45.11033.1

)10101(22

8

243

==

=

==

Note: therelativeamplitudeof

Edecreased,

and

Bincreasedandthusthepolarizedwave

becamemoreellipsoidal 25JDW,ECEFall2009


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SnellsLaw Neglectabsorptionandemission

Lightinterfacingwithasurfaceboundarywill

reflectbackintothemediumandtransmit

throughthesecondmedium

Transmittedwaveiscalledrefractedlight

Theanglesi,r,andtdefinethedirectionof

thewavenormaltotheinterface.

Thewavevectorsaredefinedaski,kr,andkt

Atanyinterface,I=r

SnellsLawStates

n2

z

y

O

i

n1

Ai

ri

Incident Light BiA

r

Br

t t

t

Refracted Light

Reflected Light

kt

At

Bt

BA

B

A

A

r

ki

kr

A light wave travelling in a medium with a greater refractive index (n1> n2) suffersreflection and refraction at the boundary.

1999 S.O. Kasap,Optoelectronics(Prentice Hall)

( )

( ) 12

2

1

sin

sin

n

n

v

v

t

i ==

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TotalInternalReflection

Ifn1>n2thentransmittedangle>incidenceangle.

Whent=90o,thentheincidenceangleiscalledthecriticalangle

Wheni>cthereis

notransmittedwaveinmedium

Totalinternalreflectionoccurs

anevanescentwavepropagatesalongtheboundary(i.e.highlosselectricfield

propagatingalongthesurface)

n2

i

n1 > n2

i

Incidentlight

t

Transmitted(refracted) light

Reflectedlight

kt

i>cc

TIRc

Evanescent wave

k i k r

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending othe incidence angle with respect to c, which is determined by the ratio of the refractive

indices, the wave may be transmitted (refracted) or reflected. (a) i< c (b) i= c (c) i> cand total internal reflection (TIR).


( )1

2sin n

nc =

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FresnelsEquations(1)

AmplitudeReflectionandTransmissionCoefficients

TransverseElectricField(TE)wavesifEi,Er,andEt

TransverseMagnetic

Field

(TM)

waves

ifEi//,

Er//,

and

Et//

Incidentwave

ReflectedWave

TransmittedWave

BoundaryConditions

Etangential(1)=Etangential(2)

Btangential(1)=Btangential(2)

Applyingtheboundaryconditions tothe

equationsaboveyieldsamplitudesof

reflectedandtransmittedwaves. TheseequationswerefirstderivedbyFresnel

)(

)(

)(

rktj

tot

rktj

ror

rktj

ioi

t

r

i

eEE

eEE

eEE

r

r

r

r

r

r

=

=

=

k i

n2

n1 > n

t=90 Evanescent wave

Reflected

waveIncident

wave

i r

Er,//

Er,Ei,

Ei,//

Et,

(b) i> cthen the incident wave

suffers total internal reflection.However, there is an evanescentwave at the surface of the medium.

z

y

xinto paperi r

Incidentwave

t

Transmitted wave

Ei,//

Ei,Er,//

Et,//

Et,

Er,

Reflected

wave

k t

k r

Light wave travelling in a more dense medium strikes a less dense medium. The plane ofincidence is the plane of the paper and is perpendicular to the flat interface between thetwo media. The electric field is normal to the direction of propagation . It can be resolvedinto perpendicular () and parallel (//) components

(a) i< cthen some of the waveis transmitted into the less densemedium. Some of the wave isreflected.

Ei,


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Definen=n2/n1astherelativerefractiveindexofthesystem

