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Analytical averaging of cross-sections for randomly oriented layered particles in the modified T -matrix method
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Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 111–122 www.elsevier.com/locate/jqsrt
Analytical averaging of cross-sections for randomly oriented layered particles in the modified T-matrix method Victor G. Farafonova, Marina S. Prokopjevab, Vladimir. B. Il’inb, a
St. Petersburg State University of Aerocosmic Instrumentation, Bol.Morskaya 67, 190000 St. Petersburg, Russia b Astronomical Institute, St. Petersburg State University, Bibliotechnaya Pl 2, Universitetskij pr. 28, 198504 St. Petersburg, Russia
Abstract Two modifications of the T-matrix method developed earlier for layered scatterers are extended by a procedure of analytical averaging of cross-sections for randomly oriented particles. One of the approaches uses spheroidal coordinates and field expansions in terms of the spheroidal wave functions and is practically equivalent to the separation of variables method formulated for spheroids. The applicability of the suggested approaches is discussed, and some numerical results are presented. r 2004 Elsevier Ltd. All rights reserved. Keywords: Light scattering; Inhomogeneous nonspherical particles
1. Introduction Many natural scatterers are nonspherical and inhomogeneous. In specific cases this inhomogeneity is well represented or approximated by a layered structure. For instance, the physical model of evolution of cosmic dust grains developed by Greenberg and his co-workers is associated with grains having several layers of different chemical compositions (see [1,2] for more details).
Corresponding author. Tel.: +7-812-184-2243; fax: +7-812-428-7129.
E-mail address: [email protected] (V.B. Il’in). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.004
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On the other hand, a layered structure is the simplest kind of particle inhomogeneity, with other possible structures being a random distribution of materials in a scatterer or their distribution in the form of macroscopic inclusions. The problem of light scattering by a nonspherical layered particle can be solved by different methods, but most of them such as coupled dipole methods, finite difference methods, etc. impose strong requirements on computer memory and speed, making computations demanded by applications impossible. For some simplified shapes of scatterers, the T-matrix method (TMM) and the separation of variables method (SVM) make use of the scattering geometry and are hence much faster than other approaches. Therefore, both methods are useful for extensive modeling of the optical properties of nonspherical inhomogeneous scatterers having size greater or comparable with the wavelength of the incident radiation, and several versions of the TMM and SVM have been developed for layered scatterers (see, e.g., [3,4] and references therein). In applications, one usually needs to average optical characteristics of an ensemble of scatterers over their orientations and sizes. A possibility of analytical averaging of these characteristics was first discussed in [5]. Such a procedure so far developed only for the standard version of the TMM [6,7] allows an efficient evaluation of the orientation-averaged characteristics of ensembles of randomly oriented axisymmetric scatterers [8]. Note that although in the work [9] the T matrix was obtained within the SVM for spheroids, it was then transformed into the T matrix arising out of the standard TMM and further analytical averaging of the matrix was performed using the usual formulae. In this paper we suggest a procedure of analytical averaging of scattered field characteristics of randomly oriented layered nonspherical particles within two approaches: the modified TMM formulated in spherical and spheroidal coordinates. The basic equations are given in Section 2. In Section 3 we discuss the applicability ranges of the approaches and present illustrative calculations performed for Greenberg’s layered model of cosmic dust grains.
2. Basic equations Analytical averaging of cross-sections over all possible orientations of a nonspherical inhomogeneous scatterer is incorporated in two approaches. The first one is a reformulation of the standard TMM (see, e.g., [8]) with the specific scalar potentials. The second combines this reformulation with the use of spheroidal coordinates and the spheroidal wave functions in the expansions of the potentials. So it is actually a version of the separation of variables method for spheroids (see discussion in [3]). Both approaches, hereafter called the modified TMM in spherical and spheroidal coordinates, have been extended for multilayered scatterers in [3]. Note that the approaches can, in principle, be applied to scatterers of a rather complex shape and structure, but they use their advantages in full measure when applied to axisymmetric scatterers and to scatterers with the confocal spheroidal boundaries of the layers, respectively. 2.1. Modified TMM in spherical coordinates Let us consider scattering of a plane wave by an ensemble of randomly oriented layered axisymmetric particles.
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The geometry of such a particle is described by the equations of its layer boundaries r ¼ rðjÞ ðyÞ, where ðr; y; fÞ are the spherical coordinates connected with the particle, n is the number of the layers, and j ¼ 1; 2; :::; n. Thus, the jth layer (j ¼ 1 corresponds to the outermost layer) is confined by the surfaces r ¼ rðjÞ ðyÞ and r ¼ rðjþ1Þ ðyÞ. The fields in the layer are denoted by Eðjþ1Þ ; Hðjþ1Þ and the wavenumbers by kjþ1 (k1 is the wavenumber in the medium surrounding the particle), Eð0Þ ; Hð0Þ and Eð1Þ ; Hð1Þ represent the incident field (a plane wave propagating at the angle a to the symmetry axis of the particle) and the scattered field, respectively. In order to find the field scattered by an ensemble of the particles we solve the Maxwell equations together with the material equations and the boundary conditions (continuity of the transversal components of the fields at the layers’ boundaries and the radiation condition at infinity) for a single particle in a fixed orientation (defined by the angle a) and then perform analytical averaging of selected scattered field characteristics over all orientations of the particle. The main features of our modification of the TMM are as follows: 1. Each field is divided in two parts—an axisymmetric (A) one that does not depend on the azimuthal angle j and a nonaxisymmetric (N) one that equals zero after averaging over j: ðjÞ EðjÞ ðrÞ ¼ EðjÞ A ðrÞ þ EN ðrÞ;
ðjÞ HðjÞ ðrÞ ¼ HðjÞ A ðrÞ þ HN ðrÞ;
ð1Þ
where j ¼ 0; 1; :::; n þ 1. 2. Proper scalar potentials are chosen for each part of the fields. For instance, for the ðjÞ ðjÞ axisymmetric parts in the case of the TE mode (E ðjÞ A;r ðrÞ ¼ E A;y ðrÞ ¼ 0), we use the potentials p ðjÞ E ðjÞ cos j A;j ðrÞ ¼ p
ð2Þ
and for the nonaxisymmetric parts we employ the potentials U ðjÞ and V ðjÞ ðjÞ ðjÞ EðjÞ N ðrÞ ¼ r U ðrÞ iz þ V ðrÞ r :
ð3Þ
The magnetic fields HðjÞ A are simply derived from the Maxwell equations. For the TM mode, the potentials are likewise. 3. For each layer, the potentials are represented as ðjÞ pðjÞ ðrÞ ¼ pðjÞ 1 ðrÞ þ p2 ðrÞ;
ð4Þ
where the potential with the subscript 1 has no singularity at the origin of the coordinates and that with the subscript 2 satisfies the radiation condition at infinity. For the incident and scattered ð1Þ ð1Þ fields and the internal field in the innermost layer, we have pð0Þ ðrÞ ¼ pð0Þ 1 ðrÞ, p ðrÞ ¼ p2 ðrÞ, ðnþ1Þ pðnþ1Þ ðrÞ ¼ p1 ðrÞ and so on. All these potentials satisfy the scalar Helmholtz equation. The equation is rewritten in the form of surface integral equations using the free-space Green function. The substitution of the boundary conditions into the equations gives the basic surface integral equations of the method which is often called the extended boundary condition method.
