Chapter

The Migration of Dius e Photon Dens ityWaves through Highly

Scatter ingMe dia

When lightenters a highly scatter ingorturbid me diumsuch as a cloud milk or

ti s sue thephotons do not s imply reect back f rom or transmit through theme diumas

they would for a pane of glas s Inste ad theindividual photons scatter manytimes and

thus trace outrandom paths b efore e scaping f rom or b e ingabsorbed bytheme dium

In thi s chapter I cons ider thecollective propertie s of these scattere d photons When

theintens ityofapointsourceinaturbid me diumwithuniform optical prop ertie s i s s i

nusoidally mo dulated a macro scopic waveofphoton dens itydevelops and propagates

spher ically outwards f rom the source Although micro scopically theindividual photons

followrandom walklike tra jector ie s macro scopically a coherentphoton dens itywave

i s cre ated

After exp er imentallyver ifyingthe exi stence of these wave s I examinethe ir re

f raction atplanar interf ace s b etween me dia with dierentoptical propertie s andtheir

ref raction and diraction by ob jects with dierentoptical propertie s than thesur

roundingmedia I showthatthe p erturbation of the dius e photon dens itywavef ronts

i s capture d bystandard ref raction diraction and scatter ingmodels

Thetheoretical bas i s of thi s workder ive s f rom thephoton dius ion equation The

radiative transp ort equation i s a more accuratemodel for the migration of photons

in general butistypically diculttohandle I start thi s chapter by reviewingthe

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

as sumptions thatreduce the general transp ort equation to a dius ion equation The

clas s ical wavebehavior of DPDWs i s then cons idere d Attheendofthechapter

the bre akdown of the dius ion approximation i s cons idere d as well as higher order

approximations tothe transp ort equation

Dius ion Approximation totheTransp ort Equation

The line ar transp ort equation for photons propagatinginme dia that scatters and

ab sorb s photons i s

Z

Lr t

r Lr t Lr t Lr tf d S r t

t s

v t

Lr tisthe radiance at position r traveling in direction attime twithunits

 

of W m sr sr steradian unit solid angle The normalize d phas e function

f repre s entstheprobability of scatter inginto a direction f rom direction

v is the sp ee d of lightintheme diumand is the transp ort co ecient

t s a

where is the scatter ing co ecientand is theab sorption co ecient S r t

s a

 

is the spatial andangular di str ibution of the source withunitsofWm sr The

photon uence i s given by

Z

rt dLr t

Thephoton ux or currentdens ityisgiven by

Z

Jrt d Lr t



Boththe uence andthe ux haveunitsofWm The line ar transp ort equation

neglectscoherence andpolar ization eects Recentlyhowever Ackerson et al have

succe s sfully include d coherence eectswithin a transp ort mo del Photon

polar ization within the transp ort equation has also b een cons idere d byFer nandez and

Molinar i

Chapter Migration of Dius e Photon Dens ityWaves

Ω

r dr

Ω

Figure A schematic of the cons ervation of photons in a small elementinphas e

space Thephas e elementisat position rtime tand direction Photons scattere d

f rom all directions into direction at position r must b e cons idere d Also thescat

ter ing f rom direction andab sorption within thephas e elementmust b e cons idere d

as well as the ux of photons through thephas e element

The transp ort equation can b e thought of as a cons ervation equation for the ra

diance If weconsider a small elementinphas e space thatisasmall volume around

position r anda small solid angle around attime t s ee g the lefthandside

of eq accountsforphotons le avingthesmall element andthe r ighthandside

accounts for photons enter ingthesmall element Therstterm on the lefthandside

is thetimeder ivativeofthe radiance which equals thenumber of photons enter ingthe

element minus thenumber leaving The s econdterm accounts for the ux of photons

alongthe direction Thethird term accountsforthescatter ingandab sorption of

photons within thephas e element Photons scattere d f rom an elementinphas e space

are balance d bythe scatter inginto another elementinphas e space The balance i s

handle d bytheintegral on the r ighthandside of eq whichaccounts for photons

at position r being scattere d f rom all directions into direction The s econdterm

on the r ighthandsideisthe source of photons

Analytic solutions of the transp ort equation are dicultto obtain andnumer ical

calculations require large amounts of computational p ower Solutions typically exi st

only for s imple geometr ie s suchasplanar geometr ie s withplanewaveillumination

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

some spher ical geometr ie s and a few other sp ecial cas e s

The s e dicultie s are re duce d by cons ider ingapproximatesolutions tothe trans

p ort equation A standard approximation metho d for the transp ort equation i s known

as the P approximation Themetho d of the P approximation i s s imply

N N

to expandthe radiance phas e function and source in spher ical harmonics Y trun

lm

catingthe s er ie s for the radiance at l N The radiance and source are expanded

as

N l

X X

Lr t rtY

lm lm

l ml

and

N l

X X

S r t q rtY

lm lm

l ml

By substituting eq into eq we s ee that i s prop ortional tothephoton

uence By substituting eq into eq we s ee that are the comp onents

m

of thephoton ux The q rt are the amplitude s of the dierentangular moments

lm

of the source atposition r andtime t

For thephas e function wemakethe re asonable as sumption thatthe scatter ing

amplitude i s only dep endentonthechange in direction of thephoton andthus



X

l

f g P

l l

l

l 

X X



g Y Y

l lm

lm

ml l

where P i s a Legendre Polynomial of order l andthe s econd line i s obtained usingthe

l

standard angular addition rule Thephas e function i s normalize d so that g

Notethat g is theaverage co s ineofthe scatter ingangle

The P approximation i s quite good when thealbedo c isclose

s s a

tounitythephas e function i s not to o ani sotropic eg g butthi s dep ends

on theoptical propertie s andthe sourcedetector s eparation i s large compare d to

g Within the P approximation the radiance can b e wr itten as

s

Lr t rt Jrt

Chapter Migration of Dius e Photon Dens ityWaves

Similarly thephoton source can b e wr itten as

S r t S rt S rt

where S rtand S rt are re sp ectively the monopole i sotropic and dip ole mo

mentsofthe source

Ins erting eq and eq into eq andintegratingover yields

rt rt rJrt S rt

a

v t

Ins erting eq and eq into eq multiplyingby andintegratingover

yields

Jrt Jrt rrtS rt

a

s

v t

where g isthereduce d scatter ing co ecient

s

s

Weobtain the P equation bydecoupling eq and eq for rt

D rt rt rt

D r rtv rt

a a

t v t v t

D S

vS rt D rS rt

v t

D v isthephoton dius ion co ecient Theab sorption co ecient i s dropped

s

f rom thephoton dius ion co ecienttokeep the s et of approximations cons i stent That

is the P approximation i s valid when thealbedo is close tounityandthe scatter ing

i s not highly ani sotropic andthus Thi s has b een di scus s e d in gre ater detail

a

s

byFurutsu andYamada The scatter ing co ecientand scatter ing ani sotropydo

not explicitly app e ar in the P equation andsub s equently the dius ion equation but

instead appear together as thereduce d scatter ing co ecient Thi s interplay b etween

the scatter ing co ecientand ani sotropyto pro duce an eective scatter ing co ecient

i s known as the s imilar ity relation

Thestandard photon dius ion equation i s obtained when theunderlined terms

in eq are dropped Droppingthe dip ole momentofthe source i s justie d

byassuming an i sotropic source Thi s as sumption i s usually supp orted by tre ating

