For measuring the size distribution and the concen-tration of clouds of particles with diameters in themicrometer range, one of the most popular and pow-erful techniques is based on the inversion of extinc-tion spectra. In this framework, the transmittanceof a beam through the cloud is measured at severalwavelengths. Then, with the Lorenz–Mie theory~assuming spherical particles! used to compute theextinction cross section of the particles and theBouguer–Lambert–Beer law used to take into ac-count the particle concentration and the path of thelight through the cloud, the recorded spectra are in-verted to retrieve the particle distribution and con-centration. The method, however, is limited to thesituation in which all the forward detected light doesnot interact with particles. This assumption is oftendissatisfied, in particular because of the finite size ofdetectors.
M. Czerwinski [email protected]! and J. Mroczkare with the Chair of Electronic and Photonic Metrology, Wrocławniversity of Technology, ulica B. Prusa 53y55, 50-317 Wrocław,oland. T. Girasole, G. Gouesbet, and G. Grehan are with theaboratoire d’Energetique de Systemes et Procedes, Unite Mixtee Recherche, Centre National de la Recherche Scientifique 6614,omplexe de Recherche Interprofessionnel en Aerothermochimie,
nstitut National des Sciences Appliques et Universite de Rouen,.P. 8, 76131 Mont Saint Aignan Cedex, France.Received 23 December 1999; revised manuscript received 4 Au-
1514 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
When the concentration of particles and the size ofthe cloud are small enough that only direct light andlight that experiences only one scattering event arecollected ~single light scattering!, the essential pa-ameter is the collecting solid angle and the finite-ize effect of detectors can be analytically corrected.When the concentration of particles and the width
f the cloud are large enough that the contribution ofight scattered more than once is not statisticallyegligible on detectors, we face multiple-scatteringonditions. Under such circumstances, there is noirect way to retrieve the contribution of direct lightrom the total collected radiation.
Although the principles of multiple scattering oflectromagnetic radiation were first established on arm theoretical basis almost 50 years ago,1 multiple
scattering still remains a complex phenomenon witha large number of practical applications. In certainmedia such as multiphase flows, suspensions in liq-uids, or fuel injectors, multiple scattering may be thepredominant process of radiative transfer and one ofthe most relevant physical phenomena to be used fordiagnostic purposes.
Handling the phenomenon of multiple light scat-tering can be approached in different ways. Amongthem, the statistical Monte Carlo method relies ontracking the behavior of the light emitted from asource viewed as a decomposition of small light pen-cils ~here called photons, but it is essential to notehat the name photon has little to do with quantumight theory!, characterized by their direction andntensity. The method is accurate, but time-onsuming, because its validity crucially depends on
ctTmt
aflspeisct
hSSSahtbpd
et
the number of photons emitted from the source oflight. Also, because of its statistical formulation it ishardly possible to apply any method of inversion tothe Monte Carlo approach. Conversely, the four-flux method views the cloud of particles as a contin-uous and isotropic medium that possesses averagescattering and absorption properties determined bythe particles. It is a fast analytical method, but thescattering media has to be monodimensional ~theloud is considered a medium of infinite lateral ex-ension limited by two infinite and parallel planes!.hese assumptions make it impossible to use theethod directly to simulate actual complex geome-
ries.The hybrid methods have been extended to take
dvantage of the Monte Carlo technique and the four-ux models, yet eliminating their limitations. Morepecifically, they are based on the four-flux model,ossess its advantages of simplicity, computationalfficiency, and analytical form, and, moreover, by us-ng the coefficients calculated from the Monte Carloimulations, they take into consideration the actualharacteristics of the system under study, includinghe detectors.
This paper is devoted to the description of theseybrid methods. The paper is organized as follows.ection 2 briefly summarizes the four-flux model, andection 3 is devoted to the Monte Carlo approach.ection 4 introduces the concept of hybrid methodsnd gives exemplifying results for two versions of theybrid methods for monodimensional clouds of par-icles. Section 5 presents the capability of the hy-rid methods of handling the case of arbitraryarticle size distributions. Section 6 is devoted to aiscussion of obtained results and perspectives.