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The potentials are expanded in terms of the spherical wave functions pðjÞ 1 ðrÞ ¼
1 X
1 aðjÞ 1;l j l ðk j rÞ Pl ðcos yÞ cos j;
ð5Þ
l¼1
U ðjÞ 1 ðrÞ V ðjÞ 1 ðrÞ
¼
ðjÞ 1 X 1 a X 1;ml m¼1 l¼m
bðjÞ 1;ml
j l ðkj rÞ Pm l ðcos yÞ cos mj;
ð6Þ
where j l ðkj rÞ and Pm l ðcos yÞ are the spherical Bessel and associated Legendre functions. In the ðjÞ ðrÞ; U ðjÞ expansions of pðjÞ 2 2 ðrÞ; V 2 ðrÞ the Bessel functions are replaced by the Hankel functions of the first kind. We substitute these expansions into the surface integral equations and obtain systems of the linear algebraic equations with respect to the expansion coefficients. The systems can be ð1Þ 1 rearranged to express the unknown coefficients of the scattered field expansion (að1Þ 2 ¼ fa2;l gl¼1 , ð0Þ 1 etc.) via the known coefficients of the incident field (i.e. the plane wave) expansion (að0Þ 1 ¼ fa1;l gl¼1 , ð0Þ etc.). For the TE mode (bð0Þ 1 ¼ b1;m ¼ 0), we have A ð0Þ að1Þ 2 ¼ T a1 ;
N;a ð0Þ að1Þ 2;m ¼ T m a1;m ;
N;b ð0Þ bð1Þ 2;m ¼ T m a1;m ;
ð0Þ ð1Þ A ð0Þ and for the TM mode (að0Þ 1 ¼ b1;m ¼ 0) the first equation is replaced by a2 ¼ T b1 . All the transition matrices T can be written as T A ¼ A2 A 1 1 , where ! ! ! ð1Þ
Aðn 1Þ
Aðn 1Þ
AðnÞ
Að1Þ A1 hj Ahh hj hh hj ¼ ðn 1Þ A2 Að1Þ Að1Þ Aðn 1Þ Ajh AðnÞ jj jj jj jh
ð7Þ
ð8Þ
and the elements of the matrices AðnÞ jj , etc. are integrals of the spherical wave functions and their derivatives for the nth layer (see [3] for more details). The matrices T can be also represented in the recursive form 1 ð0Þ ð0Þ ð0Þ þ A TðnÞ A þ A TðnÞ ; Tðn þ 1Þ ¼ Að0Þ jj jh hj hh
ð9Þ
where TðnÞ is the matrix for a n-layered particle and Að0Þ jj , etc. are the matrices for the ðn þ 1Þth layer added to the particle as a mantle. Now we substitute Eq. (7) along with the known expansion coefficients for the plane wave into the standard formulae for the cross-sections (see, e.g., [3]). Analytical averaging of the crosssection expressions for the TE and TM modes over all particle orientations gives ( " #) 1 1 X 1 1 X X X 4p ðmÞ ; TA T N;a in l 1 T N;b hC ext i ¼ 2 Re ll þ 2 m;ll þ 2 m;ln znl k1 m¼1 l¼m l¼1 l;n¼m
ð10Þ
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( 1 1 1 X X 4p X 2n þ 1 lðl þ 1Þ A 2 hC sca i ¼ 2 ii jþn l T ln þ 2Re k1 l;n¼1 2l þ 1 nðn þ 1Þ m¼1 i;j;l;n¼m h n n ðmÞ N;a N;b ðmÞ N;a T N;a m;li oln T m;nj þ iT m;li kln T m;nj ) i n n ðmÞ ðmÞ ðmÞ N;b N;b N;b ;
iT N;a m;li knl T m;nj þ T m;li tln T m;nj Oij
115
(11)
where the asterisk stands for the complex conjugation and zðmÞ ln
ð2n þ 1Þðn mÞ! ¼ 2ðn þ mÞ!
OðmÞ ln ¼
Z
1
0
m Pm l ðxÞ Pn ðxÞ dx;
ð12Þ
1
ð2l þ 1Þðl mÞ! ð2n þ 1Þðn mÞ! 2ðl þ mÞ! 2ðn þ mÞ!
