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LightandMatterReflection/Refraction/Polarization

MD6305Laser‐TissueInteractionsClass2

JaeGwan Kim

jaekim@gist.ac.kr ,X2220

DepartmentofMedicalSystemEngineering

Gwangju InstituteofSciencesandTechnology

Copyright.Mostfigures/tables/textsinthislecturearefromthetextbook“Laser‐TissueInteractionsbyMarkolf H.Niemz 2007”andthismaterialisonlyforthosewhotakethisclassandcannotbedistributedtoanyonewithoutthepermissionfromthelecturer.

LightandBulkMatter(tissue)

• Inopaquemedia,therefractionishardtomeasureduetoabsorptionandscattering

• Inlasersurgery,knowledgeofabsorbingandscatteringpropertiesofaselectedtissueisessentialforthepurposeofpredictingsuccessfultreatment

Iinc

Itrans

Transmittance(%)=Itrans/IincTransmittance(%)=Itrans/Iinc

lossloss

loss

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LightandBulkMatter(tissue)

• Typesofinteractions– Reflection(Fresnel’slaw)

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– Refraction(Snell’slaw)sin sin

– Scattering,Diffraction– Absorption variationintransmission

– Phaseshift– Emission

LightandTurbidSample

• Opticalpropertiesofturbidsample– Refractiveindex:n– Absorptioncoeff.:μa

– Scatteringcoeff.:μs

– Scatteringanisotropyfactor:g– ReducedScatteringcoeff.:μs´= μs(1-g)

– Totalattenuationcoeff.:μt= μs+ μa

• Optical mean free path of photons= 1/ μt

– Albedo: a=μs/μt (toascertainwhetherabsorptionorscatteringisdominantinturbidmedia)

– Transportcoeff.:μtr= μs(1-g) + μa

– Diffusioncoeff.:1/(3μtr)

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Refraction

• Refraction isthechangeindirectionofawave duetoachangeinitsspeed.

• Thisismostcommonlyobservedwhenawavepassesfromonemediumtoanotheratanyangleotherthan90° or0°

=

90 ,

Q. What is the index of this half circle glass?

Reflection

• Reflection isthechangeindirectionofawavefrontataninterfacebetweentwodifferentmedia(nisdifferent)sothatthewavefront returnsintothemediumfromwhichitoriginated.

• SpecularvsDiffuseReflection(roughness≳λ,tissue)

Plane of incidence

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Reflection

• Reflectivity: theratioofreflectedandincidentelectricfieldamplitudes

• Reflectance: ratioofthecorrespondingintensities(actuallyitmeansenergywhichisreflectivity2)

• Theelectrostaticfieldstoresenergy.Theenergydensityu (energyperunitvolume)isgivenby

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:vacuum permittivity

FresnelEquations

• DeducedbyAugustin‐JeanFresnel,describethebehavioroflightwhenmovingbetweenmediaofdifferingrefractiveindices.ThereflectionoflightthattheequationspredictisknownasFresnelreflection.

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Fresnel’sEquations

• Fresnel’sequationsdescribetherelationsforreflectivity andrefraction

• E,E’,E’’:amplitudeoftheelectricfieldvectorsofincident,reflected,andrefractedlight,respectively

• s andp denoteperpendicularandparalleltotheplaneofincidence– s:Germansenkrecht(perpendicular)

Fresnel’sEquations

• Question:isthefollowingequationcorrect?intensityoftherefracted+intensityofreflectedbeams=intensityofincidentbeam

• Itisnotbecauseintensity=power/unitarea• Thecrosssectionofrefractedbeamisdifferentfromthatofincidentandreflectedbeamsexceptatnormalincidence

• Onlythetotalenergyisconserved• Thereflectances inplaneare

,

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TrigonometricConversion

FresnelEquations

• TheamplitudesofreflectioncoefficientR andtransmissioncoefficientT are

R = and

wherer andt aretheratioofthereflected/transmittedwave’scomplexelectricfieldamplitudetothatoftheincidentwave

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FresnelEquations

• Reflectioncoefficient(Reflectance)– Ifincidentlightiss polarized,

– Ifincidentlightisp polarized

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

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

FresnelEquations

• Forthenormalincidentcase, 0

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1 whichshowstheconservationofenergy

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Mechanical wave simulation

Polarization of light

Polarization is something associated with the electrical field orientation of the light wave.