2. Four-Flux Method
This model relies on the solution of the radiativetransfer equation through the slab under study. Itallows us to calculate the transmittances of lightthrough the scattering cloud irradiated by a planewave.2–5 The cloud is assumed to be limited by twoparallel surfaces, ~I! and ~O!. The detector is repre-sented by a surface parallel to ~I! and ~O!. The planesurface ~S! placed between the slab and the detectoris used to take into account the reflectances of thedetection system ~Fig. 1!. The purpose of this model
Fig. 1. Geometry of the four-flux model.
is to relate the total light transmittances and reflec-tances of the optical properties of the disperse me-dium contained between planes ~I! and ~O!.
In the general model, the incident light consists ofa collimated beam with an infinite lateral extensionhitting the slab perpendicularly and of semi-isotropicradiation ~semi-isotropic means isotropy in a hemi-sphere!. The incident light is assumed to be mono-chromatic without any loss of generality, but thepolarization of light is not taken into account. Thewhole radiation field inside the medium is decom-posed into four beams ~four-flux!:
a collimated beam of intensity Ic~Z! propagatingtoward negative z ~intensity means the amount ofnergy flowing per unit of time and per unit of areahrough a surface perpendicular to the axis z!,
a collimated beam of intensity Jc~Z! propagatingtoward positive z,
a diffuse radiation of intensity Id~Z! propagatingtoward negative z,
a diffuse radiation of intensity Jd~Z! propagatingtoward positive z.
Note that here we use the word intensity according toa common practice in the field, but the quantitiesIc~Z!, Jc~Z!, Id~Z!, and Jd~Z! may actually also becalled fluxes, i.e., the four-flux model is written interms of energy balances obtained from integration ofthe corresponding intensities over appropriate hemi-spheres.
Planes ~I!, ~O!, and ~S! can be characterized by thecoefficients of reflection for the collimated and thediffused beams. For the sake of simplicity in thisstudy, we assume that the coefficients of reflectionare equal to zero for both the diffused and the colli-mated beams. This assumption for ~I! and ~O! isperfectly valid when the medium under study is notconfined by windows. This assumption for ~S! de-fines what we could call an ideal detector. We ex-pect that the relaxation of these assumptions wouldnot modify the essential results of this paper, butcould affect the numerical values of the empiricalcoefficients subsequently defined. It is also assumedthat no diffuse radiation enters the slab.6
The particles inside the medium are characterizedas follows:
The coefficient of absorption k is related to the lossof energy of the collimated beam through an infini-tesimally thin slab dz that is due to absorption.
The coefficient of scattering s is related to the lossof energy of the collimated beam through an infini-tesimally thin slab dz that is due to the scattering.
The forward-scattering ratio z is equal to the en-ergy scattered by the particle in the forward hemi-sphere over the total scattered energy and isproportional to Csca
f . For the collimated beam theevaluation is simple, but for a component of the dif-fuse radiation the forward hemisphere does not haveto coincide with the forward hemisphere relative tothe examined slab. For the sake of simplicity it is
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1515
tvtm
a
w
w
s
e
1
assumed that, for the diffuse radiation, forward-scattering radiation is also equal to z.
The average crossing parameter ε corresponds tohe average distance covered by diffuse radiation tra-ersing a slab of thickness dz. For our simulations,o calculate the coefficient, we used the following for-ula, proposed by Wang et al.7:
ε 5 c1 gc2 expF 1~g 2 1!~c3 1 c4 zo!
G1 c5~1 2 g!
3 exp~2zo! 1 12
1 1, (1)
where zo is the optical thickness of the slab, g is thesymmetry factor, and c1–c5 are constant values
given by c1 5 1.5504, c2 5 0.9642, c3 5 4.0329, c4 53.0635, and c5 5 0.8680.
The following coefficients are hereafter defined ascoefficients of forward light scattering for a colli-mated beam zs and for diffuse radiation εzs,
coefficients of backscattering for a collimated beam~1 2 z!s and for diffuse radiation ε~1 2 z!s,
coefficients of absorption for diffuse radiation εk.
