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Improved S-approximation for dielectric
particles ?
A.Y. Perelman a and N.V. Voshchinnikov b;
a State Forest Technical Academy, St. Petersburg, 194021 Russia
b Astronomy Department and Sobolev Astronomical Institute, St. Petersburg
University, St. Petersburg-Peterhof, 198504 Russia
(Received ............ 2000)
Abstract
The range of the applicability of the soft particles approximation (S-approximation)
has been extended by means of the procedure which may be considered as a specic
form of the analytic continuation. The initial element is represented by the standard
version of the approximation while the branch of the continuation should be chosen
in correspondence with the short wavelength asymptotic of the optical characteristic
under consideration. As for the extinction eôciency of dielectric spherical particles,
the improved version of the S-approximation fairly well describes the behaviour of
Mie curves for particles with the refractive indices up to 2.0 or more and may be
useful in order to get in the analytical form the \smoothed" Mie curves when the
ripple-type uctuations were averaged.
The leading role of the van de Hulst approximation both for the electric and
magnetic components of the extinction eôciency has been shown.
Key words: Light scattering; Extinction eôciency; van de Hulst approximation
1 Introduction
The theory of anomalous diraction was worked out by van de Hulst in [1]
and [2]. Within the framework of this theory, by the subtle physical reasoning
based on the Huygens' principle, the very eective representations for the
? To the memory of Hendrik Christoel van de Hulst.
 Corresponding author. Tel: (+7) 812/428 42 63; fax: (+7) 812/428 71 29.
Email address: nvv@dust.astro.spbu.ru (N.V. Voshchinnikov).
Preprint submitted to Elsevier Preprint 24 November 2000

eôciency factors and amplitude functions have been derived (see [1], [3]). The
nal expressions of the van de Hulst approximation (vdHA) describe the main
features of the scattering phenomenon. On the view of their elegance and
simplicity the vdHA presents a proper tool when investigating the problem
of light scattering by spherical particles exposed to a plane wave (the Mie
theory).
In this paper, we discuss the method to transform and sum the series which
represent the optical characteristics in the Mie problem. As a result, we are
able to give the mathematical substantiation of the theory of anomalous
diraction and formulate the conditions of its applicability without using any
physical hypotheses. New approximation represents the results of Mie theory
to a fairly high degree of accuracy within the wide range of the input param-
eters. The principal part of the presented approximation for the extinction
eôciency factor coincides with that of the vdHA.
Let us introduce the basic notations: x = 2a= is the size parameter (a the
radius of a sphere,  the wavelength of the incident radiation), m is the relative
refractive index,
 = 2(m 1)x; R = 2(m + 1)x; s  = sin 

;
c  = 1 cos 
 2 ; ci() =

Z
0
1 cos t
t
dt; c  = ci()
 2 :
Here,  and R is a phase shift of the central ray for transmitted and backscat-
tered radiation.
The most useful result obtained in the theory of anomalous diraction (see [3])
is the well-known expression for extinction eôciency factor of non-absorbing
spheres, namely,
Q vdHA
ext (m; x) = 2 4s  + 4c  : (1)
2 Description of methods
2.1 General denitions
The scattering characteristics in the Mie theory are expressed in terms of series
depending on the electric a 1n and magnetic a 2n coeôcients. For example, the
2

eôciency factor for extinction is given by series [3]
Q ext = 2
x 2 Re
1
X
n=1
(2n + 1)(a 1n + a 2n ) : (2)
The Mie coeôcients can be written in the form
a 1n = h 2
1n ih 1n h 3n
h 2
1n + h 2
3n
; a 2n = h 2
2n ih 2n h 4n
h 2
2n + h 2
4n
; (3)
where
8 > > > > > > > > <
> > > > > > > > :
h 1n = x 0
n
(mx) n (x) mx n (mx) 0
n
(x) ;
h 2n = mx 0
n
(mx) n (x) x n (mx) 0
n
(x) ;
h 3n = x 0
n
(mx) n (x) mx n (mx) 0
n
(x) ;
h 4n = mx 0
n
(mx) n (x) x n (mx) 0
n
(x)
(4)
and n (z);  n (z); 0
n
(z);  0
n
(z) are the Riccati-Bessel functions and their rst
derivatives.
By denition, the soft particles approximation (the S-approximation, SA) is
based on the assumption that the denominators in Eq. (3) can be replaced by
the following expressions
h 2
1n + h 2
3n 
jmjx 2
g(jmj; x)
; h 2
2n + h 2
4n 
jmjx 2
g(jmj; x)
; (5)
where
g(jmj; x) = O(1) as x !1 : (6)
The above suppositions re ect the behaviour of Wronskian for the Riccati-
Bessel functions and are justied by their asymptotic behaviour. In the long
run, the specic choice of the smoothing function g(jmj; x) should be deter-
mined by the short wavelength approximation of the optical characteristic
under consideration.
Some series that occur in the scattering theory can be summed in the SA. Here,
we shall conne ourselves to the case of non-absorbing particles [m = Re(m)].
The optical characteristics in the SA are represented as the linear combination
of functions (l)B( ), where B() are the basic functions of one variable 
3

