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Waves,Polarization,andCoherence

Lecture6

Biophotonics

JaeGwan Kim

jaekim@gist.ac.kr ,X2220

SchoolofInformationandCommunicationEngineering

Gwangju InstituteofSciencesandTechnology

Outline

ModelsofLight

LightasanElectromagneticWave

PolarizationofLight

– ReflectionandRefraction– WavePropagationthroughAnisotropicMedia

InterferenceandCoherenceofLight

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)cos()(2cos),(

tkzAt

zAtzE

Electromagnetic wave varies in space and time

δ is the phase constant

Electric field can be written as a :

scalar

or a vector )cos(),(

tkzAtzE

The direction of the electric field vector (which is not the same as the direction of light propagation!) is called thepolarization direction.

]Re[)cos( ixex Please remember:

LightasanElectromagneticWave

PolarizationofMonochromaticPlaneWaves

ConsideraplaneEMwavepropagatinginthezdirection→ willlieinthe(x,y)plane

E

)(Re),( kztjeAtzE

wherethecomplexenvelope:

yeAxeAyExEA yxj

yj

xyx ˆˆ

tAE xx cos

tAE yy coswhere

xy

Forsomez=constantthecomponentsofthefieldwillvaryas:

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1. In phase ,0

)cos(

cos

tAE

tAE

yy

xx

tAE

tAE

yy

xx

cos

cos x

x

yy E

A

AE

x

y

xA

yA

0

x

y

xA

yA

Linear equation

LinearPolarizedLight

2. , 90 degree out of phase 2

)cos(

cos

tAE

tAE

yy

xx

tAE

tAE

yy

xx

sin

cos 1

2

2

2

2

y

y

x

x

A

E

A

E

x

xA

yA

2

E

y

xA

yA

E

2

x

xE x

y

E

y

A A

yx EE For particular case of

Standard elliptical equation

CircularPolarizedLight

Left hand polarization Right hand polarization

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3. General cases

)cos(

cos

tAE

tAE

yy

xx 22

2

2

2

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

x

xA

yAy

xA

yAy

x

xA

yA y

x

xA

yA y

x

General elliptical equation

EllipticalPolarizedLight

LinearPolarizationin3DMovies

The glasses allow only one of the images into each eye.The two images are separated for each eye creating depth

Twosynchronizedprojectorsprojecttwoimagesonthescreen,eachwithadifferentpolarization(theimagesareprojectedthroughlinearpolarizers)

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ImportanceofPolarization

Polarizationplaysanimportantroleintheinteractionoflightwithmatter:

Theamountoflightreflectedattheboundarybetweentwomaterialsdependsonthepolarizationoftheincidentwave.

Theamountoflightabsorbedbycertainmaterialsispolarizationdependent

Lightscatteringfrommatterisgenerallypolarizationdependent

Therefractiveindexofanisotropicmaterialsdependsonthepolarization

Opticallyactivematerialshavethenaturalabilitytorotatethepolarizationplaneoflinearlypolarizedlight.

Thesepolarizationphenomenaareusedforbuildingimportantpolarizationdevices.

PolarizingFilter

• Apolarizingfiltercutsdownthereflections(top)andmadeitpossibletoseethephotographerthroughtheglassatroughlyBrewster'sangle althoughreflectionsoffthebackwindowofthecararenotcutbecausetheyareless‐stronglypolarized,accordingtotheFresnelequations

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FresnelEquations

• deducedbyAugustin‐JeanFresnel,describethebehavioroflightwhenmovingbetweenmediaofdifferingrefractiveindices.ThereflectionoflightthattheequationspredictisknownasFresnelreflection.

svs ppolarization

n1 n2

x

xx

y

y

y

k1

k3 k2

θi

θrθt

Reflectedwave

perpendicular polarization(or TE or s polarization, “s” easier to remember if we thinkof the arrow “slapping” the mirror)

parallel polarization(or TM or p polarization, “p” easier to remember if we thinkof the arrow “poking” the mirror)

mirror mirror

By solving a boundary value problem for the electromagnetic wave at the interface one can derive the Fresnel equations. This set of 4 equations gives the amounts of perpendicular and parallel polarized that reflected and transmitted at the interface.