RefectionandtransmissioncoefficientsforE are

RefectionandtransmissioncoefficientsforEIIare

Thesecoefficientsarerelatedbythefollowingtwoequations

FresnelsEquations(2)

ii

i

io

to

ii

ii

io

ro

nE

Et

n

n

E

Er

22

22

22

sincos

cos2

sincos

sincos

+==

+

==

ii

i

io

to

ii

ii

io

ro

nn

n

E

Et

nn

nn

E

Er

cossin

cos2

cossin

cossin

222

222

222

+==

+

==

1=+ ntr =+ tr 1

Theseequationsallowonetocalculate

theamplitudeandphasesoflight

propagatingthroughdifferentmedia

IfweletEiobereal,thenthephaseangles

ofrandt correspondtothephase

changesmeasuredwithrespecttothe

incidentwave

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InternalReflection

Lighttravelingfromamoredensemedium

intoalessdenseone (n2


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PhaseChangeinTIR

Forp


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EvanescentWaves

Whenictheremuststillbeanelectricfield inmedium2ortheboundaryconditionswill

notbesatisfied

Thefieldinmedium2isanevanescentwavethattravelsalongtheboundaryedgeatthe

samespeedastheincidentwaveanddissipatesintothe2nd medium

Thepenetrationdepthoftheelectricfieldintomedium2is

2

2

2

2

122

)(

1

1sin2

sin

),,( 2

=

=

=

=

i

iiiz

zktjy

t

n

nn

kk

eetzyE iz

attenuationcoefficient

evanescentwavevector

Et =e1

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ExternalReflection

Lighttravelingfromalessdensemedium

intoamoredenseone (n2>n1)

Atnormalincidence,bothFresnel

coefficientsforr//andr arenegative

ExternalreflectionforTMandTEat

normalincidencegeneratesa180degree

phaseshift. Thisphaseshiftisobserved

atall

angles

for

TE

waves

and

up

top for

TMwaves

Also,r// goesthroughzeroatthe

Brewsterangle,p

Atpthereflectedwaveispolarizedin

theEcomponentonly. ThusLightincidentatporhigherinangledoesnot

generateaphaseshiftinreflectionfor

TMwaves.

Transmittedlightinbothinternaland

externalreflectiondoesNOTexperience

aphaseshift

The reflection coefficients r//and rvs. angle

of incidence i for n1= 1.00 and n2= 1.44.

-1

-0.8

-0.6-0.4

-0.2

0

0.2

0.40.6

0.8

1

0 10 20 30 40 50 60 70 80 90

p r//

r

Incidence angle, i

External reflection


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Example: EvanescentWave

TIRfromaboundaryn1>n2generatesanevanescentwaveinmedium2neartheboundary

Describetheevanescentwavecharacteristicsanditspenetrationintomedium2

=

+

==

j

o

ii

i

io

to

ett

nnE

Et

2

2

1

2 sincos

cos2

)(2

sinsinzktjAyk

iot

ziitt

zt eeEtE

kkk

===

)sin(

)sin(

2

2

ttt

ttt

zktjAyk

iot

Ajykzktj

iot

eeEtE

eEtE

+

=

=

ttttt

rktj

iot

zkykrk

eEtE t

sincos

)(

+=

=

22

2

1

sin1cos

1sinsin

jA

nn

tt

it

==

>

=

ApplySnellsLawat c>i

ForTIR

Additionally,TIRallowsustocalculatet

2

2

2

2

1222

1

1sin2

=

== it

n

nnAk

Penetrationdepth

Complextransmissionvaluewith

imaginaryphaseconstant

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Example: InternalReflection

Reflectionoflightfromalessdensemedium

Waveistravelinginaglassofindexn1=1.450

Wavebecomesincidentonalessdensemediumofindexn2=1.430

WhatistheminimumincidenceangleforTIR?

Whatisthephasechangeinthereflectedwaveatanincidenceangleof85degrees?

Whatisthepenetrationdepthoftheevanescentwaveintomedium2whentheincidence

angleis85o?

( )450.1

430.1sin

1

2 ==n

nc

o

c 47.80=

( )

( )61447.1

85cos

45.1

43.185sin

cos

sin

2tan

22

22

=

=

=

o

o

i

i n

o

1.62||=

=

=

+ 2tan

1

cos

sin

2tan 22

22||

nn

n

i

i

o45.116=

m

mn

nni

7

2

62

2

2

122

108.71

/1028.11sin2

==

=

=

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Intensity,Reflectance,andTransmittance

