Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.astro.spbu.ru/staff/ilin2/INTAS2/PAP/P7_5.pdf Äàòà èçìåíåíèÿ: Fri Nov 19 16:18:23 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:05:58 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 8

Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626

www.elsevier.com/locate/jqsrt

Light scattering by multilayered nonspherical particles: a set of methods
Victor G. Farafonova , Vladimir B. Il'inb ; , Marina S. Prokopjevab
a

St. Petersburg University of Aerocosmic Instrumentation, Bol. Morskaya ul. 67, St. Petersburg 190000, Russia b Astronomical Institute, St. Petersburg University, Bibliotechnaya Pl 2, Universitetskij pr. 28, St. Petersburg 198504, Russia Received 1 June 2002; accepted 23 September 2002

Abstract An original approach to solution of the light scattering problems for axisymmetric particles was developed in our earlier papers. The approach is based on separation of the Úelds in two speciÚc parts and a proper choice of scalar potentials for each of them. Applications to homogeneous scatterers made Úrst in the framework of the separation of variables method (SVM) and later the extended boundary condition method (EBCM) led to more e cient solutions (at least in the case of the SVM) than the standard ones. The approach was recently applied to formulate new theoretical methods for multilayered axisymmetric particles. In this paper we further develop and systematically discuss the methods. One of them is a modiÚcation of the EBCM and another looking as (and wrongly called) a modiÚcation of the SVM is shown to be rather that of the EBCM formulated in spheroidal coordinates. The solutions are now presented in recursive forms. The ranges of applicability of the new methods are considered analytically for the Úrst time in the literature on layered scatterers. The theoretical methods and their program implementations are compared with others available. We note that usage of scalar potentials (a feature of our approach) allowed us consistently to realize the EBCM in spheroidal coordinates. Advantages of this approach in the case of layered spheroidal particles with the confocal layer boundaries are noted. Earlier we have extended the quasistatic approximation (QSA), being a generalization of the Rayleigh (RA) and Rayleigh-Gans approximations, to layered ellipsoids in the general case of nonconfocal layer boundaries. Here the connection between the QSA and the asymptotic of the scattered Úeld found in the framework of our SVM-like method in the limit of very large aspect ratios of spheroids is discussed. Keeping this fact in mind, the applicability regions of the QSA and RA are comparatively considered for multilayered ellipsoids. We also note that the formulations of the RA and QSA contains a quantity that can be interpreted as the average refractive index of a layered particle and thus gives a new rule of the e ective medium theory more appropriate for such scatterers. ? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Light scattering; Inhomogeneous nonspherical particles



Corresponding author. Tel.: +7-812-184-2243; fax: +7-812-428-7129. E-mail address: vi2087@vi2087.spb.edu (V.B. Il'in).

0022-4073/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S0022-4073(02)00310-2


