Monte Carlo study of light propagation in optically thickmedia: point source case
Giovanni Zaccanti
The results of a Monte Carlo study on light propagation in dense turbid media are presented. Thecalculations refer to the radiation emerging from a spherical scattering cell containing the diffusing medium(no diffusers outside the cell are considered) in whose center a point source is placed. Both the total scatteredpower emerging from the sphere and the impulse response were evaluated for a large range of optical depthsand different types of diffuser. The results pertaining to both the radiance at the surface of the scattering celland the impulse response are described by simple empirical relations. The results suggest a method ofmeasuring the albedo and the asymmetry factor of the diffusing medium. A comparison with some results ofthe diffusion approximation is also presented. Key words: Scattering, multiple scattering, Monte Carlo.
1. IntroductionThe study of light propagation in optically thick
media is of great interest in many fields such as atmo-spheric and ocean optics, optical communications, andscattering in biological tissues. One mathematicaltechnique for describing light propagation in denseturbid media is the radiative transfer theory. Thegeneral solution, however, is not known and accuratesolutions are limited to particular simple conditions.Approximate methods are generally used such as dif-fusion approximation or the discrete ordinate meth-od.' Another approximate method to describe photonmigration through dense turbid media is that recentlydeveloped by Bonner et al.2 based on a probabilisticmodel (the random-walk model).
Numerical methods, like Monte Carlo methods, arealso important both for investigating the limits of ap-plicability of the approximate solutions and to dealwith situations for which the approximate solutionsare not suitable. The main advantage of Monte Carlomethods is their ability to deal with different situa-tions regarding (1) the source and receiver characteris-tics, (2) the boundary of the medium, and (3) thescattering characteristics of the medium. Another ad-vantage is that Monte Carlo codes do not require the
The author is with University of Florence, Physics Department, 3Via S. Marta, I-50139 Florence, Italy.
use of a supercomputer, and personal or minicomput-ers with limited memory and speed can be used. Themain disadvantage is that calculation time can becomeprohibitively long.
In this paper the results of a study on propagation ofthe light emitted by an isotropic point source through adiffusing medium with a large optical depth are report-ed. The considered point source is in the center of asphere containing the diffusing medium. No diffusersare considered outside the sphere, so photons emittedfrom the surface are assumed not to return to thescattering medium (the same index of refraction out-side and inside the sphere is also assumed). The re-sults were obtained using an elementary Monte Carlocode3 and pertain to the way in which the scatteredradiation leaves the surface of the sphere. The codegives the angular distribution with respect to the nor-mal to the surface of the radiation leaving the sphere,together with the total attenuation, when an absorp-tion effect is present, and the impulse response de-scribing the temporal response, when a short pulse istransmitted. In Sec. II the calculation method is de-scribed together with the calculated quantities. Ex-amples of results are reported in Sec. III for nonab-sorbing media together with a simple empirical scalingformula that enables one to evaluate with good approx-imation the impulse response for different values ofthe optical depth. The effect of absorption is consid-ered in Sec. IV. Some considerations on the case of anonisotropic point source are reported in Sec. V.
The results obtained by Monte Carlo simulationsenable one to reach some general conclusions aboutlight propagation in dense turbid media and also sug-gest possible applications both of attenuation mea-
surements and temporal spread measurements using aspherical scattering cell with a point source in thecenter, to obtain information about the turbid mediumcontained inside, like the albedo coefficient w0 and theasymmetry parameter g of the scattering function.These two parameters, together with the optical depth,are the ones that mainly affect propagation in denseturbid media. These questions are discussed in Sec.VI, whereas Sec. VII reports a comparison with resultsgiven by the diffusion approximation for a point sourcein an infinitely extended medium. Conclusions are inSec. VIII.
11. Calculation MethodThe calculation code is based on the elementary
Monte Carlo method.3 The Monte Carlo method isbased on the same assumptions used by the transportequation. The advantages of this method are that itdoes not need extra approximations to the transportequation, it does not require the use of large comput-ers, and it is relatively simple to develop the code. Themain disadvantage is the relatively long calculationtime. All the results reported in this paper were ob-tained using a minicomputer (HP1000/E) with limitedmemory (32 Kbyte) and speed.
The diffusing medium, homogeneously distributed,is considered inside a spherical scattering cell (radiusR) in whose center the isotropic point source is placed.No diffusers are considered outside the sphere so pho-tons that leave the surface cannot return to the scatter-ing medium. (The same index of refraction inside andoutside the sphere is also assumed.) The point sourcecase is the simplest to study because of the symmetryof the problem. However, many considerations andresults obtained for this case are also valid for othertypes of source (see Sec. VI for an example).
The medium is characterized by the extinction coef-ficient a = a + a, with os and a scattering andabsorption coefficients, and by a scalar scatteringfunction p(O) (polarization is not taken into account).The single scattering albedo is denoted by wo (w0 = lar) and one defines the optical depth r = 7, + ra = asR +craR with r and a optical depths from scattering andabsorption, respectively.
An important parameter to characterize the medi-um, when propagation for large values of r, is consid-ered, is the diffusion length d. Diffusion length d isthe distance followed by the photon in the originaldirection before it has effectively forgotten its originaldirection of motion.4 For a nonabsorbing medium it is
= 1 (1)a(l -g) (
with g being the asymmetry factor of the scatteringfunction. The quantity d = Rild, representing thenumber of diffusion lengths contained in the radius ofthe sphere, is also used.
