894 J. Opt. Soc. Am. A/Vol. 19, No. 5 /May 2002 M. Elias and G. Elias
New and fast calculation for incoherent multiplescattering
Mady Elias
Centre de Recherche et de Restauration des Musees de France, Unite Mixte de Recherche 171 du Centre Nationalde Recherche Scientifique, 6 rue des Pyramides, 75041 Paris Cedex 01, France, and Universite d’Evry
Val d’Essonne, 91025 Evry Cedex, France
Georges Elias
Office National d’Etudes et de Recherches Aerospatiales, BP 72, 92322 Chatillon Cedex, France
Received August 7, 2001; revised manuscript received October 18, 2001; accepted October 25, 2001
1. INTRODUCTIONFor a long time in optics, experimental methods for mea-suring the light emerging from illuminated diffusing lay-ers have been restricted to particular configurations.These methods are the method of the integration sphere,which collects all the diffused light, and the measure-ments in the backscattering configuration. Bidirectionalmeasurements are now in common use with adjustableincident and observation angles. Thus it is necessary todevelop theoretical and numerical methods adapted tosuch a configuration. In a large number of cases, thelayer can be modeled as a partly absorbing medium of re-fractive index n in which many small particles that scat-ter and absorb light are embedded. The particle location,size, geometry, and orientation are random, and the par-ticle density is small enough that the light scattered byeach particle adds incoherently. The theory that can pre-dict the scattered light in all directions is then that of in-coherent multiple scattering.
This paper is motivated by studies on colors of works ofart; we wish to understand the effects of the different con-stituents of a painting on its color. The influence of avarnish on a colored Lambertian surface has already beenmodeled.1 The role of diffusing and absorbing pigmentsin producing color is the purpose of this study. Neverthe-less, the new calculation method presented here can beapplied to other fields that deal with a similar model.
Standard theories such as the two-flux2 and thefour-flux3 theories can be applied if the incident light isdiffused or collimated at normal incidence and if the col-lected light is measured in the whole half-space (the case
of the integration sphere). The angular distribution ofthe different fluxes is taken into account in the N-fluxmethod,4 which is widely applied, for example, to studythe influence of scattering particle size.5 But thismethod ‘‘gives no information about the azimuthal distri-bution of the scattered radiation,’’ (Ref. 4, p. 1502) be-cause it is computed in each channel u but is integratedfor all azimuthal angles f.
Starting from the same fundamental radiative transferequation,6 it is possible to imagine an extended N-fluxmethod where all the angles u and f of the collected beamwould be discriminated. Such an idea will lead to the so-lution of a system of very large dimensions, in which theproblems related to avoiding overflow and sampling closeto critical angles (grazing angle u5p/2, total reflectionangle) would be even more complex. A more effectivemethod is the discrete ordinate method,7 where samplingin the angle f is avoided by taking advantage of an ex-pansion of the phase function on spherical harmonics. Aset of equations is then derived, one for each azimuthalnumber. However, sampling at the u angle is main-tained.
A new and exact resolution of the radiative transferequation is presented here that solves the whole problem,i.e., leads to the scattered flux expressed as function ofthe two observer angles u and f and avoids the previousdisadvantages.
In Section 2 the theory of multiple scattering is re-called, leading to an integro-differential system. In theSection 3 the proposed technique is presented. It con-sists of two steps: The first step is to insert an auxiliary
2002 Optical Society of America
M. Elias and G. Elias Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. A 895
function from which the scattered light, which is the so-lution of an integral equation, can be computed. The sec-ond step is to expand the auxiliary function in terms ofthe spherical harmonics, giving rise to a set of integralequations on the coefficients that are well suited for nu-merical resolution. It appears that the boundary condi-tions are included in the linear operator of the integralequation, so each boundary condition is associated with aspecific equation. As an illustration, in Section 4 themethod is applied to numerical simulations of maps of thelight scattered by a thick refractive layer with a directivephase function.