600 V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626

1. Introduction Various natural scatterers are known to have internal structure, and hence the problem of light scattering by inhomogeneous particles is of large interest in di erent scientiÚc Úelds--astrophysics, atmosphere and ocean optics, biophysics, etc. In many cases inhomogeneous dust grains can be well represented by multilayered particles. For instance, quasilayers of di erent composition should form during dust grain evolution in interstellar medium. Particles with many relatively thin layers can be used to model grains with a radially changing refractive index, appearing, for instance, in the case of fractal-like, u y aggregates typical of interplanetary media. The model with a large number of very thin layers of several cyclically changing materials opens an e cient way to study e ects of composition inhomogeneity expected in cosmic dust grains. We mention only astrophysical applications, but similar examples can be found in other Úelds as well. To simulate light scattering by layered particles, when shape e ects are of small importance, one usually utilizes the model of multilayered spheres that is based on relatively simple and e ective algorithm [1]. This algorithm appears to be applicable practically in the whole range of parameter values after a small improvement. Situation with layered nonspherical particles is much more complex, despite many methods allow one to get solution to the corresponding light scattering problem. In principle, nonspherical scatterers of any structure can be treated by the methods using the representation of the scattering problem in the form of volume integral equation (e.g., the widely used discrete dipole approximation, DDA), the Únite di erence time domain methods, etc. [2]. The back sides of this universality are strong demands for computer memory and speed which often make computations required by applications impossible. Layered scatterers could be also treated by the separation of variables method (SVM) and the extended boundary condition method (EBCM) which can better involve the scattering geometry and hence are much faster for some kinds of particle shapes. As a result the methods could allow really extensive calculations, but are not yet well developed--till now detailed consideration was mainly restricted by core-mantle spheroids for the SVM [3­5] and core-mantle axisymmetric particles for the EBCM [6]. Nonspherical particles with three and more layers were studied mainly theoretically, i.e. without computations (see [7,8] for the SVM and [6,9­11] for the EBCM, with an exception being [6] where some illustrative calculations for a three-layered spheroid were done). A universal EBCM-like computer code for layered particles was recently presented in [12], but the paper does not contain results for multilayered particles, estimates of e ciency of the code, and its comparison with others (like the DDA one). It is important here also to distinguish the standard EBCM approach (see, e.g. [13]) from the approach used in [11,12] and other works cited there. In the former expansions of the Úelds in terms of the spherical wave functions considered in one coordinate system are utilized, in the latter the Úelds are expanded in terms of Únite linear combinations of the spherical wave functions, with combinations being related to di erent coordinate systems having origins distributed on a closed surface. There is a principal distinction between such single and multipole expansions (di erent basis) and hence the approaches di er in many important aspects--applicability range, e ciency, etc. In this paper we modify and investigate the standard EBCM approach used in a great number of works (see [2] for a review), and hereafter EBCM means this approach whereas the approach used in [11,12] is called the discrete source method (see [14] for more details). Besides [11,12] there were many other modiÚcations of the EBCM aimed at overcoming its probably main defect--inability to treat scattering by particles of large eccentricity (see [2] for a


V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626 601

review). A perspective way to solve this problem in the case of spheroids could be formulation of the EBCM in spheroidal coordinates (i.e. with expansion of the Úelds in terms of the spheroidal wave functions), which would better correspond to the scattering geometry than spherical coordinates and functions. For acoustic wave scattering, that was actually done in [15], but for electromagnetic wave scattering, it is impossible in the standard formulation of the EBCM (e.g. [13]) because of nonorthogonality of the spheroidal vector wave functions [16]. The fact that the standard EBCM is not appropriate (expansions of the Úelds are divergent) for scatterers of large eccentricity is well known, but based mainly on the results of calculations [2,17]. Earlier analytic investigations of the EBCM were summarized in [19] (see also [18,20]). Besides other important results the paper [19] formulates the condition of validity of the Rayleigh hypothesis on convergence of the Úeld expansions everywhere up to the scatterer boundary. Obviously, this condition is the necessary one for applicability of the EBCM in the near zone. A general condition of convergence of the series in the EBCM for the far zone was recently obtained in [21]. This condition is however too abstract to be easily applied to concrete cases. In cite [21] after rather large e orts it was only demonstrated that the condition was satisÚed for any ellipsoid. Less abstract analytic investigations of the applicability ranges of EBCM-like methods were made for perfectly conducting and dielectric homogeneous scatterers in [22,23], respectively. These papers give applicability (convergence) conditions for the far zone in the form which allows simple application to Chebyshev particles, spheroids and so on. Note that in contrast to the EBCM another popular approach--the SVM for spheroidal particles was analytically studied in detail long ago [24]. For layered particles, the methods were, however, never analyzed. Even for such fast methods as the SVM and EBCM, calculations of light scattering by multilayered nonspherical particles are very time consuming. Therefore, various approximate methods (see, e.g. [25] for a review) can be useful in applications. However, one of the most widely used approximations--the Rayleigh approximation (RA) has the same problem as the EBCM--it does not work well for particles whose shape strongly di ers from the spherical one. A generalization of the RA--the quasistatic approximation (QSA) apparently avoids this problem for homogeneous spheroidal particles [26]. The QSA was recently extended to multilayered ellipsoids [27], but its applicability range was not discussed, which is rather typical of approximations available for inhomogeneous nonspherical particles. In this paper we consider a set of exact and approximate methods to calculate the light scattering by multilayered nonspherical (mainly axisymmetric) particles. The corresponding light scattering problem and a general approach used to Únd exact solutions to the problem are presented in Sections 2 and 3, respectively. Section 4 describes the suggested modiÚcation of the EBCM for such particles and gives recursive forms of the solution. Another exact method--a modiÚcation of the EBCM in spheroidal coordinates is introduced in Section 5. The next section shows the connection of this method with the QSA and discuss other tightly connected approximations. Section 7 contains analytic estimates of the applicability ranges of the exact and approximate methods and comparison with results of numerical calculations. Conclusions are drawn in the last section. Let us introduce the notations of the versions of exact methods we mention hereafter and give several general remarks on their relationship. The standard EBCM approach (see, e.g. [13]) is denoted by sEBCM; our modiÚcation of the sEBCM [28] by mEBCM and our generalization of the mEBCM for multilayered axisymmetric particles by gmEBCM; the standard SVM approach for spheroids [29] by sSVM; our modiÚcation of the sSVM [30] by mSVM and our generalization of the mSVM for


