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8/13/2019 Optoelectronics Ch1 KASOP
1/53
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]8/13/2019 Optoelectronics Ch1 KASOP
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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
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AdditionalSupportiveMaterial
Thebreadthofthiscourseislargerthanasingletextbook
Certainsectionswillhaveaddedmaterialpresentedinclassfromthefollowingtwo
textbooks
Photonicsreadsmorelikeanencyclopediathanatextbookbuthassomeniceapplicationsanddiagrams
Optoelectronicscoverscurrentdeviceconceptsbutlacksontheory
StudentswillbetestedonmaterialfromKasap
FundamentalsofPhotonics
ISBN13: 9780471358329
JohnWiley&Sons
Optoelectroncics:infraredVisibleUV
DevicesandApplications2nded.
ISBN: 9781420067804CRCPress
3
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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.html8/13/2019 Optoelectronics Ch1 KASOP
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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
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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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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).
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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 =
15
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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/8/13/2019 Optoelectronics Ch1 KASOP
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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==
20
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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)
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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)
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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
===
==
24
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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).
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
( )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,
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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Example:AntireflectionCoatingsonSolarCells
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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Example:AntireflectionCoatingsonSolarCells
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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
IntroductiontoDiffraction
_______________
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IntroductiontoDiffraction
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
where D is the diameter of the aperture51
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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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
The rectangular aperture of dimensions a b on the leftgives the diffraction pattern on the right.
a
b
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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.
1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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