Relative(%)intensityofthereflectedlighttravelingthroughthemediaReflectance

Relative(%)intensityofthetransmittedlighttravelingthroughthemediaTransmittance

Sumofthetransmittanceandreflectanceinanyconservedsystemmustequal1

2

2

2

== rE

E

Rio

ro

2

||

1

22

||1

2

||2

|| tn

n

En

EnT

io

to

==

2

||2||

2

||

|| rE

E

Rio

ro==

2

1

22

1

2

2

== t

n

n

En

EnT

io

to

2

21

21||

+

===

nn

nnRRR

( )22121

||

4

nn

nnTTT

+===

1=+TR36

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Example:InternalandExternalReflection

Lightpropagatesatnormalincidencefromair,n=1,toglasswitharefractiveindexof1.5.

Whatisthereflectioncoefficientandthereflectancew.r.t totheincidentbeam?

Lightpropagatesatnormalincidencefromglass,n=1.5,toairwitharefractiveindexof1.0.

Whatisthe

reflection

coefficient

and

the

reflectance

w.r.t to

the

incident

beam?

Whatisthepolarizationangleoftheintheexternalreflectionfortheairtoglassinterface

describedbythefirstquestionabove? Howwouldonemakeapolaroid device(devicethat

polarizeslightbasedonthepolarizationangle)?

2.05.11

5.11

21

21

=+

=+

== nn

nn

rr 04.02

|||| ==rR or4%

2.015.1

15.1

21

21 =+

=

+

==

nn

nnrr 04.0

2

|||| ==rR or4%

( )

o

p

pn

n

3.56

5.1tan1

2

=

==

Atanincidenceangleof56.3o thereflectedlightwillbepolarizedwith

anEtotheplaneofincidence. Transmittedlightwillbepartially

polarized. Byusingastackof Nglassplates,onecanincreasethepolarizationofthetransmittedlightbyafactorofN

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ReflectanceatDifferentAnglesofIncidence

Lightpropagatesat30o incidencefromair,n=1,toglasswitharefractiveindexof1.5. What

isthereflectioncoefficientandthereflectancew.r.ttotheincidentbeam?

24.0414.1866.0

414.1866.0

21

21 =+

=

+

==

nn

nnrr 058.0

2

|||| ==rR or5.8%

943.0cos

866.0cos

cos

cos

sinsin

22

11

2

1

=

=

=

=

=

t

i

t

i

ti

nn

nn

n

n

Snells Law

replace

t=19.5o

ii

ii

io

ro

n

n

E

Er

22

22

sincos

sincos

+

==

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Example:AntireflectionCoatingsonSolarCells

Whenlightisincidentonasemiconductoritbecomespartiallyreflected

Thisisimportantbecauseitthetransmittedlighttravelingintothesolarcellisabsorbedand

convertedtoelectricalenergy

AssumetherefractiveindexofSiliconis3.5between700 800nm

Calculatethereflectanceofthesiliconsurfaceinair

Thismeansthereisa30.9%lossinefficiencyevenbeforethelightentersthesiliconsolarcell

IfonecoatsthesolarcellwithathinlayerofelectricmaterialsuchasSi3N4(siliconnitride)

thathasanintermediaterefractiveindexof1.9,thenwecanreducetheloss

or30.9%309.05.31

5.312

=

+

=R

d

Semiconductor of

photovoltaic device

Antireflection

coating

Surface

Illustration of how an antireflection coating reduces thereflected light intensity

n1 n2 n3

AB


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Notethatinorderforthisconcepttoworkthethicknessoftheantireflectivelayermustbe

matchedtothewavelengthofthelighttransmitted

Wearedealingwithexternalreflection,thusreflectedlightoffofallnormalinterfacesis

180o outofphasewithincidentlight

Phasematchingmustbeaccomplishedbetweenthelightreflectingfromtheair/coating

interfaceandthelightreflectedfromcoating/siliconinterface

Thephasedifferenceinthesystemisequivalenttok2(2d) wherek2=n2k=2n2/

Phasematchingoccurswhenk2(2d)=m

Thusthethicknessofthecoatingmustbemultiplesofthequarterwavelengthoflight

propagatingthroughit.