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

collimate d source s as i sotropic source s di splace d one transp ort me an f ree pathintothe

scatter ingme diumfromthecollimate d source Theassumption for droppingtheother

terms i s b e st s een in the f requencydomain where thetimedep endence of the source i s

taken as expi t When theintens ityofthe source i s s inusoidally mo dulated then

thephoton uence b ecomes r expi t Thetimeder ivative s can then b e replace d

by i andtherestoftheunderlined terms can b e ignore d when D v Thi s

as sumption i s equivalentto v thatisthe scatter ing f requency must b e much

s

larger than themodulation f requency

Given the s e as sumptions we arr iveatthephoton dius ion equation for rt

rt

vS rt D r rtv rt

a

t

Notethatinthe f requencydomain thephoton dius ion equation can b e rewr itten as

the Helmholtz equation

v

r k r S r

AC

AC

D

where thewavenumber is complex ie

v i

a

k

AC

D

Dius e Photon Dens ityWaves

When the source of photons in a turbid me diumisintens itymodulate d eg S rt

S rS r expi t then thephoton uence will o scillateatthesame f re

DC AC

quency Thi s small butmeasurable travelingwavedisturbance of the lightenergy

dens ity i s referre d to as a dius e photon dens itywave

Dius e photon dens itywave s are scalar damp e d travelingwaves These traveling

waves arise formally in any dius ive system thatisdriven by an o scillating source

suchasinheat conduction andchemical wave s Fi shkin andGratton for

example have calculated the lightenergy dens ity Urt within an optically

dens e homogenous me dia in the pre s ence of a mo dulate d p oint light source atthe

Chapter Migration of Dius e Photon Dens ityWaves

or igin They then us e d the re sultandthe pr inciple of sup erp o s ition toder ivethelight

energy dens ityinthe pre s ence of an ab sorbing s emiinniteplane The o scillatory

part of thesolution for an innite homogenous dens e random media withanintens ity

mo dulated point source i s

vS

AC

rt expik r expi t

AC

Dr

S is the source mo dulation amplitude D v isthephoton dius ion co ef

AC

s

cientintheturbid mediumwhere v is the sp ee d of lightinthemediumand

s

is thereduce d scatter ing co ecient is theangular mo dulation f requencyandthe

wavenumber k is given by

s

v i

a

k

D

v i

a



exp tan

D v

a

v

a

 

i co s sin tan tan

D v v

a a

This is not the only solution for k however it i s thesolution whichsati se s the

phys ical condition thatthe amplitude i s exp onentially attenuated rather than growing

ie theimaginary part of k i s gre ater than zero Thi s particular solution i s obtained

by extractingthe fromtherestofthe equation on the s econdline An analogous

equation for k can b e foundusingthefollowingapproach still requir ingtheimaginary

part of k tobegreater than zero

k x iy

k x y ixy

v

a

x y

D

xy

D

v v

u u

r

u u

v

a

B C C B

t t

i k

A A

D v v

a a

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Notethatthesolution for an intens itymodulated point source eq i s a

spher ical wave with a complex wavenumb er The complex wavenumber indicates that

thewave i s exp onentially attenuated andhasawell dened wavelength amplitude and

phas e at all p oints Qualitatively thi s wavelength corre sp onds tothe ro otme ansquare

di splacement exp er ience d byatypical photon dur inga single mo dulation p er io d It

can b e altere d by mo difying D or Thi s wave do e s not on average transp ort

a

anyenergyThenet dius iveenergy transp ort arises in the DC ie nono scillating

part of the pro ce s s

To exp er imentallyver ify the exi stence of dius e photon dens itywaves we used

the exp er imental system de scr ib e d in s ection togenerateandme asure a mo dulated

photon uence Theme asurementswere madeinatank containing a highly scatter ing

emuls ion known as Intralipid Me asurementsofthephas e and amp

litudeofthe dius e photon dens itywave DPDW were me asure d with re sp ect tothe

source ateach p oint on a cm square gr id The dimens ions of thegridwere small

compare d withthe dimens ions of thetank so thattheme dium is a good approximation

of an inniteme dium

The re sults for an Intralipid concentration of are exhibite d in g Con

stantphas e countours are shown atdegree intervals aboutthe source Notice that

the contours are circular andthatthe ir radii can b e extrap olate d backtothe source

Thephas e shift andthe quantitylnjr rj are plotte d as a function of radial di s

AC

tance f rom the source in the ins et of g The relationship s are line ar as exp ected

and giveustherealandimaginary partsofthe dius e photon dens itywavenumb er

From these me asurementswededuce thewavelengthofthe dius e photon dens ity

wave cm Theequations for therealandimaginary partsofthewavenumber

canbesolve d for thereduce d scatter ing co ecientandab sorption co ecientofthe

me dium ie



k

r



tan tan

a

v k i

Chapter Migration of Dius e Photon Dens ityWaves

k k

r i

s

a

v

Here k and k are re sp ectively therealandimaginary partsofthewavenumb er

r i

 

Us ing eq and eq wendthat cm and cm for

a

s



Intralipid at C Thephoton ab sorption can b e attr ibute d almo st entirely towater

Interaction withFreeSpace Boundar ie s

Dius e photon dens itywave s propagating in innitehomogeneous me dia are spher ical

waves If theturbid me dium i s not inniteorhomogeneous then thewave f ronts are

di storted Here I cons ider homogeneous me dia thathavea boundary b etween the

turbid me dia andme dia which do not scatter light Micro scopicallythe p erturbation

of the dius e photon dens itywave arises from photons e scapingintothe nonscatter ing

medium When a photon cro s s e s theboundary f rom theturbid me diumintothenon

scatter ingmedium there i s no mechanism for changingthe direction of thephoton to

returnittotheturbid me dium except for Fre snel reections attheboundaryThis

photon e scap e re duce s thenumber of photons in thewavefront thus re ducingthe

amplitudeandalter ingthephas e Generally thelongpath lengthphotons are more

likely to escape reducingthemean path lengthandthe DPDWwavelength Thus the

phas e tends to incre as e b ecauseofthe pre s ence of a f reespace b oundary

Within the dius ion approximation the exact b oundary condition for an index

matche d f reespace b oundary i s thatthe comp onentofthe ux normal totheinter

f ace p ointingfromthe nonscatter ingme diumintotheturbid me dium must b e zero

Sp ecically

D

J r r n rr

in

v

wheren is the normal totheboundary p ointingaway f rom theturbid mediumand r is

on theboundary Thi s b oundary condition i s known as the zero partial ux b oundary

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications ln | r AC | Phase (degrees)

Distance (cm) Position (cm)

Position (cm)

Figure Constantphas e contours shown as a function of position for a homo

geneous solution of Intralipid The contours are shown in degree intervals

Ins et Theme asure d phas e shift circle s andlnjr rj square s are plotted as a

AC

function of radial di stance f rom the source So

Chapter Migration of Dius e Photon Dens ityWaves

Φ(z) Free-Space Turbid Medium

z=-z b z=0

Figure Schematicofthe extrap olate d zero b oundary condition

condition Attheboundaryweme asure theoutward comp onentofthe ux

D

J r r n rr

out

v

D

r n rr

v

The s econdlineisder ived from the condition that J on theboundary and

in

shows thatwhatweme asure on theboundary i s prop ortional tothe uence andthe

comp onentofthe ux normal totheboundary

Generally it i s dicultto obtain analytic solutions of the dius ion equation us ing

the zero partial ux b oundary condition Instead theapproximate extrap olated zero

boundary condition is used Thi s require s the uence tobezeroatadistance of

f rom theactual b oundaryFor example for a s emiinnitemedium withthe

s

boundary at z andtheturbid me diumat zthe extrap olate d zero b oundary

condition require s z where z Thi s extrap olation di stance

b b

s

come s f rom a line ar extrap olation of the uence attheboundary tothe zero crossing

p oint s ee g It i s argue d that z gives better agreement withthe

b

s

photon transp ort equation Here I us e z toremain cons i stent

b

s

withthe recent literature eg

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Detector Image -1 z=-2z b - z tr ρ

