Spatial Sampling by Diffuse Photons

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Spatial sampling by diffuse photons

Aubrey A. Cox and Douglas J. Durian

The contribution of some region in an opaque multiple-scattering sample to the detected signal isconsidered. Because diffusion theory gives only the total photon concentration and not the fraction ofthat which ultimately reaches the detector, it must be supplemented. We show how to do so by makingfurther use of the assumption that photon migration is Markovian. This procedure is illustrated forillumination–detection geometries and scattering parameters of interest for diffusing-light spectrosco-pies. Specifically, we explore slab geometries with plane-wave illumination and detection as well as asemi-infinite sample with point illumination and detection. For the former the photon behavior as afunction of slab thickness, scattering anisotropy, absorption, and boundary reflectivity is predicted andshown to compare well with Monte Carlo random-walk simulations. © 2001 Optical Society of America

OCIS codes: 170.5280, 170.7050, 030.5620.

5,6 7

t

1. Introduction

Diffusing-light spectroscopies are powerful opticaltools for probing opaque media in which photonmigration can accurately be described as a diffusionprocess. Diffuse-transmission spectroscopy re-veals structural properties in terms of the absorp-tion length la and the transport mean-free path1 l*equal to lsy~1 2 g!, where ls is the scattering length

and g is the average cosine of the scattering angle#.iffusing-wave spectroscopy reveals dynamicalroperties in terms of the mean-squared displace-ent,2,3 ^Dr2~t!& of scattering sites in a time interval

t. These techniques have been applied to manyforms of soft condensed matter ranging from colloi-dal suspensions to foams, emulsions, and granularmedia. An implicit assumption is that the struc-tures and the dynamics of the materials are spa-tially homogeneous. It is therefore important toevaluate the contribution of potentially inhomoge-neous regions, for example, near a boundary, to thedetected signal. Similar issues arise in media suchas a biological tissue with tumor or trauma in whichspatial variation of the scattering or the absorptionis to be explicitly imaged.4 In that field conceptsrelated to spatial sampling include the statistics of

The authors are with the Department of Physics and Astronomy,University of California, Los Angeles, California 90095-1547.D. J. Durian’s e-mail address is [email protected].

Received 3 November 2000; revised manuscript received 16April 2001.

0003-6935y01y244228-08$15.00y0© 2001 Optical Society of America

4228 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

the depth probed, the photon-hitting density, thephoton-path density distribution,8 and the photon-measurement density function.9

Here, we address the spatial-sampling problemby making straightforward use of diffusion theory.Specifically, we take advantage of the fact that mi-grating photons are Markovian, meaning that theyscatter and take random steps in a manner inde-pendent of the details of their previous history, suchas the time spent in the sample. Prior studies in-voked sophisticated mathematical tools, such as ex-act treatment of random walks on a lattice,5,6

multipoint–multitime Green’s functions,7 and per-urbation methods.8,9 Although such tools are im-

portant for the imaging of inhomogeneous mediawith complicated boundaries and illumination–detection geometries, they are not necessary for thediffusing-light spectroscopies of primary interesthere.

In this paper, we begin with a discussion of theingredients of diffusion theory. We then exam-ine spatial sampling for the plane-in–plane-outillumination–detection configuration for a constant-thickness slab of scattering medium and comparepredictions with extensive random-walk computersimulations. In the final section, we consider thepoint-in–point-out illumination–detection geometryfor a semi-infinite medium. A secondary point ofthis paper is to emphasize that it is now possible toconstruct diffusion theories that accurately model thetransport in thin samples ~L , 10l*! in which theeffects of ballistic propagation, scattering anisotropy,and boundary reflections are all especially important.

iEt

ta

td

c

bb~tsoa

tta

stmpap

2. Diffusion Theory

The key approximation of diffusion theory is that thephotons in a volume element travel with equal prob-ability in all directions. Given this, a complete de-scription can be made in terms of the number densityof diffusing photons, f~r, t!. For short times anddistances and in cases of strong absorption, the dif-fusion approximation breaks down, but it can be rem-edied to a large extent by the replacement of thediffusion equation by the following telegrapher’sequation10:

D0¹2f 5 a

]2f

]t2 1 ~1 1 2ama9!]f

]t1 ma9~1 1 ama9!f.

