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