The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density
The amplitude of the coherent backscattering intensity peak is computed for a medium composed of densely packed, randomly positioned particles. The cyclical component of the Stokes reflection matrix at exactly the back scatter ing direction is expressed in terms of the ladder component, and the ladd...
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| Published in: | Кинематика и физика небесных тел |
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| Date: | 2010 |
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Головна астрономічна обсерваторія НАН України
2010
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| Cite this: | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density / M.I. Mishchenko // Кинематика и физика небесных тел. — 2010. — Т. 26, № 3. — С. 3-14. — Бібліогр.: 29 назв. — англ. |
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| citation_txt | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density / M.I. Mishchenko // Кинематика и физика небесных тел. — 2010. — Т. 26, № 3. — С. 3-14. — Бібліогр.: 29 назв. — англ. |
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| description | The amplitude of the coherent backscattering intensity peak is computed for a medium composed of densely packed, randomly positioned particles. The cyclical component of the Stokes reflection matrix at exactly the back scatter ing direction is expressed in terms of the ladder component, and the ladder component is rigorously computed by numerically solving the vector radiative transfer equation. The effect of packing density is accounted for by multiplying the single-scattering Mueller matrix by the static structure factor computed in the Percus — Yevick approximation. It is shown that increasing packing density can substantially reduce the amplitude of the copolarized coherent backscattering peak, especially for smaller particles, and can make it significantly lower than 2. The effect of packing density on the amplitude of the cross-polarized peak is significantly weaker.
Рассчитывается амплитуда обратного когерентного пика в интенсивности для среды, состоящей из плотно упакованных случайно расположенных частиц. Циклический компонент матрицы отражения в представлении Стокса в направлении точно назад выражен через лестничный компонент. Последний рассчитывается путем численного решения векторного уравнения переноса. Эффект плотной упаковки учитывается путем умножения матрицы Мюллера однократного рассеяния на статисческий структурный фактор, рассчитанный в приближении Перкуса—Йевика. Показано, что увеличение плотности упаковки может значительно ослабить параллельно поляризованный когерентный пик обратного рассеяния и уменьшить его амплитуду до значений, значительно меньших 2. Влияние плотной упаковки на амплитуду поперечно поляризованного пика оказывается намного более слабым.
Розраховано амплітуду зворотного когерентного піку інтенсивності для середовища, що складається із щільно упакованих випадково розташованих частинок. Циклічний компонент матриці відбиття у поданні Стокса у напрямку точно назад виражено через драбинний компонент. Останній розраховується шляхом числового розв’язку векторного рівняння переносу. Ефект щільної упаковки враховано шляхом множення матриці Мюллера однократного розсіяння на статичний структурний фактор, розрахований у наближенні Перкуса — Євіка. Показано, що збільшення щільності упаковки може значно послабити паралельно поляризований когерентний пік зворотного розсіяння та зменшити його амплітуду до значень, значно менших від 2. Вплив щільної упаковки на амплітуду поперечно поляризованого піку виявився набагато слабкішим.
|
| first_indexed | 2025-12-07T18:06:47Z |
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| fulltext |
ÏÐÎÁËÅÌÛ ÀÑÒÐÎÍÎÌÈÈ
UDC 523 + 523.4
Michael I. Mishchenko
NASA Goddard Institute for Space Studies
2880 Broadway, New York, New York 10025, USA
Main Astronomical Observatory of the National Academy of Sciences of Ukraine
27 Akad. Zabolotnoho St., Kyiv 03680, Ukraine
The amplitude of the coherent backscattering
intensity peak for discrete random media:
effect of packing density
The amplitude of the coherent backscattering intensity peak is computed for
a medium composed of densely packed, randomly positioned particles. The
cyclical component of the Stokes reflection matrix at exactly the back scat -
ter ing direction is expressed in terms of the ladder component, and the lad -
der component is rigorously computed by numerically solving the vector
radiative transfer equation. The effect of packing density is accounted for
by multiplying the single-scattering Mueller matrix by the static structure
factor computed in the Percus — Yevick approximation. It is shown that
increasing packing density can substantially reduce the amplitude of the
copolarized coherent backscattering peak, especially for smaller particles,
and can make it significantly lower than 2. The effect of packing density on
the amplitude of the cross-polarized peak is significantly weaker.