2. SCATTERING EQUATIONWe consider a layer of thickness H limited by two planes(Fig. 1). The incident light consists of collimated beamscoming from directions ui with intensity Ii
inc and may alsoinclude diffuse light with intensity I inc(u). The refrac-tive index of the substrate is n, and the substrate’s ab-sorption coefficient is a. An assembly of particles embed-ded homogeneously in the layer acts as a continuousdiffusing medium with an absorption coefficient K and ascattering coefficient S. Light of direction u1 is scatteredin direction u2 following the phase function p(u1 , u2).The particles’ distribution is assumed to be random inshape, size, and location, so the phase function is rota-tionally invariant, i.e., is a function of cos g5 u1 • u2 . This is also the case when the particles arespherical: The phase function is then analytically com-puted by using Mie theory.8 In summary, the intensity I1of a beam propagating over a distance dl along its direc-tion u1 decreases as
dI1 5 2~a 1 K 1 S !I1dl,
and the intensity dI2 scattered by the beam inside thesolid angle dV2 around direction u2 is
dI2 5 SI1dlp~cos g!dV2/4p,
so the normalization of the phase function is*p(u1 , u2)dV2 5 4p.
Fig. 1. Geometry of the problem. Incident collimated light,with ui direction, enters a layer with refractive index n and ab-sorption coefficient a, containing diffusing particles with absorp-tion and scattering coefficients K and S. Collimated and diffuselight propagate in the medium. The diffuse and collimated lightis collected in the direction uf .
Inside the layer, the light is the sum of collimatedbeams with directions uj and intensity Ij(Z) (after refrac-tion of the incident collimated beams at the interface andpossible reflection on the bottom) and of diffuse light withintensity I(Z, u). A direction is characterized by its unitvector u with polar angles (u, f). The direction is calledascending if 0,u,p/2 (in the direction of increasing Z)and descending otherwise. To enhance this feature, wedenote I1(u, Z) for the first case and I2(u, Z) for the sec-ond one.
Following Chandrasekhar,6 the equation of transfer forthe diffuse light is
cos udI~u, Z !
dZ5 2~K 1 S 1 a!I~u, Z !
1S
4pE I~u1, Z !p~u, u1!dV1
1S
4p (j
I j~Z !p~u, uj!.
For the sake of simplicity, distances will be normalizedby using the total extinction coefficient: z 5 (K 1 S1 a)Z and h 5 (K 1 S 1 a)H. The normalized scat-tering coefficient is then q 5 S/(K 1 S 1 a). Thetransfer equation becomes
cos udI~u, z !
dz5 2I~u, z ! 1
q
4pE I~u1 , z !p~u, u1!dV1
1q
4p (j
I j~z !p~u, uj!.
For the collimated light, which does not gain energy fromother directions, the transfer equation is simplycos ujdIj /dz 5 2Ij with straightforward integration.
Rather than working directly on the transfer equation,we now follow Mudgett and Richards,4 introducing the el-ementary flux (in absolute value)
w~u, z ! 5 ucos uuI~u, z !.
With inclusion of Wj(z) 5 ucos ujuIj(z), the diffusion equa-tion then splits into
dw1~u, z !
dz5 2
w1~u, z !
ucos uu1
q
4pE w~u1 , z !
ucos u1up~u, u1!dV1
1q
4p (j
Wj~z !
ucos u jup~u, uj!~0 , u , p/2!,
dw2~u, z !
dz5
w2~u, z !
ucos uu2
q
4pE w~u1 , z !
ucos u1up~u, u1!dV1
2q
4p (j
Wj~z !
ucos u jup~u, uj!~p/2 , u , p!.
(1)
The N-flux equation is derived from Eq. (1) by consideringthe mean flux Fi integrated over a channel i with a Du iaperture:
Fi 51
2pE
f50
2p EDui
w~u, f !sin ududf.
896 J. Opt. Soc. Am. A/Vol. 19, No. 5 /May 2002 M. Elias and G. Elias
As recalled in Section 1, azimuthal information is thenlost. This is why a new method is proposed here to solveEq. (1), which, moreover, does not use the channel con-cept.