(1)

Here, length is measured in units of l*, time is mea-sured in units of l*yc, ma9 5 l*yla is the dimensionlessabsorption coefficient, D0 5 1y3 is the dimensionlessdiffusion coefficient, and a 5 D0 is the constant thatis needed to obtain the correct ballistic-propagationspeed c. Other values of a have been recommended,ncluding most simply a 5 0, but the coefficients inq. ~1! cannot otherwise be changed without ruining

he long-time diffusion limit or self-consistency11,12

~e.g., the agreement between predictions for continu-ous illumination and the time average of a delta-function pulse illumination!. To complete theheory self-consistently, one should take the bound-ry conditions to be10

0 5 F1 1 zen̂ z ¹ 1 aSma9 1]

]tDGf~r, t!ubound, (2)

and the emerging flux of photons should be computedas

J~r, t! 5 ~D0yze!f~r, t!ubound. (3)

In Eqs. ~2! and ~3!, n̂ points perpendicularly awayfrom the sample, and ze is a number known as theextrapolation-length ratio. The ratio’s value de-pends on certain averages of the angle-dependentboundary reflectivity and can be found experimen-tally by measurement of the angular distribution ofdiffusely transmitted light.13 Note that the ordi-nary diffusion equation and the extrapolation bound-ary conditions, plus Fick’s law, are all recovered inthe limit of long-time and weak absorption in whichthe terms involving a are negligible. Also note thathis system of equations applies in other than d 5 3imensions just by one’s taking D0 5 1yd; for d 5 1

this description of transport is exact.For a given sample and illumination geometry the

resultant photon concentration f~r, t! can be com-puted from Eqs. ~1! and ~2!; the observable flux ofphotons can then be computed from f~r, t! and Eq.~3!. But what about the spatial sampling? Morespecifically, how many of the photons f~r, t!dV in thevolume element at ~r, t! ultimately reach the detec-tor? This number is simple to compute after onerecalls that photon diffusion is Markovian, indepen-

dent of prior history. Evidently, the desired quan-tity is f~r, t!P~r!, where P~r! is the probability that aphoton created at r will diffuse to the detector. If fwere computed by the averaging of the point responseover a source distribution, as is usual, then knowl-edge of P would already be available. Because spa-tial sampling thus depends doubly on the diffusionapproximation, it is further important that the im-plementation be as accurate as possible, as in Eqs.~1!–~3!. We now demonstrate two important geom-etries.

3. Slab Geometry

The most common geometry for diffusing-light spec-troscopies is one in which the sample has a constantthickness L, is essentially infinite in the other twodirections, and is subjected to a continuous beam oflight at normal incidence. Illumination and detec-tion are arranged so that there is no discriminationagainst photons on the basis of their lateral migra-tion within the slab. For example, an expandedbeam can be brought in, and a lens can be used tocollect light that emerges from a point; more simply,an unconditioned beam can be brought in, and thediffuse light can be collected onto a detector with nointervening optics at all. Such configurations areoften referred to as plane-in–plane-out because thegeometry is effectively one dimensional, with the pho-ton density f~z! versus the depth z constituting aomplete description.

Because the source of diffusing photons is providedy scattering out of the incident ballistic beam, let’segin by considering the concentration of ballisticunscattered! photons versus the depth. With depthhis beam must decay exponentially because of bothcattering and absorption. Photons that reach thepposite side of the slab can reflect with some prob-bility Rb, so, in general, there are two ballistic

beams: one in the incident 1z direction and one inhe 2z direction. Summing over multiple reflec-ions gives the concentration of these ballistics beamss

f ~b1!~ z! 5 f0

11 2 F2 exp@2z~1yls 1 1yla!#, (4)

f~b2!~z! 5 f0

F1 2 F2 exp@2~L 2 z!~1yls 1 1yla!#, (5)

where F 5 Rb exp@2L~1yls 1 1yla!# is the probabilitythat a photon will cross the sample and reflect and f0is the strength of the source as it enters the sample~which, of course, is 1 2 Rb times less than thestrength of the source as it hits the sample!.

