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2016-09-11
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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
12
: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
4
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
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