Lorenz–Mie scattering is assumed to be the actualphysical phenomenon occurring inside the mediumunder study, i.e., the particles are assumed to beperfectly spherical. We then have8
k 5 NCabs 5 N~Cext 2 Csca! 5 NCext~1 2 a!, (2)
s 5 NCsca 5 aNCext, (3)
here Csca is the Lorenz–Mie scattering cross section,Cabs is the Lorenz–Mie absorption cross section, Cextis the Lorenz–Mie extinction cross section, and a isthe single-scattering albedo, which is defined as
a 5Csca
Cext5
Qsca
Qext5
sk 1 s
, (4)
here Qsca and Qext are the Lorenz–Mie efficiencyfactors for scattering and extinction, respectively.The values of Csca, Cext, and Csca
f are given by8–11
Csca 5 Sl2
2pD (n51
`
~2n 1 1!$uanu2 1 ubnu2%, (5)
Cext 5 Sl2
2pD (n51
`
~2n 1 1!$Re~an 1 bn!%, (6)
Cscaf 5
12
Csca 2l2
2p 5(n51
`
o (m51
` ~2n 1 1!~2m 1 1@m~m 1 1! 2 n~n 1
1 (n51
`
o (m51
`
o2n 1 1
n~n 1 1!
2mm~m
516 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
where o and e designate summations restricted to oddand even subscripts, respectively, and
0!! 5 1!! 5 1,
n!! 5 ~n 2 2!!!n.
To compute the value of the asymmetry factor weused the formula given by Debye,12 which describesthe parameter g in terms of the usual Lorenz–Miecattering coefficients an and bn:
g 54
a2Qsca(n51
Fn~n 1 2!
n 1 1Re~an*an11 1 bn*bn11!
12n 1 1
n~n 1 1!Re~an*bn!G , (8)
in which the scattering coefficients are
an 5cn~a!cn9~b! 2 mcn~b!cn9~a!
jn~a!cn9~b! 2 mcn~b!jn9~a!, (9)
bn 5mcn~a!cn9~b! 2 cn~b!cn9~a!
mjn~a!cn9~b! 2 cn~b!jn9~a!, (10)
where cn are Ricatti–Bessel functions, jn are Hankelfunctions, a 5 @~pd!yl#, where d is the particle diam-ter, b 5 ma, m is the relative complex refractive
index of the particles, and l is the wavelength.We then consider the differential equations that
describe flux-energy balances along an infinitesimalslab of geometric length dz,
dIc
dz5 ~k 1 s!Ic, (11)
dJc
dz5 2~k 1 s!Jc, (12)
dId
dz5 εkId 1 ε~1 2 §!sId 2 ε~1 2 §!sJd
2 ~1 2 §!sJc 2 §sIc, (13)
dJd
dz5 2εkJd 2 ε~1 2 §!sJd 1 ε~1 2 §!sId
1 ~1 2 §!sIc 1 §sJc, (14)
and solve them to obtain final expressions for thelight transmittances through the slab under study.
The total transmittance calculated from the four-flux method in general consists of three components:
~21!n1m21y2 n!!~m 2 1!!!~n 2 1!!!m!!
Re~an am* 1 bn bm*!
11!
~21!n1m21y2 n!!m!!~n 2 1!!!~m 2 1!!!
Re~an bm*! 6 , (7)
!
1!#1
1
trtp
s
sd
c~
lc
dl
collimated–collimated transmittance, which corre-sponds to the transmittance of light entering andleaving the medium in collimated form ~tCC!;
collimated–diffused transmittance, which corre-sponds to the transmittance of the light entering theslab in collimated form and leaving it as diffuse ra-diation ~tCD!;
diffused–diffused transmittance, which corre-sponds to the transmittance of the light entering andleaving the media in diffused form ~tDD!.
In this work we assumed that the only radiation en-tering the medium under study is the collimated ra-diation so the third contribution is zero. Thereforethe total light transmittance ttot is
ttot 5 tCC 1 tCD. (15)
The transmittance tCD corresponds to the radiationthat underwent the collisions with particles insidethe slab and eventually reached the plane detectorwithin the framework of the four-flux model. tCD isa theoretical transmittance that does not depend onthe geometry of the detection system because, accord-ing to its definition, all the flux of the emerging ra-diation is collected when we are integrating over theforward hemisphere. However, this transmittancecrucially depends on the dimensions of the measuringsystem in an actual experiment because a certainamount of diffuse radiation does not reach the detec-tor. Conversely, the transmittance tCC does not de-pend on the geometry of the measuring system. Itdepends on only the physical properties of the dis-persed medium and on the width of the slab. Underthe assumption that the reflectances of the enclosingplanes are equal to 0, it is equal to the transmittanceevaluated from the Bouguer–Lambert–Beer law.