and the coeôcients (l) are the weight functions. The products (l)B() are
computed at  = , l = m and  = R, l = m. The function B() is called
the basic one if
B() = O( k ) ; k  0 ( ! 0)
and B( ) = B() in the case of non-absorbing particles. Every weight func-
tion (l) is the linear combination of l k (k are integers) with the constant
coeôcients satisfying the condition ( 1) = 0. The SA is named of the stan-
dard form if
(1) 6= 0
for all the weight functions. This form of the SA is often convenient to analyse
the results obtained.
As a preliminary, we put
g(m; x) = 1 : (7)
From Eqs. (3), (5) and (7) it follows that
a 1n = h 2
1n
mx 2 ; a 2n = h 2
2n
mx 2 (8)
in the considered version of the SA.
2.2 S-approximation in the standard form
Using the assumption given by Eq. (8) we can sum the series for electric and
magnetic components
 1 =
1
X
n=0
(n + 0:5)a 1n ;  2 =
1
X
n=0
(n + 0:5)a 2n : (9)
After some reductions based on the addition theorem for the Bessel functions
(see [4]) it is possible to nd
8 > > > > > > > <
> > > > > > > :
 1 = x 2
32m 2

b(m)

b 2
0 (m)H() + 2b 0 (m)H 1 ()
x 2 + H 2 ()
8x 4

b( m)

b 2
0 (m)H(R) + 2b 0 (m)H 1 (R)
x 2 + H 2 (R)
8x 4

;
 2 = x 2
8 [b(m)H() b( m)H(R)] ;
(10)
4

where
8 > > > > > <
> > > > > :
H() = 2 4s  + 4c  ;
H 1 () =  2 [2s  + 2c  + ci( )] ;
H 2 () =  4 (1 + 2s  + 6c  + 8c  )
(11)
and
b(m) = (m + 1) 2
4 ; b 0 (m) = m 2 + 1 : (12)
Using the expressions
h 2
j 0 = x 4 (m 2 1) 2
2
h
c  + c R + ( 1) j 1 s =2 s R=2
i
; j = 1; 2 ; (13)
Eqs. (2) and (8) { (10), the standard form of the SA for the extinction eôciency
factors can be derived
Q SA
ext (m; x) = (m)H() +  1 (m)H 1 ()
x 2 +  2 (m)H 2 ()
x 4

( m)H(R) +  1 ( m)H 1 (R)
x 2 +  2 ( m)H 2 (R)
x 4
!
(14)
+ 3 (m)H 3 () +  3 ( m)H 3 (R) :
Here, H( ), H 1 () H 2 () and H 3 () =  2 c  are the basic (van de Hulst)
functions, and
8 > > > > > <
> > > > > :
(l) = (l 4 + 6l 2 + 1)b(l)
8l 2 ;  1 (l) = (l 2 + 1)b(l)
4l 2 ;
 2 (l) = b(l)
64l 2 ;  3 (l) = b(l)
l
(15)
are the weight functions. Substituting Eq. (15) into Eq. (14) we have
Q SA
ext (m; x) = 1
8m 2

(
b(m)
" 
m 4 + 6m 2 + 1

H() + 2b 0 (m)H 1 ()
x 2
+ H 2 ()
8x 4
#
b( m)
" 
m 4 + 6m 2 + 1

H(R) + 2b 0 ( m)H 1 (R)
x 2
+ H 2 (R)
8x 4
#)
(16)
5

1
m
[b(m)H 3 () + b( m)H 3 (R)] :
With the aid of the relation (see [5])
ci() = + ln 
sin 

+ cos 
 2 ; (  1); (17)
where is the Euler constant, and Eqs. (9) { (13) we can nd the limits char-
acterizing the short wavelength approximations of the electric and magnetic
components in the SA
8 > > > > > > > > > > <
> > > > > > > > > > :
lim
x!1
"
2
x 2 Re
1
X
n=1
(2n + 1)a 1n
#
= 1
4
"
b 2
0 (m) + (m 2 1) 2
m
+
b 0 (m)(m 2 1) 2
m 2
ln
    m 1
m+ 1
   