ÀÌÏ˲ÒÓÄÀ ÇÂÎÐÎÒÍÎÃÎ ÊÎÃÅÐÅÍÒÍÎÃΠϲÊÓ ²ÍÒÅÍ ÑÈ -
ÍÎÑÒ² ÄËß ÂÈÏÀÄÊÎÂÈÕ ÄÈÑÊÐÅÒÍÈÕ ÑÅÐÅÄÎÂÈÙ: ÅÔÅÊÒ
Ù²ËÜÍί ÓÏÀÊÎÂÊÈ, ̳ùåíêî Ì. ². — Ðîçðàõîâàíî àìïë³òóäó
çâîðîòíîãî êîãåðåíòíîãî ï³êó ³íòåíñèâíîñò³ äëÿ ñåðåäîâèùà, ùî
ñêëàäàºòüñÿ ³ç ù³ëüíî óïàêîâàíèõ âèïàäêîâî ðîçòàøîâàíèõ ÷àñòèíîê.
Öèêë³÷íèé êîìïîíåíò ìàòðèö³ â³äáèòòÿ ó ïîäàíí³ Ñòîêñà ó íàïðÿì -
êó òî÷íî íàçàä âèðàæåíî ÷åðåç äðàáèííèé êîìïîíåíò. Îñòàíí³é ðîç -
ðàõîâóºòüñÿ øëÿõîì ÷èñëîâîãî ðîçâ’ÿçêó âåêòîðíîãî ð³âíÿííÿ ïå ðå -
íîñó. Åôåêò ù³ëüíî¿ óïàêîâêè âðàõîâàíî øëÿõîì ìíîæåííÿ ìàò ðè ö³
Ìþëëåðà îäíîêðàòíîãî ðîçñ³ÿííÿ íà ñòàòè÷íèé ñòðóêòóðíèé ôàê -
òîð, ðîçðàõîâàíèé ó íàáëèæåíí³ Ïåðêóñà — ªâ³êà. Ïîêàçàíî, ùî
3
ÊÈÍÅÌÀÒÈÊÀ
È ÔÈÇÈÊÀ
ÍÅÁÅÑÍÛÕ
ÒÅË òîì 26 ¹ 3 2010
© M. I. MISHCHENKO, 2010
4
M. I. MISHCHENKO
çá³ëüøåííÿ ù³ëüíîñò³ óïàêîâêè ìîæå çíà÷íî ïîñëàáèòè ïàðàëåëüíî
ïîëÿðèçîâàíèé êîãåðåíòíèé ï³ê çâîðîòíîãî ðîçñ³ÿííÿ òà çìåíøèòè
éîãî àìïë³òóäó äî çíà÷åíü, çíà÷íî ìåíøèõ â³ä 2. Âïëèâ ù³ëüíî¿ óïà -
êîâêè íà àìïë³òóäó ïîïåðå÷íî ïîëÿðèçîâàíîãî ï³êó âèÿâèâñÿ íàáàãàòî
ñëàáê³øèì.
ÀÌÏËÈÒÓÄÀ ÎÁÐÀÒÍÎÃÎ ÊÎÃÅÐÅÍÒÍÎÃÎ ÏÈÊÀ Â ÈÍÒÅÍÑÈÂ -
ÍÎÑÒÈ ÄËß ÑËÓ×ÀÉÍÛÕ ÄÈÑÊÐÅÒÍÛÕ ÑÐÅÄ: ÝÔÔÅÊÒ ÏËÎÒ -
ÍÎÉ ÓÏÀÊÎÂÊÈ, Ìèùåíêî Ì. È. — Ðàññ÷èòûâàåòñÿ àìïëèòóäà
îáðàòíîãî êîãåðåíòíîãî ïèêà â èíòåíñèâíîñòè äëÿ ñðåäû, ñîñòî -
ÿùåé èç ïëîòíî óïàêîâàííûõ ñëó÷àéíî ðàñïîëîæåííûõ ÷àñòèö. Öèê -
ëè÷åñêèé êîìïîíåíò ìàòðèöû îòðàæåíèÿ â ïðåäñòàâëåíèè Ñòîêñà â
íàïðàâëåíèè òî÷íî íàçàä âûðàæåí ÷åðåç ëåñòíè÷íûé êîìïîíåíò.
