The properties of light propagation in scattering me-dia are important in many areas of physics and en-gineering such as remote sensing of theatmosphere,1,2 studies of interstellar dust,3 industrialproduction monitoring,4 and medical diagnostics.5The Boltzmann transport equation describes the pho-ton transport in the presence of scattering. In thestrongly scattering regime, the Boltzmann transportequation reduces to the diffusion equation, which alsodescribes heat conduction in solids, diffusion of neu-trons in condensed media, and gas diffusion. Near-infrared ~NIR! light is strongly scattered inside mostbiological tissues, so the diffusion equation has beenwidely employed in studies focused on NIR imagingand spectroscopy of living tissue. These applica-tions to medical diagnostics have raised much inter-est in recent years because of a number of promisingpreliminary results and because of their significantpotential.6 Optical measurements in biological tis-
The authors are with the Department of Physics, Laboratory forFluorescence Dynamics, University of Illinois at Urbana-Champaign, 1110West Green Street, Urbana, Illinois 61801-3080.Received 15 April 1996; revised manuscript received 4 Septem-
156 APPLIED OPTICS y Vol. 36, No. 1 y 1 January 1997
sues can be quantitative. In other words, they canprovide absolute values of the optical coefficients thatcharacterize the tissue. These coefficients are theabsorption ~ma! and the reduced scattering ~ms9! coef-ficients, both in units of inverse centimeters. It hasbeen known for a long time that absorption propertiesof blood-perfused tissues can be related to hemoglo-bin content and saturation,7 and recent studies havesuggested that scattering properties are due to themitochondrial compartment8 and may be related toblood glucose concentration.9,10To exploit the capability of optical methods to be
quantitative, it is necessary to have an accurate ex-pression that describes light propagation in tissues.The common expressions are based on the Green’sfunction for the diffusion equation and thereby as-sume that the light source is pointlike. The fact thatlight propagation is not diffusive near the source sug-gests that, to solve the diffusion equation, one shouldconsider an effective photon source characterized by aspatial extension and a source strength that coulddepend on the optical properties of the medium. Inthis work, we have investigated the properties of theeffective source ~spatial extension and strength! thatare consistent with the experimental results. Toseparate the effects of spatial extension and effectivesource strength on the detected signal, we have car-ried out a study in the frequency domain. This ap-proach employs a light source that is intensity
modulated at radio frequency ~120 MHz in thisstudy!. The directly measured parameters are theaverage intensity ~dc! and the amplitude ~ac! andphase ~F! of the light intensity wave. The phaseF isindependent of the source strength, so it can be em-ployed to investigate specifically the spatial exten-sion of the effective photon source. The sourcestrength can then be studied with the dc or ac com-ponent of the intensity. In this study, we have con-sidered the phase and the dc intensity, and we haveemployed optical fibers for both delivering and col-lecting the optical signal. Our measurements areconducted in a liquid turbid medium with opticalproperties similar to those of biological tissue in theNIR. In the theory section, Section 2, we also com-ment on the measurable quantity in our experiment.
2. Theory
A. Measurable Quantity
An optical fiber of numerical aperture N collects pho-tons whose directions of propagation are within thesolid angle 2p *0
arcsinN sin udu. Within this solid an-gle, the photon current dI collected in dV arounddirection V by a fiber of area DA located at r is givenby
dI 5 vu~r, t, V!V z ndVDA, (1)
where v is the speed of light in the medium, u~r, t, V!is the angular photon density @hnvu~r, t, V! is theradiance, where h is Planck’s constant and n is thelight frequency!, and n is the unit vector normal tothe surface of the fiber tip and pointing inside thefiber. The total collected current is
I 5 DA2p *0
arcsin N
vu~r, t, V!cos u sin u du, (2)
where the polar axis is in the direction of n. If weassume that the presence of the optical fiber does notperturb the angular photon density distribution, Eq.~2! shows that the measurable quantity is the radi-ance integrated over the solid angle of collection ofthe fiber. In the diffusion approximation, the radi-ance is given by the sum of an isotropic term and asmaller directional flux:
hnvu~r, t, V! 5hnvU~r, t!