z=-z b z=0 z=z

Source +1 tr

Figure Schematic of source andimage source positions for a s emiinnitemedium

For a s emiinnitemedium thesolution of the dius ion equation withthe extra

polate d zero b oundary condition i s e as ily obtained byusingimage source s Thatis

an image of the re al source i s formed by reection of therealsourceabouttheplaneof

the extrap olated zero boundary s ee g Notethatcollimate d source s are usually



approximate d as i sotropic p oint source s which are di splace d a di stance z l

tr

s

f rom thecollimate d source Given the source andimage source conguration shown

in g thesolution of the dius ion equation for a s emiinniteme dium witha

collimate d source on therealboundary i s

q q

z z vS exp ik z z z vS exp ik

d tr d tr b

d d

q q

z

d d

D z z D z z z

d tr d tr b

d d

The source i s at and z while thedetector i s at and z z A us eful

d d

form of eq i s when thedetector i s on therealboundary z and z

d tr

Under these conditions eq re duce s to

h i

vS exp ik

ik z z z

d b tr

b

D

If there is a mismatchintheindice s of ref raction b etween theturbid me diumand

f reespace then the exact b oundary condition i s not the zero partial ux b oundary

condition b ecaus e photons are b e ing reected attheinterf ace backintotheturbid

Chapter Migration of Dius e Photon Dens ityWaves

medium In thi s cas e the exact b oundary condition i s

D D

r n rr R r R n rr J r

j in

v v

where R and R are re sp ectively the reection co ecientforthe i sotropic uence and

j

the reection co ecient for the ani sotropic ux They are given by

Z

R sin co s R d

Fre snel

Z

R sin co s R d

j Fre snel

where R istheFre snel reection co ecient for unp olar ize d light

Fre snel

n co s n co s

in out

R

Fre snel

n co s n co s

in out

n co s n co s

in out

when

c

n co s n co s

in out

when

c

Theangle of incidence i s given with respect totheboundary normal the ref racted

angle isgiven by n sin n sin and n and n are re sp ectively theindex

in out in out

of ref raction ins ideandoutsidetheturbid medium Thi s condition i s calle d the partial

ux b oundary condition The partial ux b oundary condition can b e re duce d toan

extrap olate d zero b oundary condition where

R

ef f



z l

b

R

ef f

where

R R

j

R

ef f

R R

j

This boundary condition i s de scr ib e d in detail by Haskell et al and Aronson

Ref raction and Diraction of Dius e Photon Dens ityWaves

In thi s s ection I pre s ent exp er iments which illustratethe ref raction and diraction

of dius e photon dens itywaves I demonstratethatthe ref raction of these waves at

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

planar interf ace s i s well described bySnells Law In addition I demonstratethat

s imple diractiveand ref ractivemodels can b e us e d tounderstandthe scatter ingof

these waves byab sorptiveand di sp ers ive ob jectsembedde d in an otherwi s e uniform

system

Ref raction atplanar interf ace s

Fig demonstrates the ref raction of these wave s in three ways A planar b ound

ary has b een intro duce d s eparatingthelower me dium withIntralipid concentration

c andlight dius ion co ecient D f rom theupper me dium withIntralipid

l l

concentration c andlight dius ion co ecient D In g contours of

u u



constantphas e are drawn every for the propagation of the DPDWfromthelower

me diumtotheupper medium The contours b elowtheboundary are thehomogenous

me dia contours without reectiont hey are obtaine d b efore the partition i s intro

duce d intothesample Thecontours abovetheboundary are der ived from the dius e

photon dens itywave s transmitted intothe le s s concentrated me dium As a re sultof

thedetector geometrythe clo s e st approachtothe partition i s aboutcm

We exp ect a number of general re sults First thewavelengthinthelessdens e

me dium cm should b e gre ater than thewavelengthofthediusephoton

u

dens itywaveintheincidentme dium cm Thi s was ob s erved Theratio

l

of thetwowavelengths should equal theratio of the dius ional indice s of ref raction

q q

of thetwome dia Sp ecically we s ee as exp ected that D D c c

u l l u l l u

thi s relation holds when ab sorption i s negligible Furthermore wewould exp ect that

theapparent source position S as viewe d f rom within theupper medium should

i

b e shifted from the re al source position S cm bya factor

o l u

as pre dicted bySnells law for paraxial wavesThisiswhatwendwithin the

accuracy of thi s me asurement Us ingthe radii f rom the full contour plotsweseethat

theapparent source p o s ition i s shifte d f rom cm to cm f rom the

planar interf ace

Finallyg explicitly demonstrates Snells law for dius e photon dens ity

Chapter Migration of Dius e Photon Dens ityWaves

upper

t

A Position (cm)

lower i

Position (cm)

Figure Constantphas e contours in degree intervals as a function of position

showingthepropagation of a dius e photon dens itywave acro s s a planar b oundary

that s eparate s concentrated Intralipid f rom Intralipid S source position

o

S apparent source position A p oin tonboundary angle of incident ray

i i t

angle of ref racted ray Thesolid line s are obtaine d directly f rom data Thedot

dashe d line s are obtained byinterp olation over large di stance s and are drawn to

showthe irregular itie s at large angle s

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

wave s Thi s can b e s een byfollowingthe ray f rom S tothepointAattheboundary

o

andthen intotheupper medium The ray in thelower me diummake s an angle



with re sp ect tothe surf ace normal Theupp er ray i s constructe d in the

i

standard way b etween theapparent source p o s ition S through the p ointA onthe

i

boundaryandintothemediumabovetheboundary It i s p erp endicular tothe



circular wavef rontsinthe less dens e me diumandmake s an angle with

t

re sp ect totheboundary normal Within the accuracy of the exp er iment we s ee that

sin sin sothatSnells law accurately de scr ib e s the propagation

i t l u

of dius e photon dens itywave s acro s s theboundaryThewavef ronts b ecome quite

di storted when the source ray angle excee ds degree s The s e irregular itie s are a

cons equence of total inter nal reection diraction and spur ious b oundary eects

Ref raction and diraction by spher ical inhomogeneitie s

Here I pre s entme asurements of dius e photon dens itywavef rontdistortions that

ar i s e when these wave s are p erturbed by purely ab sorptive or dispersivehomogen

eous sphere s In general onewould exp ect b oth ref ractiveand diractive processes

to aect thewavef ronts Unfortunately our intuition f rom conventional opticsisof

limited applicability s ince wemust workinthene ar eld Me asurementsofwavef ront

di stortions f rom purely ab sorbing sphere s are re asonably well de scr ib e d byasimple

diraction mo del wherebythe dius e photon dens itywaveisscattere d byanab sorbing

di sk of the samediameter The pure di sp ers ive cas e i s qualitatively dierent Here

a ray optic mo del works well for scatterers character ize d by a larger light dius ion

co ecient relativetothatofthe surroundingturbid mediumbut a diractivemodel

i s require d under theopposite conditions

The diraction of DPDWs byab sorptive sphere s i s illustrate d in g The

contours of constantphas e andamplitude are plotte d for a DPDWtravelingindier

ent concentrations of Intralipid and diracting around a cm diameter ab sorptive

sphere The sphere was saturate d with ink so thatthe f raction of incident light trans



mitted through thesphere was b elowthedetection limit of Nevertheless the

Chapter Migration of Dius e Photon Dens ityWaves

Phase Contours Phase Contours 6 6 λ = 22.2 cm λ = 15.4 4 4

2 2

0 a 0 cm

-2 -2

-4 -4

-6 -6 024 6 8 10 12 024 6 8 10 12

Amplitude Contours Amplitude Contours 6 6 λ = 22.2 λ = 15.4 4 4

2 2

0 0 cm

-2 -2

-4 -4

-6 -6 024 6 8 10 12 024 6 8 10 12

cm cm

Figure The diraction of a dius e photon dens itywaveby a spher ical ab sorb er

with a diameter of cm The light source i s atthe or igin and generates a wave

witha wavelength of cm in theplotsonthe left anda wavelength of cm in

theplotsontheright Our exp er imental theoretical re sults are thesolid dashed

curves Thephas e contours are plotted every degree s andthe amplitude contours



are plotted in decre as ingintervals of e

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

P A 1 R2 R1 P S Absorbing S P a 2 Disk

(a) (b)

