Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.astro.spbu.ru/staff/ilin2/INTAS2/PAP/P7_3.pdf Äàòà èçìåíåíèÿ: Fri Nov 19 16:18:16 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:05:40 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 13

Light scattering by a core­mantle spheroidal particle
Victor G. Farafonov, Nikolai V. Voshchinnikov, and Vadim V. Somsikov

A solution of the electromagnetic scattering problem for confocal coated spheroids has been obtained by the method of separation of variables in a spheroidal coordinate system. The main features of the solution are i the incident, scattered, and internal radiation fields are divided into two parts: an axisymmetric part independent of the azimuthal angle and a nonaxisymmetric part that with integration over gives zero; the diffraction problems for each part are solved separately; ii the scalar potentials of the solution are chosen in a special way: Abraham's potentials for the axisymmetric part and a superposition of the potentials used for spheres and infinitely long cylinders for the nonaxisymmetric part . Such a procedure has been applied to homogeneous spheroids Differential Equations 19, 1765 1983 ; Astrophys. Space Sci. 204, 19, 1993 and allows us to solve the light scattering problem for confocal spheroids with an arbitrary refractive index, size, and shape of the core or mantle. Numerical tests are described in detail. The efficiency factors have been calculated for prolate and oblate spheroids with refractive indices of 1.5 0.0i, 1.5 0.05i for the core and refractive indices of 1.3 0.0i, 1.3 0.05i for the mantle. The effects of the core size and particle shape as well as those of absorption in the core or mantle are examined. It is found that the efficiency factors of the coated and homogeneous spheroids with the volume-averaged refractive index are similar to first maximum. Key words: Light scattering, nonspherical particles, inhomogeneous particles. © 1996 Optical Society of America

1. Introduction

In the last several years, significant progress in the theoretical study of light scattering by small particles has been discerned. The methods developed make it possible to calculate the optical properties of arbitrarily shaped particles with anisotropic optical properties and inclusions.1­5 However, the major part of the numerical results deals with simple axisymmetric particles such as homogeneous spheroids or finite cylinders. Such models of particles should be useful as a solution to various problems in atmospheric and ocean optics, astrophysics, biophysics, etc. In most cases of practical interest, it is necessary to average over particle sizes and orientations. Then the computational method must be fast to get accurate nuV. G. Farafonov is with the St. Petersburg State Academy of Aerocosmic Instrumentation, St. Petersburg 190000, Russian Federation. N. V. Voshchinnikov is with the Astronomical Institute, St. Petersburg University, St. Petersburg 198904, Russian Federation. V. V. Somsikov is with the Pulkovo Observatory, St. Petersburg 196140, Russian Federation. Received 10 May 1995; revised manuscript received 27 February 1996. 0003-6935 96 275412-15$10.00 0 © 1996 Optical Society of America 5412 APPLIED OPTICS Vol. 35, No. 27 20 September 1996

merical results for polydispersions of aligned nonspherical particles. In this paper we present the solution to the problem of light scattering by core­mantle two-layered spheroidal particles, which makes it possible to consider inhomogeneous nonspherical particles with shapes that vary from needles prolate spheroids to disks oblate spheroids . The problem is solved by the method of separation of variables in a spheroidal coordinate system. This method was used to obtain a solution to the light scattering problem for spheres and infinitely long circular cylinders that are both homogeneous and core­mantle in structure.6­9 For homogeneous spheroids, the problem was worked out by Asano and Yamamoto10 and Farafonov.11 In Ref. 11 the special basis for expansion of the electromagnetic fields was introduced. Numerical results12,13 indicate that the computational efficiency of this solution is higher than that of Asano and Yamamoto.10 Here, similar ideas are used to solve the light scattering problem for core­mantle spheroids. Note that core­mantle spheroids have been previously considered by Onaka14 who extended the approach of Asano and Yamamoto.10 Some results were published in Ref. 14 for dielectric particles with refractive indices mcore 1.66 0.0i and mmantle ~ ~


1.33 0.0i. Cooray and Ciric15 studied the phase function and radar cross sections for prolate layered spheroids. Their solution is based on the method of separation of variables developed by Sinha and MacPhie16 for conducting spheroids. Coated spheroids have also been considered in Refs. 17 and 18 where the extended boundary conditions T-matrix method was applied. The results were done for small atmospheric and biological particles. Some calculations of the optical properties of optically soft core­mantle spheroids were made on the basis of the Rayleigh­ Hans­Debye approximation see the discussion in Ref. 19 .
2. Formulation of the Problem

ity i, k0 2 0 is the wave number in vacuum, r is the radius vector, 1 and 2 are the radial coordinate values at the surfaces of the mantle and the core, respectively. The magnetic fields can be obtained from electrical fields using Maxwell's equations H 1 ik E,
0

E

1 ik

H.
0

We consider a plane electromagnetic wave with an arbitrary polarization propagating at an incident angle to the rotational axis of a spheroid or the z axis; see Fig. 1 . This wave can be represented as a superposition of two components: a TE mode E H
0

The problem of electromagnetic light scattering by a core­mantle particle can be solved in the prolate and oblate spheroidal coordinate systems , , that are connected with the Cartesian system x, y, z in the following way20,21: x y z d 2 d 2 d 2
2

iy exp ik1 x sin
1 1

z cos

,

0

ix cos

iz sin z cos ; (7)

1 1

12

1 1

212

cos , b TM mode sin , (1)

exp ik1 x sin

2

12

212

E H

0

ix cos
1 1

iz sin

exp ik1 x sin z cos .

z cos

, (8)

,

0

iy exp ik1 x sin

where 1, , 1, 1 , and 0, 2 for a prolate coordinate system and 0, , 1, 1 , and 0, 2 for an oblate coordinate system; d is the focal distance. The upper sign corresponds to the prolate system and the lower to the oblate system. Note that the confocal spheroids correspond only to the same spheroidal system. Then the thickness of the coating is variable. Let the time-dependent part of the electromagnetic field be exp i t and E i , H i and E i , H i be the vectors of the electric and magnetic fields and their tangential components, respectively. The subscript i takes a value of 0, 1, 2, or 3, which refer to the field of the incident radiation, the field of the scattered radiation, and the radiation fields inside the mantle and the core of a particle. Then the scattering problem is formulated as follows i 1, 2, 3 :
2

Here ix, iy, iz are the unit vectors in the Cartesian coordinate system. The main features of the solution that we used are see also Ref. 13 : i All the fields are divided into two parts: E
i

E1i

E2i ,

H

i

H1i

H2i ,

i

0, 1, 2, 3, (9)

E

i

ki2E E

i i 1 1

0, 0, E H E H 0,
2 2
1

(2) (3) , ,
2

E H

0 0

E H E H

(4) (5)

2 2

3 3

lim r
r3

E1 r

ik1E

1

(6)

i ik0 is the wave number in a medium where ki with complex permittivity i and magnetic permeabil-