4p23hnvD4p
¹U z V, (3)
where U~r, t! is the photon density @hnvU~r, t! is thefluence rate#, andD is the diffusion coefficient definedas 1y~3ms9!, where ms9 is the reduced scattering coef-ficient. If we define uF 5 arcsin N ~uF , py2! to bethe acceptance angle of the detector fiber and if U }@exp~2kr!#yr, then the photon current collected by a
fiber pointing toward the source ~n 5 r! is given by
I 5 DAvU2
3 F14 ~1 2 cos 2uF! 1 D1 1 krr
~1 2 cos3 uF!G . (4)
The ac amplitude Iac is given by the amplitude of I@and the dc intensity is Idc 5 Iac~v 5 0!#, and thephase F is given by the argument of I. In the spher-ically symmetric case, ¹U is directed along r. Con-sequently, the flux term has its maximum valuealong r, whereas it is 0 in directions perpendicular tor. In other words, if the detector fiber points towardthe source ~n 5 r!, the radiance is
hnvu~r, t, r! 5hnvU~r, t!
4p23hnvD4p
u¹Uu, (5)
and in directions u and f the radiance is
hnvu~r, t, u! 5 hnvu~r, t, f! 5hnvU~r, t!
4p(6)
and is proportional to the photon density. Becausein the diffusion approximation the directional flux issmall compared with the isotropic term, we can con-clude that the detected signal is approximately pro-portional to the photon densityU~r, t! also in the casen 5 r. This discussion is in agreement with Haskellet al.,11 although it deviates from the conclusions ofother authors who consider the photon flux~2vD¹U z n! ~Ref. 12! or the photon current density13to be the measurable quantity. The photon flux de-scribes the net flow of photons through a given sur-face, but an optical fiber is sensitive only to part of thesolid angle, i.e., to a partial current. Liu et al.14 havealso experimentally shown that for the case n5 u, themeasured quantity is proportional to the photon den-sity rather than to the photon flux. As a conclusiveremark, we also note that in the case n5 u the photonflux is 0, whereas the experimentally measured sig-nal is comparable with that measured in the case n5r.
B. Diffusion Equation
The diffusion equation for photon transport in a mac-roscopically homogeneous turbid medium is15
]U~r, t!]t
2 vD¹2U~r, t! 1 vmaU~r, t! 5 q~r, t!,
(7)
where q~r, t! is the source term. The frequency-domain Green’s function for an infinite medium isgiven by16
UG~r, v! 51
4pvDexp~2kr!
r, (8)
where k2 5 ~vma 2 iv!y~vD! and v is the angularmodulation frequency of the source intensity.
1 January 1997 y Vol. 36, No. 1 y APPLIED OPTICS 157
C. Spatially Extended Photon Source
The solution to the diffusion equation @Eq. ~7!# for aspatially extended photon source, described by thesource term q~r!, is given by the convolution betweenq~r! and the Green’s function:
In spherical coordinates, with the polar axis along r,the convolution integral becomes
Uhs~r, v! at r2 .. g2 is thus the same as that of theGreen’s function, i.e., exp~2kr!yr.
Fig. 1. Source fiber that delivers the light to the sample and thedetector fiber that collects the optical signal. The effective photonsource is distributed in a hemisphere ~shaded! centered at the tipof the source fiber and has a radius g.
q~r! p UG~r, v! 5 *0
2p
df9 *0
p
du9 sin u9 *0
`
dr9r92q~r9, u9, f9!1
4pvDexp@2k~r2 1 r92 2 2rr9 cos u9!1y2#
~r2 1 r92 2 2rr9 cos u9!1y2 . (10)
The photon densities for a number of spherically sym-metric photon sources q~r! are reported in Appendix 1.The anisotropic emission of an optical fiber with a
large numerical aperture ~;1! can be modeled by aneffective photon source spatially distributed within ahemisphere of radius g centered at the fiber tip. Inthis way, we account for the directional emission inhalf of the solid angle ~see Fig. 1!. The scatteringcenters inside the medium act as effectively isotropicpoint sources, as has already been suggested,17 sothat we can apply Eq. ~10! to findUhs~r, v! ~where thesubscript hs stands for hemisphere!. The sourceterm q~r9, u9, f9! is written as
q~r9, u9, f9!
5 H3Seff~v!y2pg3,0,
for 0 # r9 # g and 0 # u9 # py2for r9 . g or py2 , u9 # p
,
(11)
whereSeff~v! is the total effective source strength, i.e., thenumber of photons per unit time emitted in the wholevolume of the hemisphere. The result for Uhs~r, v! atr. g and u 5 0 ~i.e., along the fiber axis! obtained by thecomputation of the integral in Eq. ~10! is
Uhs~r, v! 5Seff~v!UG~r, v!3k3g3
3 (F1 1 krS1 1g2
r2D1y2GexpHkrF1 2
2 S1 1g2
r2D1y2GJ 2 kr 1 ~kg 2 1!exp~kg!) .