Ïîñëåäíèé ðàññ÷èòûâàåòñÿ ïóòåì ÷èñëåííîãî ðåøåíèÿ âåêòîðíîãî
óðàâíåíèÿ ïåðåíîñà. Ýôôåêò ïëîòíîé óïàêîâêè ó÷èòûâàåòñÿ ïóòåì
óìíîæåíèÿ ìàòðèöû Ìþëëåðà îäíîêðàòíîãî ðàññåÿíèÿ íà ñòàòè -
÷åñêèé ñòðóêòóðíûé ôàêòîð, ðàññ÷èòàííûé â ïðèáëèæåíèè Ïåð -
êóñà—Éåâèêà. Ïîêàçàíî, ÷òî óâåëè÷åíèå ïëîòíîñòè óïàêîâêè ìîæåò
çíà÷èòåëüíî îñëàáèòü ïàðàëëåëüíî ïîëÿðèçîâàííûé êîãåðåíòíûé ïèê
îáðàòíîãî ðàññåÿíèÿ è óìåíüøèòü åãî àìïëèòóäó äî çíà÷åíèé, çíà -
÷èòåëüíî ìåíüøèõ 2. Âëèÿíèå ïëîòíîé óïàêîâêè íà àìïëèòóäó ïî -
ïåðå÷íî ïîëÿðèçîâàííîãî ïèêà îêàçûâàåòñÿ íàìíîãî áîëåå ñëàáûì.
INTRODUCTION
Coherent backscattering of light by discrete random media has been
intensively investigated during the last two decades both experimentally
and theoretically [2, 13, 16, 22, 26]. Moreover, it has been shown that this
phenomenon can be observed not only in laboratory conditions but also in
nature in the form of the photometric and polarization opposition effects [1,
10, 14, 17—20]. The primary theoretical tool for computing the
backscattering intensity peak has been the diffusion approximation [2].
However, although the diffusion approximation rather accurately predicts
the angular profile of the backscattering peak, it cannot be used to compute
the amplitude of the peak, i.e., the ratio of the intensity at the center of the
peak to the incoherent background intensity. An additional complexifying
factor is that accurate computations of the amplitude of the backscattering
peak must explicitly take into account the vector nature of light since
polarization effects have been shown to be extremely important in coherent
backscattering [8, 15, 16].
Reflection of polarized light by a discrete random medium can be fully
described by a 4´4 Stokes reflection matrix. In [8], Saxon’s reciprocity
principle [21] was used to derive a rigorous relationship between the
cyclical and ladder components of the reflection matrix at exactly the
backscattering direction. Two factors make this relationship very useful.
First, in the derivation of this relationship the vector nature of light has been
fully taken into account. Second, the ladder component of the reflection
matrix can be rigorously computed by solving the vector radiative transfer
equation (VRTE) with one of the well established numerical techniques [7,
16]. Both spherical and nonspherical scattering particles can be treated [16].
Therefore, this relationship can be used to compute the cyclical component
of the reflection matrix in the center of the backscattering peak and, thus,
the amplitude of the peak. This approach has been used in [5, 6, 9, 12, 16] to
extensively study the properties of the coherent backscattering effect for
different representations of polarization.
Numerical solutions of the VRTE used in [5, 6, 9, 12, 16] imply
sparsely distributed, “independently scattering” particles and hence are not
necessarily applicable to densely packed media, for example owing to
correlations among particle positions. Therefore, it is the aim of this paper
to extend the approach developed in [8] to media with nonzero packing
density and to examine the effect of packing density on the amplitude of the
coherent backscattering peak. Unlike the rigorous and sophisticated
approach pursued in [23—25], we will use a rather simple and inherently
approximate approach based on the so-called structure factor formalism.