3. NEW METHOD FOR SOLVING THESCATTERING EQUATIONThe procedure for solving the scattering equation is firstto introduce the auxiliary function
f~u, z ! 5 E w~u1 , z !
ucos u1up~u, u1!dV1 (2)
and to transform the integro-differential system [Eq. (1)]into a linear integral equation for the auxiliary function.Numerically, this integral equation is not simpler to solvethan the initial system but has the advantage of avoidingdivergent terms for u→p/2. Moreover, it is possible to ex-pand the auxiliary function on the spherical harmonics.The integral equation then splits into a set of integralequations for the coefficients, each of them depending onthe single variable z. Each of the integral equations ap-plies to the coefficients that share the same azimuthal in-dex. It will be shown that each of these equations maybe transformed into a linear system of moderate size, eas-ily solved in a short CPU time even on a personal com-puter.
A. Integral Equation for the Auxiliary FunctionWith
S~u, z ! 5 (j
Wj~z !
ucos u jup~u, uj!,
the formal solution of Eq. (1) is
w1~u, z ! 5 w1~u, 0!exp~ 2 z/ucos uu!
1q
4pE
0
z
@ f~u, s ! 1 S~u, s !#
3 exp@ 2 ~z 2 s !/ucos uu#ds,
w2~u, z ! 5 w2~u, h !exp@~z 2 h !/ucos uu#
1q
4pE
z
h
@ f~u, s ! 1 S~u, s !#
3 exp@~z 2 s !/ucos uu#ds. (3)
Inserting Eqs. (3) into Eq. (2) leads to the integral equa-tion for the auxiliary function.
Before solving, boundary conditions have to be takeninto account. For example, let us assume that no dif-fused light enters the layer at z 5 0 and that the bottom,at z 5 h, is covered by a material with energetic reflec-tion coefficient R(u) (defined for 0 , u , p/2). The twoboundary conditions are
w1~u, 0! 5 0,
w2~u, f, h ! 5 R~p 2 u!w1~p 2 u, f, h !.
Notice that when w1(u, f, h) is reflected with 0,u,p/2,it leads to w2(u8, f 8, h) with u8 5 p 2 u and f 8 5 f.
Using the first boundary condition in the first relation ofEqs. (3), one obtains for z 5 h and p/2 , u , p
w1~p 2 u, f, h !
5q
4pE
0
h
@ f~p 2 u, f, s ! 1 S~p 2 u, f, s !#
3 exp@2~h 2 s !/ucos uu#ds.
With the second boundary condition, Eqs. (3) become
w1~u, f, z ! 5q
4pE
0
z
@ f~u, f, s ! 1 S~u, f, s !#
3 exp@2~z 2 s !/ucos uu#ds,
w2~u, f, z ! 5q
4pexp@~z 2 h !/ucos uu#R~p 2 u!
3 E0
h
@ f~p 2 u, f, s ! 1 S~p 2 u, f, s !#
3 exp@2~h 2 s !/ucos uu#ds 1q
4p
3 Ez
h
@ f~u, f, s ! 1 S~u, f, s !#
3 exp @~z 2 s !/ucos uu#ds.
Such a formulation allows us to write the integral equa-tion on f(u, z).
All kinds of common boundary conditions can behandled this way, and any specific boundary conditiontherefore leads to a specific integral equation.
To illustrate the method, the previous case is not con-tinued; instead, the two following particular situationsare developed. The first one is as simple as possible:The incident light is restricted to a single collimated beamcoming from direction ui(u i , f i) with intensity Ii
inc . Therefractive index of the vehicle is unity, so that no refrac-tion occurs at z 5 0. The bottom of the layer is nonre-flecting (or transparent). The second example is similar,but the refractive index is different from unity.
1. Example 1With a refractive index n 5 1 and no reflection from thebottom, the collimated light in the layer is only ascending:
W~z ! 5 W inc exp~2z/cos u i!,
S~u, z ! 5 W inc exp@2z/cos u i!p~u, ui!/cos u i .