(12)
We note that in the limit g 3 0, Uhs~r, v! reduces toSeff~v!UG~r, v! as it should. It is also noteworthythat in the limit r3 `, or r2 .. g2, Uhs~r, v! becomesproportional to UG~r, v! with a constant of propor-tionality independent of r. The r dependence of
158 APPLIED OPTICS y Vol. 36, No. 1 y 1 January 1997
3. Experimental Methods
A. Strongly Scattering Material
The strongly scattering material employed in ourstudy is a 9-L liquid suspension containing water,Liposyn 20%, and black India ink. Liposyn is a fatemulsion that acts as a weak absorber and a strongscatterer for NIR light. The reduced scattering co-efficient of the medium is proportional to the concen-tration ~vyv! of Liposyn, where the proportionalityfactor is ;2.2 cm21y% ~vyv! @or, in terms of percent-age of solids content ~s.c.!, ;11 cm21y% ~s.c.!#. Inour study, we have investigated reduced scatteringcoefficients ranging from 5 to 22 cm21 ~with ma 50.035 cm21!. Black India ink is employed to varythe absorption coefficient in the range 0.035–0.14cm21 ~with ms9 5 16 cm21!. The 9-L strongly scat-tering suspension is placed in a rectangular containerof sides 32 cm 3 26 cm 3 12 cm.
B. Frequency-Domain Spectrometer
The light source is a laser diode emitting at 780 nm~Sharp LT023!. Its output, which is amplitude mod-ulated at a frequency of 120 MHz, is coupled to aplastic optical fiber with a core diameter of 2 mm anda numerical aperture of 0.5 ~uF 5 30°!. Intensitymodulation of the laser diode is accomplished by sup-plying a dc current of 50 mA ~which selects a workingpoint just above threshold! and a superimposed 120-MHz oscillating signal. As a result, the averageemitted power is;3mW and the modulation depth isclose to 100%. The radio frequency signal is pro-vided by a frequency synthesizer ~Marconi 2022A!that is phase locked to a second frequency synthe-sizer ~Marconi 2022A!. This latter synthesizer mod-ulates the gain of the photomultiplier tube ~PMT!detector ~Hamamatsu R928! at a frequency of 120MHz 1 400 Hz. The signals from both frequencysynthesizers are amplified by radio frequency ampli-fiers ~ENI Model 403LA!. The beating between thedetected signal at 120MHz and the response function
of the PMT at 120 MHz 1 400 Hz gives rise to aharmonic component at 400 Hz in the PMT outputsignal. This low-frequency component is processedto yield the frequency-domain parameters dc ~aver-age intensity!, ac ~amplitude of the intensity wave!,and F ~phase of the intensity wave! by the use ofpreviously described methods.18
C. Measurement Procedure
The experimental arrangement is shown in Fig. 2.The tips of the source and the detector fibers areplaced deep in the suspension such that we work inan effective infinite geometry. The two fibers faceeach other as shown in Fig. 1. We used two identical9-L containers, one for media of different absorptioncoefficients, and one for media of different reducedscattering coefficients. We investigated seven me-dia withma ranging from 0.035 to 0.14 cm21 ~andwithms9 5 16 cm21! and five media with ms9 ranging from5 to 22 cm21 ~and with ma 5 0.035 cm21!. For eachmedium we measured the intensity signal at source–detector separations ranging from 0 ~fiber tips in con-tact! to 2 cm, at 0.1-cm increments. To perform themeasurement in the required intensity dynamicrange ~;105!, we employed neutral density filters atthe PMT detector ~whose high voltage supply waskept constant throughout the experiment!.
4. Results
A. Determination of ma and ms9
In a previous publication,19 we proposed a multidis-tance measurement protocol for the absolute deter-mination of ma and ms9 in optically dense media.This protocol consists of acquiring data at a number~at least two! of different source–detector separa-tions, from which one can determine the slopes of the
Fig. 2. Experimental arrangement. The laser diode ~LD! emit-ting at 780 nm is intensity modulated at 120 MHz by a frequencysynthesizer ~Synth 1! by means of amplifier A1. The DC Biasselects a working point for the laser diode just above threshold.The tips of the source and the detector optical fibers are immersedin the sample to mimic an infinite geometry. The two fibers faceeach other, and the volume of the sample is ;9 L. The gain of thePMT detector is modulated at 120 MHz 1 400 Hz by synthesizer 2~Synth 2! bymeans of amplifier A2. Neutral-density filters ~F! areused to attenuate the detected signal at smaller source–detectorseparations. The data acquisition card in the computer and thetwo frequency synthesizers are phase locked ~Synch!.
