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SCATTERING AND POLARIZATION PROPERTIES
OF THE
NON­SPHERICAL PARTICLES
Nikolai V. Voshchinnikov
Astronomy Department and Sobolev Astronomical Institute, St. Petersburg University,
198504 St. Petersburg­Peterhof, Russia
ABSTRACT
The albedo and dichroic polarization of spheroidal
particles are studied. For the description of the phase func­
tion two asymmetry parameters g jj and g? characterizing
the anisotropy in forward/backward and left/right direc­
tions are introduced and calculated. The consideration is
based on the solution to the light scattering problem by the
Separation of Variables Method (Voshchinnikov and Fara­
fonov, 1993).
1. INTRODUCTION
In many scientific and engineering applications pro­
late and oblate spheroids are appropriate models for real
particles. We consider the light scattering by homoge­
neous spheroids using the Separation of Variables Method
(SVM). The optical properties of prolate spheroids of var­
ious aspect ratios a=b for several refractive indices m are
calculated and the results for the particles of the same vol­
ume are compared.
2. GENERAL DEFINITIONS AND METHOD
A spheroid (ellipsoid of revolution) is obtained by
the rotation of an ellipse around its major axis (prolate
spheroid) or its minor axis (oblate spheroid). The ratio
of the major semiaxis a to the minor semiaxis b (i.e. the
aspect ratio a=b) characterizes the particle shape which
may vary from a nearly spherical one (a=b  1) to a
needle or a disk (a=b  1).
We assume that an incident plane wave has the
wavelength . Let  denote the angle between the
propagation direction and the rotation axis of the spheroid
(0 ô    90 ô ).
For the axial propagation ( = 0 ô ), there is no
polarization of transmitted radiation due to symmetry. If
 6= 0 ô , two cases of polarization of the incident radiation
have to be considered: the electric vector ~
E is parallel (TM
mode) or perpendicular (TE mode) to the plane defined
by the spheroid's rotation axis and the wave propagation
vector.
The size parameter is given by
xV = 2rV

; (1)
where r V is the radius of the sphere whose volume is equal
to that of the spheroid. The radius r V for prolate spheroids
is defined as
r 3
V = ab 2 : (2)
One usually calculates the efficiency factors Q =
C=G which are the ratio of the corresponding cross­
sections C to the geometrical cross­section G of the prolate
spheroid (the area of the particle's shadow)
G() = b a 2 sin 2  + b 2 cos 2 

1=2
: (3)
In order to compare the optical properties of the parti­
cles of different shapes it is convenient to consider the ra­
tios of the cross­sections for spheroids to the geometrical
cross­sections of the equal volume spheres, C=r 2
V . For a
prolate spheroid they can be found as
C
r 2
V
=
[(a=b) 2 sin 2  + cos 2 ] 1=2
(a=b) 2=3 Q : (4)
The albedo of a particle can be calculated from the
extinction and scattering cross­sections
 =
C sca
C ext
: (5)
In general, the radiation scattered by aligned
spheroidal particles has an azimuthal asymmetry that pro­
vokes a non­coincidence of the directions of the radia­
tion pressure force and of the wave­vector of incident ra­
diation (Voshchinnikov, 1990; Il'in and Voshchinnikov,
1998). Another consequence of the azimuthal asymme­
try is the anisotropy of the phase function in the left/right
direction. The geometry of the phase function in for­
ward/backward and left/right directions may be charac­
terized by two asymmetry parameters g jj and g? , respec­
tively. Expressions for them can be found from the consid­
eration of radiation pressure (Voshchinnikov, 1990; Il'in
and Voshchinnikov, 1998)
g jj =
A (m; xV ; a=b; ) cos  + B (m; xV ; a=b; ) sin 
Q sca (m; xV ; a=b; )
;
(6)

Figure 1: Normalized scattering cross­sections for elongated spheroidal particles in dependence on size parameter 2a=.
Transformation to the parameter xV is given by the expression xV = (a=b) 2=3 (2a=)
g? =
A (m; xV ; a=b; ) sin  B (m; xV ; a=b; ) cos 
Q sca (m; xV ; a=b; )
;
(7)
where Q sca is the scattering efficiency factor, and the co­
efficients A and B are
A = K (x V ; a=b; )
Z 2
0
Z 
0
i(; ') cos  sin dd';
(8)
B = K (x V ; a=b; )
Z 2
0
Z 
0
i(; ') sin 2  cos 'dd':
(9)
In Eqs. (8) -- (9), K is a parameter, i(; ') the dimension­
less intensity of scattered radiation (phase function). From
a symmetry consideration, it is clear that for spheroids
g? = 0 if  = 0 ô or 90 ô . Note that in all cases the follow­
ing inequality is valid
1 
q
g 2
jj + g 2
?  1 : (10)
The dichroic polarization efficiency is defined by the
extinction cross­sections for TM and TE modes
P
 =
C TM
ext
C TE
ext
C TM
ext + C TE
ext

100% : (11)
This ratio describes the efficiency to polarize light trans­
mitted through an uniform slab consisting of non­rotating
particles of the same orientation.
The optical properties of spheroidal particles can be
determined by various methods of light scattering theory
(see Mishchenko et al., 2000 for a review). We use the
SVM's solution developed by Farafonov and numerical
code based on it (see Voshchinnikov and Farafonov, 1993
for more details). A comparison of different numerical
codes and benchmark results can be found in the paper of
Voshchinnikov et al. (2000).
The main problem of the SVM for spheroids is the
difficulties with computations of the spheroidal wavefunc­
tions. Especially it is related to very elongated particles
with sizes larger than the wavelength because the standard
expansions of the prolate spheroidal wavefunctions in se­
ries of the Legendre functions do not converge. Farafonov
and Voshchinnikov (2000, in preparation) have considered
new expansion of the prolate wavefunctions that opens a
possibility to calculate the optical properties of particles
with a=b  1. Firstly, this method was applied to calcula­
tions of the radial wavefunctions with the index m = 1 that
allows us to study the case of axial propagation of radia­
tion. Some results are shown in Fig. 1 for non­absorbing
spheroids with the refractive index m = 1:7 + 0:0i. As
the extinction and scattering cross­sections were the same
(with 6 and more digits), the values presented are expected
to be correct.
3. NUMERICAL RESULTS
We present some results illustrating the behaviour of
the optical properties of prolate spheroidal particles in a
fixed orientation.