#
;
lim
x!1
"
2
x 2 Re
1
X
n=1
(2n + 1)a 2n
#
= m :
(18)
By virtue of Eqs. (2) and (18) we have
S(m) = lim
x!1
Q SA
ext (m; x) = b 2
0 (m)
2m
+ b 0 (m)(m 2 1) 2
4m 2
ln
    m 1
m+ 1
    : (19)
2.3 S- and van de Hulst approximations
The SA can be used in order to derive the various approximations known in
the light scattering theory. As for the vdHA, it is convenient to use the SA
in its standard form. Let us illustrate this fact in the case of the extinction
eôciency factor.
The vdHA is known to be valid if
m 1  1 ; (20)
x  1 ; (21)
while the phase shift  = 2(m 1)x being arbitrarily xed (see [3]). The
condition of Eq. (20) for the SA is realized by substitution of m = 1 into the
weight functions given by Eq. (14). On the basis of Eqs. (12) and (15) we
obtain
Q SA
ext = H() + H 1 ()
2x 2
+ H 2 ()
64x 4 : (22)
6

Then, the condition of Eq. (21) for the SA is realized by retaining in Eq. (22)
its principal term as x tends to the innity. Hence, we have Q SA
ext = H(),
where  adopts the arbitrary value.
2.4 Non-standard form of the S-approximation
In [6] the extinction eôciency factor for dielectric particles has been derived
in the following form
Q SA
ext (m; x) = 1
2m
"
(m 2 + 1) 2 + !(m; ) !( m; R)
2m
#
; (23)
where
!(m; ) = a(m)ci() + a 0 (m)c  + a 1 (m)s  + a 2 (m)c  (24)
and
8 > > > > > > > > <
> > > > > > > > :
a(m) = (m 2 1) 2 (m 2 + 1) ;
a 0 (m) = 2(m 2 1) 2 (m 1) 2 ;
a 1 (m) = (m + 1) 2 (m 4 2m 3 2m 2 2m + 1) ;
a 2 (m) = a 0 (m) a 1 (m) :
(25)
This non-standard [since a(1) = a 0 (1) = 0] form of the SA written in terms
of the basic functions  0 = 1, ci( ), c  , s  and c  is suitable to show that the
SA for the extinction eôciency factors turns into the Rayleigh-Gans approxi-
mation if
m 1  1 and   1 : (26)
To prove this fact, we have to keep the basic functions up to B() = O( k )
and the weight functions up to (m) = O((m 1) k ) with k  2 as  and
m 1 become zero. But the main thing is that the SA in the form given by
Eqs. (23) { (25) presents the conventual tool for the numerical calculations.
It will be also used in the new version of the SA discussed below.
7

Fig. 1. Values of parameter x at which the eôciency factors in the S-approximation
[Eqs. (23) { (25)] are calculated with an accuracy of 5 %. The dashed curve shows
the power law approximation described by Eq. (27).
2.5 Improved form of the S-approximation (ISA)
The extinction factors calculated using the Mie theory and the S-approximation
[Eqs. (23) { (25)] are compared in Fig. 1. It shows the range of the size pa-
rameter x in which both methods yield the results coinciding within 5 %. If
1:1  m  1:5 this condition is satised if
0  x < x(m) ; x(m)  290 m 11 : (27)
To improve the SA it needs to consider more carefully the short wavelength
asymptotic of the amplitude functions. In the correspondence with Eq. (19)
we replace Eq. (8) by the following one
a 1n = h 2
1n
mx 2 g(m; x) ; a 2n = h 2
2n
mx 2 g(m; x) ; (28)
where
g(m; x) = 1 f(m; x) + 2
S(m) f(m; x) ; (29)
f(m; x) = exp
"
0:01 exp(4m) x(m)
x
#
: (30)
8

Thus, the improved form of the S-approximation for the extinction eôciency
factors can be written as
Q ISA
ext (m; ux(m)) =
"
1 S(m) 2
S(m)
exp

0:01 exp(4m)
u
!#
Q SA
ext (m; ux(m)):(31)
Here, u = x=x(m) is the size parameter in the scale of x(m), the expressions for
Q SA
ext (m; x) and S(m) are given by Eqs. (23) { (25) and Eq. (19), respectively.
3 Numerical results and discussion
We calculated the extinction eôciency factors Q ext using the Mie theory and
dierent approximations for the several refractive indices in the range from
m = 1:1 to m = 2:0. Such refractive indices cover the interval of m typical for
biological particles, sea water suspensions, water and water ice in the visible
region and various silicates. The results are shown in Figs. 2 and 3.
As follows from Fig. 2, both the ISA and the vdHA deviate signicantly from
Mie curves for very small size parameters x and m > 1:5. In this case the
condition given by Eq. (21) is not satised and the Rayleigh approximation
must be used. For larger size parameters (and phase shifts) the ISA and the
vdHA begin to work being the dierence between the exact theory and ap-
proximations reduces with the growth of  (see Fig. 3). Note that in all cases
the extinction produced by the vdHA is smaller than that of the Mie theory
while the ISA reproduces very well both the position and the height of maxima
and minima on the extinction curve. The latter is explained by the correction
described by Eqs. (29) { (31).
However, the S-approximation cannot reproduce the small-scale ripple struc-
ture superimposed on the major oscillations as seen on the Mie extinction
curves. These ripple-like uctuations result from the resonances of virtual
modes and are much better seen for the dielectric particles with large re-
fractive indices (Figs. 2 and 3). The SA is based on the assumption that the
denominators of the Mie coeôcients are replaced by some expressions indepen-
dent of the subscript n [see Eqs. (3) and (5)]. Therefore, it cannot reproduce
the ripple structure but may be considered as the method of the construction
of the smoothed Mie curves.
Figure 4 illustrates the relative accuracy of the van de Hulst and improved
S-approximations given by the equation
" = Q Mie
ext
Q vdHA; ISA
ext
1 : (32)
9