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multilayered particles by gmSVM. It should be added that the SVM approach with expansions of the Úelds in terms of the spherical wave functions is equivalent to the sEBCM as was shown in [16]. By analogy, the sSVM, i.e. the standard SVM for spheroids using the spheroidal wave functions for expansions, is equivalent to the sECBM formulated in spheroidal coordinates (sEBCMsc hereafter), i.e. with expansions of the Úelds in terms of the spheroidal wave functions. Accordingly, the mSVM and mEBCMsc as well as the gmSVM and gmEBCMsc are generally equivalent too (see Section 5 for more details). Note that the SVM approach can be applied not only to spheroids (spheres and inÚnitely long cylinders) but generally to particles of arbitrary shapes [31]. It should be also noted that in the literature the EBCM was very often called the T -matrix method (TMM). The transition (T ) matrix relates the expansion coe cients for the scattered Úeld with those for the incident one and depends only on particle parameters--size, shape, etc. Therefore, the T matrix is a very convenient characteristic in some cases (e.g., when one considers an ensemble of particles) and in several recent papers (e.g. [32­34]) this matrix is derived by methods di erent from the EBCM. As a result, the usage of the term TMM looks now a bit confusing and in this paper we call the approach EBCM.

2. Formulation of the problem We consider scattering of a plane wave incident at a n-layered axisymmetric particle. The particle geometry is deÚned by the layer surface equations r ( j) (á)=0; j =1; 2;:::;n; (1)

where á is one of the angles of the spherical coordinate system (r; á; ') connected with the particle. The surfaces further called Sj are assumed to have no common points. The Úelds in the j th layer conÚned by the surfaces r ( j) (á) = 0 and r ( j+1) (á) = 0 are denoted by ( j +1) ~ ( j +1) ~ E ;H . Thus, j = 1 corresponds to the outermost layer (a mantle of the particle) and j = n ~ ~ to the innermost layer (a core). As in the case of a homogeneous particle (n = 1), E (0) ; H (0) and ~ (1) ; H (1) are the incident and scattered Úelds, respectively. ~ E An arbitrary polarized plane electromagnetic wave, incident at the angle to the symmetry axis of the particle, can be represented by a superposition of the waves of two kinds: (a) TE mode ~r ~ E (0) (~ )= -iy exp [ik1 (x sin + z cos )]; ~ H
(0)

(~ )= r

1 1

(~ cos - ~ sin ) exp [ik1 (x sin + z cos )]; ix iz

(2)

(b) TM mode ~r ix iz E (0) (~ )=(~ cos - ~ sin ) exp [ik1 (x sin + z cos )]; ~ H
(0)

(~ )= r

1 1

~ iy exp [ik1 (x sin + z cos )];

(3)


V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626 603

where (~ ; iy ; ~ ) are the unit vectors of the Cartesian coordinate system, whose z -axis coincides with ix ~ iz the symmetry axis of the particle, ~ =(x; y; z ), 1 and 1 are the dielectric permittivity and magnetic r permeability outside the particle, k1 is the wavenumber. The problem is to solve Maxwell's equations for each layer ( j =1; 2;:::;n) 1~ ( ( E~j) (~ )= - â H~ j) (~ ); r r i j k0 ~r E ( j) (~ ) â nj = E ~~ ~ H
( j) ( j +1) ( H~ j) (~ )= r