3.1 Albedo
The integral scattering properties of particles are char­
acterized by their albedo. This quantity depends on the
particle size and, in general, on the particle shape. Figure 2
shows the size dependence of the albedo for spheroids with
m = 1:3 + 0:05i and m = 1:7 + 0:7i and the aspect ratios
a=b = 2 and 10. The calculations were made for pro­
late particles and  = 0 ô (we adopt that the incident radi­
ation is non­polarized). In this case in comparison with
others (oblate spheroids, oblique incidence of radiation)
the largest deviations of the ratio (spheroid)=(sphere)
from unity occur (see Voshchinnikov et al., 2000 for dis­
cussion).
Figure 2: Albedo of spheroidal particles normalized rela­
tive to albedo of spherical particles with the same refractive
index
It is seen that the albedo for large non­spherical par­
ticles becomes close to that of spheres. Our calculations
made for particles with different absorption show that
the distinction of the albedo for spheres and spheroidal
particles remains rather small (within  20 %) if the ratio
of the imaginary part of the refractive index to its real part
k=n >  0:2 0:3.
3.2 Asymmetry parameter
Another characteristic of scattered radiation is the
asymmetry parameter describing the spatial distribution
of scattered radiation around a particle. Usually the
anisotropy in forward/backward direction is only consid­
ered. However, the radiation scattered by any aligned
non­spherical particle possesses also an anisotropy in the
right/left direction in the case of oblique incidence of radi­
ation (see Fig. 3).
As it is seen from Fig. 4 both asymmetry factors g jj and
g ? change with a=b and . The values of radial asymmetry
factor g jj decrease with a growth of  when the path of
radiation reduces from 2a ( = 0 ô ) to 2b ( = 90 ô ). The
transversal asymmetry factor g? can be rather large and
even exceeds the radial one. Because the geometry of light
Figure 3: Geometry of light scattering by a prolate
spheroid. The wave­vector of incident non­polarized ra­
diation forms the angle  = 45 ô with the rotation axis of
the spheroid. Short­dashed curve shows the angular distri­
bution of the scattered radiation (phase function). The part
of scattered radiation which is symmetric relative to the
direction of incident radiation is plotted by long­dashed
curve. The radial and transversal asymmetry factors and
their values are indicated
scattering by very elongated spheroids approaches that of
infinite cylinders, 1 such particles scatter more radiation
``to the side'' than in forward direction.
The size dependence of asymmetry factors is plotted
in Fig. 5. It shows that the variations of g jj (x V ) for
spheroids with a=b = 2 are rather similar to those of
spheres. However, the radiation scattered by spheroids has
a noticeable azimuthal dependence which is absent for
spherical particles. If xV >  4 the azimuthal anisotropy of
scattered radiation reduces and g ? drops.
3.3 Polarization
If a volume contains aligned non­spherical particles,
the initially non­polarized incident radiation will be
partially polarized after having passed the volume. The
simplest and at the same time extreme case of particles'
alignment is the perfect alignment of non­rotating particles
(picked fence orientation). The maximum polarization
usually occurs when the major axes of the particles are
perpendicular to the direction of the incident radiation
( = 90 ô ).
The behaviour of the polarization efficiency P= for
non­absorbing and absorbing spheroids is shown in Fig. 6.
It is clearly seen that a relatively large particles produce
no polarization independent of their shape. For absorb­
ing particles, it occurs at smaller xV values than for non­
absorbing particles. This effect should depends on the
1 In this case the scattered radiation forms the conical surface with the
opening angle 2.

Figure 4: Angular dependence of the asymmetry factors of
scattered radiation
imaginary part of the refractive index as well.
4. CONCLUSIONS
The albedo of large non­spherical particles exhibits
only a weak dependence on the particle shape if the ratio
of the imaginary part of the refractive index to its real part
k=n >  0:2 0:3.
The radiation scattered by aligned spheroidal particles
has an azimuthal asymmetry and its geometry may be
described by two asymmetry parameters g jj and g?
showing the deviations from the symmetric scattering in
forward/backward and left/right directions, respectively.
The transversal asymmetry factor g? can be rather large
and even exceeds the radial one, therefore, very elongated
spheroids scatter more radiation ``to the side'' than in
forward direction.
Particles larger than a certain minimum size do not po­
larize the transmitted radiation independent of their shape.
ACKNOWLEDGEMENTS
The author is thankful to V.G. Farafonov for discus­
sion and to V.B. Il'in for valuable comments. This work
was financially supported by the INTAS foundation (grant
Open Call 99/652).
Figure 5: Size dependence of the asymmetry factors of
scattered radiation
Figure 6: Size dependence of the polarization efficiency
REFERENCES
Il'in, V.B., and N.V. Voshchinnikov, 1998: Radiation pres­
sure on non­spherical dust grains in envelopes of late­
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Mishchenko, M.I., J.W. Hovenier, and L.D. Travis (eds.),
2000: Light Scattering by Nonspherical Particles.
Academic Press, 690 pp.
Voshchinnikov, N.V., 1990: Radiation pressure on
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Voshchinnikov, N.V., and V.G. Farafonov, 1993: Optical
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