Download Phase function

2015-04-01
LightandMatter
Rayleigh Scattering_Phase Function
MD6305Laser‐TissueInteractions
Class4
JaeGwan Kim
[email protected] ,X2220
DepartmentofMedicalSystemEngineering
Gwangju InstituteofSciencesandTechnology
Copyright.Mostfigures/tables/textsinthislecturearefromthetextbook“Laser‐Tissue
InteractionsbyMarkolf H.Niemz 2007”andthismaterialisonlyforthosewhotakethis
classandcannotbedistributedtoanyonewithoutthepermissionfromthelecturer.
ModelsofLightScattering
whereπDp isthecircumferenceofa particle
andλisthewavelengthofincidentradiation
• α<<1:Rayleighscattering(smallparticlecompared
towavelengthoflight,I1/λ4)
• α≈1:Miescattering(particleaboutthesamesizeas
wavelengthoflight)
• α>>1:Geometricscattering(particlemuchlarger
thanwavelengthoflight)
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RayleighScattering
• Rayleighscatteringreferstothescatteringoflight
offofthemoleculesoftheair,andcanbeextendedto
scatteringfromparticlesuptoaboutatenthofthe
wavelengthofthelight
MieScattering
• Forparticlesizeslargerthanawavelength,Mie
scatteringpredominates.Thisscatteringproducesa
patternlikeanantennalobe,withasharperand
moreintenseforwardlobeforlargerparticles.
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Rayleighvs MieScattering
Rayleigh’sLawofScattering
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RayleighScattering
• Rayleighscatteringiselasticscattering same
valuesofk andλ asincidentlight
1. WeneglectedabsorptionduringtheRayleighlaw
derivation
2. Wealsodidnotconsiderthespatialextentof
scatteringparticles(Rayleighonlyworksforthe
particleswhichhasmuchsmallersizethanthe
wavelength) needtoapplyMiescatteringtheory
RayleighvsMieScattering
• Theyaredifferentintwoimportantrespects
1. Miescatteringshowsaweakerdependenceon
wavelength(~
with0.4
0.5)whileRayleigh
scattering(~ )
2. Miescatteringmostlyoccursintheforwarddirection,
whereasRayleighscatteringis~1
 forward
andbackwardscatteredintensitiesarethesame
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Brillouin Scattering
• Brillouin scattering:inelasticscattering
– Arisesfromacousticwavespropagatingthrougha
mediumandthuscausesinhomogeneities oftheR.I.
– Itcanhaveeitherhigherorlowerfrequenciesbecause
scatteringparticlesaremovingtoward(orawayfrom)the
lightsource consideredasanopticalDoppler effect
– Thisbecomesimportantonlyduringthegenerationof
shockwaves
Léon Nicolas Brillouin (French: [bʁilwɛ̃], bree‐u‐we(n)); August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory.
However,inRealTissue…
• Photonsaremostlyforwardscattering(Rayleighscattering
cannotbeapplied)
• WavelengthdependencyisstrongerthanpredictedbyMie
theory
• Thus,neitherRayleighnorMietheorycompletelydescribe
scatteringintissues
• Definingaprobabilityfunction p(θ)ofaphotontobe
scatteredbyanangleθ isconvenienttofitwithexperimental
data
•
, :phasefunctionofaphotontobescatteredfrom
direction into ,
• Ifp(θ)isnotdependentonθ isotropicscattering,
otherwise,anisotropicscatteringoccurs
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ProbabilityFunction
• Coefficientofanisotropyg isdefinedby
Arc length
cos
where
sin
isaprobabilityfunctionand
istheelementarysolidangle.
• Bydefinition,g representstheaveragevalueofcos
cos
• g rangesfrom0.7~0.99inbiologicaltissues 8o~45o
isalsocalledasphasefunction.
•
PhaseFunction
• Phasefunctionisnormalizedby
1
4
1
– Since theprobabilityofphotonscatteredintosomewherewithin4
steradianis1,theintegralof
in4 is4 .
• Severaltheoreticalphasefunction
–
–
–
–
proposed
Henyey‐Greensteinfunction
Rayleigh‐Gans function
δ‐Eddington function
Reynoldsfunction
’
• Amongthese,Henyey‐Greensteinfunctionisthebestfitwith
theexperimentalobservations
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Henyey‐GreensteinFunction
• ThiswasintroducedbyHenyey andGreenstein(1941)andisgivenby
1
1
4 1
2 cos /
• Itiscommonpracticetoexpressasthefunction cos
1
1
2 1
2 cos /
• Thisismathematicallyveryconvenienttohandlebecauseitisequivalent
to
cos
where
•
2
1
cos
aretheLegendrepolynomials
!
1
,∑
ModifiedHenyey‐GreensteinFunction
• Insomecase,acompositefunctionofanisotropictermu and
theHenyey‐Greensteinfunctionfitbettertoexperimental
data.
• Themodifiedfunctionisgivenby
1
1
1
4 1
2 cos /
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Henyey‐GreensteinFunction
• Itdescribesthedominantprocessofscatteringin
theforwarddirectionverywell
• Fig.2.9describesanexperimentaldataforan80μm
sampleofaortictissue
TurbidMedia
• Sofar,weconsideredtheoccurrenceofeitheronly
absorptionorscattering
• Inmosttissues,bothofthemarepresent turbid
media
• Totalattenuationcoefficient
• Themeanfreepathofturbidmediais
1
1
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OpticalAlbedo
• Itisveryconvenienttodefinetheopticalalbedoby
• Whena=0,attenuationisexclusivelyduetoabsorption,
whereasa=1,attenuationisonlyfromscattering
OpticalDepth
• Inturbidmedia,definingopticaldepthd isgivenby
where denotesthesegmentofopticalpathands
isthetotallengthofopticalpath
• Whenthemediumishomogeneous,i.e. is
constant,
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Advantageofaandd
• Insteadof and ,aanddaredimensionless
parameters
• However,aanddcontainssameinformationas
and
• Inturbidmedia,thenormalizationofthephase
functionshouldbealteredto
,(0
•
•
1)
/
cos
∑
2n
1
cos
Reduced ScatteringCoefficient
• Reducedscatteringcoefficientisveryusefulwhen
discussingthediffusionapproximationandisgiven
as
1
• Sinceforwardscatteringalonewillnotcause
attenuationofintensity,thereducedtotal
attenuationcoefficientisgivenby
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