Z
1
m Pm n ðxÞ Pl ðxÞ dx: 1 x2
1
ð13Þ
ðmÞ ðmÞ The integrals oðmÞ ln ; kln ; tln which are similar have appeared in the expressions of the cross-section for a particle in a fixed orientation [3]. Note that the scattering problem for the axisymmetric and nonaxisymmetric parts of the fields can be solved independently and hence for nonabsorbing particles the energy conservation law and the optical theorem make the corresponding parts of the averaged extinction and scattering A N N A N cross-sections equal, i.e. hC A ext i ¼ hC sca i, hC ext i ¼ hC sca i, where hC i and hC i denote the terms in (10), (11) that include the matrices T with the superscripts A and N, respectively. This fact is useful for testing computer codes and checking numerical results. At last, one must distinguish hC TE;TM i actually given by Eqs. (10) and (11) and hCiTE;TM being the cross-sections for the modes of the polarized incident radiation. For the systems of axisymmetric particles considered, the physical meaning of the parameters leads to hCiTE = incident radiation hCiTM , while hC TE i = hCiTE þ DCahC TM i = hCiTM DC. For non-polarized C ¼ 12ðC TE þ C TM Þ and hence hCi ¼ h12ðC TE þ C TM Þi ¼ 12 hC TE i þ hC TM i .
2.2. Modified TMM in spheroidal coordinates This approach can be efficiently applied to randomly oriented particles with the confocal spheroidal boundaries of the layers. The main features of the above-described TMM modification are maintained here with the only change—the spherical coordinates ðr; y; jÞ are replaced by the spheroidal ones ðx; Z; jÞ. Hence the layer boundary equations become x ¼ xj , where xj are some constants and j ¼ 1; 2; :::; n. Accordingly, the potentials are expanded in terms of the spheroidal wave functions. For example, in the prolate spheroidal coordinate system Eqs. (5) and (6) are changed for pðjÞ 1 ðrÞ ¼
1 X l¼1
ð1Þ aðjÞ 1;l R1l ðcj ; xÞ S 1l ðcj ; ZÞ cos j;
ð14Þ
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U ð1 jÞ ðrÞ V ð1 jÞ ðrÞ
¼
ð jÞ 1 X 1 a X 1;ml m¼1 l¼m
jÞ bð1;ml
Rð1Þ ml ðcj ; xÞ S ml ðcj ; ZÞ cos mj;
ð15Þ
where Rð1Þ ml ðcj ; xÞ are the radial spheroidal functions of the first kind, S ml ðcj ; ZÞ the angular spheroidal functions with the normalization factors N ml ðcj Þ [10], and cj ¼ kj ðd=2Þ, where d is the focal distance of the spheroidal surfaces. Their confocality just means that all the layer boundaries have the same foci. For the potentials with the subscript 2, the functions Rð1Þ ml ðcj ; xÞ must be ð3Þ replaced by the radial spheroidal functions of the third kind Rml ðcj ; xÞ. For the oblate spheroidal coordinates, one must change the arguments in the radial functions to ð icj ; ixÞ and those in the angular ones to ð icj ; ZÞ. The transition matrices can be introduced in this approach as well. They differ from the corresponding T matrices used in Section 2.1 as the spheroidal wave functions instead of the spherical ones are use in the expansions. The relation between the ‘spheroidal’ and ‘spherical’ T matrices is discussed in [9]. The ‘spheroidal’ T matrices can be calculated both in iterative and recursive ways given by Eqs. (8) and (9), respectively. The differences between the ‘spherical’ and ‘spheroidal’ cases appear only in the elements of the matrices AðjÞ jj , etc. The corresponding surface integrals are considered here for the surfaces x ¼ xj , and hence all the radial spheroidal functions and their derivatives disappear from the integrals. For example, for the TE mode we have (to simplify hereafter we assume the magnetic permeability mj ¼ 1 for any j) n o ð jÞ AðjjjÞ ¼ W ð jÞ Rð jÞ S ð jÞ S ð jÞ R~ ð16Þ Pð jÞ ; ðjÞ where W ðjÞ ; RðjÞ ; R~ ; PðjÞ are the following diagonal matrices: n o1 ð3Þ W ðjÞ ¼ icj ðx2j f Þ Rð1Þ ðc ; x Þ R ðc ; x Þ d ; j j j j nl 1l 1l n;l¼1 n o1 0 ð1Þ RðjÞ ¼ Rð1Þ ; 1l ðcj ; xj Þ=R1l ðcj ; xj Þ dnl n;l¼1 n o 1 0 ðjÞ ð1Þ ; R~ ¼ Rð1Þ 1l ðcjþ1 ; xj Þ=R1l ðcjþ1 ; xj Þ dnl n;l¼1 n o1 ð1Þ ; PðjÞ ¼ Rð1Þ 1l ðcjþ1 ; xj Þ=R1l ðcjþ1 ; xjþ1 Þ dnl
ð17Þ
n;l¼1
where for the prolate and oblate spheroidal systems f¼ 1 and 1, 1 respectively, and dln is the Kronecker symbol. The elements of the matrices SðjÞ ¼ snl ðcj ; cjþ1 Þ n;l¼1 are integrals of products of the angular spheroidal functions and can be written as series including the coefficients of the expansions of these functions in terms of the associated Legendre functions Z 1
1
1 S 1n ðcj ; ZÞS 1l ðcjþ1 ; ZÞ dZ snl ðcj ; cjþ1 Þ ¼ N 1n ðcj ÞN 1l ðcjþ1 Þ
1
1 ¼ N 1 1n ðcj ÞN 1l ðcjþ1 Þ
1 0 X
k¼0;1
1l d 1n k ðcj Þd k ðcjþ1 Þ
2 ðk þ 2Þ! ; 2k þ 3 k!