3. Monte Carlo Method
In this paper, the Monte Carlo method is imple-mented under the following light-scattering assump-tions13: steady state, quasi-elastic scattering, nomultiple scattering in an infinitesimal slab, the lightis described by its scalar intensity ~no account forpolarization, no interference!, and the medium understudy is considered homogeneous and isotropic.Processing multiple light scattering in terms of singlescattering relies on the description of volume elementby use of three parameters: its single-scattering al-bedo a, its extinction coefficient kext, and its phasefunction P~u! by use of Lorenz–Mie scattering theo-ry.8,14
The basic idea of the model is to track the trajectoryof what we call photons. The trajectory of each pho-ton from the time that it leaves the light source untilit is captured by the detector or lost ~either by ab-sorption or by leaving out the medium without reach-ing the detector! is stochastically reconstructed inaccordance with a set of probability densities. Asthe number of photons increases, the statistical de-scription of the physical problem approaches the de-terministic exact solution.
A specific probability density is attached to eachclass of events that can occur during the life of aphoton:
For each point of the source and for each direction,a number of photons are launched according to theluminance of this point of the source in this direction,given by the indicatrix source.
The probability of the photon’s being absorbed orscattered between the distance l and l 1 dl is ex-pressed by kext exp~2kextl !dl, where kext 5 NCext.Cext represents the extinction cross section and N ishe particle number density. Then, if r represents aandom number tossed in the range @0,1#, the dis-ance covered by the photon before a collision with aarticle is given by
l 5 2log rkext
. (16)
The probability for a photon to be absorbed by aparticle is represented by its albedo a. If r is anotherrandom number in the range #0,1#, the photon is ab-orbed if r , a; otherwise the photon is scattered.The direction of propagation of the photon after
cattering is deduced from the phase function andefined by two scattering angles, u and w. New di-
rection is calculated from the phase function P~u!.By using two other random numbers r and r9 in therange #0,1#, we have
w 5 2pr, r9 5
*0
u
P~u!dV
*0
p
P~u!dV
, (17)
where V is the solid angle.
A computer program named POLYSCAT, written inFORTRAN 77, allows one to take into account any arbi-trary design of the measuring system, different me-dia, various optical characteristics of obstacles ~wallsan be reflective or absorbant!, sources, and detectorslocations, sizes and apertures!.
Moreover, the program records the number of col-isions of each collected photon. This allows us toompute the following transmittances: tMC0, corre-
sponding to the ratio of the number of photons tra-versing the medium without any collision over thetotal number of launched photons ~this transmittance
irectly corresponds to the Bouguer–Lambert–Beeraw!, and tMCm, corresponding to the ratio of the
number of photons captured by the detector after oneor more collisions inside the medium over the totalnumber of launched photons. The total transmit-tance calculated by the Monte Carlo approach is
tMC 5 tMC0 1 tMCm. (18)
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1517
mt
i
t
p
t
1
4. Hybrid Methods
To match the advantages of the approaches pre-sented in Sections 2 and 3, we have expanded hybridmethods that assemble the advantages of simplicityand computational efficiency of the four-flux modeltogether with the accuracy and generality of theMonte Carlo approach.
We stipulate that the total transmittances calcu-lated from the four-flux model or from the MonteCarlo approach are expressed as the summation oftwo contributions: ~1! from collimated–collimatedtransmittance ~tCC! and collimated–diffused trans-
ittance ~tCD! in the four-flux model and ~2! tMC0 andMCm in the Monte Carlo approach. We also note
that the first contribution of each set of contributionsactually identifies and does not depend on the geo-metric properties of the measurement setup; bothdirectly correspond to the transmittance of theBouguer–Lambert–Beer law.15,16
The second contributions ~tCD and tMCm! corre-spond to the photons that experienced at least oneinteraction with the particles. However, tMCm takesinto account the effects associated with the full three-dimensional ~3-D! geometry of the problem understudy.