Calculations were carried out assuming a nonab-sorbing medium (a = 0,wo = 1). However, as shown inSec. IV, it is possible to obtain the results for any valueof a from the one pertaining to o- = 0. So the distance
covered between two subsequent scattering events andthe scattering angle are randomly chosen, using stan-dard rules, with probability laws determined by a, andp(O) only. Each trajectory was evaluated until it inter-sected the surface of the sphere.
Because of the symmetries of the problem, the man-ner in which the photons emerge does not depend ontheir position over the surface of the sphere. So, apartfrom the fraction exp(-,r,), representing the unscat-tered power that rapidly vanishes when r, grows, allthe considered trajectories give a positive contributionto the calculations and affect the results with the sameweight. Therefore, it is sufficient to evaluate a rela-tively small number of trajectories to have statisticallyreliable results. About 30,000 trajectories were con-sidered in our simulations.
For each trajectory having at least one scatteringevent inside the sphere, the following quantities areevaluated: (1) the angle 4 between the trajectory andthe normal to the surface of the sphere in the pointwhere it intersects the surface; (2) the optical length 1= o-sp (only due to scattering) of the trajectory, p beingthe geometric length inside the sphere; (3) the numberof scattering events K undergone by the photon. Theknowledge of the angular distribution and of the geo-metrical lengths with which the photons emerge fromthe sphere enables one to evaluate both the receivedpower and the temporal distribution for any opticalreceiver. The number of scattering events is evaluat-ed to obtain statistical information on the number ofinteractions between light and scattering medium in-side the sphere.
The values of X, 1, and K pertaining to each trajec-tory are classified in histograms and are used to obtainthe functions fl(XO), fl(l), and f 2(K) that represent theprobability density of the considered quantities.
The quantity fl(40)d represents the probability thata scattered photon emerges within the solid angle dQaround a direction forming an angle 4 with the normalto the surface. So the shape of f3(4))/coso is similar tothe one of the radiance of a generic surface elementfrom scattered radiation. From the histograms thefunction fl(4)) is approximated as
(i < 0k • '+ 1) = 2r(cosoi - cosoi+I) E
Lii=1
(2)
where Li is the number of trajectories intersecting thesurface of the sphere with 0 encompassed between Xiand 0i+1 and I is the number of considered intervals.In our simulations 200 equally spaced intervals be-tween 0 and 90° were considered. The term 2r(cosoi- cos)i+i) is the solid angle subtended between 0i and4)i+i.
The estimate of the probability density functionfl(l) is obtained as
with Mi as the number of trajectories with opticallength between i and i+1 and I as the number ofconsidered intervals. After the first Monte Carlo sim-ulations it was found that practically all the values of Iwere smaller than Imax, with max given by the empiricalrelation Imax = Ts + 3 rsrd for all considered values of 8
and rd. Two hundred equally spaced intervals be-tween Imin = r, and 1max were considered in our simula-tions. For all considered cases the fraction of photonswith 1 > max does not exceed the value of 1/2000 for rd= 6 and 1/6000 for rd = 50. The quantity fi(1)dlrepresents the probability that a photon, scattered atleast once inside the sphere, follows a trajectory withan optical length between and I + dl.
The estimate of the probability density functionf 2(K) is obtained as
f2(Ki < K <Kijl = K KNI' (4)Ki1 Ni
Ni
with the K integer 21. In Eq. (4) Ni is the number oftrajectories having a number of scattering events Kbetween Ki and Ki+i and I is the number of consideredintervals. In all the numerical simulations practicallyall the values of K were smaller than Kmax = Ts + 3 rsrdand I was taken as the minimum value between Kma,and 200. The quantityf 2(K)AKrepresents the proba-bility that a photon, scattered at least once inside thesphere, undergoes a number of scatterings between Kand K + AK before going out.
The functions fi(l) and f2(K) refer to all the scat-tered photons emerging with any value of q. To inves-tigate whether I and K depend on the angle , the codealso evaluates the functions fi(1) and f2(K), in a similarway tofi(1) and f2(K), but considering only the photonsemerging with 0 < 150.
The first- and second-order statistical moments(0)6i,i+1 and (2) i6i+J for the trajectories emergingbetween any considered angular interval 0i,0i+1 arealso evaluated. The numerical results (see Sec. III.Cfor an example) showed that for values of Td greaterthan 6 (diffusers smaller than the wavelength) or 3(spheres larger than the wavelength) Al(l) and f2(K) arealmost indistinguishable from fi(l) and f 2(K), respec-tively. The results concerning the statistical momentsalso showed that (')<iti+1 and (J2)0i,0+1 are practicallyindependent of the angular interval for rd 2 6. Sofi(l), although it is evaluated considering all the scat-tered photons emerging with any angle, is also repre-sentative of the photons emerging within any particu-lar angular interval.
From the knowledge of fl(4O) it is possible to evaluatethe scattered power received by any receiver. Forexample, a receiver with a field of view a and area Aplaced on the surface of the sphere containing a nonab-sorbing medium of optical depth -r, receives the dif-fused power Pdr given by
and vanishes rapidly when rs grows.The shape of fl(4) depends on r and on the scatter-
ing function. However, the numerical results (see Sec.III.0) show that, when rd grows, f3() reaches a limitingform independent of p(O). So the value given by Eq.(5) tends to a constant value.