Six fates are possible for the photons that enter theample: Either ballistically ~before scattering out ofhe ballistic beams! or diffusely, they may be trans-itted, backscattered, or absorbed. For ballistic

hotons the escape probabilities are given by Eqs. ~4!nd ~5! when evaluated at the boundary and multi-lied by ~1 2 Rb!; the absorption probability can be

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4229

~b1!

f

astc

wi~Ra

ts

ntbtdcd

4

found by one’s carrying out the integral *@f ~z! 1~b2!~z!#dzyla across the sample:

Tb 51 2 Rb

1 2 F2 exp@2L~1yls 1 1yla!#, (6)

Bb 5~1 2 Rb! F

1 2 F2 exp@2L~1yls 1 1yla!#, (7)

Ab 51

~1 2 F!~1 1 layls!$1 2 exp@2L~1yls 1 1yla!#%.

(8)

The value of Tb is a direct measure of the ability to seethrough the slab by use of unscattered photons. Thefraction of incoming photons that are scattered musteventually be diffusely transmitted, backscattered, orabsorbed; therefore the fraction can be found from1 2 ~Tb 1 Bb 1 Ab!, or alternatively by one’s carryingout the integral *@f ~b1!~z! 1 f~b2!~z!#dzyls across thesample:

Td 1 Bd 1 Ad 51

~1 2 F!~1 1 lsyla!

3 $1 2 exp@2L~1yls 1 1yla!#%. (9)

Note that the differentials in the integrals just aboveEqs. ~6! and ~9! are dzyla and dzyls, respectively.This is because the probability for a photon to beneither scattered nor absorbed in transversing a dz is1 2 dz~1yla 1 1yls! 5 exp@2dz~1yla 1 1yls!#.

All six probabilities ~ballistic and diffuse transmis-sion, backscattering, and absorption! are importantfor thin samples with strong absorption. The ne-glect of such possibilities has, in the past, lead toexaggerated claims regarding the breakdown ofdiffusion approximations for thin samples as well asattempts to fix the damage by, for example, theintroduction of a thickness-dependent diffusioncoefficient or extrapolation-length ratio as a fittingparameter, as was described in Ref. 14. Exag-gerated claims have also arisen from a lack of

Tzp5

@1 1 ~D02 1 D0Dfze!ma9yD0#sinh~ zpÎa

@1 1 ~D02 1 ze

2!ma9yD0#sinh~L9Î

Bzp5

@1 1 ~D02 2 D0Dfze!ma9yD0#sinh@~L9

@1 1 ~D02 1 ze

2!ma9yD0#s

f ~d1!~ z! 5f0

~d1!

D0H$~1 1 D0m9a!sinh@ zÎa#yÎ

$~1 1 D0m9a!sinh@~L9 2 z

f ~d2!~ z! 5f0

~d2!

D0H$@~1 1 D0m9a!sinh@ zÎa#yÎ

$~1 1 D0m9a!sinh@~L9 2 z!


self-consistency or from improper implementa-tion of boundary conditions, as was described inRef. 15.

Let us now turn to the diffuse photons. The firststep is to find the concentration profiles for photonsscattered out of either ballistic beam at a particulardepth. As is evident from the above discussion, thestrength of the diffuse source created by the scatter-ing out from the plus or the minus beam betweendepths zp and zp 1 dzp is simply f0

~d6! 5 f~b6!~zp!dzpyls. For time-independent problems Eq. ~1! impliesthat the diffuse density must be composed of termslike exp~6z=a! with a 5 ma9~1yD0 1 ma9!. The re-sult must, by Eq. ~2!, extrapolate to zero at a distancezey~1 1 D0ma9! outside the sample. And the overallproportionality constant must, by Eq. ~3!, give thedvertised diffuse-transmission or diffuse-back-cattering probabilities. It is simple to verify thathese requirements are satisfied by the following con-entration profiles:

here L9 5 Lyl*. The four transmission probabil-ties that appear as factors in expressions ~10! and11! are not determined by Eqs. ~1!–~3! alone.ather, they are set by the size of the discontinuityt zp per unit source, Df 5 @f ~d6!~zp