Fig. 1. Scattering geometry for a prolate spheroid with a confocal core­mantle structure. The space is divided into three parts: the outer medium, the mantle, and the core. The scattered field in the far-field zone is represented in the spherical coordinate system. The origin of the coordinate system is at the center of the spheroid whereas the z axis coincides with its axis of revolution. The angle of incidence is the angle in the x­z plane between the direction of incidence and the z axis. 20 September 1996 Vol. 35, No. 27 APPLIED OPTICS 5413


where E1i , H1i are the axisymmetrical parts independent of the azimuthal angle , whereas the integration of the nonaxisymmetrical parts E2i , H2i over gives zero. ii The scalar potentials for the solution are chosen in a special way: Abraham's potentials for the axisymmetric part and the superposition of potentials used for spheres and infinitely long cylinders for the nonaxisymmetric part. The independent solution to the first and second parts of Eqs. 9 is possible due to the commutation of the operator corresponding to the diffraction problem and the operator Lz Ref. 11 . Thus, the problem under consideration can be uncoupled relative to the azimuthal angle , i.e., each component of Fourier expansion including axisymmetric components E1i , H1i can be found separately.
3. Solution to the Axisymmetric Problem

E12
l1

al2 R11 c2, l cl
2

R13 c2, l bl2 R11 c2, l

S1l c2,

,

H12
l1

dl2 R13 l E
3 1 l1

c2, c3,

S1l c2, S1l c3, , ,

,

(14)

al3 R

1 1l

H13
l1

bl3 R11 c3, S1l c3, l

(15)

If the electromagnetic field does not depend on the azimuthal angle , we can introduce Abraham's potentials21,22: hE, hH. (10) Using these potentials, the expressions for all the components of vectors E and H can be obtained: E E where h d 2
2 2 2 12

where ci kid 2 i 1, 2, 3 , Sml ci, are the prolate angular spheroidal functions with normalization coj efficients Nml ci , and Rml ci, are the prolate radial spheroidal functions of the jth kind.21 The formulation of the problem see Eqs. 2 , 3 , and 6 is consistent with the fields described by Eqs. 10 ­ 15 . One can determine the unknown coefficients ali , bli , cli , and dli by solving the systems for the boundary conditions Eqs. 4 and 5 after substitution of the field expansions. For the incident radiation we obtain the following coefficients see Ref. 13 : a TE mode Eqs. 7 al0 2ilN1l
2

i1 k0 h h i1 k0 h h ,

,

H H

i k
0

1 hh i k0 1 hh
2 12 2

, , (11)

c1 S1l c1, cos

, bl0

0;

(16)

b TM mode Eqs. 8 al
0

0,

bl

0

2il



1 1

N

2 1l

c1 S1l c1, cos

.

(17)

1 d 2

,
2

h 11

d 21
2

2

,

h

12

are Lame metric coefficients of the prolate upper ´ sign or oblate low sign spheroidal coordinate system. The azimuthal components of vectors E and H can be written using the spheroidal functions with the index m 1 Ref. 21 . Let us consider a prolate spheroid. Then the components of the fields corresponding to the incident, scattered, and internal inside the mantle and the core of the particle radiation are E
0 1 l1

Equations 10 and 11 show that Abraham's potentials can be found separately from each other. Therefore, the potential is nonzero only for the bl2 bl3 TE TM mode, which means that bl1 2 1 2 3 dl 0 for the TE mode and al al al cl2 0 for the TM mode. Let us substitute Eqs. 10 and 11 into the boundary conditions Eqs. 4 and 5 . In the case of a TE-type wave we have
0 1 1 2

1
1

0

1
2

2
1

, (18)

al0 R bl R
l1 1 0

1 1l

c1, c1, c1, c1,

S1l c1, S1l c1, S1l c1, S1l c1,

,
2 3

H E

0 1

1 1l

, , ,

(12)

1
2

2

1
3

3
2

. (19)

1 1 l1

al R bl1 R
l1

3 1l

H11
5414

3 1l

(13)

Now let us substitute Eqs. 12 ­ 15 into the expressions for radiation fields Eqs. 18 and 19 . The and intelatter multiplied by N1n 1 c2 S1n c2, grated over from 1 to 1 can be rewritten in matrix

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Vol. 35, No. 27

20 September 1996


form parameter ~ is equal to 1 for the prolate sphef roids and 1 for oblate spheroids :
2 1 2 1 2 1 2 3 2 3 2 2 2 2 2 1 2 1

a TE mode E
i 2 i 2

Uii 1 i ik

z

Vir , Uii
z

1 1

1

I

2 1

~ f ~ f ~ f ~ f

2 1 2 1

~ ~

3

1

c2, c2, c1,

1

Z Y

2

H

Vir ;

(23)

0

1

I

2

b TM mode E
i 2

3

3

1

c2, c1 1 I

3

c1, ~

1

1

1

F, H
i 2 2

0

1 iik

Uii
0 z

z

Vir , (24)
i i

Uii

Vir .

2 3

2

1

1

c2,

2

P1Z
2

One can write the scalar potentials U and V using the spheroidal wave functions U V U V U
0 m 1l m 0 1 amlRml c1, Sml c1,

1

2

I

~ f

2 3

~

3

1

c2,

2

P3Y

0, (20)

cos m , cos m , cos m , cos m , (26) (25)

0 m 1l m 1 m 1l m 1 m 1l m 2

0 1 bmlRml c1,

Sml c1,

where Z zl z
l 2 0 l j 1 2

zl
2

j 1

,

Y

2

yl

2 1

,

F

0

fl

0 1

,

1 3 amlRml c1, Sml c1,

al1 R13 c1, l al R cl2 R
0 m nl 1 1l

1 1 1 1 m

N1l c1 , N1l c2 , N1l c2 , N1l c1 , ,
nl m 2

c2, c2, c1,

1 3 bmlRml c1, Sml c1,

yl f

3 1l 1 1l

2 1 amlRml c2, m 1l m 2 3 ml ml

al R

ci, cj
j

ci, cj ,

cR

c2,

Sml c2,

cos m ,

ci, Pj ~ ~

j Rml ci,

j Rml ci, 2 1

,

V

2

2 1 bmlRml c2, m 1l m 2 3 ml ml

Pj c2,
j Rml

1

j Rml c2, nl m 2 1

dR

c2 ,

Sml c2,

cos m , cos m , cos m .

(27)

c2,
1 3

, c3, c2 , c1, c2 , (21) U V
3 m 1l m 3 m 1l m 3 1 bmlRml c3, Sml c3, 3 1 amlRml c3,

1 3

c2, c3 c2, c1
nl m

c3, c1,

Sml c3,

(28)

I

is the unit matrix.

m The expression for integral nl ci, cj is given in the Appendix. The coefficients al1 that describe the scattered radiation can be used to determine the cross sections for extinction and scattering, the scattering matrix, etc. They can be found from the following equation:

For the TE mode, the coefficients that describe the incident radiation are equal to see Ref. 13 a
0 ml

4il 1 N k1

2 ml

c

1

Sml c1, cos sin

,

b

0 ml

0.