From the knowledge of fi(l) the impulse responseg(t) for the total diffused power emerging from thesphere can be obtained. If the instant at which thesource emits a delta function pulse is considered as t =0, the photons that follow trajectories of optical lengthI come out after a time t = l/(o.c) with c as the speed oflight in the medium surrounding the diffusers. So theimpulse response g(t) for the whole diffused poweremerging from the sphere is related to f 1(I) by
g(t) = [1 - exp(-,r)] -r cf(l), I = tc-r5R. (7)
So if a pulse Pe(t) is transmitted by the sourcethroughout a nonabsorbing medium, the whole dif-fused power emerging from the sphere Pd as a functionof t is given by
(8)Pd(t) = Pe(t')g(t - t')dt',
whereas the whole unscattered power is
Po(t) = Pe(t - to) exp(-Tr),
with to = Ric.The function fi(l), and thus g(t), refers to all the
scattered photons emerging with any angle. However,it was said previously that fi(') is also representative ofthe photons emerging only within a particular angularinterval if rd 2 6. So for large values of rd the scatteredpower received from any particular receiver when anisotropic source emits a delta pulse will have the sameshape as g(t).
To present the results, it is useful to consider theratio t/to(to = R/c) instead of t. The functiongl(t/to) isthus introduced so that
glyt-) = [1 - exp(-,r)]T1 (1), I = t (10)
Ill. Numerical Results for Nonabsorbing Media
A. Characteristics of the Considered Diffusing Media
The Monte Carlo code was used for a large range ofvalues of r, for the three scattering functions shown inFig. 1. The scattering functions, calculated with Mietheory, refer to polystyrene spheres with diameters (4))of 0.002, 0.33, and 15.8 Aum (curves a, b, and c, respec-tively) suspended in water at the He-Ne wavelength (X= 0.633 ,m). The calculation code was based on thesubroutine BHMIE reported by Bohren and Huff-man.5 Spheres with 4) = 0.002 and 0.33 ,um wereconsidered monodispersed, whereas for large spheres a
0 (degrees)Fig. 1. Phase functions used in Monte Carlo simulations. Thecurves pertain to polystyrene spheres suspended in water at the He-Ne wavelength with sizes 0.002, 0.33, and 15.8 gm (curves a, b, and c,
respectively).
Gaussian polydispersion with a standard deviation of2.9 Am was considered.
Spheres with 4 = 0.002 m are very small withrespect to X and have a scattering function like that forRayleigh scattering, with an asymmetry factor g al-most equal to zero (g = 0.00003). Spheres with 4) =15.8 Mm are large with respect to X and p(O) presents alarge forward peak. For these spheres g = 0.914.Spheres with 4) = 0.33 um have a size near to X and g =0.703. The diffusion length d comes out to 1/or, 3.37/o-,, and 11.5/i 8 for = 0.002,0.33, and 15.8 Am, respec-tively.
For the three types of sphere considered, calcula-tions were carried out for a large range of opticaldepths. Results are reported for Td ranging from rd =3 (for all types of sphere) to -rd = 100,50, and 30 for 4) =0.002,0.33, and 15.8 Mm, respectively. The values of rscorresponding to the maximum value of Td consideredfor the three types of sphere are: -r, = 100,168, and 347for 4 = 0.002, 0.33, and 15.8 m, respectively. Adifferent maximum value of Ird for which calculationswere carried out was chosen for the three types ofsphere because the calculation time increases almostproportionally to the product Tsrd (see Sec. III.C).
In this section only the results for nonabsorbingmedia are presented. In Sec. IV examples are alsogiven for diffusing absorbing media.
B. Results for fl(k)The function A(0) multiplied by d represents the
probability that a scattered photon emerges from thesphere within the solid angle d2 around a directionforming an angle s with the normal to the surface, so0(0)/cos(o) represents the shape of the radiance of a
0. 0 -
0 30 60 900 (degrees)
Fig. 2. Monte Carlo results for fl(-O) vs 0 (continuous lines) pertain-ing to different values of Td. For any value of rd, the curves /l(¢) [Eq.(11)] with n = 1.5 (dashed lines) and n = 2 (dotted lines) are also
shown. Results pertain to 0.33-Am spheres and to = 1.
generic surface element from scattered radiation. Asan example, results pertaining to k(0) at different val-ues of -rd are reported in Fig. 2 for spheres with b = 0.33Am. This figure shows that there are small variationsin f3(o) when Td varies in the range of values considered.Simulations for spheres with 4) = 0.002 and 15.8 Mmshowed results very similar to the ones pertaining to )= 0.33Am when curves referring to the same value Of Trdwere considered. For 'rd 2 6, fl(o) is well approximat-ed by a function like
A(0) = 2 cosno,27r (11)
with 1.5 n 2. To underline this result, for anycurve of (0) the curves (O) with n = 1.5 and n = 2 arealso plotted. The figures show that, when Td grows,f(0) tends to A(0) with n = 1.5 corresponding to aradiance proportional to cos0 50. The reason for whichthe function (0) was considered proportional to(cosO)n is that, when the optical depth of the mediumincreases, the surface of the sphere can be expected toact as a Lambertian diffuser, and the function ,B(0)pertaining to a Lambertian diffuser is proportional tocoso.