Figure a In the diraction mo del the sphere i s replace d byanab sorbingdiskof

thesame diameter a cm whichliesinaplanethrough the center of thesphere

R is thedistance f rom the source S toapoint A in the diraction planedashed line

and R is thedistance f rom A totheimage p oint P Here wetakethe zaxi s tobe

normal tothe diraction plane andwe let the diraction plane coincide withthe xy

plane ie z Thewavef rontat P i s calculated byintegratingthestandard Kircho

equation over the diraction plane b In the ray mo del thewavef ront i s calculated by

determiningthephas e and amplitude of rays which are ref racted through a spher ical

lens

wavef rontsontheother s ideofthe sphere are detected These wavef ronts are formed

bythe diraction of thewave aroundthe sphere

Here I havemodele d thi s eect in a s imple way In themodel I replace d the

sphere byatotally ab sorbingdiskofthe same diameter Thediskwas chosen to lie

inaplane containingthe center of the sphere with surf ace normal p ointinginthez

direction The diraction f rom thi s di sk can b e calculate d us ingthestandard Kircho

construction

Z

expik R i kz

p

dx dy R x y z

p p p

i R kR

S

The construction i s depicte d in g a Here R isthe complex amplitude

of thephoton uence in theplaneofthe di sk R is the lengthofthevector f rom

the source at position R x y z to a p oint A x y z on

s s s s

the diraction plane R is the lengthofthevector goingfromA tothedetection

p oint R x y z The Greens function i s der ive d f rom thepoint source

p p p p

solution for dius e photon dens itywave s in an innitehomogeneous me diumsothat

k i s complex Sp ecicallythe Greens function for thi s problem i s der ive d f rom

Chapter Migration of Dius e Photon Dens ityWaves

a superposition of Greens function solutions of the Helmholtz equation Ichose

a sup erp o s ition tosati sfy Dir ichlet b oundary conditions on the diraction planeat

G R R

D

z Therefore eq i s der ive d f rom theintegral of R dxdy over

z

the diraction plane with G R R expik R R expik R R where

D

R R A R R Aand R i s just theimage of R reected aboutthe

p p

p p

diraction plane

The exp er imental theoretical re sults are thesolid dotte d curve s in g The

s imple mo del approximates theme asure d wavef rontdistortion re asonably well Note

thatthere are no f ree parameters in the t Themodel app e ars totthe exp er imental

re sultsbetter for bigger ratio s of dius e photon dens itywavelengthto ob ject diameter

Of cours e thefunction R intheplaneofthediskisonlyapproximately correct

asaresultofshadowingand diraction bythe f rontportion of the sphere A s imilar

eect will mo dify the scattere d wave Thi s eect i s exp ected to b e larger as the

wavelengthdecre as e s as ob s erve d in g Nevertheless themodel capture s the

qualitativephys ics of the scatter ing

The constantphas e contours solid line arisingfromthescatter ingofanon

absorptive sphere are shown in g TheIntralipid surroundingthesphere had the

same concentration in b oth exp er iments butthe concentration of Intralipid ins idethe

sphere was e ither lesser g a or gre ater g b than the surroundingme dium

The ob s erved patter ns are dierent The s e eects can b e approximated using a ray

optics mo del in the rst cas e and a diraction mo del in the s econdcase

In the ray optic mo del thescatterer i s tre ated like a spher ical lens with a dierent

dius ional index of ref raction than the surroundingme dium The bas ic ide a of the

mo del i s depicte d in g b The complex wave amplitudeiscalculate d f rom the

amplitudeandphas e for p oints alongthe rays emergingfromthe source Someofthe

rays were ref racted through the sphere others were not Thi s mo del ignore s multiple

scatter inginthe sphere s ince thewave s are heavily damped

Again we do not exp ect themodel to give p erfect quantitative agreementwith

theme asurements s ince diraction eects are omitted However when the rays trans

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

6 6

4 4

2 2

0





0





cm

-2 -2

-4 -4

-6 -6 024 6 8 10 12 024 6 8 10 12 cm cm

(a)







(b)

Figure Thescatter ing of a dius e photon dens itywaves by purely di sp ers ive

sphere s a TheIntralipid concentration within the spher ical shell i s le s s

than the surroundingmedium b TheIntralipid concentration i s gre ater than

the surroundingmediumFor b oth the surroundingIntralipid i s the same the source i s

lo cated atthe or igin thesphere has a diameter of cm andiscentere d at x cm

y cm Thephas e contours are drawn every degree s for theexperimental solid

lines andtheoretical dashed line s re sults Thetheoretical re sultswere calculated in

a bythe ray mo del andinbbythe diraction mo del

mitted through thesphere are attenuated less than the rays outsideofthe sphere we

would exp ect diraction eectstobenegligible Thi s i s the cas e when the sphere has

asmaller concentration of Intralipid than the surroundingme diumandthe exp ected

behavior was ob s erve d s ee g a For nearaxisraysthemodel also pre dicts

an apparent source position at z cm Thi s i s e as ily ver ie d bystandard ray

s

construction technique s

The ray method does not workwell for dens e sphere s Thedens e sphere acts more

likeanab sorb er s ince thediusephoton dens itywave i s s ignicantly attenuate d up on

travelingthrough thesphere For thi s re ason one might exp ect the purely diractive

mo del discussed earlier toworkbetter Indee d thi s i s whatwas ob s erve d s ee g b

Chapter Migration of Dius e Photon Dens ityWaves

Scatter ing of Dius e Photon Dens ityWaves

The previous s ection showed that dius e photon dens itywave s are di storted bythe

pre s ence of optical inhomogeneitie s Thedegree of di stortion i s determined bythe

characteristics of theinhomogeneitysuchasits position shap e s ize and scatter ing

andabsorption propertie s We saw that in some cas e s the p erturbation can b e mo dele d

us ing a s imple diraction or ray optic mo del A b etter theory for the ob s erved per

turbation i s de s irable for many re asons In particular the s imple mo dels di scus s e d

in the previous s ection only work for sp ecic dierence s in theoptical prop ertie s and

sp ecic sourcedetector positions relativetothe inhomogeneityThat i s the ob ject

must b e place d b etween the source anddetector anditmust b e highly ab sorbing

relativetothebackgroundorhavea smaller scatter ing co ecientwithnoab sorption

contrast Since the Helmholtz equation i s known tode scr ib e the transp ort of DPDWs

in a piecewi s e homogeneous me dia we exp ect that an exact solution exi sts

for the scatter ing of DPDWs by spher ical ob jects Thesolutions will b e

s imilar to andsimpler than thetheory of Mie scatter ing often us e d in optics

In thi s s ection I der ivetheanalytic solution of the Helmholtz equation for a piece

wi s e homogeneous system cons i stingofaspher ical ob ject comp o s e d of one highly

scatter ingmediumembedde d in a s econd highly scatter ingme dium of innitespatial

extent Thi s solution i s e as ily extended to s emiinniteme dia us ingthe extrap olated

zero b oundary condition Theanalytic solution i s compare d with ex

perimental datainorder to assess thetheorys pre dictivepower and a s imple invers e

lo calization algor ithm i s demonstrated todeterminethe s ize andlocation of a spher ical

ob ject Finallythetheory i s extended to include more complex problems in imaging