THEORY AND COMPUTATIONS
Let the dis crete scat ter ing me dium be a ho mo ge neous semi-in fi nite slab
com posed of sparsely and ran domly dis trib uted par ti cles. The slab is il lu mi -
nated by a quasi-mono chro matic par al lel beam of light of in fi nite lat eral ex -
tent in ci dent in the di rec tion of the unit vec tor $n0 = {q 0 ³ p/2, j 0}, where q 0
is the cor re spond ing ze nith an gle mea sured from the pos i tive di rec tion of
the z axis and j 0 is the cor re spond ing az i muth an gle mea sured from the po s -
i tive di rec tion of the x axis in the clock-wise sense when look ing in the pos i -
tive di rec tion of the z axis (Fig. 1). In what fol lows, we will as sume for sim -
plic ity that j 0 = 0. The Stokes col umn vec tor has four Stokes pa ram e ters as
its com po nents: I = [ ]I Q U V T where T stands for “trans posed”. Let R b be
the 4´4 Stokes re flec tion ma trix for ex actly the back scat ter ing di rec tion
$nb= {q = p – q 0 , j 0 = p}. This ma trix yields the spe cific Stokes col umn vec -
tor of the back scat tered light as fol lows:
~
Ib =
1
0 0
p
m R Ib , (1)
where m q0 0= -cos and I 0 is the Stokes column vector of the incident beam
[16].
Under the simplifying assumption of a macroscopically isotropic and
mirror-symmetric particulate medium, the backscattering matrix R b has the
following block-diagonal structure [7, 16]:
5
THE AMPLITUDE OF THE COHERENT BACKSCATTERING PEAK
R b =
R R
R R
R R
R R
b b
b b
b b
b b
11 12
12 22
33 34
34 44
0 0
0 0
0 0
0 0 -
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
. (2)
In accordance with the microphysical theory of coherent backscattering
by sparse discrete random media [13, 16], the backscattering Stokes matrix
R b can be decomposed as follows:
R R R Rb b b
M
b
C= + +1 , (3)
where R b
1 is the con tri bu tion of the first-or der scat ter ing, R b
M is the dif fuse
com po nent con sist ing of all the lad der terms of scat ter ing or ders (n ³ 2), and
R b
C is the cu mu la tive con tri bu tion of all the cy cli cal terms. The ma tri ces R b
1
and R b
M can be found by nu mer i cally solv ing the VRTE [16]. Then the ma -
trix R b
C can be de ter mined from the fol low ing ex act re la tion de rived in [8,
16]:
R b
C =
R R
R R
R R
R R
b
C
b
M
b
M
b
C
b
C
b
M
b
M
b
C
11 12
12 22
33 34
34 44
0 0
0 0
0 0
0 0 -
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
, (4)
where
6
M. I. MISHCHENKO
Fig. 1. Right-handed laboratory coordinate system with origin at the upper boundary of a
semi-infinite particulate slab. The direction of light propagation is specified by a unit vector $n = {q,
j}. The slab is illuminated by a quasi-monochromatic parallel beam of light incident from above in
the direction of the unit vector $n 0
R R R R Rb
C
b
M
b
M
b
M
b
M
11 11 22 33 4405= + - +. ( ), (5)
R R R R Rb
C
b
M
b
M
b
M
b
M
22 11 22 33 4405= + + -. ( ), (6)
R R R R Rb
C
b
M
b
M
b
M
b
M
33 11 22 33 4405= - + + +. ( ), (7)
R R R R Rb
C
b
M
b
M
b
M
b
M
44 11 22 33 4405= - + +. ( ). (8)
Once all the components of the backscattering Stokes matrix are
computed, they can be used to calculate the amplitude of the coherent
backscattering peak for different states of polarization of the incident and
scattered beams [8, 16]. Specifically, if the incident light is fully linearly
polarized in the vertical direction, then the amplitudes of the copolarized
and cross-polarized peaks are expressed in terms of the backscattering