1~ ( â E~j) (~ ) r i j k0

(4)

with the boundary conditions at each layer surface (~ ) â ~ r n
j ~ S r
j

~ (~ ) â nj = H r ~

( j +1)

(~ ) â nj r ~

(5)

and the radiation condition at inÚnity (r ) lim r 9E (1) (~ ) r (1) r - ik1 E~ (~ ) 9r
~

=0;

lim r

(1) r 9H~ (~ ) (1) r - ik1 H~ (~ ) 9r

=0:

(6)

Here j and j are the dielectric permittivity and magnetic permeability, respectively, k0 = !=c is the wavenumber in vacuum, ! the radiation frequency, c the velocity of light in vacuum, nj the ~ outward normal to the j th layer surface Sj , ~ the radius-vector, r = |~ |. We assume that the time r r dependence of the electromagnetic Úelds is given by the factor exp(-i!t ). 3. Description of the approach To Únd solution to the described problem, we apply the approach earlier suggested for homogeneous axisymmetric scatterers [24]. The main features of the approach are as follows: 1. All the Úelds are divided in two parts--an axisymmetric one that does not depend on the azimuthal angle ' and an nonaxisymmetric one whose averaging over ' gives zero ~r ~( r ~( r E ( j) (~ )= E Aj) (~ )+ E Nj) (~ ); ~ H
( j)

~ (~ )= H r

( j) A

~ (~ )+ H r

( j) N

(~ ); r

(7)

where j = 0; 1;:::;n + 1. The possibility of such a representation of the Úelds was considered in [35]. The light scattering problems for the parts can be solved independently. Such a separation is possible because of commutation of the operator corresponding to the di raction problem and the operator Lz = 9= 9' (see [35] for more details). Thus, the problem under consideration can be uncoupled relative to the azimuthal angle ', i.e. each component of the Fourier expansion can be found separately. 2. Proper scalar potentials are chosen for each of the Úeld parts. For the axisymmetric parts, we apply the potentials analogous to the Abraham potentials for spheroids which are known to simplify solutions for such particles [36]. For the nonaxisymmetric parts, we utilize combinations of the Debye potentials used in solutions for spheres and z -components of the Hertz vectors used for inÚnitely long cylinders. The approach was found highly e cient for solution of the light scattering problem for spheroids by the separation of variables method [30]. Note that only the nonaxisymmetric parts and their


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potentials were used in the modiÚcation of the approach in [8]. Such a variant has more simple formulation, but computationally may be less favorite.

4. ModiÚcation of the EBCM The described approach was applied to Únd solution to the light scattering problem for homogeneous particles in the framework of the EBCM in [28,37,38]. We have also generalized that solution (mEBCM) for the case of multilayered axisymmetric scatterers [10]. Here we brie y describe the main steps of the generalized solution to be able to reveal and discuss di erences of the suggested modiÚcation of the EBCM and other methods. For the sake of simplicity, solutions for the axisymmetric and nonaxisymmetric parts are considered separately, though they are similar in many details. 4.1. Axisymmetric parts of the Úelds To construct the scalar potentials to these parts inside each layer we use the azimuthal components of the Úelds
() p( j) (~ )= EAj;' (~ ) cos '; r r ( q( j) (~ )= HAj) (~ ) cos '; r ;' r

(8)

where j =0; 1;:::;n + 1 and the potentials p; q satisfy the scalar Helmholtz equation r r q( j) (~ )+ kj2 q( j) (~ )=0 with kj = equations:
j j k0

(9)

. Other components of the Úelds can be expressed via the potentials using Maxwell's 1 9(sin áq( j) ) 9(rq( j) ) p( j) -1 ; ; i j k0 r sin á cos ' 9á i j k0 r cos ' 9r cos ' 9(sin áp( j) ) 9(rp( j) ) q( j) 1 -1 ; ; i j k0 r sin á cos ' 9á i j k0 r cos ' 9r cos '