(18)
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where the apostrophe means that k has the same evenness as ðn lÞ. We take into account that Smn ðc; ZÞ ¼
1 0 X
m d mn k ðcÞPkþm ðZÞ:
ð19Þ
k¼0;1
The matrices T for the nonaxisymmetric parts of the TE mode and for both parts of the TM mode are constructed analogously. The expressions for the averaged cross-sections (10) and (11) hold here as well if one replaces ðmÞ the associated Legendre functions by the angular spheroidal functions in all integrals zðmÞ ln , oln , ðmÞ ðmÞ ðmÞ kln , tln , and Oln as follows: Z 1
1
1 zðmÞ ! N ðc ÞN ðc Þ S ml ðc1 ; ZÞ S0mn ðc1 ; ZÞ dZ ml 1 mn 1 ln
1
1 ¼ N 1 ml ðc1 ÞN mn ðc1 Þ
1 0X 1 0 X
ðmÞ mn d ml i ðci Þ d j ðcj Þzij ;
(20)
i¼0;1 j¼0;1
where zðmÞ ij is the integral (12) and the expansion (19) is used.
3. Discussion and numerical results 3.1. Comparison with earlier works As far as we know analytical averaging of the scattered field characteristics for randomly oriented particles was performed so far only within the standard version of the TMM (see [8] and references therein). The orientation-averaged extinction and scattering cross-sections were found to be equal to sums of the traces and the norms of the T-matrices, respectively. It is interesting to compare these well-known formulae with those obtained in Section 2. In the axisymmetric parts represented by the first terms in Eqs. (10) and (11) we also get the trace and practically the norm of the matrix T A . The analogy can be explained by the fact that in this case we actually solve a scalar scattering problem using an orthogonal basis. In the nonaxisymmetric parts (see the terms including the matrices T N;a ; T N;b ) our expressions are more complicated than the trace and the norm. It is a result of our solution of the vector scattering problem using a nonorthogonal basis (see Eq. (3)). If the basis had been orthogonal, we would have had the diagonal matrices zðmÞ ¼ ði=2ÞE, kðmÞ ¼ 0, oðmÞ ¼ tðmÞ ¼ vE, and O ¼ v 1 E, where v is a constant and E the unit matrix. It is worth noting that for nonpolarized incident radiation, the expression (10) simplifies due to ðmÞ symmetry properties of the matrices T N;b m and z " # 1 1 X 1 X X 4p A N;a ð21Þ hC ext i ¼ 2 Re T~ ll þ 2 T~ m;ll ; k1 m¼1 l¼m l¼1 where the matrices T~ ¼ 12ðT TE þ T TM Þ. Now only the traces of the T-matrices are present, and moreover because of the symmetry information on the scatterer involved in the matrices T N;b m
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becomes unimportant. For the scattering cross-section hC sca i, a similar simplification does not i given by Eq. (11). occur and one should use hC TE;TM sca 3.2. Applicability of the approaches The applicability of both TMM modifications used in Section 2 has been analytically and numerically investigated for single perfectly conducting particles in [11]. The conclusions made in that paper are also valid for dielectric homogeneous particles and for any version of the TMM involving surface integral equations and monopole field/potential expansions in terms of wave functions. The results of [11] can be formulated as follows. The use of the TMM in the near-field zone (up to the particle surface) and the far-field zone is mathematically correct under different conditions. In the first case the Rayleigh hypothesis about the convergence of the field (potential) expansions everywhere up to the scatterer border must be satisfied. In the second case the condition is essentially weaker (see the condition 3 and its discussion in [3]). Forpexample, the Rayleigh hypothesis is valid for spheroids if and only if the aspect ratio ffiffiffi a=bo 2, while the second condition is satisfied always. In other words, one can use the TMM to calculate such characteristics of the scattered radiation in the far-field zone as cross-sections, scattering matrix elements, etc. for a homogeneous spheroid with any aspect ratio (however, such calculations can require very large computational efforts). For scatterers of other shapes, e.g. the Chebyshev particles, the situation is less simple, and in the far-field zone the method is mathematically correct only in a certain region of the particle shape parameters. These results are generalized for layered scatterers in [3]. It is found that while the applicability in the whole near-field zone still needs the Rayleigh hypothesis to be valid, the condition for the farfield zone becomes more complex and involves information on all layer boundaries. As a result one can utilize the TMM to calculate, e.g., cross-sections off a particle with spheroidal boundaries of the layers only if these surfaces are confocal. Otherwise there appears a certain region of the semiaxes of the boundaries where the TMM fails to be mathematically correct and hence one will encounter divergence of the solution. Note that the conditions formulated in [3] allows one to determine the applicability range for scatterers not only with spheroidal, but also with more complex layer boundaries. Obviously, the theoretical applicability range of the analytical averaging procedure is generally equal to that derived for single layered scatterers. However, important differences appear when we consider numerical problems which make the real range narrower. First, we find that the modified TMM in spherical coordinates (and probably any similar approach) generally gives the values of the orientation-averaged cross-sections with the accuracy of 4–5 orders higher than the accuracy of the cross-sections for the particle in a fixed orientation. As a result, the numerical instability limiting the applicability of the TMM (in particular for spheroids of large aspect ratios, see, e.g., [8]) arises in the averaging procedure at much larger ratios (at least in the modification of the TMM under consideration). Second, in order to derive the orientation-averaged cross-sections within the modified TMM in spheroidal coordinates, one does not need to compute the angular spheroidal functions but only (see Eqs. (16)–(19)). This circumstance extends the coefficients of their expansions d mn k significantly the practical applicability range of this TMM modification (and the equivalent separation of variables method for spheroids) because this range is mainly limited by the problems of numerical calculations of the spheroidal functions.
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It is also worth mentioning the complementary nature of the approaches described in Section 2. The modified TMM in spherical coordinates can be applied to randomly oriented particles with quite different shapes of the layer boundaries but has relatively narrow range of the applicability especially for particles of large eccentricity. The modified TMM in spheroidal coordinates can efficiently treat only layered particles with the confocal spheroidal boundaries but their eccentricity can be very large.