Stokes matrix as
z vv =
R R R R R
R R R
b b b
M
b
M
b
M
b b b
M
11
1
22
1
11 12 22
11
1
22
1
11
2 4 2+ + + +
+ + + 2 12 22R Rb
M
b
M+
, (9)
z hv =
R R R R R R
R R R
b b b
M
b
M
b
M
b
M
b b b
11
1
22
1
11 22 33 44
11
1
22
1
1
- + - - +
- + 1 22
M
b
MR-
, (10)
respectively. In the case of fully circularly polarized incident beam, we
have for the amplitudes of the helicity-preserving and opposite-helicity
peaks:
z hp
b b b
M
b
M
b b b
M
b
R R R R
R R R R
=
+ + +
+ + +
11
1
44
1
11 44
11
1
44
1
11 4
2 2
4
M
, (11)
z oh
b b b
M
b
M
b
M
b
M
b b
R R R R R R
R R
=
- + + - -
-
11
1
44
1
11 22 33 44
11
1
44
1 + -R Rb
M
b
M
11 44
. (12)
For spherical particles R Rb b22
1
11
1= and R Rb b44
1
11
1= - , which implies that
z vv
b
b b
M
b
M
b
M
R
R R R R
= -
+ + +
2
2
2 2
11
1
11
1
11 12 22
, (13)
z hv
b
M
b
M
b
M
b
M
R R
R R
= -
-
-
1 33 44
11 22
, (14)
z hp º 2, (15)
z oh
b
M
b
M
b b
M
b
M
R R
R R R
= +
-
+ -
1
2
22 33
11
1
11 44
. (16)
To solve numerically the VRTE, one needs to know single-scattering
characteristics of particles comprising the medium. The single scattering of
polarized light by a particle is described by the 4´4 Stokes scattering matrix
F [16]. For spherical particles, the elements of the scattering matrix depend
on the particle refractive index and size parameter x (defined as x = 2p lr / ,
7
THE AMPLITUDE OF THE COHERENT BACKSCATTERING PEAK
where r is particle radius and l is the wavelength in the surrounding me -
dium) as well as on the scattering angle Q (i.e., the angle between the
incidence and scattering directions). The (1, 1) element of the scattering
matrix is called the differential scattering cross section and is often denoted
as dC dsca / W. The scattering cross section C sca is obtained by integrating
the differential scattering cross section over all scattering directions:
C d
dC
d
sca
sca= ò W
W
4p
. (17)
The quantity
p
C
dC
dsca
sca( )Q
W
=
4p
(18)
is called the phase function and satisfies the normalization condition
1
4
1
4
p
p
d pò =W Q( ) . (19)
Finally, the asymmetry parameter (or the mean cosine of the scattering
angle) is defined as
ácosQñ =
1
4
4
p
p
d pW Q Q( )cosò . (20)
The stan dard tech niques for com put ing the sin gle-scat ter ing ma trix
(e.g., the con ven tional Lorenz — Mie the ory for spher i cal scat ter ers and the
T-ma trix method for nonspherical par ti cles [16]) for dis crete ran dom me dia
im ply that par ti cles are widely sep a rated and their po si tions are in de pend ent
of each other [13, 16]. How ever, in densely packed me dia both con di tions
are vi o lated. In par tic u lar, spa tial cor re la tions be tween par ti cles may be -
come im por tant. As has been sug gested in [29], in the first ap prox i ma tion
the cor re spond ing mod i fi ca tion of the sin gle-scat ter ing ma trix can be ac -
counted for via mul ti ply ing F by the so-called static struc ture fac tor S which
de pends on the fill ing fac tor f (i.e., the frac tion of the scat ter ing vol ume oc -
cu pied by the par ti cles) and the prod uct
u x= 4 2sin( / )Q . (21)
The computation of the static structure factor using the Percus—Yevick
approximation for hard, impenetrable, monodisperse spheres is described,
e, g., in [11]. Figure 2 shows the structure factor computed for several
values of the filling factor. It is seen that increasing packing density results
in an angular redistribution of the scattered intensity which is especially
pronounced at u < 10.