straight lines ln~rIdc!, ln~rIac!, and F as a function ofr. The Green’s function of Eq. ~8! provides expres-sions for these slopes that enable one to calculate maand ms9. As we comment in Appendix 1, a spheri-cally symmetric light source that is confined at r # gdoes not change the expressions for these slopes atr . g. The spherically asymmetric source distribu-tion of Eq. ~11! also does not change the expression forthese slopes at r2 .. g2. An important consequenceis that a measurement of ma and ms9 based on theslopes of ln~rIdc!, ln~rIac!, and F at large r is notaffected by the details of the source spatial distribu-tion. We have thus determined ma and ms9 of ourmedia by applying the multidistance measurementprotocol to the data collected at r between 1.5 and 2.0cm.
B. Spatial Extension of the Effective Photon Source
Because the slopes of the straight lines discussed inSection 3 are not affected by the source spatial dis-tribution and by the source strength, we have focusedour attention on the intercepts. In particular, thephase intercept is not affected by the source strength,so that the phase intercept data give informationspecifically on the source spatial extent. Figure 3shows the experimental phase intercepts as a func-tion of ma @Fig. 3~a!# and ms9 @Fig. 3~b!#. We observethat the phases are relative to the phase measured atr 5 0, i.e., with the source and the detector fiber tipsin contact. For comparison, we also report the phaseintercept predicted by the integrated radiance @Eq.~4!# calculated with uF 5 30° by using UG~r, v! @Eq.~8!# and Uhs~r, v! @Eq. ~12!#. In Uhs~r, v! we set
g 51
Re@k~ma, ms09!#
51
S32 mams09D1y2FS1 1v2
v2ma2D1y2
1 1G1y2 , (13)
where ms09 5 16 cm21 is a constant. This value of g,which reduces to 1y~3mams09!
1y2 for the dc intensity, isof the order of the diffusion length and ranges from0.4 to 0.8 cm for the media considered in this study.The diffusion length characterizes the spatial scale ofthe photon density decay. It also gives an estimateof the distance from the actual source over which thediffusive regime is established. This distance islarger than the photon mean free path between ef-fectively isotropic scattering events, which is given by1yms9. We note here that g, the extension of theeffective source, is a function of ma and v, but sur-prisingly not of ms9 @ms09 in Eq. ~13! is a constant#.The constant value of ms09 5 16 cm21 used in Eq. ~13!provides a good agreement between our model andexperiment, even in the case of media that have dif-ferent values of ms9 @see Fig. 3~b!#. We also observethat Uhs~r, v! with g given by Eq. ~13! accuratelydescribes not only the dependence of the experimen-tal phase intercept on ma and ms9, but also its absolutevalue.
1 January 1997 y Vol. 36, No. 1 y APPLIED OPTICS 159
Fig. 3. Intercept of the phase-versus-r straight line as a function of ~a! ma, ~b! ms9. In ~a!, ms9 is 16 cm21, whereas in ~b!, ma is 0.035 cm
21.The symbols are the experimental values obtained from the phase data taken at r ranging between 1.5 and 2 cm. The curves are the phaseintercepts predicted by diffusion theory for a point source ~dashed curve! and for the effective extended source described in the text ~solidcurve!.
