1. No category

2015-04-01
LightandMatter
Reflection/Refraction/Polarization
MD6305Laser‐TissueInteractions
Class2
JaeGwan Kim
jaekim@gist.ac.kr ,X2220
DepartmentofMedicalSystemEngineering
Gwangju InstituteofSciencesandTechnology
Copyright.Mostfigures/tables/textsinthislecturearefromthetextbook“Laser‐Tissue
InteractionsbyMarkolf H.Niemz 2007”andthismaterialisonlyforthosewhotakethis
classandcannotbedistributedtoanyonewithoutthepermissionfromthelecturer.
LightandBulkMatter(tissue)
• Inopaquemedia,therefractionishardtomeasure
duetoabsorptionandscattering
loss
Iinc
Transmittance(%)=Itrans/Iinc
loss
loss
Itrans
• Inlasersurgery,knowledgeofabsorbingand
scatteringpropertiesofaselectedtissueisessential
forthepurposeofpredictingsuccessfultreatment
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LightandBulkMatter(tissue)
• Typesofinteractions
– Reflection(Fresnel’slaw)
1
– 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 (to ascertain whether absorption or scattering
is dominant in turbid media)
– Transportcoeff.:μtr= μs(1-g) + μa
– Diffusioncoeff.:1/(3μtr)
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Electric Field
• Coulomb’slaw
– Themagnitudeoftheelectrostaticforceofinteractionbetweentwo
pointchargesisdirectlyproportionaltothescalarmultiplicationof
themagnitudesofcharges andinverselyproportionaltothesquareof
thedistancebetweenthem.
– Theforceisalongthestraightlinejoiningthem.Ifthetwocharges
havethesamesign,theelectrostaticforcebetweenthemisrepulsive;
iftheyhavedifferentsign,theforcebetweenthemisattractive.
Coulomb constant
Refraction
• Refraction isthechangeindirectionofawave due
toachangeinitsspeed.
• Thisismostcommonlyobservedwhenawave
passesfromonemediumtoanotheratanyangle
otherthan90° or0°
Q. What is the index of this half circle glass?
=
90 ,
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Reflection
• Reflection isthechangeindirectionofawavefront
ataninterfacebetweentwodifferentmedia(nis
different)sothatthewavefront returnsintothe
mediumfromwhichitoriginated.
• SpecularvsDiffuseReflection(roughness≳λ,tissue)
Plane of incidence
Reflection
• Reflectivity: theratioofreflectedandincident
electricfieldamplitudes
• Reflectance: ratioofthecorrespondingintensities
(actuallyitmeansenergywhichisreflectivity2)
• Theelectrostaticfieldstoresenergy.Theenergy
densityu (energyperunitvolume)isgivenby
1
2
:vacuum permittivity
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FresnelEquations
• DeducedbyAugustin‐JeanFresnel,describethe
behavioroflightwhenmovingbetweenmediaof
differingrefractiveindices.Thereflectionoflight
thattheequationspredictisknownasFresnel
reflection.
Fresnel’sEquations
• Fresnel’sequationsdescribetherelationsfor
reflectivity andrefraction
• E,E’,E’’:amplitudeoftheelectricfieldvectorsof
incident,reflected,andrefractedlight,respectively
• s andp denoteperpendicularandparalleltothe
planeofincidence
– s:Germansenkrecht(perpendicular)
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Fresnel’sEquations
• Question:isthefollowingequationcorrect?
intensityoftherefracted+intensityofreflected
beams=intensityofincidentbeam
• Itisnotbecauseintensity=power/unitarea
• Thecrosssectionofrefractedbeamisdifferentfrom
thatofincidentandreflectedbeamsexceptat
normalincidence
• Onlythetotalenergyisconserved
• Thereflectances inplaneare
,
TrigonometricConversion
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FresnelEquations
• TheamplitudesofreflectioncoefficientR and
transmissioncoefficientT are
R =
and
wherer andt aretheratioofthereflected/transmittedwave’s
complexelectricfieldamplitudetothatoftheincidentwave
FresnelEquations
• Reflectioncoefficient(Reflectance)
– Ifincidentlightiss polarized,
– Ifincidentlightisp polarized
• Transmissioncoefficient
Ts =1‐ Rs,Tp =1‐ Rp
• Iftheincidentlightisunpolarized,
R=(Rs +Rp)/2
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FresnelEquations
• Forthenormalincidentcase,
0
4
1 whichshowstheconservationofenergy
PolarizationofLight
Mechanical wave simulation
Polarization of light
Polarization is something
associated with the electrical
field orientation of the light
wave.
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PolarizationofLight
• Polarizationoflightisdefinedintermsofthetrace
patternoftheelectricfieldvectorasafunctionof
time.Ittellsusinwhichdirectiontheelectricfield
oscillates
• Thetracepatternofelectricalfieldvectorinalight
waveis…
– Predictable:
Fullypolarizedlight
– Unpredictable:
Unpolarized light
– Partialpredicable:
Partiallypolarizedlight
FullyPolarizedLight
• Lightwhichhasitselectricvectororientedinapredictablefashionwith
respecttothepropagationdirection,isfullypolarized.
Visible light: ν = (4.3~7.5)x1014Hz
Three-dimensional representation
of polarized light
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Unpolarized Light
• Naturallyproducedlight– sunlight,lightfromalightbulb,
firelight,lightfromfireflies– isunpolarized.
• Unpolarized lightcanberepresentedasanelectricfieldthat
frommomenttomomentoccupiesrandomorientationsin
thexy‐plane
y
z
x
Linear
polarized light
LightasanElectromagneticWave
• Electromagneticwavevariesinspaceandtime
• Electricfieldcanbewrittenasa:

z

scalar E ( z, t )  A cos2 (  t )     A cos(kz  t   )





vector E ( z, t )  A cos(kz  t   )
δ: the phase constant , k: propagation constant, ω: angular wavenumber
• Thedirectionoftheelectricfield
vector(whichisnotthesameas
thedirectionoflightpropagation!)
iscalledthepolarizationdirection.
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PolarizationofMonochromaticPlaneWaves


E ( z, t )  A cos(kz  t   )
ix
Please remember: cos( x)  Re[e ]

E willlieinthe(x,y)plane
ConsideraplaneEMwavepropagatinginthezdirection→



E ( z , t )  Re  A e j (t  kz ) 


wherethecomplexenvelope:
Atz=constant,thecomponentsofthefieldwillvaryas:
E x  Ax cost  E y  Ay cos t   

where    y   x

A  E x xˆ  E y yˆ  Ax e j x x  Ay e
j y

y
LinearPolarizedLight
1.
  0, 
In phase
 E x  Ax cos t

 E y  Ay cos(t   )
 E x  Ax cos t
 
 E y   Ay cos t

Ey  
Ay
Ax
Ex
Linear equation
Ay
y
Ay
y
x
x
Ax
 0
Ax
 
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CircularPolarizedLight
2.
 

, 90 degree out of phase
2
 E x  Ax cos t

 E y  Ay cos(t   )
 E x  Ax cos t


 E y   Ay sin t

2
2
Ey
Ex
 2 1
2
Ax
Ay
Standard elliptical equation
For particular
Ay case of
y
x
x
E
E