6! 2f ~d6!~zp

7!#yf0~d6!. For L . l* this discontinuity is

given by D0Df 5 g, which is the average cosine ofhe scattering angle; for thinner samples it islightly smaller.16 This is the only place that scat-

tering anisotropy explicitly enters into the problemoutside of l* 5 lsy~1 2 g!. Physically the disconti-

uity arises because, when the scattering is aniso-ropic, more diffuse photons will be found justeyond the source point, where they scatter out ofhe ballistic beam. This process is discussed inetail in Ref. 16 and can also be seen in the exactontinuum description of transport in a one-imensional system.17 In terms of the discontinu-

ity at the source point the diffuse-transmission andthe diffuse-backscattering probabilities are

1 ~ ze 1 D0Df!cosh~ zpÎa!

Îa 1 2ze cosh~L9Îa!, (12)

!Îa#yÎa 1 ~ ze 2 D0Df!cosh@~L9 2 zp!Îa#

L9Îa!yÎa 1 2ze cosh~L9Îa!. (13)

ze cosh@ zÎa#%Bzpz # zp

yÎa 1 ze cosh@~L9 2 z!Îa#%Tzpz $ zp

, (10)

ze cosh~ zÎa#%TL2zpz # zp

Îa 1 ze cosh@~L9 2 z!Îa#%BL2zpz $ zp

, (11)

!yÎa

a!y

2 zp

inh~

a 1

!Îa#

a 1

Îa#y

tsdcf

Edbtrpsguec

p

~pbd

lt

tts

a

Note that TL2zpis given by one’s substituting L9 2 zp

for zp in the expression for Tzp, and the method is

similar for BL2zp.

The next step is to average Eqs. ~12! and ~13! overhe source distribution. The strength of the pointources was noted above to be proportional to theifferential dzp; therefore the total diffuse-photononcentration field is found by one’s carrying out theollowing integration over zp:

f~d!~ z! 5 *0

L9

f ~d1!~ z! 1 f ~d2!~ z!. (14)

valuating the result at z 5 0, L9 gives the totaliffuse-backscattering and diffuse-absorption proba-ilities, respectively, through Eq. ~3!. Even thoughhe integrals are of just exponential functions, theesults are cumbersome and are not given here. Inractice, it is simple to load the above formulas into aymbolic-manipulation program, carry out the inte-rals exactly, and use the results for numerical eval-ation; we expect that the examples shown providenough guidance to make this unnecessary. In thease of very strong anisotropy, D0Df 5 g 3 1, the

results are simple enough to quote. There will be noballistic transmission, backscattering, or absorptionbecause all incoming photons will immediately bescattered out of the beam at z 5 0. The diffuseconcentration, transmission, and backscattering canthus be found by one’s taking zp 3 0 in expressions~10!, ~12!, and ~13!, respectively:

The discontinuity in concentration at z 5 0 disap-ears for the general case of D0Df , 1 and is replaced

with a maximum near ^zp& 5 ~l*yls 1 l*yla!21, whichis the average penetration depth.

The above predictions can be compared with MonteCarlo random-walk simulations. This technique is astraightforward means of solving the transport prob-lem exactly and is limited by only counting statistics,provided that the wavelength of light is small com-pared with the transport mean-free path. Our sim-ulation methods are standard and were described inseveral of our earlier papers.10,12,13,16 As a severetest, we illustrate in Fig. 1 the results for a thin slab,L9 5 Lyl* 5 3, of a material with scattering anisot-

