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COHERENT OPPOSITION EFFECTS FOR SEMIíINFINITE
DISCRETE RANDOM MEDIUM IN THE DOUBLEíSCATTERING
APPROXIMATION
V.P.Tishkovets \Lambda , P.V.Litvinov \Lambda\Lambda , M.V.Lyubchenko \Lambda
\Lambda Astronomical observatory of Kharkov National University
Sumskaya St.,35, Kharkov, 61022, Ukraine
Eímail: tishkovets@astron.kharkov.ua
\Lambda\Lambda Institute of Radio Astronomy of NASU
Chervonopraporna St.,4, Kharkov,61002, Ukraine
Eímail: litvinov@ira.kharkov.ua
J. Quant. Spectrosc. Radiat. Transfer
(Received 19 March 2001)
Abstract í The rigorous equations of the theory of multiple scattering of light by a layer
of disordered medium have been used in the doubleíscattering approximation for semiíinfinite
medium to determine the influence of the particle properties on the coherent opposition effects.
The effects were found to be strongly dependent on concentration of scatterers in the medium.
The polarization opposition effect is more sensitive to the properties of the scatterers than
the photometric opposition effect. The interference of waves could result in the negative
polarization at the backscattering direction as well as in the positive polarization.
Pages: 7
Figures: 3
Proposed running head: Coherent backscattering by random media
1

1 Introduction
Coherent backscattering of light by disordered medium result in the wellíknown backscattering
enhancement phenomenon which manifests itself as a narrow intensity peak centered at the
exact backscattering direction [1,2]. In case of unpolarized incident wave this effect can be
accompanied by the polarization opposition effect that may be shown as a narrow peak of
negative polarization at a small phase angle [3,4] (the angle between the reflected beam and
the backscattering direction). Both of these effects are typical for many atmosphereless cosmic
bodies, laboratory samples of powderílike media and rough surfaces [1í6]. Therefore they can
be an informative characteristics in various optical applications.
Theoretical description of coherent backscattering of light by disordered medium is too
complicated even in the case of the Rayleigh scatterers. The exact vector theory of coherent
backscattering for semiíinfinite medium composed of noníabsorbing pointílike scatterers has
been obtained only recently [7,8]. Numerical calculations of the polarization opposition effect
for such medium are in good agreement with the laboratory data [9]. However the dependence
of the opposition effects on the properties of the scatterers (size parameter, real and imaginary
parts of refractive index) is studied insufficiently.
The equations for coherent and incoherent (diffuse) components of reflection matrix for a
layer of disordered medium composed of chaotically oriented arbitrary scatterers have been
obtained recently in Ref.10. These equations are too complicated for numerical calculations.
We use these equations in doubleíscattering approximation to consider the influence of particle
properties on the opposition polarization effects for semiíinfinite medium.
2 Basic equations
In the case of beam normally incident on a rarefied, homogeneous and isotropic layer of discrete
random medium the reflection matrix S pnïÚ was presented in Ref.10 as a sum of three matrices
S pnïÚ = S (nc)
pnïÚ + S (co)
pnïÚ + S 1
pnïÚ (p; n; ï; Ú = \Sigma1). Matrix S (nc)
pnïÚ corresponds to the diffuse part
of scattered radiation and can be written as:
S (nc)
pnïÚ = n 0
k
X
L
d L
M0N0 (#)
Z kZ 0
0
exp(2z Im('')
cos # )ff (z)(pn)(ïÚ)
L dz: (1)
Here matrix S (nc)
pnïÚ is normalized to a unit of surface area, n 0 is the concentration of particles in
the medium, k = 2‹=Ö; Ö is the wavelength of the incident radiation, d L
M0N0 (#) is the Wigner
function [11], # is the scattering angle (the angle between the incident beam and the scattering
direction), Z 0 is the thickness of the medium layer, M 0 = Ú \Gamma n; N 0 = ï \Gamma p, '' is the complex
effective refractive index of the medium. Here and hereafter the CP írepresentation [12,13] (the
basis of the circular polarization) is used.