Z

1

c1, c2 Z

2

Y

2

F0.

(22)

For the TM mode, the transformation of the infinite system Eqs. 20 and 21 can be performed by the replacements i 3 i, i 3 i and al j 3 bl j , cl2 3 dl2 . In order to obtain the corresponding systems for an oblate spheroid one must use the standard replacements c 3 ic d 3 id , 3 i and oblate spheroidal functions instead of prolate functions.
4. Solution to the Nonaxisymmetric Problem

0 For the TM mode, the coefficients aml have the opposite sign and the multiplicand 1 1 see Eqs. 7 , 8 , 23 , and 24 . Let us introduce the vector wave functions see Ref. 20

M

a ml ml

a,

N

a ml

1 k

Mmla,

The second terms in Eqs. 9 can be represented in the following form:

where a is either the unit vector of the Cartesian system or the radius vector, ml are the scalar wave functions that represent the solutions to the scalar Helmholtz equation in the corresponding curvilinear coordinate system. As follows from Eqs. 23 ­ 28 , our approach is equivalent to the choice of the vector spheroidal wave functions Mmlz, Nmlz i.e., a iz and
20 September 1996 Vol. 35, No. 27 APPLIED OPTICS 5415


Mmlr, Nmlr i.e., a r as the basic functions. Note that a r for spheres and a iz for infinitely long cylinders.23 The representation of the radiation fields as described by Eqs. 23 ­ 28 is consistent with the for-

mulation of the problem given by Eqs. 2 , 3 , and 6 . The boundary conditions Eqs. 4 and 5 are used to find the unknown coefficients. We insert Eqs. 23 into the boundary conditions. After mathematical transformations we obtain see Ref. 13

U U 1 1
1 0

0

U U U
1

1

d V 2 ~ f ~ f d V 2 d V 2
1 1

1

U

2

d V 2 U
2

2

~ f ~ f U
2

d V 2
2

2

U

0

1

1

2 U 1
2

2

d V 2

,
2

U

0

U

1

d V 2

d V 2
2 2

1

c22 1 c12

~ f
3

U

2

~ f

d V 2

2
1

(29)

U U 2 U
2

2

d V 2 ~ f ~ f d V 2 d V 2
2

2

U

3

d V 2 U
3

~ f ~ f U

d V 2
3

3

2

2

3 U 1
2

3

d V 2

.
2

2

d V 2 d V 2
3

1

c22 1 c32

2 2

f

U

2

~ f

d V 2

2

1
3

U

3

2

(30)

In the case of the TM mode, the analogous expressions can be obtained from Eqs. 29 and 30 by the replacements i 3 i and i 3 i. Let us assume that i 1 and 1 1. Then for the TM mode transformed Eqs. 29 and 30 can be written in the following form: U U
0 0

U d V 2
1

1

U

2

U

1

~ f

U
1

2

~ f

d V 2

2

V 1 1 U
0

V 1 2

2

, U 1
2

U

1

d V 2

1

d V 2
2 2

2

c22 1 c12

~ f

U

2

~ f

d V 2

2
1

(31)

5416

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Vol. 35, No. 27

20 September 1996


U U
2

2

U

3

~ f

d V 2

2

U
2

3

~ f

d V 2

3

V

V 1 2

3

U

2

d V 2

2

1

c2 c3

2 2

1
2

2

. U
2

~ f

d ~ f V 2

2

1 3

U

3

d V 2

3
2

(32)

In addition to previous definitions see Eqs. 21 , we introduce the following new ones: Z
m l m l m l m l m l m l m l j

ci, cj K ci, cj ci, cj

m nl m nl m nl

ci, cj ci, cj ci, cj

m m m

, , . (33)

zj

m ,l m ,l 1 1 ml 2 2 ml 2 2 ml 0 m m

, ,
3 ml 3 ml 3 ml 3 ml 1 1 ml 1

Y F

j

yj fl
1 1 1 1 1 1 1

m ,l m m m

,

Xj z1, z2, y1, y2, x1, x2, f

xj

m

,

1 ml

a

R R R

c1, c1, c2, c2, c2, c2, c1,
1

Nml c1 , Nml c1 , Nml c2 , Nml c2 , Nml c2 , Nml c2 , Nml c1 Rml
1

c1b c

1 ml

c1d c1b

R

1 ml

a

Rml R Rml
1

1 ml

a

The expressions for the integrals on the angular spheroidal functions nl m , nl m , nl m are given in the Appendix. Let us substitute Eqs. 25 ­ 28 into Eqs. 29 ­ 32 , then multiply them by Nmn 1 c2 Smn c2, cos m and integrate over from 1 to 1 and over from 0 to 2 . Using the orthogonality of functions cos m , after some mathematical transformations see Ref. 13 , we obtain a series of systems of equations m 1, 2, . . . : a TE mode c1,
1

4i

l1

Nml

c

Sml c1, cos sin
~ ~ c2, c2,

,
2 1 2 1
1 1

1

3

1

1

1

2 1 2 1 2 1 2 1

1 1

3

X Y

1

~ f ~ f
1

~ ~

3

1

c2, c2,
3

1

1 1 c1, 1 1
1

3

X Y

2

1

3

3

1

1

3

1

3

3

1

3

2

c2, c1
3 1

c1,
1

1

Fm, X Y
2

~ ~

3

1

c2, c2,

1

1

1 1

3

X Y

1

1

~ ~

c2, c2,
3

~2 f 1 ~ f
1

3

3

3

1

1

3

1

1

3

3

1

2 1

3

2

c2, c1
2

c1, ~ f

c1,
1

1

Fm,
2

~

1

1

c2,

2

2

2 3
2

1 ~

1

P1X c2,
2

1

~ 2 3

1

c2,
1

2 3

1

1

P1 X ~ f

2

1

3

2

2

1

P3Y
2

1

~

1

3

c2, 1

2 3 P1X
1

1 ~ ~ 1

1

P3Y
1

0,

~

1

1

c2,

2

2

2 3 2 3
3

1

2

1

c2, c2,
2

2

~ f 2 3 1 P3Y ~

1 c2,

1

P1X ~ f

2

1

3

2

2

1

1

2

1

2

2 3

1

P3Y

0,

(34)
5417

20 September 1996

Vol. 35, No. 27

APPLIED OPTICS


where
1 3 2 1

I I ~ ~
1

2 1

~ ~

1 3

~~ f ~ f~
2

1 3

Q c2, Q c2, Q c2, Q c2,
2 2

2 1

, , 1
2 2

Z2

c1, c2 2 1 1

2 I 1
1

2 1
1

1 X1

2 1

Q c2, Y1 ;

1

X

2

Y2 (37)