( EAj) (~ )= r ( HAj) (~ )= r

; : (10)

Easy to see that the potentials p and q are connected with the TE and TM modes, respectively. The equations for p are generally similar to those for q, and below we mention the TE mode and the potentials p only if an essential di erence appears. For each layer, we represent the scalar potentials as sums
( ( q( j) (~ )= q1j) (~ )+ q2j) (~ ); r r r

(11)

( ( where q1j) has no peculiarity at the origin of the coordinate system, q2j) satisÚes the radiation (n+1) (n+1) condition at inÚnity. Note that q1 = q(n+1) , q2 = 0 and Eq. (11) can be considered for (1) (1) j =1; 2;:::;n + 1, if one uses the following notations: q1 = q(0) , q2 = q(1) .


V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626 605

The introduced quantities allow one to write the surface integral equation equivalent to the scalar wave equation (see, e.g. [39]) in a more convenient way: r; r 9G (kj ;~ ~ ) 9q( j) (~ ) r - G (kj ;~ ~ ) q (~ ) r r; r 9n 9n
( j)

S

dS =

( r r -q1j) (~ ); ~ Dj ; ( q2j) (~ ); r

j

~ R3 \ D j ; r

(12)

where G (k; ~ ~ ) r; r 1; 2;:::;n). Then a result we shall The boundary p
( j)

=p

(

9p( j) = 9nj q
( j)

=q

(j

9q( j) = 9nj

is the free-space Green function, Dj the domain conÚned by the surface Sj ( j = the potential for a layer can be expressed via the potential for the previous one. As get the solution in the form, where quantities for each layer are totally separated. conditions (5) for the potentials can be rewritten as follows: j +1) ; ( j +1) rá 9p 1 j j ( j +1) 1 - ctgá p + -1 ; 9nj r 2 j +1 j +1 r 2 + rá ; (13) +1) ; ( j +1) rá 1 j 9q j ( j +1) 1 - ctgá q + -1 ; 9nj r 2 j +1 j +1 r 2 + rá
~ S r
j

where rá is the derivative After the substitution of 9G (kj q( j+1) (~ ) r 9 Sj

of rj (á) with respect to the spherical angle á, and j =1; 2;:::;n. the conditions (13) in Eq. (12) one gets ;~ ~ ) j 9q( j+1) (~ ) r; r r 1 j -1 - + n 9n j +1 j +1 (r )2 +[(r )á ]2 q
( j +1)

(r )á â 1- ctgá r

(~ ) G (kj ;~ ~ ) r r; r



dS =

( r r -q1j) (~ ); ~ Dj ; ( q2j) (~ ); r

~ R3 \ D j : r

(14)

The scalar potentials p; q are expanded in terms of the spherical wave functions
( p1j) (~ ) r ( q1j)



(~ ) r

=

( a1j) ;l

( j) l=1 b1;l

1 jl (kj r ) Pl (cos á) cos ';

(15)

1 where jl (kj r ) are the spherical Bessel functions, Pl (cos á) the associated Legendre functions, ( j) ( j) j =1; 2;:::;n + 1. The potentials p2 ;q2 are expanded in the same way, but jl (kj r ) are replaced by the Hankel functions of the Úrst kind h(1) (kj r ). The spherical harmonic expansion of the free-space l Green function is well known (see, e.g. [40]). All the expansions are substituted into the surface integral equations (14). Due to orthogonality of the spherical wave functions, one gets two inÚnite systems of algebraic equations relative to the


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expansion coe cients of the scattered and internal Úeld potentials. Solution of the systems gives the coe cients for the scattered Úeld, ~ (1) = {a(1) } , in the form typical of the EBCM a 2;l l=1 ~ (1) = A2 A a
-1 (0)
1

~; a

(16)

where ~ (0) = {a(0) } are known coe cients for the incident Úeld (a plane wave). The matrices A1 a 1;l l=1 and A2 are determined as follows: (n (n -Ahj-1) - A(n-1) -A(1) - A(1) -Ahj) A1 hj hh hh ··· = : (17) (n (n (n A2 A(1) A(1) Ajj -1) Ajh-1) Ajj ) jj jh Here A( j) are the matrices whose elements are surface integrals of products of the spherical wave functions and their derivatives calculated for the j th layer, for instance
(j (Ahj) )ln =

i(2l +1) 2l(l +1)