To smooth the statistical fluctuations in the numeri-cal results, any point reported in Fig. 2 was obtainedwith an averaging procedure over five data, corre-sponding to 2.5°. The larger statistical fluctuations inthe range of small values of arise from the smallnumber of photons exiting within the correspondingsmall solid angle.
1,KFig. 3. Monte Carlo results for the functionsfl(l) andf2 (K) pertain-ing to 0.002- m spheres and to different values of Td are reported.The functions represent respectively the probability density for theoptical length followed by the photons, and for the number ofscattering K undergone inside the spherical cell of radius R = rdld.
WO = 1.
. 0044
R
' . 0022
.-4-
. 00000 1200
I,K2400
Fig. 4. Same as Fig. 3 but for 15.8-Am spheres.
C. Results for f1(I and f2(K)
Examples of results pertaining to the functions fi(l)and f2(K), representing the probability density that ascattered photon comes out from the sphere followinga trajectory with optical length I (only from scattering)and after a number of scattering events K are reportedin Figs. 3 and 4. To underline the similarity betweenthe functions fi (l) and f 2(K) both functions are report-ed in the same figure with the same scale for 1 and K.In Figs. 3 and 4 the results pertaining to Td = 6, 9, 15,and 20 are reported for the spheres with 4) = 0.002 and15.8 ,m, respectively.
Figures 3 and 4 show that fl(l) and f2(K) broadenwhen Td increases. An important result shown bythese figures is that for rd > 6 for 0.33 and 15.8-Mmspheres and -rd 2 9 for b = 0.002 Am, the curves per-
qK< 1 > ) 2000
1000
500
250
1005025
K=2 100
0 1 1 2<I >K
Fig. 5. Probability density functions q<(l/(l)K) for a photon mov-ing in an infinitely extended nonabsorbing medium for differentvalues of K. The curves show that the distributions become narrow-er and narrower around the peak value l/K 1, when K increases.
taining to fi(l) and f 2(K) become practically indistin-guishable. The largest differences are for I < r, forwhich fl(l) = 0. To give an explanation of these re-sults, the probability density qK() that a photon whichundergoes K scattering events has covered an opticaldistance 1 was considered. For a photon moving in aninfinitely extended medium, the resulting probabilityfor first-order scattering is
q1 (l) = exp(-1). (12)
For second-order scattering, q2(l) can be obtained as
q2(1) = J q1(1')q 1(1 - l')dl' = exp(-1). (13)
By iterating the procedure used for second-order scat-tering the following is obtained:
rI iK-1
qK(I) = qK-( 1')q1(1 - l')dl' = exp(-). (14)
The statistical moments for this distributions comeout as
and mode 1maxK, the value of 1 corresponding to thehighest probability, is given by
Ima.K =K -1. (16)
Note that, for large values of K, (lm)K K (lI)K comesout and this suggests that qKW() should have a narrowdistribution around the mean value (I)K = K. Figure5 reports the probability density function
qK( ) = KqK() (17)
obtained by Eq. (14) considering the variable l/(I)Kinstead of I for different values of K. The figure shows
Fig. 6. Mean value (K) of the number of scattering events under-gone by the photons inside the spherical cell of radius R = rd1d vs Tdfor a nonabsorbing medium. Marks refer to the Monte Carlo resultswhereas continuous curves were obtained by using the empirical
relation given by Eq. (18). Similar results hold for (I).
that, when K increases, q(l/( Il)K) becomes a narrowerand narrower distribution around the value maxK/()K= (K - 1)/K 1. So for large values of K there is asmall probability of having values of I far from thevalue ()K = K. These considerations account for thesimilarity between the curves f(l) and f 2(K) that be-comes practically indistinguishable for large values ofTd-
Figures 3 and 4 show that the number of scatteringevents that a photon undergoes within the sphere rap-idly increases when rs and and increase. The meanvalue (K) for the number of scatterings undergone bythe photons inside the sphere is well approximatedfrom the following simple empirical relation when rd 26:
(K) = 0.50TrTd. (18)
The comparison between Monte Carlo results and thevalues given by Eq. (18) is shown in Fig. 6. Equation(18) also gives a good approximation for the meanvalue of the trajectory optical depth () because (1)(K) for the considered values of rd.
The results for (K) enable one to foresee the calcula-tion time necessary for the Monte Carlo code in differ-ent situations. The calculation time increases almostproportionally to the total number of scattering pointsevaluated, so for large values of rd it increases propor-tionally to the product rsTd. Then the calculation timerapidly grows when Tr and d grow. With our MonteCarlo code and our minicomputer, -20 min are suffi-cient to evaluate 30,000 trajectories for the minimumvalues of r and d considered ( = Td = 3), but 10days are necessary to evaluate the same number oftrajectories when Istd 10,000 (the maximum value ofTs
Td we considered).
. 016
008~~~~~~~~
0. 0000 400 e e00
Fig. 7. Monte Carlo results for the probability density functionsfl(l) and Ilf). Where the function fl(l) refers to the optical lengthsfollowed by all the photons before leaving the sphere, A(I) refers tothe photons exiting with angles t < 150 only. Spheres with -Ii = 0.33
ham; Wo = 1.