f~d!~ z! 5f0

D0HzeBd

$~1 1 D0m9a!sinh@~L9 2 z!Î

Td 51 1 z

@1 1 ~D02 1 ze

2!ma9yD0#sinh~L

Bd 5@1 1 ~D0

2 2 ze!ma9yD0#sinh~L9

@1 1 ~D02 1 ze

2!ma9yD0#sinh~

Ad 5 1 2 ~Td 1 Bd!,

ropy, absorption, and boundary reflections that areall strong: g 5 2y3, ma9 5 l*yla 5 0.1, and R 5 0.5independently of angle!, respectively. The upperlot in Fig. 1 shows the photon concentration in theallistic beams; the simulations exactly obey the pre-ictions of Eqs. ~4! and ~5! by construction. The

ower plot in Fig. 1 shows the diffuse-photon concen-ration profile based on 4.5 3 108 random walks. As

a check on the simulations, we also collected a histo-gram of locations at which random walks wereabsorbed because such histograms should be propor-tional to the diffuse-photon concentration. Indeed,the two simulation curves agree, and both are in goodaccord with the prediction of Eq. ~14!. The diffuse-ransmission and diffuse-backscattering probabili-ies are also in good accord with predictions. Theimulated values are multiplied by zeyD0 and plotted

at z 5 0 and z 5 L9, respectively, in Fig. 1. Evidentlythese products agree well with the end points of thepredicted profile, as expected from Eq. ~3!. De-

Fig. 1. Ballistic- and diffuse-photon concentrations plotted versusthe depth z into a slab of thickness L 5 3l*. The optical param-eters are R 5 0.5, g 5 2y3, and ma9 [ layl* 5 0.1. The solid circlest z 5 0 and z 5 L are the predicted diffuse-photon concentrations

based on Eq. ~3! and the simulation results for diffuse backscat-tering and transmission, respectively. The good agreement im-plies that the predicted backscattering and transmissionprobabilities are accurate.

z 5 0a#yÎa 1 ze cosh@~L9 2 z!Îa#%Td z $ 0

, (15)

e

9Îa!yÎa 1 2ze cosh~L9Îa!, (16)

Îa!yÎa 1 ~ ze 2 1!cosh~L9Îa!

L9Îa!yÎa 1 2ze cosh~L9Îa!, (17)

(18)


s

adddthctb

4

spite numerous claims to the contrary, it is possible toaccurately model transport in thin slabs, even withstrong absorption; the key is to keep careful track ofthe six possible fates of incoming photons and to usea self-consistent diffusion theory.

Now we address the spatial-sampling problem.The number of those photons at z that are ultimatelytransmitted ~backscattered! is given by the totalnumber of photons f~d!~z! times the probability of aphoton that is created at z to be transmitted ~back-cattered!. This outcome is simply f~d!~z!Tzp5z,

where the transmission probability is given byEq. ~12!; the approach is similar for backscattering.Here the discontinuity at zp 5 z must be set to zero,i.e., Df 5 0, because the diffuse photons that arecreated at z do not come from a beam but rather fromall surrounding directions. These predictions maynow be compared with simulations.

The photon behavior plotted versus the samplethickness L is shown in Fig. 2. The results are for

Fig. 2. Diffuse-photon concentration and the spatial samplingplotted versus the depth zyL into the slab for several slab thick-nesses, as labeled. The boundary reflectivity is that of a water–air interface ~ze 5 1.6769!. Other system parameters are g 5 0nd ma9 5 0. The dashed curve represents the predicted totaliffuse-photon concentration; the dotted curve represents the pre-icted spatial sampling for backscattered photons; the dotted–ashed curve represents the predicted spatial sampling forransmitted photons; the solid curves represent the correspondingistograms from Monte Carlo random-walk simulations; the solidircles at z 5 0 and z 5 L are the predicted diffuse-photon concen-rations based on Eq. ~3! and the simulation results for diffuseackscattering and transmission, respectively.


isotropic scattering, g 5 0, with no absorption, ma9 50, and with boundary reflections given according toFresnel’s law for a water–air interface. For thickslabs, f~d!~z! approaches a line, and the spatial sam-pling for transmission approaches a parabola thatpeaks at Ly2. The spatial sampling for backscatter-ing is concentrated near the incident boundary andapproaches a parabola with a minimum of zero at z 5L. The agreement between similation and predic-tion becomes better and better for thicker samplesbut is still very good for thin samples. Most of theerror is in regard to the spatial sampling experiencedby the backscattered photons.