As follows from Eqs. (13) and (16), the copolarized and opposite-
helicity amplitudes depend explicitly on the (1, 1) element of the single-
scattering component of the Stokes backscattering matrix R b11
1 . If this
component is neglected, which is the case in the framework of the diffusion
8
M. I. MISHCHENKO
approximation, the copolarized amplitude becomes equal to 2. Therefore, it
is the deviation of Rb11
1 from zero that makes the copolarized amplitude
smaller than 2. The single-scattering component of the Stokes back -
scattering matrix is in turn proportional to the backscattering phase function
p(180°). Specifically, in the case of a semi-infinite medium [7, 16],
R
p
b11
1 180
8
=
°v
p
( )
, (22)
where v is the single-scattering albedo (for nonabsorbing media considered
below, v = 1).
Fig ure 3, a shows the back scat ter ing phase func tion P = p(180°) vs. size
pa ram e ter x com puted for monodisperse spher i cal la tex par ti cles in wa ter
(rel a tive re frac tive in dex 1.195) for sev eral val ues of the fill ing fac tor rang -
ing from 0 to 0.4. In ad di tion, Fig. 3, b shows the ra tio R of p(180°) for
densely packed par ti cles (f > 0) to that for widely sep a rated par ti cles (f = 0).
It is seen that the ef fect of par ti cle cor re la tions can be strong and is es pe -
cially pro nounced for par ti cles with size pa ram e ters smaller than about 2,
and that in creas ing pack ing den sity al ways in creases p(180°). Note, how -
ever, that the phase func tion for Ray leigh par ti cles (x << 1) re mains in tact
with in creas ing pack ing den sity. In ter est ingly, p(180°) is an os cil lat ing
rather than a monotonically de creas ing func tion of size pa ram e ter, and,
thus, one should not ex pect a monotonic de pend ence of the copolarized am -
pli tude z vv on x [see Eq. (13)]. For com par i son, Fig. 3, c shows the asym me -
try pa ram e ter vs. size pa ram e ter for the same val ues of the fill ing fac tor.
One sees that for fill ing fac tors not ex ceed ing 0.4 and size pa ram e ters larger
than about 1.5, ácosQñ is a monotonically in creas ing func tion of size pa ram -
e ter. There fore, we have to con clude that there is no di rect cor re la tion be -
tween the asym me try pa ram e ter on one hand and the back scat ter ing phase
func tion (and, thus, the copolarized am pli tude) on the other hand.
Fig ures 4, a—c show the copolarized, z vv , cross-po lar ized, z hv , and op -
po site-helicity, z oh , am pli tudes vs. size pa ram e ter com puted for a semi-in fi -
9
THE AMPLITUDE OF THE COHERENT BACKSCATTERING PEAK
Fig. 2. Static struc ture fac tor S for dif -
fer ent val ues of the fill ing fac tor f
nite scat ter ing me dium which is com posed of monodisperse spher i cal par ti -
cles with the in dex of re frac tion 1.195 and is il lu mi nated per pen dic u larly to
its bound ary. For both sparsely and densely packed par ti cles, the VRTE was
rig or ously solved us ing the tech nique de scribed in [3, 16]. For densely
packed par ti cles, the sin gle-scat ter ing char ac ter is tics were mod i fied by
means of the static structure factor.
It is clearly seen from Figs 4, a—c that in creas ing pack ing den sity af -
fects all the am pli tudes. The ef fect is es pe cially strong for small size pa ram -
e ters and is much more no tice able for z vv and, to a lesser de gree, for z oh be -
cause these am pli tudes de pend on the sin gle-scat ter ing com po nent Rb11
1 ex -
plic itly [Eqs. (13) and (16)].
Not sur pris ingly, lo cal max ima of z vv (Fig. 4, a) ex actly fol low lo cal
min ima of the back scat ter ing phase func tion p(180°) (Fig. 3, a). Since in -
10
M. I. MISHCHENKO
Fig. 3. Back scat ter ing phase func tion P =
p(180°), the ra tio R of the back scat ter ing
phase func tion for densely packed par ti cles
(f > 0) to that of sparsely dis trib uted par ti cles
(f = 0), and the asym me try pa ram e ter ácosQñ
vs. size pa ram e ter for monodisperse spher i -
cal par ti cles with re frac tive in dex 1.195 and
dif fer ent val ues of the fill ing fac tor f
creas ing pack ing den sity al ways in creases the back scat ter ing phase func -
tion, it al ways re duces the copolarized am pli tude and, for small par ti cles,
can even make it lower than the Ray leigh-limit value of 1.752 [16]. On the
other hand, the size-pa ram e ter dependences of the am pli tudes z hv and z oh do
not quite fol low that of Rb11
1 . Ap par ently, this can be ex plained by the im -
plicit de pend ence of the ra tio z hv on the sin gle-scat ter ing ma trix and by a
com pli cated size pa ram e ter de pend ence of the di ag o nal el e ments of the lad -
der com po nent R b
M [see Eqs. (14) and (16)].