The principle of the hybrid method is then to as-sume that tMCm is proportional to tCD, which can bewell understood when we consider that both factorsdepend on the geometry of the measuring system. Itis expected that the differences between tCD andtMCm will decrease with the size of the particles owingto the more peaked forward distribution of the scat-tered radiation displayed when the particle size isincreased, a fact that will be corroborated by ourresults ~see Figs. 9–14! in Subsection 4.B!. Thechallenge is then to evaluate the proportionality co-efficients that depend on wavelength, the 3-D geom-etry of the source of light, the properties of the cloudof particles, and the detector.
We also assumed that the validity range of thesecoefficients is large enough in such a way that only afew coefficients have to be evaluated for practicalapplications.
We then have the ansatz
tMC 5 tCC 1 KtCD. (19)
Many numerical experiments have been carried outfor different wavelengths ~ranging from 4 3 1027 to9 3 1027 m! for different diameters of particles ~rang-ng from 1027 to 3 3 1026 m! and for various number
densities, and the associated coefficients K~l, d, N!have been evaluated. We could then show certaindependencies between the values of the coefficient Kversus tCD ~hybrid method 1! or of K versus tMC ~hy-brid method 2!.
All the calculations included in this paper weremade for water particles in the air and the followinggeometric parameters of the system ~Fig. 2!:
light-source diameter: d1 5 0.002 m,thickness of the examined medium: Z 5 0.1 m,
518 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
distance between the outer plane of the slab andhe detector: d3 5 0.01 m,
diameter of the detector: d2 5 0.02 m,half-angle of aperture of the detector: a 5 30°.
A. Hybrid Method 1
In hybrid method 1, in which the coefficient K de-ends on tCD@K 5 h~tCD!#, we calculate the value of
tCD for the given wavelength, particle diameter, andconcentration. Figures 3–8 exemplify the relation-ship between K and tCD for particle diameters rang-ing from 0.4 3 1026 to 0.8 3 1026 and for l 5 0.4 31026 m and l 5 0.6 3 1026 m.
In these figures multiple-scattering situations arerepresented by the linear parts of the graphs belowthe dashed lines. For these data, the ratio of tCCover tCD is smaller then 0.15. It means that theassociated particle concentrations are rather largeand that the beam entering the cloud of particles in
Fig. 2. Experimental geometry for the Monte Carlo simulations.
Fig. 3. Relationship between the coefficient K and the transmit-ance tCD ~l 5 0.4 3 1026 m, d 5 0.4 3 1026 m!.
tc
iwle
t
t
collimated form essentially reaches the detector asdiffuse radiation. Points above the dashed line cor-respond to relatively weak concentrations of particleswith the consequence that the influence of tCD on theotal light transmittance is negligible enough whenompared with that of tCC. Equation ~19! then
shows that, for the data, the value of K is of littlenfluence. Using the points below the dashed line,e then obtained the following mathematical corre-
ation, which is valid for all the concentrations in thexamined range:
K 5 B exp~tCDA!, (20)
where A and B are constants for the wavelengths andthe diameters given.
Table 1 presents the exemplifying values of thecoefficients A and B for the particle diameters in therange from 0.4 3 1026 to 0.8 3 1026 m and wave-lengths from 0.4 3 1026 to 0.6 3 1026 m.
Fig. 7. Relationship between the coefficient K and the transmit-ance tCD ~l 5 0.4 3 1026 m, d 5 0.8 3 1026 m!.
Fig. 8. Relationship between the coefficient K and the transmit-ance tCD ~l 5 0.6 3 1026 m, d 5 0.8 3 1026 m!.
Fig. 4. Relationship between the coefficient K and the transmit-tance tCD ~l 5 0.6 3 1026 m, d 5 0.4 3 1026 m!.
Fig. 5. Relationship between the coefficient K and the transmit-tance tCD ~l 5 0.4 3 1026 m, d 5 0.6 3 1026 m!.
Fig. 6. Relationship between the coefficient K and the transmit-tance tCD ~l 5 0.6 3 1026 m, d 5 0.6 3 1026 m!.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1519
Table 1. Some Exemplifying Values of A and B ~Hybrid Method 1!