The photon behavior plotted versus the scatteringanisotropy, i.e., versus the average cosine g of thescattering angle, is shown in Fig. 3. The results arefor a fairly thin slab, L 5 10l*, with no absorption, ma95 0, and with boundary reflections given according toFresnel’s law for a water–air interface. As the scat-tering anisotropy is increased, the peak in f~d!~z!moves to smaller values of z in fairly good accord withpredictions, even though the transmission and the

Fig. 3. Diffuse-photon concentration and the spatial samplingplotted versus the depth z into the slab for several degrees ofscattering anisotropy, as labeled. The boundary reflectivity isthat of a water–air interface ~ze 5 1.6769!. Other system param-eters are L 5 10l* and ma9 5 0. Dashed curve: predicted totaldiffuse-photon concentration; dotted curve: predicted spatialsampling for backscattered photons; dotted–dashed curve: pre-dicted spatial sampling for transmitted photons; solid curves:corresponding histograms from Monte Carlo random-walk simu-lations; solid circles at z 5 0 and z 5 L: predicted diffuse-photonconcentrations based on Eq. ~3! and the simulation results fordiffuse backscattering and transmission, respectively.

wtti

sa

backscattering probabilities do not change apprecia-bly.

The photon behavior plotted versus the absorptionma9 is shown in Fig. 4. The results are for a fairlythin slab, L 5 10l*, with isotropic scattering, g 5 0,and with boundary reflections given according toFresnel’s law for a water–air interface. As absorp-tion increases, the concentration on the transmissionside decreases, as does the transmission probabilityitself. And the curve that represents the spatialsampling experienced by the transmitted photons be-comes progressively flatter, as eventually all photonsare absorbed except the so-called snakelike photonsthat travel a nearly straight path across the slab.As before, agreement between prediction and simu-lation captures the trends quantitatively to a highdegree.

Finally, the photon behavior plotted versus theboundary reflectivity R ~taken independently of an-gle! is shown in Fig. 5. The results are for a fairlythin slab, L 5 10l*, with isotropic scattering, g 5 0,and no absorption, ma9 5 0. As the reflectivity in-creases, the curve that represents the concentrationof photons in the slab builds up and, along with thatfor the spatial sampling, becomes progressively flat-ter. The extrapolation length also grows in accordwith Eq. ~2!. The agreement between prediction and

Fig. 4. Diffuse-photon concentration and the spatial samplingplotted versus the depth z into the slab for several levels of ab-orption, as labeled. The boundary reflectivity is that of a water–ir interface ~ze 5 1.6769!. Other system parameters are L 5

20l* and g 5 0. The same curve and symbol representations thatwere used for Fig. 2 are used here.

simulation is good and becomes even better for higherreflectivity. In general, diffusion theories improveas the number of scattering events experienced by atypical photon increases. Proper treatment ofboundary reflections is crucial to achieving the gainin accuracy that is due to the extra scattering causedby reflections.

4. Semi-Infinite Geometry

The most common geometry for biomedical imaging–dosing applications is one in which the sample isessentially semi-infinite with a planar interface andis subjected to illumination by a tightly focused beamof light at normal incidence. All photons enteringthe sample are therefore eventually either backscat-tered or absorbed. The detection of backscatteredlight is accomplished through a lens or an opticalfiber and is performed versus the radial distance fromthe illumination point. Such a configuration is in-herently three dimensional with none of the simpli-fying symmetry of the slab geometry.

The diffusion theory of Eqs. ~1!–~3! applies equallyell to the semi-infinite geometry. Implementing

he proper boundary conditions and averaging overhe source distribution requires straightforward butnvolved mathematical analysis.18 Instead, we fol-

low a common procedure and use the method of im-ages to make the concentration vanish at ze outside

Fig. 5. Diffuse-photon concentration and the spatial samplingplotted versus the depth z into the slab for several angle-independent reflectivities, as labeled. The system parameters areL 5 10l*, g 5 0, and ma 5 0. The same curve and symbol repre-sentations that were used for Fig. 2 are used here.