(a)
(b) Fig. 1. Extinction cross-sections of randomly oriented prolate particles in the 3 mm (a) and 10=20 mm (b) band regions. The results for three-layered spheroids with the different middle layer material (‘sil’ means silicate, ‘org’ organics, and ‘ice’ water ice) are shown by short-dashed lines and those for homogeneous spheroids with the refractive indices correspondingly derived using the EMT by longdashed lines. The radius of the sphere having the volume equal to that of the spheroids is 1:44 mm, the aspect ratios of all layer boundaries a=b ¼1.4.
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3.3. Some numerical results For the sake of illustration, we have calculated the orientation-averaged extinction crosssections for layered analogs of cosmic dust grains used in the model of Greenberg [1,2]. This physical model assumes that in diffuse interstellar clouds silicate grains are coated with a layer of carbonaceous organic refractory, while in dense molecular clouds these grains are further coated with an outer ice mantle. There are also evidences for a nonspherical shape of the grains.
(a)
(b) Fig. 2. Extinction cross-sections of randomly oriented prolate spheroids with different aspect ratios a=b in the 3 mm (a) and 10=20 mm (b) band regions. The refractive index was derived using the EMT for the equal volume fractions of silicate, organics and ice. The radius of the sphere having the volume equal to that of the spheroids is 1:44 mm.
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In Fig. 1 we present the extinction cross-sections calculated for randomly oriented three-layered prolate spheroidal particles with a silicate core, an organics layer and an ice mantle. The refractive indices for the astronomical silicate, refractory organics and ice were taken from [12]. All layer boundaries have the aspect ratio a=b ¼ 1:4, i.e. are not confocal and hence the method from Section 2.1 was employed. The size of the equivolume spherical particles was selected to be 1:44 mm (when the volume fractions of all layers are equal, the radius of the core is 1 mm, that of the layer 1:26 mmÞ. For particles of the smaller size ðt0:5 mmÞ, small scatterer size approximations (e.g., the special version of the Rayleigh approximation for layered ellipsoids with the nonconfocal boundaries of layers from [13]) are valid [14] and hC ext i are nearly proportional to the particle size, i.e. the band shape is independent of it. For particles as large as 5–10 mm, the 10 and 20 mm silicate bands disappear, as opposed to spectra observed for the interstellar medium. These spectra are often explained using the Effective Medium Theory (EMT) which defines the mean refractive index for an inhomogeneous scatterer and approximates the optical properties of the scatterer by those of a homogeneous particle of the same shape. Therefore in Fig. 1 we also show the results obtained for randomly oriented homogeneous spheroids with the refractive index derived using the standard (Bruggeman) rule of the EMT. Fig. 2 demonstrates structure effects on the 3 mm water ice and 10 and 20 mm silicate bands. To show the role of the middle layer, we give curves for particles in which organics is replaced by silicate or ice. The figure also illustrates the shape effects by considering randomly oriented spheroids of different aspect ratios (those with the large aspect ratios were treated by the method from Section 2.2). One can see that the shape and structure effects are different and both effects can be important in astrophysical applications.
4. Conclusions We have demonstrated that analytical averaging of the extinction and scattering cross-sections for randomly oriented layered scatterers is also possible within the nonstandard versions of the TMM, including the SVM for spheroids. Although the SVM is only efficient for scatterers with the confocal spheroidal boundaries of layers, it is applicable to particles of high eccentricity. The suggested analytical averaging procedure for the SVM-like modification of the TMM from Section 2.2. has an even wider applicability range than the SVM for single particles, since only the radial spheroidal functions need to be calculated in this case. Thus, this procedure is likely to provide a unique way to quickly get exact results for ensembles of randomly oriented spheroids of high eccentricity and, together with the similar procedure for the modified TMM from Section 2.1, they present a set of tools useful for applications of models of layered nonspherical particles.
Acknowledgements The authors thank A. Li for providing the optical constants of astronomical silicate, organics, and ice used in the calculations. The work was partly supported by the Russian Ministry of
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Education (grant E02-11.0-8). M.P. and V.I. acknowledge the support within the Russian federal programs for Astronomy and for prominent scientific schools (grant 1088.2003.2).
References [1] Greenberg JM. Interstellar dust. In: McDonnell JAM, editor. Cosmic dust. New York: Wiley; 1978. p. 187–256. [2] Greenberg JM, Li A. Tracking the organic refractory component from interstellar dust to comets. Adv Space Rev 1999;24:497–504. [3] Farafonov VG, Il’in VB, Prokopjeva MS. Light scattering by multilayered nonspherical particles: a set of methods. JQSRT 2003;79–80:599–626. [4] Gurwich I, Kleiman M, Shiloah N, Oaknin D. Scattering by an arbitrary multi-layered spheroid: theory and numerical results. JQSRT 2003;79–80:649–60. [5] Varadan VV, Varadan VK. Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers. Phys Rev D 1981;21:388–94. [6] Mishchenko MI. Extinction of light by randomly oriented non-spherical grains. Astrophys Space Sci 1990;164:1–13. [7] Khlebtsov NG. Orientational averaging of light scattering observables in the T-matrix approach. Appl Opt 1992;31:5359–65. [8] Mishchenko MI, Hovenier JW, Travis LD. Light scattering by nonspherical particles. San Diego: Academic Press; 2000. [9] Schulz FM, Stamnes K, Stamnes JJ. Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T-matrix computed in spheroidal coordinates. Appl Opt 1998;37:7875–96. [10] Komarov VI, Ponomarev LI, Slavyanov SY. Spheroidal and Coulomb spheroidal functions. Moscow: Nauka; 1976. [11] Il’in VB, Loskutov AA, Farafonov VG. Modification and investigation of T-matrix method for plane wave scattering by a perfectly conducting axisymmetric body. Electromagnetic scattering for multilayered spheres. J Comp Math Math Phys 2004;44:349–69. [12] Li A, Greeberg JM, Zhao G. Modelling the astronomical silicate feature: I. On the spectrum subtraction method. Mon Not R Astron Soc 2002;334:840–6. [13] Posselt B, Farafonov VG, Il’in VB, Prokopjeva MS. Light scattering by multilayered ellipsoidal particles in the quasistatic approximation. Meas Sci Technol 2002;13:256–62. [14] Somsikov W, Voshchinnikov NV. On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared. Astron Astrophys 1999;345:315–20.
Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 111–122 www.elsevier.com/locate/jqsrt
Analytical averaging of cross-sections for randomly oriented layered particles in the modified T-matrix method Victor G. Farafonova, Marina S. Prokopjevab, Vladimir. B. Il’inb, a
St. Petersburg State University of Aerocosmic Instrumentation, Bol.Morskaya 67, 190000 St. Petersburg, Russia b Astronomical Institute, St. Petersburg State University, Bibliotechnaya Pl 2, Universitetskij pr. 28, 198504 St. Petersburg, Russia
Abstract Two modifications of the T-matrix method developed earlier for layered scatterers are extended by a procedure of analytical averaging of cross-sections for randomly oriented particles. One of the approaches uses spheroidal coordinates and field expansions in terms of the spheroidal wave functions and is practically equivalent to the separation of variables method formulated for spheroids. The applicability of the suggested approaches is discussed, and some numerical results are presented. r 2004 Elsevier Ltd. All rights reserved. Keywords: Light scattering; Inhomogeneous nonspherical particles
1. Introduction Many natural scatterers are nonspherical and inhomogeneous. In specific cases this inhomogeneity is well represented or approximated by a layered structure. For instance, the physical model of evolution of cosmic dust grains developed by Greenberg and his co-workers is associated with grains having several layers of different chemical compositions (see [1,2] for more details).
Corresponding author. Tel.: +7-812-184-2243; fax: +7-812-428-7129.
E-mail address: [email protected] (V.B. Il’in). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.004
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On the other hand, a layered structure is the simplest kind of particle inhomogeneity, with other possible structures being a random distribution of materials in a scatterer or their distribution in the form of macroscopic inclusions. The problem of light scattering by a nonspherical layered particle can be solved by different methods, but most of them such as coupled dipole methods, finite difference methods, etc. impose strong requirements on computer memory and speed, making computations demanded by applications impossible. For some simplified shapes of scatterers, the T-matrix method (TMM) and the separation of variables method (SVM) make use of the scattering geometry and are hence much faster than other approaches. Therefore, both methods are useful for extensive modeling of the optical properties of nonspherical inhomogeneous scatterers having size greater or comparable with the wavelength of the incident radiation, and several versions of the TMM and SVM have been developed for layered scatterers (see, e.g., [3,4] and references therein). In applications, one usually needs to average optical characteristics of an ensemble of scatterers over their orientations and sizes. A possibility of analytical averaging of these characteristics was first discussed in [5]. Such a procedure so far developed only for the standard version of the TMM [6,7] allows an efficient evaluation of the orientation-averaged characteristics of ensembles of randomly oriented axisymmetric scatterers [8]. Note that although in the work [9] the T matrix was obtained within the SVM for spheroids, it was then transformed into the T matrix arising out of the standard TMM and further analytical averaging of the matrix was performed using the usual formulae. In this paper we suggest a procedure of analytical averaging of scattered field characteristics of randomly oriented layered nonspherical particles within two approaches: the modified TMM formulated in spherical and spheroidal coordinates. The basic equations are given in Section 2. In Section 3 we discuss the applicability ranges of the approaches and present illustrative calculations performed for Greenberg’s layered model of cosmic dust grains.
2. Basic equations Analytical averaging of cross-sections over all possible orientations of a nonspherical inhomogeneous scatterer is incorporated in two approaches. The first one is a reformulation of the standard TMM (see, e.g., [8]) with the specific scalar potentials. The second combines this reformulation with the use of spheroidal coordinates and the spheroidal wave functions in the expansions of the potentials. So it is actually a version of the separation of variables method for spheroids (see discussion in [3]). Both approaches, hereafter called the modified TMM in spherical and spheroidal coordinates, have been extended for multilayered scatterers in [3]. Note that the approaches can, in principle, be applied to scatterers of a rather complex shape and structure, but they use their advantages in full measure when applied to axisymmetric scatterers and to scatterers with the confocal spheroidal boundaries of the layers, respectively. 2.1. Modified TMM in spherical coordinates Let us consider scattering of a plane wave by an ensemble of randomly oriented layered axisymmetric particles.
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The geometry of such a particle is described by the equations of its layer boundaries r ¼ rðjÞ ðyÞ, where ðr; y; fÞ are the spherical coordinates connected with the particle, n is the number of the layers, and j ¼ 1; 2; :::; n. Thus, the jth layer (j ¼ 1 corresponds to the outermost layer) is confined by the surfaces r ¼ rðjÞ ðyÞ and r ¼ rðjþ1Þ ðyÞ. The fields in the layer are denoted by Eðjþ1Þ ; Hðjþ1Þ and the wavenumbers by kjþ1 (k1 is the wavenumber in the medium surrounding the particle), Eð0Þ ; Hð0Þ and Eð1Þ ; Hð1Þ represent the incident field (a plane wave propagating at the angle a to the symmetry axis of the particle) and the scattered field, respectively. In order to find the field scattered by an ensemble of the particles we solve the Maxwell equations together with the material equations and the boundary conditions (continuity of the transversal components of the fields at the layers’ boundaries and the radiation condition at infinity) for a single particle in a fixed orientation (defined by the angle a) and then perform analytical averaging of selected scattered field characteristics over all orientations of the particle. The main features of our modification of the TMM are as follows: 1. Each field is divided in two parts—an axisymmetric (A) one that does not depend on the azimuthal angle j and a nonaxisymmetric (N) one that equals zero after averaging over j: ðjÞ EðjÞ ðrÞ ¼ EðjÞ A ðrÞ þ EN ðrÞ;
ðjÞ HðjÞ ðrÞ ¼ HðjÞ A ðrÞ þ HN ðrÞ;
ð1Þ
where j ¼ 0; 1; :::; n þ 1. 2. Proper scalar potentials are chosen for each part of the fields. For instance, for the ðjÞ ðjÞ axisymmetric parts in the case of the TE mode (E ðjÞ A;r ðrÞ ¼ E A;y ðrÞ ¼ 0), we use the potentials p ðjÞ E ðjÞ cos j A;j ðrÞ ¼ p
ð2Þ
and for the nonaxisymmetric parts we employ the potentials U ðjÞ and V ðjÞ ðjÞ ðjÞ EðjÞ N ðrÞ ¼ r U ðrÞ iz þ V ðrÞ r :
ð3Þ
The magnetic fields HðjÞ A are simply derived from the Maxwell equations. For the TM mode, the potentials are likewise. 3. For each layer, the potentials are represented as ðjÞ pðjÞ ðrÞ ¼ pðjÞ 1 ðrÞ þ p2 ðrÞ;
ð4Þ
where the potential with the subscript 1 has no singularity at the origin of the coordinates and that with the subscript 2 satisfies the radiation condition at infinity. For the incident and scattered ð1Þ ð1Þ fields and the internal field in the innermost layer, we have pð0Þ ðrÞ ¼ pð0Þ 1 ðrÞ, p ðrÞ ¼ p2 ðrÞ, ðnþ1Þ pðnþ1Þ ðrÞ ¼ p1 ðrÞ and so on. All these potentials satisfy the scalar Helmholtz equation. The equation is rewritten in the form of surface integral equations using the free-space Green function. The substitution of the boundary conditions into the equations gives the basic surface integral equations of the method which is often called the extended boundary condition method.