Q c2,

1

2

I I Q c,

~ ~

1

~ K c2, c2 , f ~ K c2, c2 , f
1

b TM mode Z
1

1
1 1 2 2

3

1

3

1

3

c1, c2 X1

Y1

Fm,

Z2

c1, c2 X

2

Y2 . (38)

c2, c2 , b TM mode

I

~ f

c, c

; (35)

If we turn to the spherical particles 1 , 23 d 3 0, and 1 d 2 r1, 2 d 2 r2 are the radii c2, c2 , X1 Y
1 3

1 1

~ ~

3 3

1 3 1

1 1

~ f ~ f c1 ,

~ ~
1 1

3 3 1

1 3

c2, c2 ,
1

1 1

X2 Y
2

c2, c1 1 ~

c1,

Fm, X1 X

~ ~

3

1

c2 ,

1

2 1 2 1

3 1

2

~ K c2, c2 f ~ f c2, c2

1

3

1

c2,

1

1I 2 1 1

1

~ ~ ~

~ f
3 1 1 3 1 2 2

2

~ ~

3

3

c2,

1

~ K c2, c2 f ~ f
1

Y1 Y2 Fm, P1X2

1

3

3

c2,

1

2 1
2 1

1I c2, c1
2

~ f
3 1 2

1

c2, c2 c1 , c2,
2 2 2

3

c1, ~ f ~

1

1

2 2

~

1 3

1

c2,
2

P1X1 ~ f ~ ~ ~
1

1 3 2

1

~

1

c2 ,

P3Y1 2 3 1

~

1

c2,

P3Y

0, P1X
1

~ ~

1

1

c2,

2

2 2

~ K c2, c2 f ~ f c2, c2

2

1

1

c2,

2

2 3
2

1I 2 3 1 ~

~ f
1 2 2 1 2 2 2

2

P1X P3Y

2

~ ~

1

3

c2, 2 3

~ K c2, c2 f c2, c2 P3Y2

1

2

1

3

c2,

2

1I

~ f
1 2 2

2

~ f

0.

(36)

The coefficients for the field of the scattered radiation can be found from the following expressions: a TE mode Z c1, c2 ~ f
5418

1

I 1

2 1
1

1

2 1

Q c2, Y2

1

X

1

Y1

2 1

Q c2,

1

X

2

Fm,
20 September 1996

of confocal spheres , then the systems of equations give the exact solution. Its difference from the standard Mie solution is in the choice of distinct scalar potentials and the uncomfortable coordinate system with the z axis not coinciding with the direction of the propagation of the plane wave. The proposed solution allows us to investigate analytically the systems of the linear algebraic equations and the convergence of the expansions for the internal and scattered radiation fields. The final-

APPLIED OPTICS

Vol. 35, No. 27


conclusions for core­mantle spheroids coincide with those for homogeneous spheroids see the discussion in Ref. 11 . The main ideas can be briefly described as follows. Let us consider an arbitrary prolate or oblate spheroid that does not degenerate into a needle 1 1 for the prolate spheroidal coordinate system or a disk 1 0 for the oblate system . Using the asymptotic expansions for spheroidal functions and integrals nl m , nl m , etc. in the case of large indices l and n, the systems of linear algebraic equations Eqs. 20 , 34 , and 36 can be transformed into the following form: z where n
1 2 n 1n1

the first or second index denotes the polarization of an incident or scattered wave. If it is equal to 1 or 2 , the electric vector of the beam is perpendicular or parallel to the corresponding reference plane. In the far zone r 3 in the spherical coordinate system r, , or 3 , 3 cos , i 3 i in the spheroidal system , we obtain see Ref. 13 T
11 l1 1 k1amlSml c1, cos 1 ibmlS ml

i lal1 S1l c1, cos
m 1l m

i

l1

c1, cos

sin

cos m , sin m ,

(41) (42)

z

2n1 i1

z

gniz

i

fn,

(39)

T12
m 1l m

1 i lbml

mSml c1, cos sin mSml c1, cos sin

1, 2, . . . , z0 p 1,
i1

0 and gni const n,
n1

f

2 n

.

T21
m 1l m

1 i lbml

sin m ,

(43)

These systems are quasi-regular, i.e., the sum of the absolute values of coefficients on the right-hand side of Eq. 39 becomes smaller than unity beginning ~ from some number n N:
1 2 i1

T

22 l1

i lbl1 S1l c1, cos
m 1l m ml 1 k1amlS

i

l1

c1, cos sin cos m . (44)

ib S

1 ml

ml

c1, cos

gni

p ~

1.

The systems given by Eq. 39 have a unique solution , which can be deterin space l2 i.e., ·n 1 zn 2 mined by the reduction method this means that we can solve the truncated systems . The obtained field of the scattered radiation see Eqs. 13 and 26 belongs to the space L2 S , where S is the surface of a confocal spheroid larger than or equal to the given one. Asano24 and Barber25 indicated that the solution of Asano and Yamomoto and the T-matrix method lead to the numerical instability that occurs when one solves the system of linear equations for the axisymmetric part when c 17. For the T-matrix method it has been shown that the use of extended precision in the calculations permits enlargement of the limits to some extent.26 In the approach described, such a defect is absent because Eqs. 20 can be rewritten as Eq. 39 but with the coefficients 1 0. 2
5. Characteristics of Scattered Radiation

Using the amplitude matrix, we can find the scattering matrix27 that allows us to combine the Stokes parameters of incident radiation with the scattered radiation, which gives complete information about scattered radiation. Elements of the scattering matrix are not the amplitude functions, but the products of them, the so-called dimensionless intensity functions of scattered radiation iij Tij 2. If the incident radiation is nonpolarized, the intensity of scattered radiation can be calculated according 1 to the formula i , i i12 i21 i22 . The 2 11 integral characteristics of scattered radiation cross sections of a particle for extinction Cext, scattering Csca, and absorption Cabs are determined as27 Cext 4 Re A 1 , i k12 1 k1 2 Cext A
4 0 0°

,

(45)

Csca C

12

d,

(46) (47)

Let us assume that the incident radiation propagates in the x­z plane of the Cartesian coordinate system, the scattering plane contains the Oz axis the rotational axis of a spheroid , and the wave vector of scattered radiation see Fig. 1 . The relation between the fields of the incident and scattered radiation is determined by the amplitude matrix E E
1 1

abs

Csca,

1 exp i k1r ik1r

k1r

T22 T12 E T21 T11 E

0 0

,

(40)

where the subscripts of the dimensionless intensity functions Tij are made in accordance with the rule:

where A 1 is the amplitude of the electrical field for scattered radiation, i 0 is the unit vector that defines the polarization of incident radiation, d is an element of the solid angle, and is the scattering angle angle between the directions of the incident and scattered waves . The angle is connected with angles , , and by the relation cos cos cos sin sin cos . Equations 41 ­ 47 can be used to compute various cross sections that are usually normalized by the viewing geometric cross section of a spheroid the
20 September 1996 Vol. 35, No. 27 APPLIED OPTICS 5419


area of the particle shadow : G b1 a12 sin2 b12 cos2 G a1 a12 cos2 b12 sin2
12 12

for a prolate spheroid,

(48)

where a2 and b2 are the major and minor semiaxes of the core. The radial coordinates 2 and 1 that define the inner and outer boundaries of a spheroidal particle surface of the core and mantle are connected with the corresponding semiaxes as a1,2 b1,2 a1,2 b1,2
2 2 12

for an oblate spheroid, (49)
1,2

1
12

for a prolate spheroid,

(55)

where a1 and b1 are the major and minor semiaxes of a coated spheroid. Then the corresponding efficiency factors Q C G can be written as follows:
1,2

Q

4
ext

a1,2 b1,2

1

for an oblate spheroid.