The expansion coefficients ff (z)(pn)(ïÚ)
L are determined from the system of equations [10]:
ff (z)(pn)(ïÚ)
L = ¼ (pn)(ïÚ)
L exp(\Gammaœ z ) + 2‹n 0
k 3
X
qq1
¼ (pq)(ïq1 )
L
\Theta
Z X
l
ff (ae)(qn)(q 1 Ú)
l exp(\Gammaœ ae )d L
M0N (!)d l
M 0 N (!) sin !d!dae; (2)
2

where ae; ! are the polar coordinates of integration point with respect to the point z, the angle
! (0 ß ! ß ‹) is counted off from the backscattering direction, œ x = 2Im('')x, N = q 1 \Gamma q
(q; q 1 = \Sigma1). Coefficients ¼ (pn)(ïÚ)
L are the expansion coefficients of the scattering matrix
elements for isolated particle in series of the Wigner D function [12]. For spherical particles
they can be written as:
¼ (pn)(ïÚ)
L 1
= 1
4
X
Ll
(2L + 1)(2l + 1)ha (pn)
L a \Lambda(ïÚ)
l iC L1M0
L\GammanlÚ C L1N0
L\Gammaplï ; (3)
where a (pn)
L = a L + pnb L , a L ; b L are the Mie coefficients [14], angle brackets denote averaging
over particle properties, symbols C denote the ClebshíGordan coefficients [11], the asterisk
denotes complex conjugation. The Eqs.(1), (2) are one of forms of the wellíknown vector
radiative transfer equation.
Expression for describing of coherent part of scattering radiation has the form
S (co)
pnÚï + S 1
pnÚï = n 0
k
Z kZ0
0
G (z)(pn)(ïÚ) exp[\Gamma2zIm('')(1 \Gamma
1
cos # )]dz; (4)
where S (co)
pnÚï corresponds to the coherent part of scattering radiation, matrix S 1
pnÚï corresponds
to some part of the incoherent radiation. Matrices S (co)
pnÚï and S 1
pnÚï are normalized similarly to
the matrix S (nc)
pnïÚ . Matrix G (z)(pn)(ïÚ) is defined by the equation [10]:
G (z)(pn)(ïÚ) = ht pn t \Lambda
Úï i + 2‹n 0
k 3
X
qq1 LM
i (pq)(ïq1 )
LM
Z X
lm
fi (ae)(qn)(q 1 Ú)
lm
\Theta d L
MN (!)d l
mN (!) exp(\Gamma~œ ae )Fm\GammaM (ae; #; !)dae sin !d!: (5)
Here t pn is the amplitude scattering matrix for isolated particle,
i (pq)(ïq 1 )
L 1 M 1
=
X
Ll
(2L + 1)(2l + 1)
4 ha (pq)
L a \Lambda(ïq 1 )
l iC L1M1
Lm1 l\Gammaï C L1N
L\Gammaqlq 1
(\Gamma1) l d L
m1 \Gammap (#);
Fm (ae; #; !) =
X
l
i \Gammal (2l + 1)j l (2ae cos #
2 )d l
m0 ( #
2 )d l
m0 (!);
~
œ ae = aeIm('')
`
2 \Gamma cos !(1 \Gamma
1
cos #
)
'
+ iae cos ![Re('') \Gamma 1](1 + 1
cos #
); (6)
where m 1 = M 1 +ï, j l (x) is the spherical Bessel function. Coefficients fi (z)(pn)(ïÚ)
LM are determined
from the systems of equations [10]:
fi (z)(pn)(ïÚ)
LM = (\Gamma1) L i \Lambda(Ú ï)(np)
LM + 2‹n 0
k 3
X
qq 1
¼ (pq)(ïq1 )
L
Z X
lm
fi (ae)(qn)(q 1 Ú)
lm
\Theta d L
MN (!)d l
mN (!) exp(\Gamma~œ ae )Fm\GammaM (ae; #; !)dae sin !d!: (7)
Matrix S (co)
pnïÚ describes the interference of conjugate pairs of waves propagating along the
same trajectories in a medium but in reversal directions [1í7] including loopílike trajectories
[10]. The case when these loopílike trajectories degenerate to polylines was considered in
Ref.10 as a part of incoherent radiation which was described by matrix S 1
pnïÚ . But more
detailed consideration shows that this part of scattered radiation is coherent part. Therefore
S 1
pnïÚ should correspond to single scattering (see the first term in the right side of the Eq.(5)).
3

Matrices (1) and (4) are written in the CP írepresentation. The transformation from CP to
LP írepresentation (the basis of linear polarization) can be found in Refs.12,13. For example,
using determination of the Stokes parameters in Ref.12, the components of the diffuse reflection
matrix R (nc)
ij can be written as:
R (nc)
11 = U
X
pn
S (nc)
pnpn ; R (nc)
21 = \GammaU
X
pn
S (nc)
pn\Gammapn ; R (nc)
22 = U
X
pn
S (nc)
pn\Gammap\Gamman ;
R (nc)
33 = U
X
pn
S (nc)
pn\Gammap\Gamman i p\Gamman ; R (nc)
43 = U
X
pn
S (nc)
pnp\Gamman i p\Gamman+1 ; R (nc)
44 = U
X
pn
S (nc)
pnpn i p\Gamman : (8)
Here U = \Gamma1=2k 2 cos #. Similar relations can be written for the components of the coherent
reflection matrix R (co)
ij .