1
B. Comparisons
Figures 9–14 present the comparisons between thetransmittances calculated by different methods: theMonte Carlo method, the classical method based onthe Bouguer–Lambert–Beer law, the classical four-flux method, and hybrid method 1.
Hybrid method 1 provides correct values of trans-
Fig. 9. Transmittance as a function of particle concentration, d 50.4 3 1026 m, l 5 0.4 3 1026 m.
Fig. 10. Transmittance as a function of particle concentration,d 5 0.4 3 1026 m, l 5 0.6 3 1026 m.
520 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
mittances, as we can see from its agreement with theMonte Carlo method. Once the values of A and Bare obtained, the accuracy of the method is advanta-geously complemented by its extreme computationalefficiency, allowing us to use it as a routine technique.Of course, the Bouguer–Lambert–Beer law is notvalid for large particle concentrations ~for example, in
Fig. 11. Transmittance as a function of particle concentration,d 5 0.6 3 1026 m, l 5 0.4 3 1026 m.
Fig. 12. Transmittance as a function of particle concentration,d 5 0.6 3 1026 m.
Fig. 11, the discrepancy can reach 10 orders of mag-nitude!.
C. Hybrid Method 2
In hybrid method 2, the coefficient K is determinedversus the transmittance tMC @K 5 f ~tMC!#. It ismore adaptable to practical situations in which, forthe given l, tMC is the actual measurable quantity.
igures 15–20 present exemplifying relationships be-ween the values of the coefficient K and the trans-
mittance tMC for particle diameters ranging from0.4 3 1026 to 0.8 3 1026 m and for l 5 0.4 3 1026 mand l 5 0.6 3 1026 m.
The relationships exhibited in the figures are wellfitted by the correlation
K 5 CtMCD, (21)
where the coefficients C and D are constants for fixedand d.
Fig. 13. Transmittance as a function of particle concentration,d 5 0.8 3 1026 m, l 5 0.4 3 1026 m.
Fig. 14. Transmittance as a function of particle concentration,d 5 0.8 3 1026 m, l 5 0.6 3 1026 m.
Table 2 presents exemplifying values of the coeffi-cients C and D for particle diameters in the rangefrom 0.4 3 1026 to 0.8 3 1026 m and wavelengths inthe range from 0.4 3 1026 to 0.6 3 1026 m. Obvi-ously, because K is computed as a function of tMC, inthis hybrid method, the values of tMC found with thehybrid method and the original ones obtained fromMonte Carlo simulations are always in good agree-ment.
5. Hybrid Methods in the Case of Polydispersions
The next step is to evaluate how the hybrid methodsbehave in the presence of arbitrary particle size dis-tributions. To examine this issue, we use the coef-ficients A and B ~hybrid method 1! or C and D ~hybridmethod 2! estimated in Section 4 to determine lighttransmittances for clouds containing particles of dif-ferent diameters. Of course, as the coefficients werecalculated for only discrete values of particle diame-
Fig. 15. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.4 3 1026 m, d 5 0.4 3 1026 m!.
Fig. 16. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.6 3 1026 m, d 5 0.4 3 1026 m!.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1521
alues
1
ters, the particle size distribution has to be presentedin the form of a histogram.
In this paper we consider the case of symmetricunimodal distribution as defined in Table 3. Forthis distribution, transmittances were computed forthree wavelengths, l 5 0.4 3 1026 m, l 5 0.75 3 1026
Fig. 17. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.4 3 1026 m, d 5 0.6 3 1026 m!.
Fig. 18. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.6 3 1026 m, d 5 0.6 3 1026 m!.
522 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
m, and l 5 0.9 3 1026 m in the range of numberdensities from 5 3 1015 to 5 3 1016 particlesym3.For each number density we calculated the partialnumber density of each class of particle diameter,and for these partial number densities we calculatedthe partial transmittances thybi
associated with the
Fig. 19. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.4 3 1026 m, d 5 0.8 3 1026 m!.
Fig. 20. Relationship between the coefficient K and the transmit-tance tMC~l 5 0.6 3 1026 m, d 5 0.8 3 1026 m!.
of C and D ~Hybrid Method 2!
avelength ~m 3 1026!