To show that the probability density fi(l) does notdepend on the exit angle k when d is sufficiently large,Fig. 7 shows on the same graph the functions fi(l) andfi(l). We recall that f1(1) is the probability densityfunction referring to the trajectories for which 0 < 15°.Curves pertaining to f1 (l) are those with larger fluctua-tions. [These larger fluctuations result from the factthat, whereas all the scattered photons give a usefulresult for fi(l), only 10% of the scattered photons giveausefulresultforfi().] Figure 7 refers to l = 0.33 m.Similar results were obtained for the other spheresconsidered. Curves fi(l) and 11(l) become almost in-distinguishable when rd 2 9, 6, and 3 for P = 0.002,0.33, and 15.8 m, respectively.
D. Results for g1(tto) and a Simple Formula for ImpulseResponse
The function g(t/to) represents the impulse re-sponse for the whole diffused power going out from thesphere taking the transit time for the unscattered pow-er to as the unit time to measure t. The impulseresponse for the power received by any particular re-ceiver with a limited field of view also has practicallythe same shape as g(t/to) when rd is sufficiently large,because the probability function for does not show anappreciable dependence on for high values of Ird.
The results for g(t/to) are reported in Fig. 8 for thethree types of sphere considered. The curves pertain-ing to d = 3 were clipped for values of g(t/to) > 0.36.The figures show that curves pertaining to the samevalue of d for different types of sphere are practicallyidentical when Trd 6. This means that for pulsepropagation in dense turbid media rd is the importantparameter rather than the optical depth Tr. A similarresult is assumed by the diffusion theory.
A comparison between the shapes of the curves f(l)pertaining to different values of Ird, when each curve
0 25 t /to 50Fig. 8. Monte Carlo results for the impulse response gl(t/to) refer-ring to the whole diffused power leaving the sphere of radius R = Tdld.The transit time to for the unscattered power is taken as the unit tomeasure t. Curves refer to 0.002-, 0.33-, and 15.8-um spheres and to
the values of Td indicated near the curves; w = 1.
was plotted assuming 1max = Ts + 3 srTd, showed similarcurves. This fact suggested an attempt to find a sim-ple scaling relation enabling one to obtain the curvesfl(l) pertaining to every value of Td from a curve f (l)pertaining to a known value of rd. For this purpose anattempt was made to distort the original curves withsimple changes of variable (1 - ) to obtain, for thecorresponding probability density function h(Q),curves almost indistinguishable for various values ofTd. After a few attempts on all the Monte Carlo re-sults, it was found that, if the variable = I[3-r(Td +1)] is considered instead of 1, the probability density of4,
h() = 3
(Td + )f1(l), I = 3
(T + 1, (19)
becomes practically independent of -rs and rd when rd2 6. As an example, Fig. 9, referring to = 0.002 /im,shows the function h(Q) for several values of Td.Curves pertaining to values of d 6 are practicallyindistinguishable apart from small values of whereh(Q) = 0 when < r/3-r,(rd + 1). [This result showsthat a better value for 1.ax in Monte Carlo simulationswould be 3 s(Td + 1) instead of the value T + 3 sTdassumed in this work. The differences between thetwo formulas, however, are significant only for thesmaller values of rd considered.]
A function that approaches the curves h(Q) is
h(t) = C-3 75exp(- 0.3465) if > 0
h(f) Td
- -- rd =9
6 T \ sE-lO°
0.0 2 41
0 = ./(34 .( + 1))Fig. 9. Monte Carlo results for the probability density h(s) of = l/3
Ts(Td + 1) for various values of Td. Curves referring to different
values of rd are almost indistinguishable apart from the smallervalues of t. The figure refers to 0.002-jum spheres and w = 1.Almost indistinguishable results were obtained for 0.33- and 15.8-
um spheres.
For rd 9 (and whatever value of r) results C =0.0338.
Equation (20) was obtained by fitting the MonteCarlo curves like those in Fig. 9. The function used tofit the results was chosen with a dependence on similar to the one obtained by using the diffusionapproximation and the random walk model (see Sec.VII).
From the function h) it is possible to obtain anapproximate relation describing all the consideredfunctions fi(1), g(t), and g(t/to) for any value of r and-r. As an example, Fig. 10 reports the Monte Carloresults together with the curves obtained by
g1(tlto) = [1 exp(-,r)] 3(+|) t1 3 (d + 1)]
X ex 1 04(rd + 1) o
=0, t < to .
t > to,
(21)
A good agreement between gl(t/to) obtained by MonteCarlo simulations and by the empirical relation wasfound. In Fig. 10 the results for = 0.002 gm werereported. Analogous results were obtained for 0.33-and 15.8-Arm spheres.