Plane of Incidence

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s vs ppolarization

• x– perpendicular( ┴)componentofpolarization(transverseelectric(TE)orspolarization‐ fromGermansenkrecht)

• y– parallel(//)componentofpolarization(transversemagnetic(TM)orppolarization)

FresnelEquations

• Reflectioncoefficient– Ifincidentlightisspolarized,

– Ifincidentlightisppolarized

• TransmissioncoefficientTs =1‐ Rs,Tp=1‐ Rp

• Iftheincidentlightisunpolarized,R=(Rs +Rp)/2

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FresnelEquations

• TheamplitudesofreflectioncoefficientRandtransmissioncoefficientTare

R= and

whererandtaretheratioofthereflected/transmittedwave’scomplexelectricfieldamplitudetothatoftheincidentwave

Brewster’sAngle

• Anangleofincidenceatwhichlightwithaparticularpolarizationisperfectlytransmittedthroughatransparentdielectricsurface,withnoreflection.

• Whenunpolarized lightisincidentatthisangle,thelightthatisreflectedfromthesurfaceisthereforeperfectlypolarized

polarizer

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Brewster’sAngle,CriticalAnglen1< n2 – external reflection(ex: reflection from air to glass)

Brewster’s angle – the incidence angleat which the parallel polarized waveis not reflected

1

21tann

nB

n1> n2 – internal reflection(ex: reflection from glass to air)

Critical angle – the incidence anglefor which the refraction angle is 900

(for θ>θc all the incident light is totally reflected)

1

21sinn

nc

Brewster,CriticalAngleApplic.

Forθ>θc→totalinternalreflection→usedforlightpropagationinopticalfibers

θB

A Brewster window transmits TM (parallel) polarized light with no reflection loss (used in lasers cavities)

Polarizer-a device which converts an unpolarizedbeam into a beam with single polarization state

If unpolarized light is incident on a surfaceat Brewster angle, the reflected light is linearly polarized with the electric vector perpendicularto the plane of incidence (the parallel componentis not reflected) → polarization by selective reflection

Partiallyp‐polarizeds‐polarized

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• Linerpolarizer– Absorptivepolarizer:theunwantedpolarizationstatesareabsorbedbythedevice

• Crystals:tourmaline,herapathite• PVA plasticwithaniodinedopingisstretchedduringthemanufacturingprocess

• Wire‐gridpolarizer:– Paralleltothewireisreflectedwhiletheperpendiculartothewireistransmitted

– Theseparationdistancebetweenthewiresmustbelessthanthewavelength oftheradiation,andthewirewidthshouldbeasmallfractionofthisdistance.

– Thismeansthatwire‐gridpolarizersaregenerallyonlyusedfor microwaves andforfar‐ andmid‐infrared light.

Polarizer

Polarizer

• Linerpolarizer– Beam‐splittingpolarizer:theunpolarized beamissplitintotwobeamswithoppositepolarizationstates

• Polarizationbyreflection

• Birefringent polarizer• Thinfilmpolarizer:glasssubstratesonwhichaspecialopticalcoatingisappliedcausinganinterferenceeffects

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Birefringence

• Ananisotropic crystalexhibitsdifferentrefractiveindicesfordifferentpolarizationcomponentsofthelight→whenlightrefractsatthesurfaceofananisotropiccrystal(quartzorcalcite),thetwopolarizationsrefractsatdifferentangles,beingspatiallyseparated(birefringence ordoublerefraction).

• Usually,twocementedprismsmadeofanisotropic(uniaxial)crystalsindifferentorientationsareusedtoobtainpolarizedfromunpolarized light.

OpticalAxis

• An opticalaxis isalinealongwhichthereissomedegreeof rotationalsymmetry inan opticalsystemsuchasa cameralens or microscope.

• Foran opticalfiber,theopticalaxisisalongthecenterofthe fibercore,andisalsoknownasthefiberaxis.