In special case of the exact backscattering (# = ‹) the Eqs.(6) yields [10]:
Fm (ae; ‹; !) = ffi m0 ; i (pn)(ïÚ)
LM = \Gamma(\Gamma1) L ¼ (pn)(ïÚ)
L ffi MN0 : (9)
In this case, as follows from the Eqs.(2),(4) and (5), S (nc)
pnïÚ = S (co)
pnÚï + S 1
pnÚï . Thus from the
Eqs.(1),(4) and (8) we can obtain Mishchenko's relations [15]:
2R (co)
11 = R (nc)
11 +R (nc)
22 \Gamma R (nc)
33 +R (nc)
44 \Gamma 2R 1
11 ;
2R (co)
22 = R (nc)
11 +R (nc)
22 +R (nc)
33 \Gamma R (nc)
44 \Gamma 2R 1
22 ;
2R (co)
33 = \GammaR (nc)
11 +R (nc)
22 +R (nc)
33 +R (nc)
44 \Gamma 2R 1
33 ;
2R (co)
44 = R (nc)
11 \Gamma R (nc)
22 +R (nc)
33 +R (nc)
44 \Gamma 2R 1
44 ; (10)
where matrix R 1
ij corresponds to single scattering.
If the size parameter of scatterers isn't too large, numerical solution of system (2) doesn't
have peculiarity. The numerical tests of the diffuse matrix (1) are in good agreement with those
from the vector radiative transfer equation. Unlike the system (2), the system (7) depends
on scattering angle #. At scattering angle # 6= ‹ numerical solution of this system is too
complicated because of frequent oscillation of the function Fm (ae; #; !). In this work we use the
doubleíscattering approximation for semiíinfinite medium. In this approach the expressions
for scattering matrices are simplified considerably. For example Eqs. (1),(2) are reduced to:
S (nc)
pnïÚ = \Gamma
n 0 cos #
2kIm('')(1 \Gamma cos #)
X
L
d L
M 0 N 0
(#)A (pn)(ïÚ)
L ; (11)
where
A (pn)(ïÚ)
L = ¼ (pn)(ïÚ)
L + ‹n 0
k 3 Im('')
X
qq1
¼ (pq)(ïq 1 )
L
X
l
¼ (qn)(q 1 Ú)
l
`Z ‹=2
0
d L
M 0 N (!)d l
M 0 N (!) cos # sin !d!
cos # \Gamma cos !
+
Z ‹=2
0
d L
M0N (‹ \Gamma !)d l
M0N (‹ \Gamma !)
sin !d!
1 + cos !
'
: (12)
The first term in the last equation corresponds to the single scattering, the second one is related
to the secondíorder scattering.
To determine the coherent matrix components in this approach, we use the expressions:
Fm (ae; #; !) =
X
l
i \Gammal (2l + 1)j l (2ae cos #
2 )d l
m0 ( #
2 )d l
m0 (!)
= i \Gammam Jm (ae sin # sin !) exp[\Gammaiae cos !(1 + cos #)];
Z 1
0
Jm (bx) exp(\Gammaax)dx = b m
p
a 2 + b 2 (a +
p
a 2 + b 2 ) m ; (m ? \Gamma1; Re(a) ? Im(b)): (13)
4

Here Jm (x) is the Bessel function. Thus from Eqs.(4) and (5) one can obtain:
S (co)
pnÚï = \Gamma
‹n 2
0 cos #
k 4 Im('')(1 \Gamma cos #)
X
LM lmqq1
i (pq)(ïq 1 )
LM (\Gamma1) l i \Lambda(Ú q 1 )(nq)
lm i \Gammajm\GammaM j B (pn))(ïÚ)
LM lm ; (14)
where
B (pn)(ïÚ)
LM lm =
Z ‹=2
0
d L
MN (!)d l
mN (!) c jm\GammaM j sin !d!
p
c 2 + f 2 (f + p
c 2 + f 2 ) jm\GammaM j
+
Z ‹=2
0
d L
MN (‹ \Gamma !)d l
mN (‹ \Gamma !)
c jm\GammaM j sin !d!
p
c 2 + f \Lambda2 (f \Lambda +
p
c 2 + f \Lambda2 ) jm\GammaM j ;
c = sin # sin !;
f = 2Im('') + cos !
`
Im('')(1 \Gamma
1
cos #
)
+ i[Re('') \Gamma 1](1 + 1
cos #
) + i(1 + cos #)
'
: (15)
As follows from the above presented formulae, at the exact backscattering direction (# = ‹)
the intensity and the interference term of the secondíorder scattered waves are numerically
equal to each other. At the other scattering angles (# 6= ‹) the matrix (14) depends strongly
on the properties of the medium.