For values of d > 6, from Eq. (21) one obtains
=0 if S < o,
with 4o = rs/3rs(rd + 1) and C a normalizing constarthat
Fig. 10. Comparison between the impulse response g(tto) ob-tained by Monte Carlo simulations and by the empirical relationgiven by Eq. (21). The curves refer to 0.002-,um spheres for differentvalues of Td and wo = 1. Virtually identical results were obtained for
0.33- and 15.8-gm spheres (see Fig. 8).
with t the value of t corresponding to the maximumof g(t/to); (t/to) the mean value of t/to, and At/1o/tothe width of the curve evaluated between the times atwhich gl(t/to) drops to 1/10 of the maximum value. Acomparison between the statistical parameters per-taining to Monte Carlo results and the values given byEqs. (22a)-(22c) showed a good agreement for tmax/toand At111o/to (differences <5% for all considered valuesof Tr and rd > 6) whereas the values of (t/to) given byEq. (22b) were 15% larger than the ones given byMonte Carlo simulations. These differences aremainly due to the higher values given by Eq. (21) for tito rd, with respect to the Monte Carlo results.
So Eq. (21) connecting gl(t/to) to -r, and Td can beused as a good approximation of g(t/to) and thus givesan approximate but simple relation to obtain informa-tion on the diffusing medium when the shape of g1(t/to)is known.
IV. Results for Absorbing MediaAs stated in Sec. II, the Monte Carlo code was used
assuming a nonabsorbing medium (a = 0 and wo = 1).However, the data obtained for a = 0 can be used toobtain the results pertaining to any value of a. If thesame number of diffusers (and thus the same value ofa,) is maintained inside the sphere and an absorptioncoefficient is introduced (due to absorption of the dif-fusing sphere or of the surrounding medium) withoutmodifying the scattering function, the trajectories fol-lowed by the photons remain unchanged with respectto a = 0, but the radiation that follows a trajectory oflength p is exponentially attenuated by a factor
I0ap = Ta
T8
So Eq. (10) describing the impulse response for thetotal diffused power coming out from the sphere filled
S0
0.995 ~~. .98
.0.9998
0. 00
0 10 t/to 20Fig. 11. Impulse response gl(t/to) referring to the whole diffusedpower leaving the sphere when a nonunitary albedo is considered.Curves refer to 15.8-,um spheres. The figure shows how an albedoslightly <1 is sufficient to cause a large attenuation and distortion in
the impulse response with respect to w = 1 when d is large.
by a nonabsorbing medium when an absorbing medi-um is considered becomes
gl(tto) = [1 - exp(-r,)]-rj 1 (l) exp(-ral/lr), I = 'r t (23)
with Ta = Raa optical depth due to absorption and fi(l)pertaining to the nonabsorbing medium with an opti-cal depth rs from scattering.
The absorption effect by the medium causes both anattenuation of the power coming from the sphere and achange of the shape of the impulse response with re-spect to the nonabsorbing case. The result is a pulsewith an attenuated amplitude and a shortened width.As an example Fig. 11 shows g(t/to) for different val-ues of w0 and two fixed values of ITd (d = 9 and 30) forspheres with - = 15.8 ,m. The figure shows that forlarge values of d a small absorption coefficient (or analbedo slightly smaller than one) is sufficient to modifytotally the propagation with respect to the case of anonabsorbing medium.
The total power (scattered and nonscattered) com-ing from the sphere when Mra Fd 0 becomes
Pt(TsTa) = Pc{exp -(,r + Ta)
+ [1 - exp(--r)] fl(l) exp (Ta )dl} (24)
with f(l) pertaining to a = 0. For a purely diffusingmedium the result is Pt(rsra = 0) = Pe, whereas for apurely absorbing medium Pt(Tr = Oia) = Pe exp(-Ta).When oma 0 and a # 0 the attenuation factor can bemuch greater than exp(+,ra) because the trajectoriesfollowed by the radiation to reach the surface of thesphere can be much larger than the radius of thesphere.
Figure 12 shows ln[Pe/Pt(rs,ra) as a function of Ta
for several values of d and for the three types of sphere
Fig. 12. Logarithm of the attenuation of the total power leaving thesphere (apparent optical depth) vs r. for different values of Td andthe three types of sphere considered. Dotted, dashed, and dotted-dashed lines were obtained by using the Monte Carlo results pertain-ing to 0.002-, 0.33-, and 15.8-,um spheres, respectively, for fl (1) in Eq.(23). Continuous curves were obtained by using for f,(l) the func-
tion derived by the empirical relation for hQ() [Eq. (20)].
considered. This quantity represents the apparentoptical depth of the sphere as obtained by measuringthe attenuation of the total transmitted power. Therange of values considered for Ta is sufficient to give alarge attenuation factor for all the considered values ofTd. The values of Pt(Ts,a) were evaluated by using inEq. (24) the Monte Carlo results for f (l). The figureshows small differences between the results pertainingto the three types of sphere for any fixed value of rd.In Fig. 12 the results obtained using in Eq. (23) theapproximated form derived from hi() [Eq. (20)] forfi(l) were also reported (continuous lines). A goodagreement between the results obtained by MonteCarlo and by the approximated function was found forthe range of values displayed in the figure.
It should be underlined as in the case of large valuesof rd that a small absorption optical depth is sufficientto almost cause the total extinction of the radiation.As an example, whereas for Ta = 1 the result is anattenuation by a factor el when Td = 0, for the samevalue of ra one has an attenuation by a factor -el0when Td = 50.