Without incident diffuse light, the two boundary condi-tions are w1(u, 0) 5 0 and w2(u, h) 5 0. Equations (3)reduce to
w1~u, z ! 5q
4pE
0
z
@ f~u, s ! 1 S~u, s !#
3 exp@2~z 2 s !/ucos uu#ds,
M. Elias and G. Elias Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. A 897
w2~u, z ! 5q
4pE
z
h
@ f~u, s ! 1 S~u, s !#
3 exp@~z 2 s !/ucos uu#ds,
and leads to the integral equation on f(u, z):
f~u, z ! 5q
4pE
f150
2p
df1ED1 1 D2
@ f~u1 , s ! 1 S~u1 , s !#
3exp~2uz 2 su/ucos u1u!
ucos u1up~u, u1!sin u1du1ds,
(4)
with
D1 : 0 , u1 , p/2 and 0 , s , z,
D2 : p/2 , u1 , p and z , s , h.
Equation (4) does not seem less complex than the initialsystem [Eq. (1)]; it still has a divergent component in theintegral operator when s → z and u1 → p/2. However, ifEq. (4) is solved numerically as it stands (which is not ourobjective), f(u, z) has to be sampled with respect to z,and Eq. (4) involves the integral
J 5 EDz
exp~2x/ucos u1u!
ucos u1udx
over a small range Dz. As J 5 @2exp(2x/cos u)#Dz , it ap-pears that the integral equation is divergence free.
2. Example 2We now consider a vehicle with refractive index n Þ 1and still a nonreflective bottom at z 5 h. The incidentcollimated light is partly reflected at z 5 0 with theFresnel reflection coefficient R(u i) for nonpolarized light.It is also partly refracted into the layer with a directionu0(u0 , f i) such that
n sin u0 5 sin u i .
The transmission coefficient is T(u i) 5 1 2 R(u i). In-side the layer we then get
W~z ! 5 W incT~u i!exp~2z/cos u0!,
S~u, z ! 5 W incT~u i!exp~2z/cos u0!
3 p~u, u0!/cos u0 .
For the diffuse light, a descending component is nowpartly reflected at z 5 0, creating an ascending compo-nent, and partly refracted in the z , 0 half-space, whichis where the incident light comes from (emerging diffuselight). The boundary condition at z 5 0 is
w1~u, f, 0! 5 R1~u!w2~p 2 u, f, 0!,
where R1 is the reflection coefficient expressed as func-tion of the internal angle rather than of the externalangle.
Returning to the integral equation and inserting theboundary conditions in Eqs. (3), we obtain the final ex-pression:
f~u, f, z ! 5q
4pE
f150
2p
df1ED1 1 D2
@ f~u1 , f1 , s !
1 S~u1 , f1 , s !#exp~2uz 2 su/ucos u1u!
ucos u1u
3 p~cos g!sin u1du1ds
1q
4pE
f150
2p Eu150
p/2 Es50
h
@ f~p 2 u1 , f1 , s !
1 S~p 2 u1 , f1 , s !#R1~u1!
3exp@2~z 1 s !#/cos u1]
cos u1
3 p~cos g!sin u1du1d. (5)
Comparing Eq. (5) with Eq. (4), we see that the first termof the right-hand side is the same. This is quite general.Particular boundary conditions add particular terms tothe first one.
B. Expansion on the Spherical HarmonicsThe next step is to expand f(u, z) on the spherical har-monics, taking advantage of the fact that the phase func-tion depends only on cos g. The phase function is first ex-panded on the Legendre polynomials,
p~cos g! 5 (n50
`
pnPn~cos g!,
and the Legendre polynomials on the spherical harmon-ics,
Pn~cos g! 5 (m50
n
a~n, m !Pnm~cos u!
3 Pnm~cos u1!cos m~f 2 f1!,
with a(n, m) 5 1 for m 5 0 and a(n, m) 5 2@(n2 m)!/(n 1 m)!# for m Þ 0. We thus get
p~u, u1! 5 (n50
`
(m50
n
a~n, m !pnPnm~cos u!
3 Pnm~cos u1!cos m~f 2 f1!.
Dealing with example 1 and locating the incidence direc-tion at f i 5 0, this expansion yields
S~u, z ! 5 Wi
exp~2z/cos u i!
cos u i(n50
`
(m50
n
a~n, m !pn
3 Pnm~cos u!Pn
m~cos u i!cos mf.