C. Source Strength of the Effective Photon Source
Once the effective source extension has been deter-mined by Eq. ~13!, the intercepts of the intensitystraight lines provide information on the effectivesource strength. Figure 4 shows the experimentalintercepts of ln~rIdc! as a function of ma @Fig. 4~a!# andms9 @Fig. 4~b!#. For comparison, we also show theintercepts predicted by the integrated radiance @Eq.~4!# calculated with uF 5 30° by using UG~r, v! @Eq.~8!# and Uhs~r, v! @Eq. ~12!#. In Uhs~r, v! we haveused the expression of Eq. ~13! for g and we have set
Seff } Sma
ms9, (14)
where S is the actual source strength. This effectivesource strength is compatible with our experimental
results for the intensity intercepts ~see Fig. 4!. Incontrast to the phase intercepts, the absolute valuesof the experimental and the predicted intensity in-tercepts cannot be compared because of the unknowneffect of the instrumental response. For this reason,the predicted intensity intercepts in Fig. 4 are shiftedby an arbitrary offset. The fact that Seff decreaseswith ms9 is reasonable. In fact, the source dimensiong does not change with ms9, and the number of pho-tons emitted per unit time by the effective source isreduced by the scattering-induced absorption. Theincrease of Seff with ma is apparently surprising. Itis the net result of two competing effects determinedby an increase in the absorption coefficient. Thefirst effect is the reduction in the number of photonsemitted per unit time by the effective source becauseof the increased absorption. The second effect is the
Fig. 4. Intercept of the ln~rIdc!-versus-r straight line as a function of ~a! ma, ~b! ms9. In ~a!, ms9 is 16 cm21, whereas in ~b!, ma is 0.035 cm
21.The symbols are the experimental values obtained from the intensity data taken at r ranging between 1.5 and 2 cm. The curves are theintensity intercepts predicted by diffusion theory for a point source ~dashed curve! and for the effective extended source described in thetext ~solid curve!.
160 APPLIED OPTICS y Vol. 36, No. 1 y 1 January 1997
Fig. 5. Symbols represent ~a! the measured phase, ~b! ln~rIdc! as functions of source–detector separation r. The medium has anabsorption coefficient ofma 5 0.035 cm21 and a reduced scattering coefficient of ms9 5 18 cm21. Consequently the radius of the hemisphericeffective photon source is g 5 0.71 cm. The continuous curves have the slopes predicted by diffusion theory and are shifted by an offsetto match the experimental data at r . g. At r , g the experimental points deviate significantly from the linear behavior predicted bydiffusion theory.
reduction in the source dimensions ~g } 1y=ma!,which confines the effective source volume closer tothe actual photon source. This second effect reducesthe probability of photon absorption within the effec-tive source volume.The inclusion of the following effects, ~1! integrated
radiance over the acceptance angle ~,py2! of the op-tical fiber, ~2! hemispheric light source with radius ggiven by Eq. ~13!, and ~3! effective source strengthgiven by Eq. ~14!, results in the following expressionfor the experimentally measured intensity at large r~r .. g! and u 5 0:
I~r, v! } SDAma
k3g3 @1 1 ~kg 2 1!exp~kg!#
3 F14 ~1 2 cos 2uF! 1 Dk~1 2 cos3 uF!G exp~2kr!r
.
(15)
5. Discussion and Conclusion
The experimental results shown in Figs. 3 and 4 showthe inadequacy of the diffusion-equation Green’sfunction in describing the dependence of the inter-cepts of dc intensity and phase on ma and ms9. Suchinadequacy cannot be resolved by modeling the lightsource ~a 2-mm-diameter optical fiber in our case!with an extended photon source of given dimensions.On the contrary, the experimental results can be re-produced by assuming that there is an effective pho-ton source whose spatial extent and strength dependon ma and ms9. We have found that the dimensions ofthe effective photon source g is of the order of thediffusion length, which represents the distance overwhich the photon migration becomes diffusive.Figure 5 shows the different r dependencies of theexperimental data at r , g and r . g for the case ma5 0.035 cm21 and ms9 5 18 cm21. In this case, the
effective source dimension g is 0.71 cm. Because thediffusion-equation Green’s function is accurate onlyat r . g, the different regime at r , g explains thefailure of the Green’s function in predicting the dcand phase intercepts. This observation has impor-tant consequences in quantitative spectroscopy ofturbid media. The measurement of the slopes of thedc and phase straight lines versus r provides accuratevalues of ma and ms9 ~Ref. 19!. However, to measurethese slopes, one needs to acquire data at multiplesource–detector separations at which different vol-umes are sampled by the photon paths. In applica-tions in which the sample is not macroscopicallyhomogeneous, this fact can represent a problem.Consequently algorithms that employ a calibrationmeasurement followed by one measurement at a sin-gle source–detector separation have been proposed.20A similar algorithm has been proposed in steady-state spectroscopy,21 in which the slope and interceptof the dc straight line are used to determine ma andms9. These algorithms are based on the diffusion-equation Green’s function and employ the interceptsas well as the slopes of the straight lines associatedwith the measured quantities. As a result of thisstudy, these algorithms are not expected to provideaccurate results, unless the calibration medium andthe sample have the same optical properties.The properties of the effective photon source ~g and
Seff! have been empirically derived here to provide anaccurate description of our experimental results.However, we have not developed a model startingfrom first principles, which explains the physical or-igin of the discrepancy between the experimentaldata and the predictions of diffusion theory. It ispossible that the acquisition geometry ~relative ori-entation of the source and the detector fibers! and thenumerical apertures of the optical fibers play a role inour observations that has not been quantified. How-
1 January 1997 y Vol. 36, No. 1 y APPLIED OPTICS 161
ever, the effects that we observed are so large thatthey cannot be ignored. Other researchers have ob-served similar effects ~nonzero phase intercepts in aplot of phase versus source–detector separation! in ageometry in which the source and detector fibers werenot facing each other.11 Their phase intercepts at100MHzwere larger than110°, and this result couldnot be explained.In conclusion, this work points out an important
limitation of the Green’s function solution ~i.e., thesolution for a point-like source! to the diffusion equa-tion for measurements of photon transport in turbidmedia. More work is required for investigating thephysical origin of this limitation and to assess theinfluence of the specific source–detector configura-tion used to measure the optical parameters of a tur-bid medium.