An Analytic Solution

Theder ivation of theanalytic solution for the scatter ing of DPDWs f rom spher ical

inhomogeneitie s b egins withthe Helmholtz equation eq In the pre s ence of a

spher ical heterogeneitythephoton uence i s foundby constructing a general solution

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

toeq outsideandinsidethe sphere andapplyingtheappropr iateboundary

conditions It i s natural toanalyze the problem in spher ical co ordinates who s e or igin

coincide s withthe center of the spher ical ob ject s ee g Thegeneral solution

outsidethe sphere is a superposition of incidentand scattere d wave s ie

out inc scatt

where

vS

AC

out

expik jr r j

inc s

Djr r j

s

 l

out

X X

vS k

AC

out out 

j k r h k r Y Y i

l s s lm

lm

l

D

l ml

is thespher ical wave cre ated bythe source and incidentonthe sphere

h i

X

out out

A j k r B n k r Y

scatt lm l lm l lm

lm

is thewavescattere d f rom the ob ject

Ins idethe sphere thegeneral solution i s

h i

X

in in

C j k r D n k r Y

in lm l lm l lm

lm

Here j xand n x are Spherical Bessel and Neumann functions re sp ectively h x

l l

l

out

are the Hankel functions of the rst kind Y are the spher ical harmonics k

lm

in

and k are the complex wavenumb ers outsideand ins idethe sphere re sp ectively r r

s

is the position of thedetector source me asure d f rom the center of thesphere and r

r isthesmaller larger of jrj and jr jTheunknown parameters A B C

s lm lm lm

D are determine d us ingthefollowingboundary conditions a must b e nite

lm

everywhere except at a source b must asymptotically approach a spher ically

out

outgoingwaveas r c the ux normal totheboundary must b e continuous

ie D r r D r r where D D isthephoton dius ion co ecient

out out in in out in

outside ins ide thesphere and d thephoton uence must b e continuous acro s s the

boundary ie at r a

in out

Chapter Migration of Dius e Photon Dens ityWaves

(a) (b) z Detector Detector

a r1,d x Object 2 r r - r 1,2 s rs Object 1 rs,1 rs,2 First Order

Source Second Order Source

Figure Tosolvethe Helmholtz equation for a spher ical b oundary it i s natural to

us e spher ical co ordinate s withthe or igin atthe center of the ob ject a The source i s

positione d on the zaxi s to exploit the azimuthal symmetry of the problem

s

andthe relevantdistance s b etween the source ob ject anddetector are indicate d in the

gure Scatter ingfrommultiple ob jects i s diagramme d in b The rst andsecond

order waves scattere d f rom the rst ob ject are illustrated bythesolid anddashed line

re sp ectivelyTherelevantdistance s are indicated in the diagram

Cons ider ingthese boundary conditions andusingthe orthogonalityrelation for the

spher ical harmonics I nd

out

vS k D xj xj y D yj xj y

AC out l in l

l l

out 

A i k z Y h

lm s

lm l

D

D xh xj y D yh xj y

out l in

l l l

B iA

lm lm

out

vS k D xh xj x D xh xj x

AC out out l

l l l

out 

C i h k z Y

lm s

l lm

D

y xj xj y D yh D xh

l in out

l l l

D

lm

out in

where x k a y k a r r z and j and h are the

s s

l l

rst der ivative s of thefunctions j and h with re sp ect tothe argument Placing

l

l

the source on the zaxi s exploitsthe azimuthal symmetry of the problem le adingto

A C for m Thedistortion of thewaveisentirely dep endentonthe

lm lm

out
out out in
in in

parameters k k k k D D r andthe ob ject

out in s

s a s a

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

radius a In general the innitesumfor converge s p ermittingthesumtobe

out

truncate d after obtainingthede s ire d preci s ion The pro cee ding calculations require



no more than terms in theseriesto obtain b etter than preci s ion whichfar

excee ds exp er imental preci s ion On a Sun Micro systems Mountain View CA Sparc

can b e calculate d time s p er s econddep endingontheabovementioned

out

parameters

For the sp ecial cas e where in theheterogeneity i s a p erfect ab sorb er I sati sfy the

zero partial ux b oundary condition

D

r r

v r

at r a Of cours e Thesolution in thi s cas e i s

in

out

k

out

j x j x

l

vS k

l

AC

out 

s

h k z Y A i

s lm

l lm

out

k

D

h x h x

l l

s

B iA

lm lm

C

lm

D

lm

Theanalytic solutions enable us toestimatetheme asurement preci s ion require d to

detect optical inhomogeneitie s The require d phas e preci s ion i s determine d f rom the

positiondep endent dierence in phas e b etween the incidentwaveandthedistorted

wave while the require d amplitude preci s ion i s found f rom the positiondep endent

ratio of j jj j Contour plotsofthephas e dierence andtheamplituderatio

out inc

indicatethe spatial positions which are mo st s ens itivetothe pre s ence of the ob ject as

well as the require d s ignaltonoi s e ratio Fig illustrates thi s spatiallydep endent



s ens itivity for a p erfectly ab sorbing sphere immers e d in a me diumwith cm

s

 

and cm These plotsshowthat phas e and amplitude preci s ion

a

i s sucent for lo calization withme asurementsmadeintheshadowwithin cm of



the ob ject of the cm diameter ab sorb er Thi s i s well within the phas e and

amplitude preci s ion available with currentdetectors Lo calization of smaller

Chapter Migration of Dius e Photon Dens ityWaves

Figure The s e s ens itivityplotsdemonstratethephas e andamplituderesolution

nece s sary tome asure a DPDWdistorted by a p erfect ab sorb er Plotte d in a i s

thephas e dierence b etween an incidentwaveandthewavedistorted by a cm

diameter ab sorb er Theratio of the amplitudeofthedistorted wave with re sp ect to

the incidentwaveisplotte d in b For these plots the surroundingmediums optical

 

characteristics are cm and cm themodulation f req i s MHz

a

s

and v cms Thedots in a repre s entthelocations where me asurements

were madeinorder tocharacter ize the ob ject

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

ab sorb ers will require b etter preci s ion A more detaile d s ignaltonoi s e analys i s that

reveals the limitstodetecting lo calizing andcharacter izing i s given in chapter

Experimental Ver ication of theAnalytic Solution

Twosets of exp er imentswere p erformed onetocheckthevalidityofthetheory and

theother toresolveobjectcharacteristics byttingthetheory to exp er imental data

In the rst s et of exp er iments the ob ject and source are xe d in theIntralipid with

a s eparation z Thephas e and amplitudeofthedistorte d DPDW are me asure d

s

bymovingthedetector to dierent p ointsonatwodimens ional gr id containingthe

source andthe center of the ob ject The s e exp er imental re sults are then compare d

tothe pre diction of eq for the given ob ject prop ertie s In the s econdsetof

exp er iments the propertie s of dierentspher ical ab sorb ers are foundbyttingthe

theory toame asurementofthedistorted wavef ront along a line Thi s was accom

pli shed by minimizingthe le ast square s theoretical t tothe exp er imental datausing

the ob ject position and radius as f ree parameters Theoptical prop ertie s of the In

tralipid were determine d b efore e ach exp er imentthrough s eparateme asurementsof

phas e and amplitudeoftheDPDW propagatinginthe innitehomogeneous system

These quantitie s were us e d in thesub s equentanalys i s

Theme asurementsindicatethattheanalytic theory accurately pre dictsthedis

tortion of the DPDW Furthermore b ecauseofthe clo s e agreement we are able to

character ize a spher ical absorber embedde d in theturbid me diumThese observations