We therefore expand f(u, z) as
f~u, z ! 5Wi
cos u i(n50
`
(m50
n
gn~m !~z !Pn
m~cos u!cos mf
(the terms in sin mf will disappear in further computa-tion).
When we insert all these expansions into Eq. (4) andusing the relationship
898 J. Opt. Soc. Am. A/Vol. 19, No. 5 /May 2002 M. Elias and G. Elias
Ef150
2p
cos m1f1 cos m~f1 2 f !df1 5 phmdm,m1cos mf,
with hm 5 2 for m 5 0 and hm 5 1 otherwise, the inte-gration over f1 decouples all azimuthal components m.After identification of the right- and the left-hand sides ofthe integral equation, there remains for each value of m(from zero to infinity) a set of equations
gn~m !~z ! 5
q
4pnhm (
n15m
` ED11D2
H @ gn1
~m !~s !
1 exp~2s/cos u i!pn1a~n1 , m !Pn1
m ~cos u i!#
3exp~2uz 2 su/ucos u1u!
ucos u1ua~n, m !Pn1
m ~cos u1!
3 Pnm~cos u1!J sin u1du1ds.
Finally, defining
K ~m !~x, n, n1! 5 a~n, m !Eu50
p/2 exp~2x/cos u!
cos u
3 Pnm~cos u!Pn1
m ~cos u!sin udu
and using the symmetry properties of the associated Leg-endre polynomials, we obtain the final equation for the co-efficients as
gn~m !~z ! 5 (
n15m
` Es50
h
H ~m !~n, n1 , z, s !
3 @ gn1
~m !~s ! 1 bn1
~m !~s !#ds for n > m, (6)
with
bn1
~m !~s ! 5 exp~2s/cos u i!pn1a~n1 , m !Pn1
m ~cos u i!,
H ~m !~n, n1 , z, s !
5q
4pnhmK ~m !~ uz 2 su, n, n1!
for 0 , s , z ,
H ~m !~n, n1 , z, s !
5q
4pnhm~21 !n 1 n1K ~m !~ uz 2 su, n, n1!
for z , s , h . (7)
Considering now example 2 with
f~u, z ! 5W incT~u i!
cos u0(n50
`
(m50
n
gn~m !~z !Pn
m~cos u!cos mf,
we see that the set of equations is similar to Eq. (6), withan additional term,
gn~m !~z ! 5 (
n15m
` Es50
h
H1~m !~n, n1 , z, s !
3 @ gn1
~m !~s ! 1 bn1
~m !~s !#ds for n > m, (8)
with
bn1
~m !~s ! 5 exp~2s/cos u0!pn1a~n1 , m !Pn1
m ~cos u0!,
H1~m !~n, n1 , z, s !
5 H ~m !~n, n1 , z, s !
1q
4pnhm~21 !m1n1U ~m !~z 1 s, n, n1!, (9)
U ~m !~x, n, n1!
5 a~n, m !Eu50
p/2 exp~2x/cos u!
cos u
3 R1~u!Pnm~cos u!Pn1
m ~cos u!sin udu.
C. Method for Numerical ResolutionIn practice, the number of coefficients pn in the phasefunction is reduced: for n larger than some value N, thecoefficients are negligible (in Mie’s model, N may be foundin Ref. 9). So if it is assumed that pn 5 0 for n . N,Eqs. (7) and (9) ensure that gn
(m)(z) 5 0 for n . N.Moreover, because Pn
m(x) 5 0 for m . n, m is also lim-ited to N. Finally, the number of systems to solve is fi-nite, as well as the summation in n1 .
For instance, if N 5 3, we get four systems:
• one for m 5 0, with 0 < n,n1 < 3;• one for m 5 1, with 1 < n,n1 < 3;• one for m 5 2, with 2 < n,n1 < 3;• one for m 5 3, with n 5 n1 5 3.