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OpticAxisofaCrystal

• It isthedirectioninwhicha ray oftransmittedlightsuffersno birefringence

• Uniaxialcrystals:thehexagonal,tetragonal,andtrigonal crystalsystemshaveoneopticaxis

• Biaxialcrystals:orthorhombic,monoclinic,andtriclinichavetwoopticaxes

• Ifthelightbeamisnotparalleltotheopticaxis,thenthebeamissplitintotworays(theordinaryandextraordinary)whenpassingthroughthecrystal.Theserayswillbemutuallyorthogonallypolarized.

CrystalStructures

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Ordinaryvs Extraordinary

• Ifunpolarized lightentersthebirefringent materialatsome angleofincidence,– thecomponentoftheincidentradiationwhosepolarizationisperpendiculartothecrystalaxis(ordinaryray)willberefractedaccordingtothestandard lawofrefraction foramaterialofrefractiveindex no,

– theotherpolarizationcomponent,theso‐calledextraordinaryraywillrefractatadifferentangledeterminedbytheangleofincidence,theorientationoftheopticaxis,andthebirefringence

BirefringentPolarizer

Nicoleprism Glan‐Thomsonprism

Glan‐Foucaultprism Glan‐Taylorprism

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BirefringentPolarizer

Ordinaryrayoro‐ray

Extraordinaryrayore‐ray

WollastonPrism

Crystalaxis

Senarmont Prism

Rochon Prism

15~45o

Malus’law

• Whenaperfectpolarizerisplacedinapolarizedbeamoflight,theintensity,I,ofthelightthatpassesthroughisgivenby

WhereIo istheinitialintensityθi istheanglebetweenθ0andθ1

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Polarizer

• Circularpolarizer(polarizingfilter)– tocreatecircularlypolarizedlightoralternativelytoselectivelyabsorborpassclockwiseandcounter‐clockwisecircularlypolarizedlight

– Polarizingfiltersinphotography– 3DGlasses

A typical wave plate is made of anisotropic materials (birefringent crystal).

There is a phase delay between the two polarization components which “see” different refractive indices of the anisotropic material

The phase difference is given by:

where L is the length of the wave plate; n1, n2-the refractive indices correspondingto the two polarization components

Lnn )(2

1221

2

→ a half wavelength, Half wave plate

→ a quarter wavelength, Quarter wave plate

Wave plate (retarder)

WaveRatarder (Waveplate)

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The light remains linear polarized, but the polarization plane will be rotated at 2θ.

The polarization plane can be rotated by different angles if the half wave plate is rotated

)cos(

cos

originyy

xx

tAE

tAE

For linear polarized light (δorigin=0 or π), after passing a half wave plate:

)0or ()or (0 origintotal

linear polarization

HalfWavePlate

When do we need to use a half wave plate?

-in an experimental set-up when the plane of polarization of a laser beam needs to be rotated

- when the laser power needs to be attenuated, a wave plate and a polarizer can be used for this purpose

details, principle of operation & videos:http://www.newport.com/content/default.aspx?id=516

HalfWavePlate

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2

A quarter waveplate shifts the relative phase by .

What is the effect of a quarter waveplate on linear and circular polarized light?

PossibleExamQuestion

PossibleExamQuestion

2

A quarter wave plate converts a linear polarization into a circular polarization and vice-versa.

2

A quarter waveplate shifts the relative phase by .

What is the effect of a quarter waveplate on linear and circular polarized light?

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Outline

ModelsofLight

LightasanElectromagneticWave

PolarizationofLight

– ReflectionandRefraction– WavePropagationthroughAnisotropicMedia

InterferenceandCoherenceofLight

Interference

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Interference

The phenomenon by which (electromagnetic) waves interact with one another

Interference is the result of the superposition principle:

i

i trEtrE ),(),(

Consider two fields (most general case:different directions, different frequencies) :

)cos(),( 111011

trkEtrE

)cos(),( 222022

trkEtrE

Interference occurs when the two fields are overlappedspatially  and temporally 

Intensity(Irradiance:energy/area/time)

),( trE

• We do not actually measure the electric field .

• We measure the intensity of the light.

• Add intensities is not always the same as adding the electric fields.

for coherent light.