were not obvious a priori for onema jor re ason thetheory i s der ived from the dius ion

equation butphoton migration i s b etter approximated by a transp ort equation In

f act s ignicant dierence s b etween the dius ion equation andthe transp ort equation

arise near sharp b oundar ie s As mentioned below evidence of the s e dierence s have

b een detected

Theme asure d di stortion of the DPDWby a p erfectly ab sorbing sphere i s shown in

g and compare d tothe pre dicte d di stortion Thi s compar i son illustrates that

theanalytic solution shows go o d agreement withthe exp er imental data

Chapter Migration of Dius e Photon Dens ityWaves

Figure The exp er imental me asurements solid line s of a DPDWdistorted bya

cm radius p erfect ab sorb er are compare d tothetheoretical pre diction dotte d lines

for the given experimental parameters Phas e contours are drawn every degree s



in a while the amplitude contours are drawn every e For thi s exp er iment the

 

optical prop ertie s of the surroundingme diumwere cm and cm

a

s

f MHz and v cms

Figure Thetsto exp er imentC and G f rom table are pre s ente d in a and

b re sp ectivelyThe exp er imental datas are compare d tothe best t solid line

The exp er imental parameters are given in table

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Table Theresultsofttingtheory to a s er ie s of exp er imental ob s ervations of

aDPDW scattere d byanab sorb er are pre s ente d in thi s table Theabsorber had

a radius of a cmandwas positioned at Z cm X cm and

exp

Y cmFor e ach exp er iment thedetector was positioned at Z and scanned

detector

from X The exp er imentswere p erforme d in dierent concentrations of



Intralipid for whichthephoton random walk step i s given by l In all exp er iments



cm f MHz and v cms

a



Exp Z Intralipid l Z X Y a

detector fit fit fit fit

s

cm cm cm cm cm cm

A

B

C

D

E

F

G

H

As an example of theutilityoftheanalytic solution a s imple le astsquare s tting

algor ithm was us e d tottheanalytic solution totheme asurementsofphas e and

amplitudeoftheDPDWto pre dict ob ject s ize andlocation Me asurementswere taken

along line s parallel tothose indicate d in g a The re sultsofthese experiments

are pre s ente d in table Fits for twoofthe s e exp er iments are shown in g

The re sultsintable showthat a t tome asurementsmadeintheshadowofthe

ob ject determines thexand y position of theab sorb er to an accuracy of cmand

the z position to cm Finallythe ob ject radius was determined to within

Chapter Migration of Dius e Photon Dens ityWaves

Figure Thetsto exp er iment C s and H s f rom table for the scatter ing

of DPDWs f rom purely scatter ing sphere s The exp er imental data are given bythe

symbols andthe best tsbythesolid lines There i s an arbitrary amplitudeand

phas e dierence b etween thetwosetsofdata Thetswere madeusingthe ob jects

optical prop ertie s and initial source amplitudeandphas e as f ree parameters The

exp er imental parameters are given in table

cm Withadecre as e in thephoton random walk step the di screpancy b etween the

determine d radius andthe known radius i s s een todecre as e Thi s trend i s a re sultof

applyingthe dius ion equation toasystem witha sharp ab sorbingboundary

Todemonstratethatthi s le astsquare s tting algor ithm can b e us e d tocharacter

ize theoptical prop ertie s of spher ical ob jects I me asure d theamplitudeandphas e of

DPDWs scattere d by purely scatter ing ob jects The ob jectswere sphere s of p oly

styrene resin with dierent concentrations of titaniumoxide TiO Themethod for

castingthe s e sphere s i s de scr ib e d in s ection Me asurementswere taken along lines

parallel tothose indicate d in g a with a cm diameter ob ject centere d atx

and y cm Theresultsofthe s e exp er iments for sphere s with dierent concentrations

of TiO are pre s ented in table

Fitsfortwoofthese experimentsareshown in g Thets agree well withthe

exp er imental data Fitswere made for sphere s with e ight dierent concentrations of

TiO and in all cas e s go o d agreementwas found Thereduce d scatter ing co ecientof

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Figure Thebesttsforthereduce d scatter ing co ecients i s graphed versus the

TiO concentration in theresinsphere s The exp ecte d line ar relationship and zero

intercept are ob s erved

the ob ject i s exp ected to incre as e line arly withthe concentration of TiO Thedatain

table showthi s trend A summary of thedetermine d re duce d scatter ing co ecient

versus TiO concentration i s shown in g Notethatthe relationship i s linear

andthatthereduce d scatter ing co ecient goes to zero as the TiO concentration goes

to zero

Scatter ing f rom Multiple Ob jects

When the sample contains two or more spher ical ob jects thedistorted wave i s calcu

lated bysumming scatter ingevents of dierent order We rst calculatethe scatter ing

of the incidentwavefromeach ob ject Thi s i s the rst order scattere d wave The rst

order scattere d waves are incidentonand cons equently scattere d bythesurrounding

ob jects re sulting in s econd order scattere d waves whose amplitudeissmaller than

the rst order wave For two spher ical ob jectsembedde d in an innitehomogeneous

Chapter Migration of Dius e Photon Dens ityWaves

Table Re sultsforthette d ob ject re duce d scatter ing co ecientversus dierent

TiO concentrations

Background TiO Concentration Ob ject

 


Exp cm cm grams TiO ml Re s in cm

a

s s

A

B

C

D

E

F

G

H

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

me dium the general solutionisofthe form



X

n n

out inc

scatt scatt

n

n

th th

is then order scattere d wavefromthei ob ject s ee g While where

scatti

the rst order waves are e as ily calculate d us ing eq thesecondorder

scatti

waves require thesolution of complex integral equations s ince the rst order

scatti

wave s are not spher ical If the rst order wave s are spher ical to a good approximation

then the s econdorder wave s can b e computed analytically us ingthe same pro ce dure

for calculatingthe rst order scattere d waves Thecondition i s only sati se d for small

ab sorbing ob jects In thi s regimewe can checkthe s ignicance of thesecondordere d

scattere d wave s f rom theratio of to Thi s ratio indicates that is

inc

scatti scatti

negligible when

v v r

ai aj sd

a a exp ik r r r r

si ij jd sd

i j

D D r r r

o o si ij jd

where i and j denotethe dierent ob jects s ee g and is the dierence in

ai

th

theab sorption co ecientbetween the i ob ject andthe background

SemiInniteMedia

In me dical imaging me asurements are typically madebyplacingthe source andde

tector on the scalp or surf ace of the bre ast Tre atingsuch a system as inniteis

obviously incorrect and will le ad to di screpancie s b etween theory and exp er iment