V. Considerations for Nonisotropic SourcesThe results presented in previous sections referred
to the isotropic point source. If a nonisotropic pointsource is considered, the functions f3(O), f1(), f2(k), andthose derived from them are not sufficient to givecomplete information on the radiation coming fromthe sphere, because the radiance varies on the surfacein a way that should be studied for any particularsource. However, the results for i(3G), f1(l), and f2(K)obtained for the isotropic source remain valid for thetotal power coming from the sphere. The power re-ceived by a receiver that does not receive all the power
a
-.. 1 1;-'
0. 00
0 9 t/to 18Fig. 13. Comparison between the impulse response g1(t/to) per-taining to the total diffused power leaving the sphere (radius R = 10cm) when an isotropic point source is placed in the center (dottedcurves) and when an infinitely thin beam source is considered (con-tinuous lines) with a receiver of area 1 cm2 and field of view semia-perture of 9° in front of the beam source. Spheres with 4 = 0.33 ,um;w= 1. To compare the results, the area of the curves was normal-
ized to one.
coming from the sphere can no longer be evaluated bysimple relations like Eq. (5). For large values of Td,
however, one can foresee that the surface of the sphereis again homogeneously illuminated, because it is suffi-cient that the radiation goes away from the source forone diffusion length to forget the original direction ofmotion. Then, if the radius of the sphere is large withrespect to the diffusion length (d >> 1), one expectsthat the results for the isotropic and nonisotropicsources become practically identical. In Fig. 13, weshow the shape of the impulse response due to diffusedpower pertaining to a receiver of area A = 1 cm2 and afield of view a = 90 placed on the surface of a sphericalscattering cell with radius R = 10 cm in whose center aninfinitely thin beam source is placed. The receiverwas considered in front of the thin beam source and thescattering cell was filled with a suspension of nonab-sorbing spheres ( = 0.33 asm). The results for twovalues of rd are reported. In the same figure the shapeof the impulse response gl(t/to) pertaining to the iso-tropic point source was also reported for the sameoptical depths (dotted lines). To compare the results,the area of the curves was normalized to one. The twocurves are almost equal at Td = 9.
The total received power was also evaluated for thereceiver considered and the thin beam source. Theresult was close to the one obtained using Eq. (5)(referring to the isotropic source).
The results for the thin beam source and the particu-lar receiver were obtained using a rather different codebased on a semi-Monte Carlo method.3
VI. Practical Applications of the Results PresentedThe results presented can be useful to design experi-
ments to measure some optical characteristics of tur-
bid media. As an example, we refer to a sphericalscattering cell with blackened walls with a small isotro-pic source (for example, a source like that described inRef. 6) placed in the center, and to an optical receiverwith area and angular field of view known, placed at ahole on the surface of the sphere. If one measures thereceived power when the scattering cell is (1) empty(Per), (2) filled with the turbid medium to be examined(Pir), and (3) a calibrated sample of purely absorbingmaterial (producing a known increase Ar,, in absorp-tion) is added to the turbid medium (P2,), it is possibleto obtain the total power Pe, P1, and P 2 coming outfrom the sphere in the various situations. So by meansof curves like the ones reported in Fig. 12 it is possible(1) to obtain the value of Ird pertaining to the turbidmedium and (2) to obtain the value of -r pertaining tothevalue of P,/Pe when the value of -r is known. So, ifthe extinction coefficient of the turbid medium isknown (for example, by a standard transmissometricmeasurement on a sample of the turbid medium7 ), it ispossible to obtain the parameters a, 0ra, and g thatdetermine light propagation at large optical depths.The measurements can also be made by using a noniso-tropic source if the value of Ird is sufficiently large. Inthis case, the way in which the radiation comes fromthe sphere does not necessarily depend on the particu-lar source.
To invert the results, Eq. (23) can be used, taking forfi(l) the approximate expression derived by Eq. (20).This measuring method can be useful to measure theoptical properties of dense liquid turbid media or ofdiffusers that can be suspended in a liquid.
Another way to obtain information about the turbidmedium is to measure the impulse response. This ispossible if the shape and the attenuation of a shortlight pulse can be measured.
VII. Comparison with the Results of Other Theories
The results obtained by Monte Carlo simulationswere compared with those obtained by using the diffu-sion approximation. In Fig. 14 the results pertainingto the attenuation of the total power leaving the spherewere reported. In the figure the curves referred to asMonte Carlo results were obtained by using in Eq. (24)the approximate empirical form for fi(l) derived fromEq. (20). The results pertaining to the diffusion ap-proximation were evaluated by means of the followingrelation:
obtained by using Eq. (9.54) of Ref. 1 for the diffuseflux Fd. Equation (25) refers to the total flux acrossthe spherical surface of radius R = Tdld centered on thepoint source placed in an infinitely extended medium,whereas the Monte Carlo results refer to the diffusingmedium inside the sphere only. Figure 14 shows thatthe differences between the results obtained with thetwo methods increase when rd decreases, showing,however, a general good agreement especially for smallvalues of r. The differences can result from the dif-
10rd 1001
00
20
15
3in .54 . Monte Carlo
Diffusionapproximation
0 I
0 1 2 a Ta 4
Fig. 14. Comparison between the attenuation of the total powerleaving the sphere, plotted vs -ra, obtained by using both the diffu-sion approximation [Eq. (9.54) of Ref. 1 and the Monte Carloresults. Results for the diffusion approximation refer to the attenu-ation of the total flux across the spherical surface of radius R = mdld
centered on the point source placed in an infinitely extended diffus-ing medium [Eq. (9.54) of Ref. 1]. Results referred to as MonteCarlo ones were obtained by using in Eq. (24) the empirical formula
for fi(l) derived from Eq. (20).
ferent boundary conditions in Monte Carlo simula-tions with respect to those used in diffusion approxi-mation, but one must also remember that assumptionsmade in the diffusion approximation are not valid forsmall values of Td.