2
Left hand polarization
y
x
E
E
Ax
A
x
 
E x  EAy y
y
Ax
A

2
Right hand polarization
EllipticalPolarizedLight
3. General cases
 E x  Ax cos t

 E y  Ay cos(t   )
2

2
2Ex E y
Ey
Ex
 2
cos   sin 2 
2
Ax Ay
Ax
Ay
General elliptical equation
Ay
y
Ay
y
Ay
y
Ay
y
x
x
x
x
Ax
Ax
Ax
Ax
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Animated Demonstration
LinearPolarizationin3DMovies
Twosynchronizedprojectorsprojecttwo
imagesonthescreen,eachwithadifferent
polarization(theimagesareprojected
throughlinearpolarizers)
The glasses allow only one of the images into each eye.
The two images are separated for each eye creating depth
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ImportanceofPolarization
Polarizationplaysanimportantroleintheinteractionoflightwithmatter:
 Theamountoflightreflectedattheboundarybetweentwomaterialsdependson
thepolarizationoftheincidentwave.
 Theamountoflightabsorbedbycertainmaterialsispolarizationdependent
 Lightscatteringfrommatterisgenerallypolarizationdependent
 Therefractiveindexofanisotropicmaterialsdependsonthepolarization
 Opticallyactivematerialshavethenaturalabilitytorotatethepolarizationplaneof
linearlypolarizedlight.
Thesepolarizationphenomenaareusedforbuildingimportantpolarizationdevices.
PolarizingFilter
• Apolarizingfiltercutsdownthereflections(top)andmadeit
possibletoseethephotographerthroughtheglassatroughly
Brewster'sangle althoughreflectionsoffthebackwindowof
thecararenotcutbecausetheyareless‐stronglypolarized,
accordingtotheFresnelequations
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svs ppolarization
Plane of Incidence
Reflected
wave x
k3
y
θr
θi
x
y
n1
θt
k2
y
perpendicular polarization
(or TE or s polarization,
“s” easier to remember if we think
of the arrow “slapping” the mirror)
k1
mirror
mirror
x
parallel polarization
(or TM or p polarization,
“p” easier to remember if we think
of the arrow “poking” the mirror)
n2
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.
s vs ppolarization
• x– perpendicular( ┴)componentofpolarization
(transverseelectric(TE)orspolarization‐ from
Germansenkrecht)
• y– parallel(//)componentofpolarization
(transversemagnetic(TM)orppolarization)
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Brewster’sAngle
• Anangleofincidenceatwhich
lightwithaparticular
polarizationisperfectly
transmittedthrougha
transparentdielectricsurface,
withnoreflection.
• Whenunpolarized lightis
incidentatthisangle,thelight
thatisreflectedfromthesurface
isthereforeperfectlypolarized
 polarizer
Brewster’sAngle,CriticalAngle
n1< n2 – external reflection
(ex: reflection from air to glass)
Brewster’s angle – the incidence angle
at which the parallel polarized wave
is not reflected
 B  tan 1
n2
n1
n1> n2 – internal reflection
(ex: reflection from glass to air)
Critical angle – the incidence angle
for which the refraction angle is 900
(for θ>θc all the incident light is totally
reflected)
n
 c  sin 1
2
n1
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Brewster’sAngle
• Airtowater(n=1.33)
• Atnormalincidence,
re lectance≠0
• Fromtheaboveeqs,
itisnotclearwhatwill
bethevaluesofRs orRp
Brewster’sAngle
• When issmall,
≅
Divide by and /
"
,
"
• Byinsertingn’=1.33,
≅
≅ 2%
• Thisiswhyweneedto
protectoureyeswhenthe
laserison
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Brewster,CriticalAngleApplic.
Forθ>θc→totalinternalreflection
→usedforlightpropagationinopticalfibers
s‐polarized
θB
Partiallyp‐polarized
A Brewster window transmits TM (parallel) polarized light with no
reflection loss (used in lasers cavities)
Polarizer-a device which converts an unpolarized
beam into a beam with single polarization state
If unpolarized light is incident on a surface
at Brewster angle, the reflected light is linearly
polarized with the electric vector perpendicular
to the plane of incidence (the parallel component
is not reflected) → polarization by selective reflection
Polarizer
• Linearpolarizer
– Absorptivepolarizer:theunwanted
polarizationstatesareabsorbedbythedevice
• Crystals:tourmaline,herapathite