PolarizationofLight

• Polarizationoflightisdefinedintermsofthetracepatternoftheelectricfieldvectorasafunctionoftime.Ittellsusinwhichdirectiontheelectricfieldoscillates

• Thetracepatternofelectricalfieldvectorinalightwaveis…– Predictable:Fullypolarizedlight

– Unpredictable:Unpolarized light

– Partialpredicable:Partiallypolarizedlight

PolarizationofLight

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• Lightwhichhasitselectricvectororientedinapredictablefashionwithrespecttothepropagationdirection,isfullypolarized.

Three-dimensional representation of polarized lightVisible light: ν = (4.3~7.5)x1014Hz

FullyPolarizedLight

x

y

z

Linear polarized light

Unpolarized Light

• Naturallyproducedlight– sunlight,lightfromalightbulb,firelight,lightfromfireflies– isunpolarized.

• Unpolarized lightcanberepresentedasanelectricfieldthatfrommomenttomomentoccupiesrandomorientationsinthexy‐plane

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• Electromagneticwavevariesinspaceandtime• Electricfieldcanbewrittenasa:

• Thedirectionoftheelectricfieldvector(whichisnotthesameasthedirectionoflightpropagation!)iscalledthepolarizationdirection.

)cos()(2cos),(

tkzAt

zAtzE

δ: the phase constant , k: propagation constant, ω: angular wavenumber

scalar

vector )cos(),(

tkzAtzE

LightasanElectromagneticWave

PolarizationofMonochromaticPlaneWaves

ConsideraplaneEMwavepropagatinginthezdirection→ willlieinthe(x,y)plane

E

)(Re),( kztjeAtzE

wherethecomplexenvelope:

tAE xx cos tAE yy cos where xy

Atz=constant,thecomponentsofthefieldwillvaryas:

)cos(),(

tkzAtzE ]Re[)cos( ixex Please remember:

yeAxeAyExEA yx jy

jxyx

ˆˆ

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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

Animated Demonstration

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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)

ImportanceofPolarization

Polarizationplaysanimportantroleintheinteractionoflightwithmatter:

Theamountoflightreflectedattheboundarybetweentwomaterialsdependsonthepolarizationoftheincidentwave.

Theamountoflightabsorbedbycertainmaterialsispolarizationdependent

Lightscatteringfrommatterisgenerallypolarizationdependent

Therefractiveindexofanisotropicmaterialsdependsonthepolarization

Opticallyactivematerialshavethenaturalabilitytorotatethepolarizationplaneoflinearlypolarizedlight.

Thesepolarizationphenomenaareusedforbuildingimportantpolarizationdevices.

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PolarizingFilter

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

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)

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’sAngle

• Airtowater(n=1.33)• Atnormalincidence,re lectance≠0

• Fromtheaboveeqs,itisnotclearwhatwillbethevaluesofRs orRp

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• When issmall, ≅

• Byinsertingn’=1.33,≅ ≅ 2%

• Thisiswhyweneedtoprotectoureyeswhenthelaserison

Brewster’sAngle

Divide by  ",and  / "

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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• Linearpolarizer– Absorptivepolarizer:theunwantedpolarizationstatesareabsorbedbythedevice

• Crystals:tourmaline,herapathite• PVA plasticwithaniodinedopingisstretchedduringthemanufacturingprocess

• Wire‐gridpolarizer:– Paralleltothewireisreflectedwhiletheperpendiculartothewireistransmitted

– Theseparationdistancebetweenthewiresmustbelessthanthewavelength oftheradiation,andthewirewidthshouldbeasmallfractionofthisdistance.

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

Polarizer

Polarizer

• Linearpolarizer– 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)crystalsindifferentorientationsareusedtoobtainpolarizedlightfromunpolarized light.

OpticalAxis

• An opticalaxis isalinealongwhichthereissomedegreeofrotationalsymmetry inan opticalsystemsuchasa cameralens or microscope.

• Foran opticalfiber,theopticalaxisisalongthecenterofthe fibercore,andisalsoknownasthefiberaxis.

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OpticAxisofaCrystal

• It isthedirectioninwhicha ray oftransmittedlightsuffersno birefringence

• Uniaxialcrystals:thehexagonal,tetragonal,andtrigonalcrystalsystemshaveoneopticaxis

• Biaxialcrystals:orthorhombic,monoclinic,andtriclinichavetwoopticaxes

• Ifthelightbeamisnotparalleltotheopticaxis,thenthebeamissplitintotworays(theordinaryandextraordinary)whenpassingthroughthecrystal.Theserayswillbemutuallyorthogonallypolarized.

CrystalStructures

Uniaxial crystals

Biaxial crystals

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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

http://www.altechna.com/product_details.php?id=877

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

HalfWavePlate