(56)

c

2 1 1

2

~ f

2 1

~ cos2 f

12

The efficiency factors can also be considered as a function of size parameter 2 a1 given by 2a
1

Re
l1

i lal 1 S1l c1, cos i
l1

k1a

1 ml

c

S

ml

c1, cos , (50) 2 a1 c

11

for a prolate spheroid,

(57)

m 1l m

ibml 1 S Qsca

ml

c1, cos 1

sin

2 1 1

1

12

for an oblate spheroid.

(58)

c12 2

2 1

~ f

2 1

~ f cos2 Re

12

al1 2N1l2 c1
l1 m ln m ln

in
m 1l mn m

l

In some cases it is useful to know the ratio of the core-to-mantle volume or the core volume to the total volume of a coated particle. For confocal spheroids, one can determine the latter ratio using the semiaxes ratios or parameters 1 and 2 as Vcore Vtotal (51)
2 2 1 2 2 1

1 1 k2amlamn* 1

c1, c1 c1, c1

ik1 b a ab
1 1* ml mn

1 1* ml mn m nl m ln

~ f ~. f

(59)

c1, c1 c1, c1 Nml c1 Nmn c1 ,

1 1 bmlbmn*

m m m where ln , ln , and ln are the integrals of angular spheroidal functions see the Appendix , the asterisk denotes the complex conjugation. Equations 50 and 51 are valid for TE and TM polarization of incident radiation but for the TM mode the coefficients al1 should be replaced by coefficients bl1 , 1 1 whereas the coefficients aml and bml can be determined from Eqs. 36 and 38 . To compare the optical properties of particles of various shape, the cross sections can be normalized by the geometric cross section of the equivolume sphere

Equation 59 also allows us to calculate the values of 1 or 2 when the ratio Vcore Vtotal is the input parameter. For a prolate spheroid, the iterative scheme can be applied in the following manner:
n 1,2 1,2 n1

Vtotal,core Vcore,total

13 2 2,1 2,1

1

,

(60)

where n 1, 2 . . . and the initial value 1,2 0 2,1. For an oblate spheroid, parameter 1,2 can be found using Newton's method 2
n 1,2 n13 1,2

Vtotal,core Vcore,total
n12 1,2

2 2,1 2,1

1 , (61)

C rv

2 1 2
1

4 1

~ cos2 f 2 ~ f 1

12 16

Q.

(52)

3

1

Here, rv1 is the radius of equal to that of a given ~ above, f 1 for prolate oblate spheroids. The sphere can be defined as rv r
5420
3
1,2

the sphere spheroidal spheroids radius of

with the volume particle and, as ~ and f 1 for the equivolume (53) (54)

0. where n 1, 2 . . . and the initial value 1,2 0 The thickness of the mantle is not constant across the surface of a confocal coated spheroid. For a ray with incident angle , is equal to a1 a2 a1 b1
2 12

sin2
2

cos2
12 2

a1,2b

2 1,2

for prolate spheroids, for oblate spheroids
Vol. 35, No. 27

3 v1,2

a1,22b

1,2

a2 b2

sin

cos

2

(62)

APPLIED OPTICS

20 September 1996


Table 1. Efficiency Factors for Extinction Qext and Scattering Qsca for Core­Mantle Spheres and Spheroids at

0°

a

Prolate Spheroid Sphere N 4 6 8 10 12 14
a

Oblate Spheroid a1 b
1

a1 b Q
ext

1

2 Q
sca

10 Q
sca

a1 b Q
ext

1

2 Q
sca

a1 b Qext 0.185 0.162 0.1637 0.163724 0.163729 0.163729

1

10 Qsca 0.178 0.169 0.1636 0.163724 0.163728 0.163729

Qsca 1.09 3.638 3.5801 3.57777 3.577749 3.577749 1.5

Qext 0.232 0.224 0.2250 0.22445 0.224454 0.224454 5, and V
core

4.15 6.419 6.4163 6.41811 6.418088 6.418089 1.3

4.03 6.445 6.4156 6.41812 6.418088 6.418089 0.0i, 2 a1

0.219 0.221 0.2252 0.22444 0.224454 0.224454 V
total

1.84 1.608 1.6369 1.63663 1.636630 1.636630

2.56 1.629 1.6365 1.63664 1.636630 1.636630

m ~

core

0.0i, mmantle ~

0.5.

for a prolate spheroid and a
1

a1 b1

2

12

cos2
2

sin2
12

a2

a2 b2

cos2

sin2

(63)

for an oblate spheroid.
6. Computational Tests

The numerical code for coated spheroids is based on a previous code for homogeneous particles see Ref. 13 . The restrictions for large particles are mainly the result of calculation difficulties with the spheroidal functions. This problem is divided into two parts: determination of the eigenvalues and a reasonable choice of the method for spheroidal function expansion. At present, an exact and effective method for determination of eigenvalues has been developed. It appears to have no restrictions and can be used to calculate the eigenvalues with parameter c as high as 100. For calculation of spheroidal functions, we use the expansions in terms of Legendre functions, Bessel functions, or solutions of the corresponding differential equations, depending on the size and shape of a particle. The computational program was examined using various tests, which included internal control as well as a comparison with known results for homogeneous spheroids and core­mantle spheres. 1 For nonabsorbing particles the efficiency factors for extinction and scattering must be equal for the m m same azimuthal index m: Qext Qsca Q ·m Q m . Then by increasing the number of terms N in sums over l and n Eqs. 50 and 51 , one should obtain the decreasing difference between these two factors, i.e., Qext m Qsca m 3 0 if N 3 . Table 1 shows the behavior of efficiency factors in the case of radiation propagating along the rotation axis of a spheroid when the sums over m contain only one term for m 1. Our calculations demonstrate that the convergence for spheroids follows that for core­mantle spheres. The efficiency factors for core­mantle spheres have been calculated using the method from Ref. 28. The convergence can be determined by the particle size 2 a1 and is independent of its shape. The latter is explained by the successful