Also,toobtainagooddegreeofdestructiveinterference,theamplitudesoftheAandB

wavesmustbecomparable. Thusweneed

This

yields

a

reflection

coefficient

between

the

air

and

coating

that

is

equal

to

that

betweenthecoatingandthesemiconductor. Inourcasen2shouldequal1.87whichis

closetothatofSi3N4at1.9.

md

n=

2

2 2

=

24nmd

312 nnn =

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Also,toobtainagooddegreeofdestructive

interference,theamplitudesoftheAandBwaves

mustbecomparable. Thusweneed

Thisyieldsareflectioncoefficientbetweentheairand

coatingthatisequaltothatbetweenthecoatingand

thesemiconductor. Inourcasen2shouldequal1.87whichisclosetothatofSi3N4at1.9.

Thusabout10%oflightisnowreflectedoffofthe

coatingsurfaceandanother10%isreflectedfromthe

silicon. Ofthesecond10%,about10%ofthatis

reflectedbackfromtheair/nitrideinterfaceontothe

siliconagain. Sothetotalgainopticalgainacquiredthroughuseoftheantireflectivecoatingisabout10%.

312 nnn =

096.09.11

9.112

=

+

=AR 0877.0

5.39.1

5.39.12

=

+

=BR

d

Semiconductor ofphotovoltaic device

Antireflectioncoating

Surface

Illustration of how an antireflection coating reduces thereflected light intensity

n1 n2 n3

AB


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Example: DielectricMirrors

Dielectricmirror stackofdielectricswithalternatingrefractiveindices

Thethicknessofeachlayeris aquarterwavelength: /(4ni)=i /4

Reflectedwavesfromtheinterfacesinterfereconstructivelytogenerateahighlyreflective

coatingoveraopticalwavelengthrangecenteredato

Thereflectioncoefficientforaparticularboundaryissimilartothatcalculatedpreviously

Thusthereflectioncoefficientsusingvaluesofi andjalternatethroughoutthemirror Afterseveralalternatingreflectances,thetransmissionbecomesexceedinglysmallandlight

isreflectedbackfromthesurfaceatvaluesnearunity(1).

n1 n2

AB

n1 n2

C

Schematic illustration of the principle of the dielectric mirror with many low and highrefractive index layers and its reflectance.

Reflectance

(nm)0

1

330 550 770

1 2 21

o

1/4 2/4


1,

2

=

+

= ij

nn

nnR

ji

ji

ij

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MultipleInterferenceandOpticalResonators

Opticalresonatorstoresenergyorfilterslightonlyatcertainfrequencies

Builtbyaligningtwoflatmirrorsparalleltooneanotherwithfreespaceinbetweenthem

ReflectionsbetweenmirrorsurfacesM1andM2leadtoconstructiveanddestructive

interferencethecavity

Thisleadstostationary(orstanding)EMwavesinthecavity

A

B

L

M1 M2 m= 1

m= 2

m= 8

Relative intensity

m

m m + 1m - 1

(a) (b) (c)

R~ 0.4R~ 0.81 f

Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflectedwaves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowedin the cavity. (c) Intensity vs. frequency for various modes. Ris mirror reflectance and

lower Rmeans higher loss from the cavity.

1999 S.O. Kasap, Optoelectronics (Prentice Hall)43

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MultipleInterferenceandOpticalResonators

Sincetheelectricfieldatthemirrorsmustbezero,wecanonlyfitintegermultiplesof

halfwavelengthsintothecavityoflengthL

Eachcavitymodeisdefinedbythemvalue,orthenumberofthehalfwavelengthsthat

constructivelyinterferingwithinthecavity.

Resonantfrequenciesarethebeatoscillationfrequenciesresonantinthecavity

Freespectralrange

Lm =2

...3,2,1=m

fm mvL

cmv ==

2

L

cvf

2

=

fmmm

vvvv == +1

A

B

L

M1 M2 m= 1

m= 2

m= 8

Relative intensity

m

m m + 1m - 1

(a) (b) (c)

R~ 0.4

R~ 0.81 f

Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflectedwaves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed

in the cavity. (c) Intensity vs. frequency for various modes. Ris mirror reflectance andlower Rmeans higher loss from the cavity.