0

2 kj2 rj h(1) (kj rj ) jn (k l

j +1 rj

)-

j j +1

kj+1 (1) h (kj rj ) jn (k kj l

j +1 rj

)

1 1 1 1 âPl (cos á)Pn (cos á) sin á + kj (rj )á sin2 á Pl (cos á)Pn (cos á)

-

j j +1

1 1 Pl (cos á)Pn (cos á) h(1) (kj rj ) jn (k l

j +1 rj

)-

j j +1

-1 d á: (18)

â(kj rj sin á - kj (rj )á cos á)h(1) (kj rj ) jn (k l

j +1 rj

1 1 )Pl (cos á)Pn (cos á)

The subscripts j; h of A( j) matrices show what radial functions are used in the corresponding places in the integrals (18)--the spherical Bessel ( j ) or Hankel (h) one. 4.2. Nonaxisymmetric parts of the Úelds The scalar potentials selected for these parts are superpositions of the vertical components of the Hertz vector U ( j) and the Debye potentials V ( j) ( j = 0; 1;:::;n). For instance, for TM mode we have ~ E
( j) N ( j) N

=-

1~ ~ â â (U i j k0

( j)

~z + V ( j)~ ); r i

(19)

~ H

~ = â (U ( j)~z + V ( j)~ ): r i

In the same way as in Section 4.1 we represent the scalar potentials for each layer as sums U
( j)

(~ )= U r

( j) 1

(~ )+ U r

( j) 2

(~ ); r

V

( j)

(~ )= V r

( j) 1

(~ )+ V r

( j) 2

(~ ): r

(20)


V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626 607

The boundary conditions for the TM mode can be written as follows: U V
( j) ( j) j)

= = =

U V

( j +1) ( j +1)

;

;
j j +1

9U ( 9n

9U ( j+1) + 9n -sin á

-1

r r sin á

á

r2 + r
( j +1)

2 á

r cos á

9U ( j+1) 9r

9V ( j+1) 9U ( j+1) + r2 + rV 9á 9r
j j +1


~ S r
j

; r cos á 9U ( j+1) 9r

:

(21)

9V ( j 9n

)

=

9V ( j+1) + 9n -sin á

-1

rá cos á - r sin á r 2 sin á r2 + r
( j +1) 2 á

9V ( j+1) 9U ( j+1) + r2 + rV 9á 9r

;

Substitution of the corresponding boundary conditions in the surface integral equations (12) leads to the integral equations well resembling those for the axisymmetric parts (see Eq. (14)). The potentials are expanded in terms of the spherical wave functions U
( j) 1

(~ ) r





( V1 j)

(~ ) r

=

( a1j) ;m

l

( j) m=1 l=m b1;ml

m jl (kj r ) Pl (cos á) cos m':

(22)

The expressions for the potentials with the subscript 2 are the same after the replacement of jl (kj r ) with h(1) (kj r ). l Substitution of the expansions into the surface integral equations gives for each m two inÚnite systems of algebraic equations relative to the coe cients of the potential expansions. Solution of the systems provides the coe cients for the scattered Úeld ~ (1) = {a(1) l } , ~ (1) = {b(1) l } (m ¿ 1) in am 2;m l=1 bm 2;m l=1 the form (16)­(17) with the only di erence--the matrices A( j) are now twice as large and have the block structure
(j Ahj) = ( j) hj; 1 ( j) hj; 2 (j Úhj;)1 (j Úhj;)2

; ; (23)

(j (j where hj;)i , Úhj;)i are the matrices whose elements are surface integrals of products of the spherical wave functions and their derivatives calculated for the j th layer (see [10] for more details). The systems arising for axisymmetric and nonaxisymmetric parts were investigated analytically and numerically in the case of homogeneous particles [23]. It was found that they had similar properties and a ected the range of applicability of the method nearly in the same way.