3 Calculations and discussion
The coherent backscattering mechanism for the effect of the negative polarization was considí
ered in details in Refs.3, 4. It was demonstrated that in case of the positive polarization for
the isolated scatterers, the interference of secondíorder scattering waves results in the negaí
tive polarization peak in the backscattering direction [3,4]. However positive polarization is a
peculiar property of the Rayleigh scatterers. The angular dependence of polarization for the
particles whose diameter is of the order of the wavelength becomes too complicated. Therefore,
the interference of secondíorder scattering waves can result in positive polarization as well as
in negative one.
We demonstrate this by calculations of the angular dependence of linear polarization degree
P (P = \GammaR 21 (#)=R 11 (#)) and normalized intensity I (I = R 11 (#)=R 11 (‹)) for semiíinfinite
medium composed of identical spherical particles under different filling factors ¦. Filling factor
is related to concentration of scatterers n 0 in the medium by the expression:
¦ = 4
3 ‹~a 3 n 0 ; (16)
where ~ a is the radius of the scatterer. The following value of the effective refractive index of
the medium is used '' = 1 + in 0 C ext =2k, where C ext is the optical extinction cross section. For
comparison we presented also the angle dependence of the linear polarization P 0 = \Gammas 21 =s 11
for isolate particles , where s ij are the Mie scattering matrix elements.
Fig.1 corresponds to the medium composed of spherical particles with ~ x = k~a = 3 and the
complex refractive index ~
m = 1:35 + 0i. Scattering matrix element s 11 for such particles is
5

very asymmetric (s 11 (# = 0 0 )=s 11 (# = 180 0 ) ' 194) and the most of radiation is scattered in
direction of angles # ! 60 0 . The polarization of the interference part of scattered radiation is
determined by behavior of P 0 in ranges of angles # ! 60 0 and # ? 120 0 . Average value of P 0 in
these scattering angle ranges is positive (see Fig.1). Therefore the interference of secondíorder
scattered waves results in the negative polarization near the opposition direction. Note, that
our data are in a qualitative accordance with the data for the semiíinfinite medium composed
of noníabsorbing Rayleigh scatterers [9].
Fig.2 corresponds to scattering particles with ~
x = 4:5 and the refractive index ~
m = 1:33+0i.
For such particles s 11 (# = 0 0 )=s 11 (# = 180 0 ) ' 162. Average value of P 0 in range of # ! 60 0
and in range of # ? 120 0 is negative. In this case the interference of the secondíorder scattered
waves results in a positive peak of polarization in the backscattering direction.
Fig. 3 shows another polarization behavior of the reflected radiation. It corresponds to
scattering particles with ~ x = 3 and the refractive index ~
m = 1:5 + 0:5i. In this case s 11 (# =
0 0 )=s 11 (# = 180 0 ) ' 350. Average polarization P 0 in range of angles # ! 60 0 is positive, and
in range of # ? 120 0 is negative. The interference of waves for semiíinfinite medium results in
more complicated dependence of the polarization.
The photometric opposition effect always manifests itself as a typical sharp peak of intensity
centered in the exact backscattering direction. Width of the peak depends significantly on the
filling factor (Figs.1í3).
Note that presented data are obtained in the assumption that the orders of scattering
higher than the second one are polarized weakly. However in some case (for example, when
Im( ~
m) ' 0) multiple scattering intensity can be considerable and influence essentially on the
polarization pattern.
It is necessary to note also the polarization behavior for rarefied media. As shown in Fig.1
the peak of negative polarization is asymmetric. Such polarization behavior differs from the
symmetric peak of negative polarization observed for many atmosphereless celestial bodies [6].
This difference is caused by nearífield effect [16]. The role of this effect increases with an
increase of the phase angle [17]. It leads to symmetric polarization curve in backscattering
direction for closely packed media.
Acknowledgements
We thank Michael I. Mishchenko for computer codes for numerical solution of the vector
radiative transfer equation. This allowed us to test our calculations of diffuse reflection matrix.
V.Tishkovets and P.Litvinov thank INTAS (grant N 1999í00652) for the support of this work.
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7

Figure captions
Fig. 1. Coherent (1, 2) and diffuse (3) components of scattering radiation for medium with
~
x = 3 and ~
m = 1:35 + 0i. Curve 1 corresponds to ¦ = 0:001, curve 2 corresponds to ¦ = 0:01.
s 21 is the Mie matrix element.
Fig.2. The same as in Fig.1, but for ~ x = 4:5 and ~
m = 1:33 + 0i.
Fig.3. The same as in Fig.1, but for ~
x = 3 and ~
m = 1:5 + 0:5i. Curve 1 corresponds to
¦ = 0:002, curve 2 corresponds to ¦ = 0:01.
8