We only add intensities when the light is incoherent.

2),(~),( trEtrI

2

2121 ),(),(),(),( trEtrEtrItrI

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Interference

The measurable quantity is the intensity!

where < > stands for time average

////////),( 21212

22

1 EEEEEEtrI

I1 I2 I12Interference term

221 /),(),(/),( trEtrEtrI

α

1E

2E

cos2121 EEEE

dot product

2021

forEE

No interference for perpendicular polarizations !Maximum interference for parallel polarizations

Interference

Assume parallel polarizations:

where

For simplicity, we’ll get rid of the time average < > and assume real fields

So,

)(011

111 tzkjeEE)(

022222 tzkjeEE

)(02

)(01

)(02

)(01

221

2 222111222111//// tzkjtzkjtzkjtzkj eEeEeEeEEEEI

cos2cos2 212102012

022

01

)(0201

202

201

212121212121

IIIIEEEE

eeEEEE tzkkjtzkkj

212121 tzkkcos2 2121 IIIII

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Interference

212121 tzkkcos2 2121 IIIII

+

inphase

...2,1,02 nforn

+

...2,1,0)12( nforn

outofphase

Constructiveinterference Destructiveinterference

+

Nointerference

...2,1,02

)12( nforn

Wave splitting

Wave combina-

tionLight source

Introduction of phase difference

Introduction of phase

difference

Key elements in an interferometer

• Light source

• Element for splitting the light into two partial waves

• Different propagation paths where the partial waves undergo different phase contributions

• Element for superposing the partial waves

• Detector for observation of the interference

Interferometer

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y

Wavefront division

))2cos(1(2)( 0 xz

DIxI

D

z

x

Young’s interferometer

Pin hole

Pin hole

Screen

D

zL

Young’sInterferometer

Amplitude division

Detector

Mirror 1

Mirror 2

Slit

x

2/11I

t

MichelsonInterferometer

2 14

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Coherence

21 III no interference → the beams are incoherent

→ the beams are coherentIf θ=0, π… and I1 = I2, I = 4I1

cos)(2 2121 IIIII

1)(0

→ the beams are partially coherent

cos2 2121 IIIII

Degree of coherence

coherentincoherent

Incoherence

21 kk

This can occur if, for ex:

21 frequencies drastically different or many waves of different frequencies( temporal incoherence )

21 kk k vectors drastically different or many waves originating from different locations ( spatial incoherence )

)( 11 tzkje

)( 22 tzkje

21 No interference!Incoherent source!

21 III no interference → the beams are incoherent

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Coherence‐spatialvs temporal

Temporal coherence – means a strong correlation (fixed phase relationship) betweenthe electric fields at one location but different times.

Temporal coherence tells us how monochromatic a source is!

two fields oscillating at the same frequency butdelayed with each other → perfectly correlated

Spatial coherence – means a strong correlation (fixed phase relationship) betweenthe electric fields arriving at the same time but different locations.

For ex, within a cross section of a beam from a laser with diffraction limited beam quality,the electric fields at different positions oscillate in a correlated way.

Spatial coherence tells us how collimated (non-divergent ) a source is!

Coherence‐spatialvstemporal

https://youtu.be/paEe‐EZ2eXc

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)4

cos1(2)( 0 x

ItI

Mirror 2

Mirror 1

Slit

x

Detector

Temporal coherence

Spatial coherence

yD

z

x

))2cos(1(2)( 0 xz

DIxI

SpatialandTemporalCoherence

Lc: coherence length, Temporal coherence is determined by the emission spectrum of the light source

cL

x

I cL

x

I

Wavelength (nm)

I

Wavelength (nm)

I

Fourier Transform.

EmissionSpectrum,TemporalCoherence

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QuantifyingCoherence

1cCoherence time - the source bandwidth

A narrow linewidth means high temporal coherence and long coherence length !

Coherence length

20~cc cL c – the speed of light, Δλ: source spectral width

Coherence Time

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References

B.E.A. Saleh, M.C. Teich, “Fundamentals of Photonics”

M. Born, E. Wolf, “Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light”