The impulse response obtained by Monte Carlo cal-culations for the diffused power leaving the sphericalsurface was also compared with that obtained by usingthe diffusion approximation in the form reported byIto.8 The results reported by Ito, however, refer to apoint source in an infinitely extended medium and toan isotropic receiver placed at a certain distance fromthe source. The pulse shape obtained by Ito [Eq. (23)of Ref. 8], using our notation, is given by
G(R, T') = )2 (3rd t ) exP(-' t 34 d to) t > 0,
=0, t < 0,(26).
for an isotropic receiver placed at a distance R = dldfrom the source. The pulses corresponding to Eq. (26)have a shape similar to those obtained by Monte Carlocalculations but are broader and have a longer tail:the peak time tm'j/to = 0.5rd (for nonabsorbing medi-um) is about twice that given by Monte Carlo results.The larger broadening obtained by the diffusion ap-proximation may mainly result from the longer trajec-tories that can be followed by the photons when theinfinitely extended medium is considered.
We note that a result similar to that given by Eq. (26)can be obtained by using a probabilistic model.2 Themodel consists of a random walk that takes place on an
infinitely extended cubic lattice, the random walk be-ing allowed to step to nearest neighbors only. Thelattice spacing (L) is chosen to be the rms distancetraveled between successive scattering events (withour notations L = V1/o-) and the scattering is assumedto be isotropic (all the scattering directions beingequally probable). The probability that a randomwalk beginning at the origin (in which an isotropicsource is placed) is at r at step K is, when K is large [Eq.(8) of Ref. 2],
PK(r) (/2 expF 3 (x2 + y2 + Z2) (27)
with xy,z integers identifying the lattice point r =(xL,yL,zL). So if R is the distance of point r from theorigin,
x2 + y2 + z2 = R2 S2
results. If we take into account that when K is largeone can identify optical length of the trajectory withthe number of scatterings undergone (see Figs. 3 and4), Eq. (27) gives, for the probability P(l,Tr) that aphoton is at an optical distance r = aoR from thesource following a trajectory with optical distance 1,the relation
P(1,,r,) A -) exp( 3 -s (28)
with A a normalizing constant. By taking into accountthat t/to = /co8R, Eq. (28) shows the same dependenceon t/to as that given by the diffusion approximation[Eq. (26)] when an isotropic phase function is assumed,so that Td = T.
Vill. ConclusionsThe results of a Monte Carlo study on light propaga-
tion in dense turbid media have been presented. Thecalculations refer to the radiation emerging from aspherical scattering cell containing the diffusing medi-um (no diffusers are considered outside the sphere) inwhose center a point source is placed. Both the totalscattered power emerging from the sphere and theimpulse response were evaluated.
The results for an isotropic source and a nonabsorb-ing medium can be summarized as:
(1) The radiance of a generic element of area on thesurface of the sphere shows a behavior as cosn, with n< 1 when rd 2 6 and seems to approach the limitingvalue n = 0.5 when rd increases.
(2) The probability density functionsfi(l) andf 2(K)pertaining to the trajectory optical length and to thenumber of scattering events undergone by the photonsinside the sphere become almost identical when rd 2 6.The mean value of K (and 1) is approached by Eq. (18),showing how the number of scattering events involvedin the propagation grows almost proportionally to TdT s
when Td increases.(3) The results pertaining to the impulse response
for the diffused power emerging from the sphere show
that g1(t/to) does not depend on the particular diffus-ing medium for a fixed value of rd = r,(1 - g). Asimple empirical relation describing the function gl(tIto) was given [Eq. (21)]. The impulse response doesnot show a dependence on exit angle s when rd 2 6.
The results when an absorption effect was presenthave also been presented. They show how a smallvalue of the optical depth due to absorption is suffi-cient to cause the absorption of practically all theemitted power when Td becomes high; so even an albe-do coefficient slightly smaller than one can be suffi-cient to produce the extinction of the emitted radia-tion.
An example of results pertaining to a nonisotropicsource has also been presented. The comparison ofthe results pertaining to the isotropic source and to asource emitting a thin collimated beam shows that theimpulse responses are almost identical even when asmall receiver having a field of view of a few degrees isconsidered in front of the thin beam source if Td issufficiently large.
The results pertaining to the attenuation and to theshape of the impulse response when an absorbing ef-fect is present, suggest practical applications for mea-surements designed to evaluate the optical propertiesof the diffusing medium.
The results obtained by Monte Carlo calculationswere compared with those obtained by using the diffu-sion approximation for an isotropic point source andan infinitely extended medium. The comparison,made on the total attenuation and on the impulseresponse, showed differences that may result mainlyfrom the fact that the Monte Carlo results refer to afinite diffusing medium.
This work was supported by Ministero dell'Univer-sita' e della Ricerca Scientifica.
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