Appendix A: Photon Densities for Some SphericallySymmetric Photon Sources
In this Appendix, we consider four spherically sym-metric photon sources described by q~r!. We indi-cate the source strength at angular frequency v withS~v! and the linear dimension of the source with g.For each of these source terms, we have derived thephoton density in the diffusion regime by applyingEq. ~10!.
1. Spherical Source with Constant Strength
The source term for a constant photon source in thesphere of radius g is
qP~r! 53S~v!
4pg3 P~ryg!, (A1)
where P~ryg! is defined to be 1 if r # g and 0 if r . g.The resultant photon density at r . g is
UP~r, v! 5 SUG~r, v!3
k3g3 @kg cosh~kg! 2 sinh~kg!#,
(A2)
where k2 5 ~vma 2 iv!y~vD! and UG~r, v! is given byEq. ~8!.
2. Spherical Source with a Strength Linearly Decreasingfrom the Center
The source term for a spherical photon source withradius g and with a strength that decreases linearlywith distance from the center is
qL~r! 53S~v!
pg3 L~ryg!, (A3)
where L~ryg! is defined to be 1 2 ryg if r # g and 0 ifr . g. The resultant photon density at r . g is
UL~r, v! 5 SUG~r, v!12k4g4 @kg sinh~kg!
2 2 cosh~kg! 1 2#. (A4)
162 APPLIED OPTICS y Vol. 36, No. 1 y 1 January 1997
3. Distributed Source with a Strength DecreasingStronger than Exponentially from the Center
The source term for a spherically symmetric sourcewhose strength decreases with r like the diffusion-equation Green’s function is
qexp~r! 5S~v!
4pg2
exp~2ryg!
r. (A5)
This source term represents an indefinitely distrib-uted source with a source strength decaying with r asexp~2ryg!yr. The characteristic decay length of thissource is g, and the characteristic decay length of theGreens’ function @Eq. ~8!# is 1yRe~k!. The photondensity that corresponds to this source term is
Uexp~r, v! 5 SUG~r, v!
expF2~1 2 kg!rgG
1 2 k2g2 .
(A6)
4. Spherical Shell
The source term for a spherical shell of radius g is
qshell~r! 5S~v!
4pg2 d~r 2 g!, (A7)
where d~r2 g! is the Dirac delta. The correspondingphoton density at r . g is
Ushell~r, v! 5 SUG~r, v!sinh~kg!
kg. (A8)
We observe that in the limit g 3 0, all four sourcedistributions 1-4 become a photon point source. Inthis limit, as expected, we have
limg30
UP 5 limg30
UL 5 limg30
Uexp 5 limg30
Ushell
5 S~v!UG~r, v!. (A9)
We also note that at distances larger than the effec-tive source extent ~i.e., at r . g for sources 1, 2, and4! the r dependence of the photon density isexp~2kr!yr. This result holds also for source 3, pro-vided that 1yRe~k! .. g. This result indicates thatfor spherically symmetric sources, the slopes of thestraight lines ln~rIdc!, ln~rIac!, and F as a function ofr are not influenced by the details of the source spa-tial distribution.
This work was conducted at the Laboratory forFluorescence Dynamics, which is supported by theU.S. National Institutes of Health, grant RR03155and by the University of Illinois at Urbana-Champaign. This research is also supported by U.S.Institutes of Health grant CA57032. We thank Al-bert Cerussi for providing the fiber-coupled,intensity-modulated laser diode.
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