Planar b oundar ie s b etween dius iveand nondius ivemedia can be modele d by re

quir ing on an extrap olate d zero b oundary a di stance z f rom the

out o

s

actual b oundary someinvestigators us e z away f rom the dius ive

o

s

me dium Multiple planar b oundar ie s can b e mo dele d byemploying

additional extrap olate d zero b oundary conditions Torstorder the extrap olated

zero amplitudeboundary condition i s satised byplacinganimage source of negat

ive amplitudeatthe position of the actual source reected aboutthe extrap olated

Chapter Migration of Dius e Photon Dens ityWaves

zero b oundaryThephoton uence i s then calculated by superimposingthe DPDWs

generated bythetwosourcesandtheir respectivescattere d waves In general one

must also cons ider an image of the scattere d waves to ensure that equals zero on

out

the extrap olated zero boundaryThese image s then cre atewaves that scatter o the

ob ject ad innitum

General Heterogeneous Me dia

In biological media theoptical inhomogeneitie s will have arbitrary shapesItisnot

possible tondanalytic solutions for general heterogeneous media Wemust therefore

re sort tonumer ical technique s There are manyapproaches tonumer ically solvingthe

dius ion equation for spatially varyingoptical propertie s including nite dierence

nite element and p erturbativemetho ds Here I pre s enta short review of p erturbative

methods

With a p erturbativemethod thesignal re achingthedetector i s cons idere d tobe

a superpositionofthe DPDWthat travelle d through a homogeneous system plus

the rst order scatter ing of DPDWs f rom optical inhomogeneitie s plus thesecond

order etc Theoptical prop ertie s of the backgroundhomogeneous medium are usually

taken tobetheaverage or mo st common optical propertie s Onegenerally divides

the region of intere st ie the region containingthe inhomogeneity intovoxels The

rst order scattere d DPDWisthen the scatter ingofthe incident DPDW f rom e ach

voxel If theoptical prop ertie s of thevoxel are the sameasthebackgroundthen no

wave i s scattered from thatvoxel Thevoxels are chosen tobesmall enough so that

thescattere d DPDW can b e line ar ize d thatistheamplitudeofthe scattere d waveis

line arly proportional tothechange in theab sorption co ecientandthechange in the

re duce d scatter ing co ecient

Oneway toder ivethe line ar ize d scattere d DPDWistotakethe limiting form of

out in

eq for small radius sphere s To le ading order in k a and k a

scatt

expik jr rj a expik jr r j

s d

r r r vS

scatt s d AC

D jr rj jr r j

out s d

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

v r rcos

a

s

ik ik

D jr rj jr r j r

out s d

sout

s

Here is the dierence in theab sorption co ecientofthevoxel and

a ain aout

background is the dierence in thereduce d scatter ing co ecient

s sin sout

k k and is theangle b etween the line joiningthe source tothevoxel andthe

out

line joiningthedetector tothevoxel Thevolume for a sphere of radius a appears in

eq If thevoxel i s not a sphere then thea must b e replace d bytheactual

volumeofthevoxel

The rst order scat r r r r islinearized byassumingthat

scatt s d

s s

tere d wavereachingthedetector i s foundbysummingthe contr ibutions f rom e ach

voxel When r r r i s linearized then a matr ix equation can b e wr itten for

scatt s d

the rst order scattere d wave Thematr ix equation i s

scatt

B C B C C B

r r M M M

s d a m

scatt

B C B C C B

B C B C C B

B C B C C B

B C B C C B

B C B C C B

r r M M M

s d a m

scatt

B C B C C B

B C B C C B

B C B C C B

B C B C C B

B C B C C B

B C B C C B

B C B C C B

A A A

M M M r r

n n nm am sn dn

scatt

C B C B

N N N

m

s

C B C B

C B C B

C B C B

C B C B

C B C B

N N N

m

s

C B C B

C B C B

C B C B

C B C B

C B C B

C B C B

C B C B

A A

N N N

n n nm

sm

th

r r isthe rst order scattere d wave for the i sourcedetector pair and

si di aj

scatt

are re sp ectively thechange in theab sorption andreduce d scatter ing co ecients

sj

of voxel j relativetothebackground Theelementsofmatr ix M andmatr ix N are

given bythe linearized vers ion of eq Sp ecically

a v expik jr r j expik jr r j

si j j di

M vS

ij AC

D jr r j jr r j D

out si j j di out

Chapter Migration of Dius e Photon Dens ityWaves

and

expik jr r j expik jr r j a

si j j di

N vS

ij AC

D jr r j jr r j

out si j j di

co s

ij

ik ik

jr r j jr r j

si j j di

sout

th th

where r is the position of the j voxel and r and r are theposition of the i

j si di

source anddetector re sp ectively

Thi s samematr ix equation can b e found directly f rom theheterogeneous dius ion

equation When theoptical prop ertie s are spatially varyingthen thephoton dius ion

equation i s

rD r rr D rr rv rr i rvS r

a

Separatingthespatially constantterms tothe lefthandsideandthespatially varying

terms totherighthandsidewe get

v

ao

r r ri r

D D

o o

v r v

a

r S r r r rr r i

ao

s s

D r D v

o

so

Thesolution tothi s equation i s after integrating r r rrbypartsand recog

s

nizingthat r k

o

Z

v r vS exp ik jr r j

a o o s

r Gr r r

D jr r j D

o s o

r

s

rr rGr r dr

so

Thi s equation i s usually solved perturbatively byassumingthat

Thi s i s known as the Bor n approximation Substitutingthi s p erturbative

expans ion into eq andcollectingterms of like order weobtain

vS exp ik jr r j

o o d s

r r

s d

D jr r j

o d s

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Intensity

Time

Power

Frequency

Figure A drawing of a puls etrain f rom a mo delo cke d las er andthe corre sp ond

ingFour ier expans ion

Z

Dr v r

a

r rGr r r r r rGr r dr r r

s d s d s d

D D

o o

Thi s equation for the rst order scattere d wave rst Bor n approximation i s identical

tothe equation thatweobtained bysummingthe limitingformoftheanalytic solution

over all voxels

TimeDomain Me asurements

The DPDWscatter ingtheory i s e as ily extended tothetime domain A puls etrain

of light propagatinginaturbid me dia can b e thoughtofasasuperposition of many

DPDWs with dierentmodulation f requencie s s ee g Thusatimeresolved

me asurementofthepropagation of a light puls e i s an e asy way todeterminethe

f requency re sp ons e of the system To calculatethe response to a puls e of light

we s imply computethescatter ingdue toeach DPDW in parallel

Chapter Migration of Dius e Photon Dens ityWaves

I computed thetemp oral evolution of a light puls e withwidth p s and p er io d

T s in an innitemedium with dierent s ize p erfect ab sorb ers The re sults

indicatethattheme asure d photon uence decre as e s as a re sultofanab sorb er but

thatthedecay rateofthe uence i s relatively unaected byits pre s ence The s e re sults

are cons i stent withthe exp er imental ob s ervations of Liu et al

Photon Migration within the P Approximation

Atoptical wavelengths b etween and nm theabsorption of photons in the

body is generally small compare d tothe corre sp onding scatter ingrate Thusamajor

condition for thevalidityofthe dius ion approximation i s sati se d Thecriter ia i s

sometime s violated in hematomas liver andother regions with large concentrations of

blood where photon ab sorption i s large A more accuratemodel of photon transp ort

i s require d tode scr ib e andanalyze photon migration through the s e systems Thi s

s ection pre s entsthe P solution of the transp ort equation which i s a more

accurateapproximation for photon transp ort than the dius ion approximation I

demonstratethe advantage s and di sadvantage s of the P approximation for analyzing

highly ab sorbing systems I ndthatthe P approximation in general p ermits a more

accuratedetermination of thereduce d scatter ing andab sorption co ecients

a

s

for highly ab sorbing systems ie or systems prob e d atmodulation

a

s

f requencie s in exce s s of to GHz In systems with highly ani sotropic scatter ing

ie hco s i determination of thereduce d scatter ing co ecientusingthe P

approximation gives value s comparable to re sults obtaine d within the dius ion P

approximation

P Theory

The transp ort equation was pre s ente d in s ection eq alongwitha de scr iption

of the P approximation metho d Here I pre s entthesolution of the P approxim

N

ation and di scus s thelimits in whichthe P approximation re duce s tothe dius ion