1999 S.O. Kasap, Optoelectronics (Prentice Hall) 44

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FabryPerotOpticalResonator

Simpleopticalcavitythatstoresradiationenergyonlyatcertainfrequencies

AssumeawaveAtravelswithinthecavityandisreflectedbackandforthaswaveB.

Thefieldandintensityofthecavityare:

Spectralwidthofthecavity:

WhereFiscalledtheFinesseofthecavitywhichistheratioofmodeseparationtospectralwidth. Thusaslossesdecrease,finesseincreases. Alsolargerfinessesleadtosharper

modepeaks

L

m

m - 1

Fabry-Perot etalon

Partially reflecting plates

Output lightInput light

Transmitted light

Transmitted light through a Fabry-Perot optical cavity.


R

RF

=

1

)(sin4)( 222

kLRRI

IEI ocavity

+==

2max )( RI

II o

=

jkLcavity

er

AE

=

2

1

...664422 ++++= kLkLjkLcavity eArjAreArAE

F

vv

f

m=

)(sin4)(

)1(22

2

kLRRI

RII incidenttrans

+

=

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ResonatorModeswithSpectralWidth

ConsideraFabryPerotopticalcavityofairlength=100micronswithmirrorsthathavea0.9

reflectance.

Calculatethecavitymodenearestto900nm.

Calculatetheseparationofthemodesandthespectralwidthofeachmode.

( )( )

( )nm

m

m

L

m

mLm

m 90.900222

1010022

22.22210900

1010022

6

9

6

=

==

=

==

( )

nmvc

vv

c

HzF

vv

R

RF

Hzm

sm

L

cvv

m

m

mm

m

f

m

fm

136.0

1003.5

8.299.019.0

1

105.1101002

/103

2

2

10

12

6

8

===

==

=

=

=

=

===

bandwidth of resonator output

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OpticalTuning

Useconceptofpenetrationdepth

associatedwithevanescentwaves

FrustratedTotalInternalReflection(FTIR)

occurswhenthethicknessofthen2

mediumisthinnerthanthepenetration

depthallowingthewavetopartially

transmitthroughmedium

YieldspartialtransmissionintoaTIR

interface Transmittedbeamcontainssomeintensity,

thusRisreducedbelow1

i

n2

n1 > n2

Incidentlight

Reflectedlight

r

z

Virtual reflecting plane

Penetration depth,

z

y

The reflected light beam in total internal reflection appears to have been laterally shifted byan amount zat the interface.

A

B


i

n2

n1 > n2

Incident

light

Reflected

light

r

When medium B is thin (thickness d is small), the field penetrates tothe BC interface and gives rise to an attenuated wave in medium C.The effect is the tunnelling of the incident beam in A through B to C.

z

y

d

n1

A

B

C


Incidentlight

Reflectedlight

i> c

TIR

(a)

Glass prism

i> c

FTIR

(b)

n1

n1n

2 n1

B= Low refractive indextransparent film (n

2)

ACA

Reflected

Transmitted

(a) A light incident at the long face of a glass prism suffers TIR; the prism deflects thelight.(b) Two prisms separated by a thin low refractive index film forming a beam-splitter cube.The incident beam is split into two beams by FTIR.

Incidentlight


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TemporalandSpatialCoherence

Partialcoherenceisdefinedbytheability

topredictthephaseofanyportionofthe

wavefromanyotherportion

Temporalcoherencemeasurestheextentatwhichtwopointsonthewaveare

separatedintime

Coherencetime:

Spatialcoherencemeasurestheextentat

whichtwopointsareseparatedonthe

waveinspace

Coherencelength:

Spectralwidth:

Example:589nmlaserwithspectralwidth

of5x1011

Hz

TimePQ

Field

Amplitude

Time

(a)

Amplitude

=1/t

Time

(b)

P Q

l = ct

Space

t

(c)

Amplitude

(a) A sine wave is perfectly coherent and contains a well-defined frequencyo. (b) A finitewave train lasts for a duration t and has a length l. Its frequency spectrum extends over= 1/t. It has a coherence time tand a coherence length l. (c) White light exhibitspractically no coherence.