• PVA plasticwithaniodinedopingisstretchedduringthe
manufacturingprocess
• Wire‐gridpolarizer:
– Paralleltothewireisreflectedwhiletheperpendiculartothe
wireistransmitted
– Theseparationdistancebetweenthewiresmustbelessthan
thewavelength oftheradiation,andthewirewidthshouldbe
asmallfractionofthisdistance.
– Thismeansthatwire‐gridpolarizersaregenerallyonlyused
for microwaves andforfar‐ andmid‐infrared light.
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Polarizer
• Linearpolarizer
– Beam‐splittingpolarizer:theunpolarized beamissplit
intotwobeamswithoppositepolarizationstates
• Polarizationbyreflection
• Birefringent polarizer
• Thinfilmpolarizer:glasssubstratesonwhichaspecialoptical
coatingisappliedcausinganinterferenceeffects
Birefringence
• Ananisotropic crystalexhibitsdifferentrefractive
indicesfordifferentpolarizationcomponentsofthe
light→whenlightrefractsatthesurfaceofan
anisotropiccrystal(quartzorcalcite),thetwo
polarizationsrefractsatdifferentangles,being
spatiallyseparated(birefringence ordouble
refraction).
• Usually,twocementedprismsmadeofanisotropic
(uniaxial)crystalsindifferentorientationsareused
toobtainpolarizedfromunpolarized light.
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OpticalAxis
• An opticalaxis isalinealongwhichthereissomedegreeof
rotationalsymmetry inan opticalsystemsuchasa camera
lens or microscope.
• Foran opticalfiber,the
opticalaxisisalongthe
centerofthe fibercore,
andisalsoknownasthe
fiberaxis.
OpticAxisofaCrystal
• It isthedirectioninwhicha ray oftransmittedlight
suffersno birefringence
• Uniaxialcrystals:thehexagonal,tetragonal,and
trigonal crystalsystemshaveoneopticaxis
• Biaxialcrystals:orthorhombic,monoclinic,and
triclinichavetwoopticaxes
• Ifthelightbeamisnotparalleltotheopticaxis,then
thebeamissplitintotworays(theordinaryandextr
aordinary)whenpassingthroughthecrystal.These
rayswillbemutuallyorthogonallypolarized.
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CrystalStructures
Uniaxial crystals
Biaxial crystals
Ordinaryvs Extraordinary
• Ifunpolarized lightentersthebirefringent material
atsome angleofincidence,
– thecomponentoftheincidentradiationwhosepolarizatio
nisperpendiculartothecrystalaxis(ordinaryray)willbe
refractedaccordingtothestandard lawofrefraction fora
materialofrefractiveindex no,
– theotherpolarizationcomponent,theso‐calledextraordin
aryray willrefractatadifferentangledeterminedbythea
ngleofincidence,theorientationoftheopticaxis,andthe
birefringence
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BirefringentPolarizer
Nicoleprism
Glan‐Thomsonprism
Glan‐Foucaultprism
Glan‐Taylorprism
BirefringentPolarizer
Senarmont Prism
WollastonPrism
Crystalaxis
Extraordinary
ray
ore‐ray
15~45o
Rochon Prism
Ordinaryray
oro‐ray
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Malus’law
• Whenaperfectpolarizerisplacedinapolarized
beamoflight,theintensity,I,ofthelightthatpasses
throughisgivenby
WhereIo istheinitialintensity
θi istheanglebetweenθ0andθ1
Polarizer
• Circularpolarizer(polarizingfilter)
– tocreatecircularlypolarizedlightoralternativelyto
selectivelyabsorborpassclockwiseandcounter‐
clockwisecircularlypolarizedlight
– Polarizingfiltersinphotography
– 3DGlasses
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WaveRatarder (Waveplate)
 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:
  1   2 
2

(n2  n1 ) L
where L is the length of the wave plate; n1, n2-the refractive indices corresponding
to the two polarization components
 


2
→ a half wavelength,
Half wave plate
→ a quarter wavelength,
Quarter wave plate
Wave plate (retarder)
HalfWavePlate
 
E x  Ax cos t


 E y  Ay cos(t   origin   )
For linear polarized light (δorigin=0 or π),
after passing a half wave plate:
 total   origin    0( or  )     ( or 0)
http://www.altechna.com/product_details.php?id=877
linear polarization
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
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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
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