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4.3. Recursive forms of the solution It is important to note that the most essential part of the solution-- Eq. (17) can be rewritten in the recursive form ( ( -A(1) - A(1) A1n+1) A1n) hh hj ; (24) = ( ( A2n+1) A(1) A(1) A2n) ; jj jh where matrix the (n Eq. the right-hand side vector contains the values of A1 ;A2 for a given n-layered particle, the corresponds to a new outer layer, and the left-hand side vector gives the values of A1 ;A2 for + 1)-layered particle formed in such a way. (24) can be rewritten in another way
n+1 (n (n =(Ajj +1) + A(1) Tn )(Ahj+1) + A(1) Tn )-1 ; jh hh

T

(25)

( ( where Tn = A2n) (A1n) )-1 : Eq. (25) is equivalent to that obtained earlier for multilayered axisymmetric particles in [6]. Although Eq. (24) looks less exiting, it should be preferable to Eq. (25) as the former ( ( needs only one matrix inversion (after multiplication of A-matrices in Tn+1 = A2n+1) (A1n+1) )-1 ) in contrast with the latter that requires inversions for each layer (to Únd each Tn starting with n =1) and one more at the last step.

5. ModiÚcation of the EBCM in spheroidal coordinates and the SVM The approach described in Sections 3 and 4 has been used in [7] to Únd the theoretical solution of the light scattering problem deÚned in Section 2 in the framework of the EBCM formulated in spheroidal coordinates (i.e. with the potentials expanded in terms of the spheroidal wave functions, etc.). Note that such a version of the EBCM for spheroids only weakly di ers from the SVM (see also [16]) as the separation of variables in the boundary conditions for spheroids is impossible and the light scattering problem in the SVM is reduced to solution of inÚnite algebraic systems similar to those arising in the EBCM with spheroidal coordinates (a di erence may appear in the boundary conditions as it will be seen below). Therefore, this variant of the EBCM can also be considered as a SVM-like method. For homogeneous and core-mantle spheroidal particles, our approach has been applied to the SVM in [24,30,5], respectively (mSVM solutions). It should be added that as far as we know Úrst the problem of electromagnetic scattering was consistently solved within the EBCM formulated in spheroidal coordinates in [7] (mEBCMsc solution). To use in the full manner the advantages of expansions in terms of the spheroidal wave function, we assume that all layer boundaries are confocal, i.e. for all j , r ( j) (á) are coordinate surfaces of the same spheroidal coordinate system ( ; à; '). It means that the major aj and minor bj semiaxes of the spheroidal surfaces conÚning the layers satisfy the condition a2 - b2 = a2 - b2 = ··· = a2 - b2 = 1 1 2 2 n n d 2
2

;

(26)

where d is the focal distance of the spheroids.


V.G. Farafonov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 79­80 (2003) 599 ­ 626 609

As in Sections 3 and 4 we divide the Úelds in the two parts and introduce the scalar potentials p; q; U; V . The scalar Helmholtz equation (9) and the equivalent surface integral equation (12) are written in the spheroidal coordinates as well as the boundary conditions (see also Eq. (39) below). The spheroidal wave functions are used in expansions of the potentials and the free-space Green function. For instance, for prolate spheroidal boundaries of layers we have (cf. Eq. (15))
( p1j) (~ ) r



q (~ ) r
( r p2j) (~ )

( j) 1

=
l=1

a b a b

( j) 1;l ( j) 1;l ( j) 2;l ( j) 2;l

R(1) (cj ; )S1l (cj ;à) cos '; 1l R(3) (cj ; ) S1l (cj ;à) cos '; 1l

(27)



q (~ ) r

( j) 2

=
l=1

(28)

where R(1); (3) (cj ; ) are the prolate radial spheroidal functions of the Úrst or third kinds, S1l (cj ;à) the 1l prolate angular spheroidal functions with the normalization factor N1l (cj ) [41], and cj = k (dj = 2). After substitution of the expansions in the equation analogous to Eq. (14), due to orthogonality of the spheroidal functions one gets an inÚnite system of linear algebraic equations like Eq. (16). The integrals in the matrix elements in the ana