19

tcefittas

w

z

0mttp

aFt

dag

Tg

4

the sample ; this approach only approximates therue boundary conditions of Eq. ~2!. Still followingommon practice, we further approximate by not av-raging over the distribution of source points. De-ning the coordinate system so that the interface is inhe z 5 0 plane means that the point source must beaken at rs 5 ~0, 0, zp! with zp 5 1y~1 1 ma9! being theverage penetration depth. The required imageource is then at r#s 5 ~0, 0, 2zp 2 2ze!. The diffuse-

photon concentration at r 5 ~x, y, z! per unit sourceis then approximately the sum of the source plus theimage terms

f~r!yf0 5exp@2ur 2 rsu~ma9yD!1y2#

4pDur 2 rsu

2exp@2ur 2 r# su~ma9yD!1y2#

4pDur 2 r# su, (19)

where D 5 D0y~1 1 D0ma9!. On the basis of expres-sion ~19! and Fick’s law the probability for a photonthat is created at r to exit through an area element atre 5 ~xe, ye, 0! is

Br3re5

z4pur 2 reu3

@1 1 ur 2 reu~ma9yD!1y2#

3 exp@2ur 2 reu~ma9yD!1y2#

1z 1 2ze

4pur# 2 reu3@1 1 ur# 2 reu~ma9yD!1y2#

3 exp@2ur# 2 reu~ma9yD!1y2#, (20)

here r# 5 ~x, y, 2z 2 2ze! is the image of r. Thespatial-sampling function for photons entering at rsand exiting at re is then just the product of Eqs. ~19!and ~20!. In addition, to an explicit dependence on

p and ze, which describe source and boundary effects,respectively, this result is somewhat different fromthat of Eq. ~13! of Ref. 8.

A single example of spatial sampling for the semi-infinite geometry is shown in Fig. 6. The systemparameters are chosen as in Ref. 8: l* 5 1 mm, ma 5.002 mm21, and a source–detector separation of 10m. The sampling is strongly concentrated near

he source and the detection points. Halfway in be-ween the maximum sampling is at a depth of ap-roximately 3 mm.

5. Conclusions

Spatial-sampling probabilities for homogeneous mediacan easily be computed by a double application of dif-fusion theory. First, compute the diffuse-photon con-centration field for a point source; if desired, averageover the distribution of source points. Next, computethe probability for photons that are created at eachposition to reach the detector and then multiply by thelocal concentration. This approach is exact in d 5 1dimensions and is very accurate in d 5 3 dimensions,s can be judged from the data shown in Figs. 2–5.urthermore, the technique is conceptually simplerhan those in previous studies.5–9 The examples for


homogeneous media that have been shown here arealso complementary to previous research in that thesource distribution, the boundary conditions, and thescattering anisotropy are all explicitly included.Such effects are especially important for fairly thinslabs, 5 , Lyl* , 15, that are typically used iniffusing-light spectroscopies. We believe that the ex-mples plotted in Figs. 2–5 provide a comprehensiveuide to practitioners of these methods.

We thank Joe Rudnick for helpful conversations.his study was supported by NASA through Micro-ravity Fluid Physics grant NAG3-1419.

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5. G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of pene-tration depth of photons remitted from irradiated tissue,” J.Mod. Opt. 36, 349–359 ~1989!.

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Fig. 6. Contour diagram for spatial sampling in the y 5 0 planefor a point-in–point-out semi-infinite geometry in which diffusephotons are created at rs 5 ~0, 0, 0.998 mm! and are detected atre 5 ~10 mm, 0, 0!. The optical parameters are l* 5 1 mm andma 5 0.002 mm21.

9. S. R. Arridge, “Photon-measurement density functions,” Appl.

1

approximation fail to describe photon transport in random
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Spatial Sampling by Diffuse Photons