0.5 0.6
C D C D
.5432 0.1752 0.5353 0.1234
.5614 0.2445 0.6 0.2421
.5084 0.2122 0.5476 0.2344
.483 0.1987 0.5839 0.2964
.2654 0.06218 0.4196 0.1236
W
00000
1
1
pdctdmwBt
0
Table 3. Examined Particle Size Distribution
corresponding class. The total transmittance ttotwas then evaluated from the partial transmittancesthybi
by
ttot 5 S 1
(i51
N
thybiD12E
FE (22)
in which N is the number of classes. Table 4 pre-sents the values of parameters E and F for the stud-ied wavelengths.
Figures 21–23 present the comparisons betweenthe total transmittances calculated according to hy-brid method 1 supplemented with Eq. ~22! and theMonte Carlo method for l 5 0.4 3 1026 m, l 5 0.75 3026 m, and l 5 0.9 3 1026 m ~similar results are
obtained for hybrid method 2 and therefore do notneed to be presented!. The figures do not presentthe classical Bouguer–Lambert–Beer results be-cause, for the concentrations under study, the asso-
Fig. 21. Comparison between the transmittance calculated fromhybrid method 1 and that from the Monte Carlo method for l 5.4 3 1026 m.
ciated values differ too much from the presentedones. For example, for l 5 0.4 3 1026 m the trans-mittances computed according to Bouguer–Lambert–Beer law varied from 0.16 3 10230 down to 0.27 30261 for 1016 particlesym3.Transmittance predictions from hybrid method 1
~or 2! are not in perfect agreement with Monte Carloredictions. Nevertheless, the slope behavior of theependence between the transmittance and the con-entration is retrieved, and the absolute values of theransmittances are close enough. The maximumifference between Monte Carlo results and hybrid-ethod predictions is smaller than by a factor of 5,hereas the values calculated according to theouguer–Lambert–Beer law typically differ by a fac-
or of ;1065, representing a dramatic improvement inany case.
Moreover, once the coefficients K are established
Fig. 22. Comparison between the transmittance calculated fromhybrid method 1 and that from the Monte Carlo method for l 50.75 3 1026 m.
Fig. 23. Comparison between the transmittance calculated fromhybrid method 1 and that from the Monte Carlo method for l 50.9 3 1026 m.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1523
2. B. Maheu, J. N. Le Toulouzan, and G. Gouesbet, “Four-flux
1
for the experimental geometry under study, a hybridmethod computes the transmittances in real timecompared with Monte-Carlo simulations. For exam-ple, the computations of the results presented in thispaper require ;0.1 s with a typical Pentium 100 PCcompared with ;8 h of Monte Carlo computationswith a HP 9000 series 700.
Furthermore, the capability of the hybrid methodsto be applied to the case of arbitrary particle sizedistributions has been successfully tested.17
6. Conclusion
Our purpose in this paper was to present an originalmethod for predicting light transmittances throughclouds of particles, in which multiple light scatteringmight be a dominant phenomenon, taking into ac-count the 3-D geometry of source, cloud, and detector.
Hybrid methods, based on the four-flux methodwith empirical coefficients evaluated from a finitenumber of Monte Carlo computations, have beendemonstrated as computationally efficient and accu-rate tools to predict light transmittances for mono-dispersions and polydispersions, especially whencompared with the classical Bouguer–Lambert–Beerlaw. The hybrid method ~namely, hybrid method 1!provides a solution to the direct problem, i.e., it allowsus to estimate light transmittances versus particleproperties and wavelength for the given source–slab–detector geometric setup.
Numerical results demonstrate that the hybridmethods converge to the results obtained from MonteCarlo simulations in a wide range of concentrations.In a companion paper, we demonstrate that the hy-brid methods are well suited to the solution of theassociated inverse problem, i.e., retrieving particlesize distributions from transmittance spectra.
The properties ~rapidity and flexibility! have to beimplemented to the inversion code in order to retrievethe particle size distribution from the spectrum oflight transmittances.