choice of scalar potentials. Note that Asano and Yamomoto10 used Debye's potentials that requires one to increase considerably the number of terms N for extremely prolate or oblate particles. 2 The values of the efficiency factors for TM and TE polarization are the same as for parallel radiation incidence i.e., if 0° . This test is not trivial since one can carry out the calculations using different expressions for each mode. 3 When summing over the azimuthal index m in Eqs. 50 and 51 , the convergence of efficiency factors should also occur. This is shown in Table 2 for radiation that propagates perpendicular to the rotation axis of a spheroid. It can be seen how the convergence process depends on particle shape. In reality, the number of terms M correlates with the volume of a particle that is proportional to parameters c1 and 1 i.e., the volume grows with a decrease of the ratio a1 b1 . Note that, as follows from Eqs. 53 ­ 58 , for the same values of c1 and a1 b1 the volume of the oblate particle is a1 b1 times larger than that of a prolate particle. 4 A detailed comparison of the optical properties of core­mantle and homogeneous spheroids has been made using different transformations of a core­ mantle particle to a homogeneous particle: i Vcore Vtotal 3 0, ii Vcore Vtotal 3 1, iii mcore ~ mmantle, ~ iv mmantle 1.0 0.0i. In all cases we found that ~ the factors for core­mantle particles became closer to those for spheroids consisting of the core or mantle material only.

Table 2. Efficiency Factors for Scattering QscaTM for Core­Mantle Spheroidsa

Prolate Spheroid M 1 2 3 4 5 6 7
a

Oblate Spheroid 10 a1 b
1

a1 b

1

2

a1 b

1

2

a1 b

1

10

1.702229 1.808355 1.808948 1.808949 1.808949 -- -- m ~
core

0.04962866 0.04962866 -- -- -- -- --
mantle

2.152832 3.317797 4.503349 4.670039 4.673185 4.673225 4.673225 0.0i, c
1

0.2755930 0.3845179 0.4001552 0.4008675 0.4008813 0.4008815 0.4008815 4, 90°, and

V

core

Vtotal

1.5 0.0i, m ~ 0.5.

1.3

20 September 1996

Vol. 35, No. 27

APPLIED OPTICS

5421


Fig. 2. Percent difference between coated spheres and coated spheroids defined by Eq. 64 : mcore ~ 1.5 0.0i, mmantle ~ 1.3 0.0i, Vcore Vtotal 0.5, a1 b1 1.0001, F, prolate spheroids, E, oblate spheroids.

Fig. 3. Scattering efficiency factors Qsca at 0° as a function of size parameter 2 a1 for the prolate coated spheroids solid curves : mcore 1.5 0.0i, mmantle 1.3 0.0i, a1 b1 2. The ~ ~ dashed curve represents the results for homogeneous spheroids with m 1.4 0.0i. ~

5 When particles are nearly spherical, the optical properties of coated spheroids and core­mantle spheres should be almost the same. We considered various absorbing and dielectric particles. Some results for nonabsorbing particles are shown in Fig. 2 where we present the relative differences in percent: Q sphere Q Csca spheroid sphere rv
2

7. Numerical Results and Discussion

sca

100%.

(64)

sca

The values of are plotted as a function of the size 2 rv for spheroidal particles with parameter xv an aspect ratio of a1 b1 1.0001 and a volume ratio of Vcore Vtotal 0.5. Then the aspect ratio of the core is constant and equal to a2 b2 1.00016. The wavelike behavior is typical only for dielectric particles. For highly absorbing particles, the values of demonstrate smooth, monotonous growth with xv. In the interval of the aspect ratios a1 b1 from 1.0001 to 1.01, the percent difference is invariant to this ratio and 5 a1 b1 1 if xv 20­25. Note also that the behavior of percent difference is independent of the particle structure ratio Vcore Vtotal . The difference between spheroidal and spherical particles occurs because of small deviations between their shapes. Because of the different path of radiation inside prolate and oblate spheroids see Eqs. 62 and 63 at parallel incidence, we obtained the interference picture in the opposite phase. When 90°, the picture remains the same, but the open circles oblate spheroids and filled circles prolate spheroids exchange places. All the calculations were performed on a PC AT486 50 computer. Using the last version of the numerical code29 January 1995 , the PC requires 1.5 min of CPU time and 1.4 Mbytes of memory to compute the efficiency factors in Table 2 oblate spheroids, a1 b1 2, m 1, . . . , 7, N 14 . If we also calculate the factors for the TE mode N 20 , the PC needs 4.5 min of CPU time and 3.3 Mbytes of memory. We also found that the computational time increases proportional to N2 N2.5 and the 2 required memory increases to N .
5422 APPLIED OPTICS Vol. 35, No. 27 20 September 1996

In this section we present the results of numerical calculations that illustrate the behavior of efficiency factors. The most intriguing feature in the case of core­mantle spheroids is the opportunity to study the influence of the particle structure i.e., core size and shape , whereas the variations of particle shape and inclination angle affect the optical properties in the same way as for homogeneous spheroids. However, because of the confocal structure, we cannot consider the particles with arbitrary core and mantle size and shape. To describe a core­mantle spheroid geometry, three different parameters can be used for example, c1, a1 b1, a2 b2 or 2 a1 , a1 b1, Vcore Vmantle . Other particle characteristics can be calculated from Eqs. 53 ­ 63 . The principal limitation for confocal spheroids is that a2 b2 a1 b1, which means that we can consider particles with a nonspherical core and an almost spherical mantle but not vice versa. Nevertheless, the remaining space of possible parameter values enlarges considerably the well-studied models of core­mantle spheres and infinitely long cylinders. Below we discuss some results for prolate and oblate spheroids with a1 b1 2 and 10. If we fix the values of Vcore Vmantle and a1 b1, then the shape of the core remains the same for all particle sizes 2 a1 . For Vcore Vmantle 0.5 and a1 b1 2 or 10 , the core aspect ratio is equal to a2 b2 2.58 or 14.09 and 3.07 or 19.78 for prolate and oblate spheroids, respectively. We considered the particles with the following refractive indices of the core and mantle: mcore 1.5 0.0i, 1.5 0.05i and mmantle ~ ~ 1.3 0.0i, 1.3 0.05i. The calculations were performed with a resolution of 0.1­ 0.2 in size parameter.
A. Nonabsorbing Spheroids

Figures 3 and 4 illustrate the size dependence of the efficiency factors for scattering Qsca for nonabsorbing spheroids with a1 b1 2 and 0°. Three values of the volume ratio Vcore Vmantle 0, 0.5, and 1.0 are considered in which two extreme cases represent the homogeneous spheroids consisting of mantle and core


Fig. 4. Same as Fig. 3 but for oblate spheroids.