Fig. 2. Extinction eôciency factors calculated on the basis of Mie theory, improved
S-approximation [Eqs. (29) { (31)], van de Hulst approximation [Eq. (1)], Rayleigh
and Rayleigh-Gans approximations. The factors are plotted versus the phase shift
 (lower x-axis) while the upper x-axis shows the corresponding size parameters x.
10

Fig. 2. (continued.)
It is seen that " reduces when  increases and the ISA may be applied for
the reliable representation of the extinction for particles with the refractive
indices as large as 2.0 and more. But in these cases the uctuations due to the
ripple structure become very large and the deviations of the Mie curves from
the \smoothed" ones given by the ISA may exceed several dozens of percents.
We also determined the range of validity of the ISA corresponding to the
relative errors j"j  1 %, 5 % and 10 %. The results presenting in Fig. 5 show
the upper limits for the size parameter x in dependence on the refractive index
m. They are plotted for the values of m  m  when the dependence of x on
m may be approximated by the following power functions
x <  240 m 15:6 ; if jQ ISA
ext =Q Mie
ext 1j  1% ; (33)
x <  160 m 9:30 ; if jQ ISA
ext =Q Mie
ext 1j  5% (34)
11

Fig. 3. The same as in Fig. 2 but now in the another scale.
12

Fig. 3. (continued.)
and
x <  270 m 9:99 ; if jQ ISA
ext =Q Mie
ext 1j  10% : (35)
If m < m  , the ISA may be used for arbitrarily large (innite) values of x.
This occurs for m  < 1:03, m  < 1:10 and m  < 1:22 if the relative error being
smaller than 1 %, 5 % and 10 %, respectively.
It should be noted that the S-approximation may be applied for the descrip-
tion of other optical characteristics besides extinction. It conserves the es-
sential analytical properties of exact solution [see Eq. (23)]. On the contrary,
both the van de Hulst and Rayleigh-Gans approximations are of dierent an-
alytical structure as compared with the exact solution. At the same time,
the known approximations of the Mie theory are derived starting from the
S-approximation by the formal use of their validity conditions, and the role
of each condition giving by Eqs. (20) and (21) can be explicitly shown. The
13

Fig. 4. Relative errors of calculations of the extinction eôciency factors in the case
of improved S-approximation and van de Hulst approximation in the comparison
with the Mie theory.
14

Fig. 4. (continued.)
analytical formulae of the S-approximation also open the wide opportunities
for applications in solution to various direct and inverse problems.
4 Conclusions
We have considered the soft particles approximation (S-approximation) which
reproduces the principal features of light scattering by large and soft spherical
particles and its improved form (improved S-approximation) based on the
correction for the short wavelength region. The S-approximation extends the
results of the van de Hulst approximation of anomalous diraction and may be
used for calculations of the optical properties of particles with the refractive
indices up to 2.0 and more.
Actually, the S-approximation restores the averaged curves of the Mie theory
15

Fig. 5. Values of parameter x at which the eôciency factors in the improved
S-approximation [Eqs. (29) { (31)] are calculated with dierent accuracy. The
dashed curves show the power law approximations described by Eqs. (33) { (35).
without the appreciable errors (see Figs. 2, 3) and may be useful in order to
get in the analytical form the \smoothed" Mie curves when the averaging of
the ripple-type uctuations was made.
The results obtained are of importance since the averaged characteristics are
just used in a number of the direct and inverse optical problems.
16

Acknowledgements
This work was supported by INTAS grant (Open Call 99/652).
References
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1946;11:pt. 1.
[2] van de Hulst HC. The solid particles in interstellar space. Rech. Astron. Obs.
Utrecht 1946;11:pt. 2.
[3] van de Hulst HC. Light scattering by small particles. New York: Wiley, 1957.
[4] Watson GN. A treatise on the theory of Bessel functions. Cambridge: Cambridge
University Press, 1958.
[5] Olver FWJ. Introduction to asymptotic and special functions. New York:
Academic Press, 1974.
[6] Perelman AY. Extinction and scattering by soft particles. Applied Optics
1991;30:475-484.
17