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The potentials are expanded in terms of the spherical wave functions pðjÞ 1 ðrÞ ¼
1 X
1 aðjÞ 1;l j l ðk j rÞ Pl ðcos yÞ cos j;
ð5Þ
l¼1
U ðjÞ 1 ðrÞ V ðjÞ 1 ðrÞ
¼
ðjÞ 1 X 1 a X 1;ml m¼1 l¼m
bðjÞ 1;ml
j l ðkj rÞ Pm l ðcos yÞ cos mj;
ð6Þ
where j l ðkj rÞ and Pm l ðcos yÞ are the spherical Bessel and associated Legendre functions. In the ðjÞ ðrÞ; U ðjÞ expansions of pðjÞ 2 2 ðrÞ; V 2 ðrÞ the Bessel functions are replaced by the Hankel functions of the first kind. We substitute these expansions into the surface integral equations and obtain systems of the linear algebraic equations with respect to the expansion coefficients. The systems can be ð1Þ 1 rearranged to express the unknown coefficients of the scattered field expansion (að1Þ 2 ¼ fa2;l gl¼1 , ð0Þ 1 etc.) via the known coefficients of the incident field (i.e. the plane wave) expansion (að0Þ 1 ¼ fa1;l gl¼1 , ð0Þ etc.). For the TE mode (bð0Þ 1 ¼ b1;m ¼ 0), we have A ð0Þ að1Þ 2 ¼ T a1 ;
N;a ð0Þ að1Þ 2;m ¼ T m a1;m ;
N;b ð0Þ bð1Þ 2;m ¼ T m a1;m ;
ð0Þ ð1Þ A ð0Þ and for the TM mode (að0Þ 1 ¼ b1;m ¼ 0) the first equation is replaced by a2 ¼ T b1 . All the transition matrices T can be written as T A ¼ A2 A 1 1 , where ! ! ! ð1Þ
Aðn 1Þ
Aðn 1Þ
AðnÞ
Að1Þ A1 hj Ahh hj hh hj ¼ ðn 1Þ A2 Að1Þ Að1Þ Aðn 1Þ Ajh AðnÞ jj jj jj jh
ð7Þ
ð8Þ
and the elements of the matrices AðnÞ jj , etc. are integrals of the spherical wave functions and their derivatives for the nth layer (see [3] for more details). The matrices T can be also represented in the recursive form 1 ð0Þ ð0Þ ð0Þ þ A TðnÞ A þ A TðnÞ ; Tðn þ 1Þ ¼ Að0Þ jj jh hj hh
ð9Þ
where TðnÞ is the matrix for a n-layered particle and Að0Þ jj , etc. are the matrices for the ðn þ 1Þth layer added to the particle as a mantle. Now we substitute Eq. (7) along with the known expansion coefficients for the plane wave into the standard formulae for the cross-sections (see, e.g., [3]). Analytical averaging of the crosssection expressions for the TE and TM modes over all particle orientations gives ( " #) 1 1 X 1 1 X X X 4p ðmÞ ; TA T N;a in l 1 T N;b hC ext i ¼ 2 Re ll þ 2 m;ll þ 2 m;ln znl k1 m¼1 l¼m l¼1 l;n¼m
ð10Þ
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( 1 1 1 X X 4p X 2n þ 1 lðl þ 1Þ A 2 hC sca i ¼ 2 ii jþn l T ln þ 2Re k1 l;n¼1 2l þ 1 nðn þ 1Þ m¼1 i;j;l;n¼m h n n ðmÞ N;a N;b ðmÞ N;a T N;a m;li oln T m;nj þ iT m;li kln T m;nj ) i n n ðmÞ ðmÞ ðmÞ N;b N;b N;b ;
iT N;a m;li knl T m;nj þ T m;li tln T m;nj Oij
115
(11)
where the asterisk stands for the complex conjugation and zðmÞ ln
ð2n þ 1Þðn mÞ! ¼ 2ðn þ mÞ!
OðmÞ ln ¼
Z
1
0
m Pm l ðxÞ Pn ðxÞ dx;
ð12Þ
1
ð2l þ 1Þðl mÞ! ð2n þ 1Þðn mÞ! 2ðl þ mÞ! 2ðn þ mÞ!