Fig. 6. Same as Fig. 5 but for oblate spheroids.

materials. The large-scale behavior of the curves in Figs. 3 and 4 is a result of the interaction between incident radiation and the radiation that passes through the particle. Because of the interference, one can obtain a series of maxima and minima usually with decreasing amplitude. For coated spheroids, the qualitative explanation of such oscillations remains similar to that for homogeneous spheroids see Ref. 24 . The large-scale phenomena are roughly described by an anomalous diffraction approximation: the maxima height depends on the particle shape and the refractive index whereas their position can be determined by the phase shift of the central ray equal to 4 lcore mcore ~ 1 lmantle mmantle 1 , where 2lcore and 2lmantle are the ~ ray paths in the core and mantle, respectively. The oscillation period can be obtained from the solution to 2 . A similar explanation is also valid at oblique incidence as seen from Figs. 5 and 6. The small deviations of the period from that predicted by anomalous diffraction can be explained by the edge phenomena. It was also found that the curves for oblate spheroids at parallel incidence resemble those for prolate particles at perpendicular incidence and vice versa Ref. 24 . A comparison of Figs. 3 and 5 and Figs. 4 and 6 gives evidence that this is typical for coated spheroids as well. Note that the positions of first maxima for prolate spheroids with a1 b1 2 are

approximately the same when we consider the factors as a function of parameter xv. For a core­mantle particle, it is possible to introduce the effective refractive index meff that can be ~ defined as a volume-averaged refractive index meff ~ mcoreVcore mmantleVmantle Vtotal. Then we obtain ~ ~ meff ~ 1.4 for the cases shown in Figs. 3­ 6. The corresponding curves that were calculated for homogeneous spheroids are plotted in Figs. 3­ 6 by the dashed curves, indicating a good correspondence of scattering factors for effective and coated particles to a first maximum. The curves of the efficiency factors for oblate particles show the small-scale ripples superimposed on the major oscillations. These ripplelike fluctuations result from the resonances of virtual modes27 and are much more obvious for oblate particles than for prolate particles. From Fig. 7, where the efficiency factors for elongated and flat particles are plotted, it can be seen that the values of Qsca for prolate and oblate spheroids differ strongly at parallel incidence. At perpendicular incidence, the behavior of the factors resembles the results presented in Figs. 3­ 6. However, the difference between prolate and oblate spheroids is greater. If we consider the cross sections instead of the efficiency factors, the values of Csca between a1 b1 2 and 10 differ by a smaller amount than those shown in Figs. 3, 4, and 7. This can be explained by

Fig. 5. Scattering efficiency factors Qsca at 90° as a function of size parameter 2 a1 for the prolate coated spheroids solid curves : mcore 1.5 0.0i, mmantle 1.3 0.0i, a1 b1 2. The ~ ~ dashed curve represents the results for homogeneous spheroids with m 1.4 0.0i. ~

Fig. 7. Scattering efficiency factors Qsca at 0° as a function of size parameter 2 a1 for the coated spheroids solid curves : mcore 1.5 0.0i, mmantle 1.3 0.0i, a1 b1 10. The dashed ~ ~ curves represent the results for homogeneous spheroids with m ~ 1.4 0.0i. The factors for oblate spheroids were multiplied by 20. 20 September 1996 Vol. 35, No. 27 APPLIED OPTICS 5423


Fig. 8. Efficiency factors for a, extinction Qext; b, scattering Qsca; and c, absorption Qabs at 0° as a function of size parameter 2 a1 for the prolate coated spheroids: a1 b1 2; 1, mcore ~ 1.5 0.05i, mmantle 1.3 0.05i; 2, mcore 1.5 0.05i, mmantle ~ ~ ~ 1.3 0.0i; 3, mcore 1.5 0.0i, mmantle 1.3 0.05i; 4, mcore ~ ~ ~ 1.5 0.0i, mmantle 1.3 0.0i. ~

Fig. 9. Same as Fig. 8 but for oblate spheroids.

the large difference in geometric cross sections G see Eqs. 48 and 49 .
B. Absorbing Spheroids

Figures 8 and 9 demonstrate how the absorption in core or and mantle influences the optical properties of prolate and oblate spheroids at parallel incidence. The general behavior of extinction factors for spheroids is similar to that for coated spheres: the damping of the interference and ripplelike structure and the convergence of efficiency factors to the limit of geometrical optics Qext 3 2, Qsca 3 1, Qabs 3 1 can be seen when the absorption grows. Note that the absorption in core or and mantle causes a decrease of the height of maxima but acts weakly on their position because of the same phase shift of the central ray. However, because of different mantle thicknesses as follows from Eqs. 62 and 63 , a1 a2 a1 0.05, and b1 b2 b1 0.4 for prolate and oblate spheroids, respectively , the presence of an absorbing core or mantle gives distinct results curves 2 and 3 in Figs. 8 and 9 . It is interesting to note that, for 2 a1 16, the prolate coated spheroid with a dielectric mantle
5424 APPLIED OPTICS Vol. 35, No. 27 20 September 1996

curve 2 in Fig. 8 absorbs approximately as much radiation as that composed of the absorbing core and absorbing mantle. This phenomenon is due to the inhomogeneity of the particle: the nonabsorbing mantle refracts the incident light, concentrates it on the absorbing core, and, thus, the total absorption efficiency increases. In the case of the nonabsorbing core and absorbing mantle, the latter screens the core that leads the optical properties of particles closer to those of the dielectric particles than to the highly absorbing ones but with a damped ripplelike structure. For the oblate coated particles considered, the absorption in core and mantle produces similar effects in extinction, scattering, and absorption see curves 2 and 3 in Fig. 9 . It is also seen that the ripplelike structure is attributed mainly to the dielectric mantle because it is generated by the resonances of virtual modes damped in the absorbing particles.27

Appendix

The expressions for integrals of products of normal¯ ized angular spheroidal functions Sml c, ¯ Nml 1 c Sml c, and their first derivatives S ml c, 1 Nml c S ml c, can be represented as a series containing the expansion coefficients drml of prolate


angular spheroidal functions in terms of associated Legendre functions Ref. 21 :
1 m nl

1 m nl

ci, cj
1

¯ S

mn

ci ,

¯ Sml cj,
1

1

2

d

ci, cj
1

¯ S

mn

ci,

¯ Sml cj,
1

d
mn ml

N dr ci dr c

1 mn

ci Nml
ml

c

j r 0,1

dr

mn

c

i

N

1 mn

ci Nml 2 2m ci,

c

j r 0,1

j

d dr dr

r2

c

j

2r

rr 1 2m 3 2r

2m
2

1

r 1 ¯ Sml cj,
1

2r
1 m nl

2m ! , r! d
mn

ml

c

j

2r mr 2r 2m c
j

1 m1 m 1 2r 2m 3 1 r 2m 2 3 2r 2m 5

ml 2

ci, cj
1

¯ S

mn

r 2r r 1

2m 2m

Nmn d dr

1

ci Nml
ml

c

j r 0,1

dr r 2m

ci
m nl

2r
1

2 2m

2m ! , r!

r1

c

j

2r

1

ci, cj
1

¯ S

mn

ci, ci, 1

¯ S

ml

cj,

1

2

ml 1

r 2m 1 cj 2r 2m 3 r 1 2m ! , r! 1
mn 2

¯ m2 S

mn

¯ Sml cj,
2

d dr
mn

2r
1 m nl

2 2m ci,

N d

1 mn

ci Nml

1

c

j r 0,1

ci dr

ml

c

j

ci, cj
1

¯ S

mn

¯ Sml cj,
1

2r

m r m 1 r 2m ! . 2r 2m 1 r!