For the numerical resolution, @0, h# is split into Q seg-ments @(i 2 1)Dz, iDz# for 1 < i < Q and h 5 QDz.Anticipating the numerical study, it appears that Dz' 1/50 is an appropriate value for accurate results. In
each segment, gn(m)(z) is assumed to be constant with the
value gn,i(m) 5 gn
(m)(zi), where zi 5 (i 2 1/2)Dz is themiddle of each segment. The same operation is per-formed on bn
(m)(z). For example 1, the discrete version ofsystem (6) is then
gn,i~m ! 5 (
n15m
N
(j51
Q
Hn,n1 ,i, j~m !
3 @ gn1 , j~m ! 1 bn1 , j
~m ! #, m < n < N, 1 < i < Q,
with
Hn,n1 ,i, j~m ! 5 E
zj2Dz/2
zj1Dz/2
H ~m !~n, n1 , zi , s !ds.
This is a square system with Q(N 1 1 2 m) unknowns.For instance, for N 5 3 and h 5 2, Q(N 1 1 2 m)5 400 for m 5 0. Q(N 1 1 2 m) reduces to 100 form 5 3. This order of magnitude is quite appropriate fora fast resolution, so that, as already noted, each element
M. Elias and G. Elias Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. A 899
of the H matrix is computable without a divergence prob-lem. The same discussion holds for example 2.
D. Further Comments
1. In a comparison of this method with the N-flux one,which integrates the flux on the azimuthal angle, it isclear that it is sufficient to solve only the m 5 0 system,rather than the N 1 1 ones.
2. A significant simplification occurs if the phase func-tion is isotropic, i.e., p(cos g) 5 1. In this case, onlyg(z) 5 g0
(0)(z) remains. For instance, in example 1, theonly equation to solve is
g~z ! 5 Es50
h
H~z, s !@ g~s ! 1 b~s !#ds,
with
b~s ! 5 exp~2s/cos u i!,
H~z, s ! 5q
2K~ uz 2 su!,
K~x ! 5 Eu50
p/2 exp~2x/cos u!
cos usin udu.
K(x) represents the exponential integral function, al-ready used in Ref. 6.
4. NUMERICAL RESULTSA FORTRAN code has been written to implement themethod, following the two examples and solving Eqs. (6)and (8). Our final aim is to compute the distribution inthe directions of the emerging scattered light, that is, thedistribution of w2(u, f, 0) after possible refraction out-side the layer. Let us(us , fs) designate the light’s direc-tion (p/2 , us , p) and ws(us) its flux. In the case ofrefraction, the relationship between the angles u and usis sin us 5 n sin u and fs 5 f. The transmission coeffi-cient for w2(u, f, 0) is T1(u) 5 1 2 R1(p 2 u) 5 T(p2 us). The light’s flux in a solid angle dV 5 sin ududfat the interface is w2(u, 0)dV, and the light istransmitted as T1(u)ws(us)dVs . Using dVs /dV5 n2 cos u/cos us , we get
ws~us , f ! 5 T~p 2 us!w2~u, f, 0!
cos us
n2 cos u.
After resolution, maps of ws(u, f ) are plotted with
X 5 A2 sin~u/2!cos f, Y 5 A2 sin~u/2!sin f.
The interest of this mapping of a hemisphere on a plane isthat it preserves solid angles:
dXdY 512 sin ududf 5
12 dV.
Without an exact solution to compare our results with,testing the accuracy of the numerical results is notstraightforward. However, an overall test can be per-formed on the total flux when q 5 1. This conditionmeans that scattering occurs only without absorption (theconservative case). A balance may then be establishedbetween the flux of the incident light and the sum of the
collimated reflected light and the total diffuse emerginglight at z 5 0, on the one hand, and the collimated lightand the total diffuse light at z 5 h, on the other hand.All the fluxes are computed independently, involving onlythe gn
(0)(z) coefficients. We performed a large number oftests, varying the refractive index, the normalized layerthickness h, the angle of incidence u i , and mainly thestep size Dz. We conclude that with Dz 5 1/50, the bal-ance is fulfilled within at least three digits, which hasbeen considered adequate.