11
THE AMPLITUDE OF THE COHERENT BACKSCATTERING PEAK
Fig. 4. Am pli tude zvv of the copolarized in ten -
sity peak (a), the cross-po lar ized am pli tude zhv
(b), and the op po site-helicity am pli tude zoh (c)
vs. size pa ram e ter for a semi-in fi nite me dium
com posed of monodisperse, ran domly po si -
tioned spher i cal par ti cles with re frac tive in dex
1.195 and dif fer ent val ues of the fill ing fac tor f
DIS CUS SION
The standard theories of radiative transfer and coherent backscattering are
based on the assumption that particles forming a discrete random medium
are widely separated and totally uncorrelated [13, 16]. The approach
pursued in this paper can only be considered to be a rather trivial
approximate “patch” intended to take into account the effect of spatial
correlations for closely spaced particles. It goes without saying that it
cannot replace a rigorous theory directly based on the Maxwell equations.
In fact, it will be very interesting to compare our approximate results with
exact computations when the latter have become available.
As mentioned above, the diffusion approximation predicts the
amplitude of the copolarized backscattering peak exactly equal to 2 [2].
However, some experimentalists have claimed that they found evidence for
an amplitude smaller than two. As a result, a discussion as to the actual
value of the copolarized amplitude has taken place (see, e.g., [4] and
references therein). In order to partially explain the discrepancy between
the prediction of the diffusion approximation and laboratory experiments,
the effect of the backscattering contribution from nonself-avoiding closed
light paths, in addition to the ladder and cyclical contributions, was studied
[27, 28].
As follows from Eq. (13), the amplitude of the copolarized peak is
never equal to 2 since, for real scattering particles, Rb11
1 is never exactly
equal to zero but rather is a positive number. Moreover, it was shown in [16]
that z vv is always smaller than 2. The degree of deviation of the copolarized
amplitude from the value 2 depends primarily on the value of the
backscattering phase function. For latex particles in water p(180°) is
usually small (except for particles with size parameters less than about 1)
because the corresponding relative refractive index (1.195) is small.
However, for larger refractive indices p(180°) can be much larger, thereby
causing copolarized amplitude values significantly smaller than 2 [9, 12].
As follows from the calculations reported in this paper, spatial
correlations among scattering particles caused by nonzero packing density
can be an additional important factor which can substantially increase the
backscattering phase function and, thus, reduce the amplitude of the
copolarized backscattering peak. The effect is especially pronounced for
small particles (but not for Rayleigh scatterers) and weakens as the particle
size parameter becomes much greater than 1. On the other hand, the
amplitude of the helicity-preserving amplitude z hp for spherical particles is
not influenced at all by increasing packing density and is identically equal
to 2 independently of the filling factor. Apparently, this can explain, at least
qualitatively, why in the measurements for latex particles in water reported
in [4] the amplitude of the helicity-preserving backscattering peak was
always equal to 2 (within the measurement accuracy), while the amplitude
of the copolarized peak was substantially smaller than 2, especially for
small particles.
12
M. I. MISHCHENKO
Finally we note that for spherical particles the equalities Rb22
1 = Rb11
1 and
Rb44
1 = -Rb11
1 do not, in general, hold. Therefore, the helicity-preserving
amplitude z hp can explicitly depend on the single-scattering component of
the Stokes backscattering matrix R b
1 [Eq. (11)] and, thus, can be
appreciably smaller than 2 [5, 6, 16] and change with increasing packing
density. Therefore, unlike the case with spherical particles, the calculation
of the helicity-preserving amplitude for nonspherical particles should
explicitly take into account the effect of packing density.