c

lt=

tcl =

t

lv

=

mml

st

6.0

102 12

=

=

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IntroductiontoDiffraction

Airyringsareadiffractionpatternclearlyvisiblewhenlightpassesthroughacircular

aperture

ThediffractedbeamdoesNOTcorrespondtotheshadowoftheaperture

Insteadthelightimagedpassedtheapertureistheresultofbothlightpassing

throughtheapertureandlightscatteredofftheedges. Thescatteredlight

generatesaninterferencepatternintheimage

Diffractedlightfromadistancegeneratestheimageinaplanerwavefront:

Fraunhofer Diffraction Diffractedlightfromanearbyapertureimagesthesurfacewithsignificant

wavefront curvature: FresnelDiffractionLight intensity pattern

Incident light wave

Diffracted beam

Circular aperture

A light beam incident on a small circular aperture becomes diffracted and its light

intensity pattern after passing through the aperture is a diffraction pattern with circularbright rings (called Airy rings). If the screen is far away from the aperture, this would be aFraunhofer diffraction pattern.

1999 S.O. Kasap, Optoelectronics (Prentice Hall) 49

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Incident plane wave

New

wavefront

A secondarywave source

(a) (b)

Another new

wavefront (diffracted)

z

(a) Huygens-Fresnel principles states that each point in the aperture becomes a source ofsecondary waves (spherical waves). The spherical wavefronts are separated by.The newwavefront is the envelope of the all these spherical wavefronts. (b) Another possiblewavefront occurs at an angle to thez-direction which is a diffracted wave.



_______________

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Lightemittedfromapointsource

Thesingleslitdiffractionequationyieldsan

intensity

Withzerointensitypointsat

( )

( )

sin2

1

sin2

1sin

)(

)(

sin

0sin

sin

ka

kaaCe

E

eyCE

eyE

jk

ay

yjk

jk

=

=

=

=

,...2,1

sin

=

=

m

a

m

D

22.1sin =

=

=

sin2

1sin)0(

sin2

1

sin2

1sin

)(

2

kacI

ka

kaCa

I

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ImageResolution

Consider2nearbycoherentsourcesare

imagedthroughanapertureofdiameterD

Thetwosourceshaveanangular

separationof.

Asthepointsgetclosertogether angularseparationbecomesnarrower

diffractionpatternsoverlapmore

AccordingtotheRayleighcriterion,thetwo

spotsarejustobservablewhenthe

principlemaximumofonediffraction

patterncoincideswiththeminimumof

another

ThisminimumisobtainedbytheangularradiusoftheAirydisk

S1

S2

S1

S2

A1

A2

I

y

Screen

s

L

Resolution of imaging systems is limited by diffraction effects. As points S1and S2get closer, eventually the Airy disks overlap so much that the resolution is lost.


The rectangular aperture of dimensions a b on the leftgives the diffraction pattern on the right.

a

b


D

22.1sin =

whereDisthediameteroftheaperture

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DiffractionGratings

BraggDiffractionCondition

Forlightincidentatanangle

Incidentlight wave

m= 0

m= -1

m= 1

Zero-order

First-orde

First-order

(a) Transmission grating (b) Reflection grating

Incidentlight wave

Zero-order

First-order

First-order

(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) Areflection grating. An incident light beam results in various "diffracted" beams. Thezero-order diffracted beam is the normal reflected beam with an angle of reflection equal

to the angle of incidence.


First order

Normal tograting planeNormal to

face

d

Bl d ( h l tt ) ti

d

z

y

Incident

light wave

Diffraction grating

One possiblediffracted beam

a

Intensity

y

m= 0

m= 1

m= -1

m= 2

m= -2

Zero-order

First-order

First-order

Second-order

Second-order

Single slitdiffraction

envelope

dsin

(a) (b )

(a) A diffraction grating withNslits in an opaque scree. (b) The diffracted lightpattern There are distinct beams in certain directions (schematic)

,...3,2,1,0

sin

=

=

m

md

,...3,2,1,0

)sin(sin

=

=+

m

md im

Optoelectronics Ch1 KASOP

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