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

approximation The P equations are not solved becaus e of incons i stencie s that arise

atboundar ie s

Expandingthe radiance Lr t phas e function f and source S r t

terms of the transp ort equation eq in spher ical harmonics andevaluatingthe

integral over d the transp ort equation i s rewr itten as

X

lm

l

r Y q

lm lm lm lm

t

v t

lm

l

th

where g note g is the co ecientforthe l moment

s l a a l

t t

l

of the normalize d phas e function For the HenyeyGreenstein phas e function g g

l

where g is theaverage cosineofthe scatter ingangle s ee appendix B When the

l l

photon scatter ing i s ani sotropic then

t t

Next wemultiply eq by Y andintegrateover Us ingthe ortho

gonality relations for the spher ical harmonics weobtain an innite s et of couple d

line ar dierential equations for that agree with Kaltenbachand Kaschke See

lm

appendix A for the s e calculations Within the P approximation themomentsgreater

than l are ignore d ie weset for l By cons ider ing higher moments

lm

of the radiance the P approximation should b e more accuratethan the dius ion ap

proximation However the P approximation will bre ak down as the ani sotropyofthe

radiance i s incre as e d by incre as ingphoton ab sorption andor the DPDWmodulation

f requency

Workinginthe f requency domain ie t i the equation for in a

homogeneous me diumis

r r rWq rXq r Yq rZq r

where

i

a a a

t t t t t t t

v v

i i i i

a

t t t

v v v v

Chapter Migration of Dius e Photon Dens ityWaves

andthe r ighthands ide of eq contains the momentsofthe source di str ibution

The co ecients W X Y and Z are given in s ection A by eq A eq A

eq A and eq A re sp ectively

Lets as sumethatthe source i s an i sotropic p oint source suchthat q for

l

l For an inniteme diumthesolution of eq i s of theform

expik r

p

r

where k is given by

p

p

k

p

Here I concentrateonthenegative ro ot The positiverootcontr ibutes tothesolution

only within a few meanfreepaths of the source Thi s solution has b een di scus s e d

previously particularly with regards tothe positiverootandtheappropr iate

boundary conditions for s emiinniteme dia

For typical parameters where the dius ion approximation i s known tobevalid

 

j j For example us ing cm cm g and

s a

we s ee that Eq can then b e expanded to rst order

giving

s

k i k

a

t

p dif f

v

This is thewell known wavenumber solution f rom the dius ion equation s ee eq

In the regimewhere the dius ion approximation i s known tobevalid eq

and eq can b e approximated as

t t

i

a

t t t

v

Thus for the s e parameters the P solution re duce s tothe dius ion solution indicating

thatthe dius ion equation i s valid when j j ie

t

j j i

a

v

t t

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

For systems that do not satisfy thi s condition it i s b elieved thatthe full solution of

the P equation would more accurately approximatephoton transp ort through the

system Furthermore f rom thi s condition we s ee thatthe limitsofvalidityofthe

dius ion equation can b e checked by incre as ing relativeto incre as ing

a s

v relativeto and decre as ingthe scatter ing ani sotropy f actor while holding

s

constant

s

Compar i son of P and Dius ion Theor ie s

Totest the us efulness of the P approximation compare d tothe dius ion approxima

tion I rst generated data for known parameters us ing a Monte Carlo computer co de

for photon transp ort in an innite homogeneous system Thecodeisexplained in

s ection andsupplie d in appendix C The Monte Carlo co dewas us e d tondthe

temp oral re sp ons e to a puls e of light injected intoa homogeneous innitemedium for

var ious optical prop ertie s andscatter ing ani sotropie s I then us e d thegenerated data

to compare dius ion theory and P as a function of themodulation f requency

a

byFour ier transformingthedata andthe scatter ing ani sotropyThe compar i son was

madebytting amplitudeandphas e dataversus the sourcedetector s eparation us ing

the P solution and dius ion solution tond and Sourcedetector s eparations

a

s

ranging f rom to cm in step s of cm were us e d

Fig di splays theoptical prop ertie s determine d f rom the MonteCarlodata

us ingthe P approximation andthe dius ion approximation versus the known ab

sorption co ecientofthemedium Re sults are plotte d for data generate d withtwo

dierent ani sotropy f actors All re sults in g are for a mo dulation f requency

of MHz Theanalysis based on the P approximation i s s ignicantly b etter than

dius ion theory atdeterminingthe correct when thescatter ing i s i sotropic ie

s

g andtheabsorption co ecient excee ds of theknown re duce d scatter ing

co ecient A s imiliar dierence i s ob s erve d for thedetermined ab sorption co ecient

For ani sotropic scatter ingg we s ee thatinnding the P approximation

s

i s not as go o d as the dius ion approximation when although P still en

a s

Chapter Migration of Dius e Photon Dens ityWaves

able s a more accuratedetermination of Thisismostlikely a re sult of a premature

a

truncation of the spher ical harmonic expans ion of thephas e function in arr ivingat

the P theory In dius ion theory the ani sotropy i s implicitlycontained in thereduce d

scatter ing co ecient while in the P theory the ani sotropy i s expressed explicitly

Mo difyingthe P theory withthe E approximation di scus s e d byStar may

improvethedetermination of Bas icallywithin the E approximation a delta

s

function i s added tospher ical harmonics expans ion of thephas e function eq

to comp ensate for thetruncation

Toinvestigatethe accuracy of the dius ion approximation andthe P approxim

ation for high mo dulation f requencie s I us e d Monte Carlo data for a system with

 

cm and cm and calculated theoptical propertie s us ingboth

a

s

approximations for f requency comp onents ranging f rom to GHz The re sults are

plotte d in g for a system with i sotropic scatter ing g a andband ani so

tropic scatter ing with g g c and d In the cas e of i sotropic scatter ing

the P approximation i s in general more accuratethan dius ion theoryalthough dif

fus ion theory i s accurateto for mo dulation f requencie s le s s than GHz For

i s more accurately determined by dius ion theory ani sotropic scatter ing however

s

up to GHz while P i s sup er ior for determining Similar trends are ob s erved for

a

dierentabsorption co ecients

Summary

Wehave s een thatthe migration of photons in highly scatter ingme dia can b e tre ated

bythephoton dius ion equation For an intens itymodulate d source the dius ion

equation pre dicts a coherentphoton dens itywaves that propagatespher ically outwards

f rom the source andthi s has b een ob s erved Although micro scopically thephotons

are individually following a random walk macro scopicallythey pro duce a coherent

intens itywave The prop ertie s of thi s intens itywave can b e understood using conven

tional optics Thi s was demonstrate d exp er imentally withthe ref raction of DPDWs

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

Figure A compar i son of dius ion theory clo s e d circle s andthe P approxima

tion op en circle s for ndingthe scatter ingtop andabsorption b ottom prop ertie s

of an innite system is presentedasafunction of the known ab sorption co ecient



of theme dium was xe d at cm The re sults for i sotropic scatter ing are

s

pre s ente d in a and b andthe ani sotropic re sults are given in c and d

Chapter Migration of Dius e Photon Dens ityWaves

Figure Optical prop ertie s determined using dius ion theory stars and P dia

monds are compare d withthe known optical propertie s solid line as a function of

themodulation f requencyThe re sults for i sotropic scatter ing are pre s ente d in a and

b andthe ani sotropic re sults are given in c and d

Boas Dius e Photon Prob e s of Turbid Media Theory andApplications

ata planar interf ace b etween two dierent scatter ingme dia andthe ref raction dif

f raction and scatter ingby spher ical inhomogeneitie s Intere stinglythescatter ingis

accurately mo dele d byananalytic solution of the Helmholtz equation andisanalog

ous to a scalar vers ion of Mie Theory for thescatter ing of electromagneticwaves from

dielectr ic sphere s Exp er imental ob s ervations demonstratethatthi s solution can b e

us e d in conjunction with a s imple imaging algor ithm tocharacter ize spher ical ob

jects Finally welooked at higher order approximations tothe transp ort equation

sp ecically the P approximation and foundthattheapplicabilityofDPDWs could

be extended to prob e highly ab sorbingme dia suchasliver andhematomas