The au thor is grate ful to P. V. Litvinov, Yu. G. Shkuratov,
V. P. Tishkovets, and E. G. Yanovitskij for many fruit ful dis cus sions. Par -
tial fund ing for this re search was pro vided by the NASA Ra di a tion Sci ences
Pro gram man aged by H. Maring.
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Re ceived March 26, 2009
14
M. I. MISHCHENKO
|
| id | nasplib_isofts_kiev_ua-123456789-73239 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-07T18:06:47Z |
| publishDate | 2010 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Mishchenko, M.I. 2015-01-06T16:47:51Z 2015-01-06T16:47:51Z 2010 The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density / M.I. Mishchenko // Кинематика и физика небесных тел. — 2010. — Т. 26, № 3. — С. 3-14. — Бібліогр.: 29 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/73239 523 + 523.4 The amplitude of the coherent backscattering intensity peak is computed for a medium composed of densely packed, randomly positioned particles. The cyclical component of the Stokes reflection matrix at exactly the back scatter ing direction is expressed in terms of the ladder component, and the ladder component is rigorously computed by numerically solving the vector radiative transfer equation. The effect of packing density is accounted for by multiplying the single-scattering Mueller matrix by the static structure factor computed in the Percus — Yevick approximation. It is shown that increasing packing density can substantially reduce the amplitude of the copolarized coherent backscattering peak, especially for smaller particles, and can make it significantly lower than 2. The effect of packing density on the amplitude of the cross-polarized peak is significantly weaker. Рассчитывается амплитуда обратного когерентного пика в интенсивности для среды, состоящей из плотно упакованных случайно расположенных частиц. Циклический компонент матрицы отражения в представлении Стокса в направлении точно назад выражен через лестничный компонент. Последний рассчитывается путем численного решения векторного уравнения переноса. Эффект плотной упаковки учитывается путем умножения матрицы Мюллера однократного рассеяния на статисческий структурный фактор, рассчитанный в приближении Перкуса—Йевика. Показано, что увеличение плотности упаковки может значительно ослабить параллельно поляризованный когерентный пик обратного рассеяния и уменьшить его амплитуду до значений, значительно меньших 2. Влияние плотной упаковки на амплитуду поперечно поляризованного пика оказывается намного более слабым. Розраховано амплітуду зворотного когерентного піку інтенсивності для середовища, що складається із щільно упакованих випадково розташованих частинок. Циклічний компонент матриці відбиття у поданні Стокса у напрямку точно назад виражено через драбинний компонент. Останній розраховується шляхом числового розв’язку векторного рівняння переносу. Ефект щільної упаковки враховано шляхом множення матриці Мюллера однократного розсіяння на статичний структурний фактор, розрахований у наближенні Перкуса — Євіка. Показано, що збільшення щільності упаковки може значно послабити паралельно поляризований когерентний пік зворотного розсіяння та зменшити його амплітуду до значень, значно менших від 2. Вплив щільної упаковки на амплітуду поперечно поляризованого піку виявився набагато слабкішим. The author is grateful to P. V. Litvinov, Yu. G. Shkuratov, V. P. Tishkovets, and E. G. Yanovitskij for many fruitful discussions. Partial funding for this research was provided by the NASA Radiation Sciences Program managed by H. Maring. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел Проблемы астрономии The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density Амплитуда обратного когерентного пика в интенсивности для случайных дискретных сред: эффект плотной упаковки Амплітуда зворотного когерентного піку інтенсивності для випадкових дискретних середовищ: ефект щільної упаковки Article published earlier |
| spellingShingle | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density Mishchenko, M.I. Проблемы астрономии |
| title | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| title_alt | Амплитуда обратного когерентного пика в интенсивности для случайных дискретных сред: эффект плотной упаковки Амплітуда зворотного когерентного піку інтенсивності для випадкових дискретних середовищ: ефект щільної упаковки |
| title_full | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| title_fullStr | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| title_full_unstemmed | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| title_short | The amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| title_sort | amplitude of the coherent backscattering intensity peak for discrete random media: effect of packing density |
| topic | Проблемы астрономии |
| topic_facet | Проблемы астрономии |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/73239 |
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