Recently the standard four-flux model has beenextended to take into account any degree of anisot-ropy of forward and backward radiation by calculat-ing the corresponding average path-lengthparameters from a multiple-scattering approachbased on solving the radiative transfer equation forthe successive scattering orders and also any degreeof asymmetry between the diffuse intensities.18–20
Such extensions could be incorporated into ourpresent approach and would likely lead to interestingvariants with, however, a price to pay, namely, a lossof simplicity. In view of the quality of the results weshall present, such sophistication does not seem to benecessary in the case of hybrid methods.
References1. S. Chandrasekhar, Radiative Transfer ~Oxford U. Press, Lon-
don, 1950!.
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models to solve the scattering transfer equation in terms ofLorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 ~1984!.
3. M. Czerwinski and J. Mroczka, “4 flux model in real dispersemedia examination: the results of computer simulation,” inProceedings on Computer Added Metrology ~Polish Academy ofSciences, Warsaw, Poland, 1997!, pp. 275–282.
4. B. Maheu, J. P. Briton, and G. Gouesbet, “Four flux model anda Monte-Carlo code: comparisons between two simple andcomplementary tools for multiple scattering calculations,”Appl. Opt. 28, 22–24 ~1989!.
5. G. Gouesbet, B. Maheu, and J. N. Letoulouzan, “Simulation ofparticle multiple scattering and applications to particle diag-nostics,” in Heat Transfer in Radiating and Combusting Sys-tems, M. G. Carvalho, F. Lockwood, and J. Taine, eds.~Springer-Verlag, New York, 1991!, pp. 173–185.
6. B. Maheu and G. Gouesbet, “Four flux models to solve thescattering transfer equation: special cases,” Appl. Opt. 25,1122–1128 ~1984!.
7. Y. P. Wang, Z. S. Wu, and K. F. Ren, “Four flux model withadjusted average crossing parameter to solve the scatteringtransfer equation,” Appl. Opt. 28, 24–26 ~1989!.
8. G. Gouesbet, B. Maheu, and G. Grehan, “Single scatteringcharacteristics of volume elements in coal clouds,” Appl. Opt.22, 2038–2050 ~1983!.
9. K. Kerker, The Scattering of Light and Other ElectromagneticRadiation ~Academic, New York, 1969!.
10. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles ~Wiley, New York, 1983!.
11. P. Chylek, “Mie scattering into the backward hemisphere,” J.Opt. Soc. Am. 63, 1467–1471 ~1974!.
12. P. Debye, “Der Lichtdruck auf Kugeln von beliebigen Materi-al,” Ann. Phys. ~Leipzig! 4, 57–136 ~1909!.
13. B. Maheu, J. P. Briton, G. Grehan, and G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geom-etry,” Part. Part. Syst. Charact. 9, 52–58 ~1992!.
14. C. Roze, B. Maheu, G. Grehan, and J. Menard, “Evaluations ofthe distance of visibility in a foggy atmosphere by Monte-Carlosimulation,” Atmos. Environ. 25, 769–775 ~1994!.
15. M. Czerwinski, J. Mroczka, K. F. Ren, T. Girasole, G. Grehan,and G. Gouesbet, “Scattered light predictions under multiplescattering conditions with application to inversion scheme,” inProceedings of the Seventh European Symposium on ParticleCharacterisation ~NurnbergMesse GmbH, Nurnberg, Ger-many, 1998!.
16. M. Czerwinski, J. Mroczka, T. Girasole, G. Grehan, and G.Gouesbet, “Hybrid method to predict scattered light transmit-tances under multiple scattering conditions,” in Proceedings ofthe Fifth International Congress on Optical Particle Sizing,~University of Minnesota, Minneapolis, Minn., 1998!.
17. M. Czerwinski, “Modelisation de la turbidite spectrale d’unmilieu multidiffusif et son application au probleme inverse,”Ph.D. dissertation ~Universite de Rouen, Rouen, France,1998!.
18. W. E. Vargas, “Generalized four-flux radiative transfer model,”Appl. Opt. 37, 2615–2623 ~1998!.
19. W. E. Vargas, “Two-flux transfer model under nonisotropicpropagating diffuse radiation,” Appl. Opt. 38, 1077–1085~1999!.
20. W. E. Vargas, “Diffuse radiation intensity propagatingthrough a particulate slab,” J. Opt. Soc. Am. 16, 1362–1372~1999!.