As an illustration, numerical simulations with refrac-tive index n 5 1.5 are presented. For the phase func-tion, a variety of models can be found in the literature.We first choose the simple directive model:
p~cos g! 5 1 1 cos g,
which requires only two Legendre polynomials. Elemen-tary scattering is then maximum in the forward directionand zero in the backward direction.
The results are computed for q 5 0.9 and h 5 10, for u ivarying from 0° to 75° every 5°. The CPU time on a per-sonal computer for computing all elements of the H1 ma-trix and solving the systems for these 16 values of the in-cident angle is 100 s. It appears that for all incidentangles, the total flux at z 5 h is approximately 1% of theincident flux. This shows that the value of h is largeenough to simulate a layer of infinite thickness. The evo-lution of the total diffused emerging flux and of the colli-mated reflected flux at z 5 0, both normalized to the in-cident flux, is plotted in Fig. 2 as a function of the incidentangle. Up to 75°, the emerging diffused flux remainsnearly constant.
Three maps for the emerging diffused light are pre-sented for u i 5 0°, 30°, and 60° (Figs. 3–5). The firstmap concerns single scattering, issued only from thesource term S(u, z). The second one concerns a higher-order scattering, and the third one is the sum of the two
Fig. 2. Total diffused emerging flux [curve (1)] and collimatedreflected flux [curve (2)] emerging at z 5 0, both normalized tothe incident flux, as a function of the incident angle u i forq 5 0.9, h 5 10, n 5 1.5 with p(cos g) 5 1 1 cos g.
900 J. Opt. Soc. Am. A/Vol. 19, No. 5 /May 2002 M. Elias and G. Elias
previous ones. In each case, the direction of the incidentlight (at f i 5 0) is plotted as a black-and-white circle.For the three incident directions, single-scattering [Figs.3(a), 4(a), and 5(a)] reflects the shape of the phase func-tion: No scattered light appears in the incident back-ward direction. Higher-order scattering looks very dif-ferent [Figs. 3(b), 4(b), and 5(b)]. The maximum isobserved near u50°, and the lack of symmetry close to thevertical axis is weak, even for u i 5 60°. The ratio be-
tween the single-scattered flux and the total-scatteredflux is equal to 3% for u i 5 0°, 4% for u i 5 30°, and 8%for u i 5 60°. Higher-order scattering dominates but issmaller at grazing incidences than at normal incidence.The overall maps [Figs. 3(c), 4(c), and 5(c)] summarizethis behavior. In this particular situation, although thephase function is far from isotropic, the diffuse emerginglight is nearly independent of the azimuthal angle what-ever the direction of the incident light. The same conclu-
Fig. 3. Angular distribution of the emerging diffused light for q 5 0.9, h 5 10, n 5 1.5, p(cos g) 5 1 1 cos g, and u i 5 0°. (a) Singlescattering, (b) higher-order scattering, (c) total scattering.
Fig. 4. Angular distribution of the emerging diffused light for the same configuration as in Fig. 3, for u i 5 30°.
Fig. 5. Angular distribution of the emerging diffused light for the same configuration as in Fig. 3, for u i 5 60°.
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Fig. 6. Angular distribution of the emerging diffused light for the same configuration as in Fig. 5, p(cos g) 5 (1 1 cos g)3/2.
sion can be drawn from Fig. 6, where a more directivephase function has been tested: p(cos g) 5 (11 cos g)3/2 for u i 5 60°.
5. CONCLUSIONA new method has been proposed for solving the equa-tions for the multiple scattering of light. It has beenshown that the introduction of an auxiliary function inthe diffusion equation together with its expansion on thespherical harmonics allows us to compute the exact angu-lar distribution of the light scattered by a diffusing me-dium. In contrast to the case of the N-flux method, theresult is expressed as a function of both angles u and f.Although the equations to be solved remain complex atfirst sight, they are actually well suited to an efficient nu-merical resolution. In the near future we will comparethe results obtained here with those of the N-flux method.Then we intend to apply our method to problems relatedto the color of works of art: glazes and the mixture ofmineral pigments.
ACKNOWLEDGMENTSThe authors thank Jacques Lafay, head of the Laboratoired’Optique des Solides of Paris VI University, for his inter-est in our work and for helpful discussions.
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