N

1 mn

ci Nml
ml

c

j r 0,1

dr 2m 2m

ci

d dr

r1

c

j

rr 2r r

1 1

ml 1

cj

m r 2m 1 2r 2m 3 r 1 2m ! , r! 1
mn 2

The prime over the summations indicates that the even odd terms only are summarized when the index n m is even odd . To check the calculations we can use the rule that the simultaneous replacements of n by l and ci by cj in integrals nl, nl, nl, and nl should not change them, m m i.e., nl ci, cj The relations beln cj, ci , etc. tween integrals also exist:
m nl m nl

2r
1 m nl

2 2m ¯ S
1

ci, cj ci, cj

m ln m ln

cj, ci cj, ci

2 2

m nl m nl

ci, cj , ci, cj
m nl

ci, cj .

ci, cj
1

mn

ci,
1

¯ Sml cj, c dr
r 0,1

d The coefficients drmn reach their maximum values for the index r n m: dn mmn 1; for the indices r n m or r n m the coefficients drmn drop rapidly. Therefore, the integrals sums considered converge rapidly. For oblate spheroidal functions the replacements drmn c 3 drmn ic and Nmn c 3 Nmn ic should be made. We are grateful to V. B. Il'in and anonymous reviewers for valuable comments. We thank M. I. Mishchenko who brought to our attention the paper of Cooray and Ciric.15 This research was made possible in part by grant NVT000 from the International Science Foundation. The calculations were carried out mainly on the PC AT 486 computer, which was purchased with funds from the Fund for East Europe Sponsorship 12161.
20 September 1996 Vol. 35, No. 27 APPLIED OPTICS 5425

Nmn d dr dr r

ci Nml
ml

j

ci

r2

c

j

2r

r 1rr m 2m 3 2r 2m
2

1

ml

cj
ml

3r m r 2r 2m cj

m1 m 2 1 2r 2m 3 2

2

m 1r 2r 2m 2 2m r 1

2m 1 r 2m 3 2r 2m 5 2m ! , r!

2r


References and Notes
1. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848 ­ 872 1988 . 2. J. L. Hage, J. M. Greenberg, and R. T. Wang, "Scattering from arbitrarily shaped particles: theory and experiment," Appl. Opt. 30, 1141­1152 1991 . 3. B. T. Draine and J. Goodman, "Beyond Clausius­Mossotti: wave propagation on a polarized point lattice and the discretedipole approximation," Astrophys. J. 405, 685­ 697 1993 ;P.J. Flatau, G. L. Stephens, and B. T. Draine, "Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure," J. Opt. Soc. Am. A 7, 593­ 600 1990 ; K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653­ 677 1994 . 4. J. I. Peltoniemi, K. Lumme, K. Muinonen, and W. M. Irwine, "Scattering of light by stochastically rough particles," Appl. Opt. 28, 4088 ­ 4095 1989 . 5. F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803­ 814 1993 . 6. G. Mie, "Beitrage zur Optik truber Medien speziell kolloidaler ¨ ¨ Metallosungen," Ann. Phys. Leipzig 25, 377­ 445 1908 . ¨ 7. K. S. Shifrin, "Light scattering by core-mantle particles," Izv. Akad. Nauk SSSR Ser. Geofiz. N 2, 15­21 1952 . 8. A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely oriented cylinders," J. Appl. Phys. 37, 3195­3203 1966 . 9. G. A. Shah, "Scattering of plane electromagnetic waves by infinite concentric circular cylinders at oblique incidence," Mon. Not. R. Astron. Soc. 148, 93­102 1972 . 10. S. Asano and G. Yamamoto, "Light scattering by a spheroidal particle," Appl. Opt. 14, 29­49 1975 . 11. V. G. Farafonov, "The scattering of a plane electromagnetic wave by a dielectric spheroid," Differential Equations Sov. 19, 1765­1777 1983 . 12. N. V. Voshchinnikov and V. G. Farafonov, "Light scattering by dielectric spheroids. I," Opt. Spektrosk. 58, 81­ 85 1985 . 13. N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19­86 1993 . 14. T. Onaka, "Light scattering by spheroidal grains," Ann. Tokyo Astron. Obs. 18, 1­54 1980 . 15. M. F. R. Cooray and I. R. Ciric, "Scattering of electromagnetic 16.

17.

18.

19. 20. 21.

22.

23. 24. 25.

26.

27. 28.

29.

waves by a coated dielectric spheroid," J. Electro. Waves Appl. 6, 1491­1507 1992 . B. P. Sinha and R. H. MacPhie, "Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence," Radio Sci. 12, 171­184 1977 . B. Peterson and S. Strom, "T-matrix formulation of electro¨ magnetic scattering from multilayered scatters," Phys. Rev. D 10, 2670 ­2684 1974 . P. W. Barber, "Scattering and absorption by homogeneous and layered dielectrics," in Acoustic, Electromagnetic and Elastic Wave Scattering--Focus on the T-Matrix Approach, V. K. Varadan and V. V. Varadan, eds. Pergamon, New York, 1980 , pp. 191­209; D.-S. Wang and P. W. Barber, "Scattering by inhomogeneous nonspherical objects," Appl. Opt. 18, 1190 ­ 1197 1979 ; D.-S. Wang, C. H. Chen, P. W. Barber, and P. J. Wyatt, "Light scattering by polydisperse suspensions of inhomogeneous nonspherical particles," Appl. Opt. 18, 2672­2678 1979 . V. N. Lopatin and F. Ya. Sid'ko, Introduction to Optics of Cell Suspension Nauka, Novosibirsk, Russia, 1987 . C. Flammer, Spheroidal Wave Functions Stanford U. Press, Stanford, Calif., 1957 . I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions Nauka, Moscow, 1976 . J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes NorthHolland, Amsterdam, 1969 . J. A. Stratton, Electromagnetic Theory McGraw-Hill, New York, 1941 . S. Asano, "Light scattering properties of spheroidal particles," Appl. Opt. 18, 712­723 1979 . P. W. Barber, "Resonance electromagnetic absorption by nonspherical dielectric objects," IEEE Trans. Microwave Theory Tech. MTT-25, 373­381 1977 . M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16 ­21 1994 . C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles Wiley, New York, 1983 . N. V. Voshchinnikov, "Dust grains in reflection nebulae. Spherical core-mantle grains," Sov. Astron. 22, 561­566 1978 . The numerical code is available on request from N. V. Voshchinnikov, e-mail: nvv@aispbu.spb.su.

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20 September 1996