Z
1
m Pm n ðxÞ Pl ðxÞ dx: 1 x2
1
ð13Þ
ðmÞ ðmÞ The integrals oðmÞ ln ; kln ; tln which are similar have appeared in the expressions of the cross-section for a particle in a fixed orientation [3]. Note that the scattering problem for the axisymmetric and nonaxisymmetric parts of the fields can be solved independently and hence for nonabsorbing particles the energy conservation law and the optical theorem make the corresponding parts of the averaged extinction and scattering A N N A N cross-sections equal, i.e. hC A ext i ¼ hC sca i, hC ext i ¼ hC sca i, where hC i and hC i denote the terms in (10), (11) that include the matrices T with the superscripts A and N, respectively. This fact is useful for testing computer codes and checking numerical results. At last, one must distinguish hC TE;TM i actually given by Eqs. (10) and (11) and hCiTE;TM being the cross-sections for the modes of the polarized incident radiation. For the systems of axisymmetric particles considered, the physical meaning of the parameters leads to hCiTE = incident radiation hCiTM , while hC TE i = hCiTE þ DCahC TM i = hCiTM DC. For non-polarized C ¼ 12ðC TE þ C TM Þ and hence hCi ¼ h12ðC TE þ C TM Þi ¼ 12 hC TE i þ hC TM i .
2.2. Modified TMM in spheroidal coordinates This approach can be efficiently applied to randomly oriented particles with the confocal spheroidal boundaries of the layers. The main features of the above-described TMM modification are maintained here with the only change—the spherical coordinates ðr; y; jÞ are replaced by the spheroidal ones ðx; Z; jÞ. Hence the layer boundary equations become x ¼ xj , where xj are some constants and j ¼ 1; 2; :::; n. Accordingly, the potentials are expanded in terms of the spheroidal wave functions. For example, in the prolate spheroidal coordinate system Eqs. (5) and (6) are changed for pðjÞ 1 ðrÞ ¼
1 X l¼1
ð1Þ aðjÞ 1;l R1l ðcj ; xÞ S 1l ðcj ; ZÞ cos j;
ð14Þ
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U ð1 jÞ ðrÞ V ð1 jÞ ðrÞ
¼
ð jÞ 1 X 1 a X 1;ml m¼1 l¼m
jÞ bð1;ml
Rð1Þ ml ðcj ; xÞ S ml ðcj ; ZÞ cos mj;
ð15Þ
where Rð1Þ ml ðcj ; xÞ are the radial spheroidal functions of the first kind, S ml ðcj ; ZÞ the angular spheroidal functions with the normalization factors N ml ðcj Þ [10], and cj ¼ kj ðd=2Þ, where d is the focal distance of the spheroidal surfaces. Their confocality just means that all the layer boundaries have the same foci. For the potentials with the subscript 2, the functions Rð1Þ ml ðcj ; xÞ must be ð3Þ replaced by the radial spheroidal functions of the third kind Rml ðcj ; xÞ. For the oblate spheroidal coordinates, one must change the arguments in the radial functions to ð icj ; ixÞ and those in the angular ones to ð icj ; ZÞ. The transition matrices can be introduced in this approach as well. They differ from the corresponding T matrices used in Section 2.1 as the spheroidal wave functions instead of the spherical ones are use in the expansions. The relation between the ‘spheroidal’ and ‘spherical’ T matrices is discussed in [9]. The ‘spheroidal’ T matrices can be calculated both in iterative and recursive ways given by Eqs. (8) and (9), respectively. The differences between the ‘spherical’ and ‘spheroidal’ cases appear only in the elements of the matrices AðjÞ jj , etc. The corresponding surface integrals are considered here for the surfaces x ¼ xj , and hence all the radial spheroidal functions and their derivatives disappear from the integrals. For example, for the TE mode we have (to simplify hereafter we assume the magnetic permeability mj ¼ 1 for any j) n o ð jÞ AðjjjÞ ¼ W ð jÞ Rð jÞ S ð jÞ S ð jÞ R~ ð16Þ Pð jÞ ; ðjÞ where W ðjÞ ; RðjÞ ; R~ ; PðjÞ are the following diagonal matrices: n o1 ð3Þ W ðjÞ ¼ icj ðx2j f Þ Rð1Þ ðc ; x Þ R ðc ; x Þ d ; j j j j nl 1l 1l n;l¼1 n o1 0 ð1Þ RðjÞ ¼ Rð1Þ ; 1l ðcj ; xj Þ=R1l ðcj ; xj Þ dnl n;l¼1 n o 1 0 ðjÞ ð1Þ ; R~ ¼ Rð1Þ 1l ðcjþ1 ; xj Þ=R1l ðcjþ1 ; xj Þ dnl n;l¼1 n o1 ð1Þ ; PðjÞ ¼ Rð1Þ 1l ðcjþ1 ; xj Þ=R1l ðcjþ1 ; xjþ1 Þ dnl
ð17Þ
n;l¼1
where for the prolate and oblate spheroidal systems f¼ 1 and 1, 1 respectively, and dln is the Kronecker symbol. The elements of the matrices SðjÞ ¼ snl ðcj ; cjþ1 Þ n;l¼1 are integrals of products of the angular spheroidal functions and can be written as series including the coefficients of the expansions of these functions in terms of the associated Legendre functions Z 1
1
1 S 1n ðcj ; ZÞS 1l ðcjþ1 ; ZÞ dZ snl ðcj ; cjþ1 Þ ¼ N 1n ðcj ÞN 1l ðcjþ1 Þ
1
1 ¼ N 1 1n ðcj ÞN 1l ðcjþ1 Þ
1 0 X
k¼0;1
1l d 1n k ðcj Þd k ðcjþ1 Þ
2 ðk þ 2Þ! ; 2k þ 3 k!
(18)
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where the apostrophe means that k has the same evenness as ðn lÞ. We take into account that Smn ðc; ZÞ ¼
1 0 X
m d mn k ðcÞPkþm ðZÞ:
ð19Þ
k¼0;1
The matrices T for the nonaxisymmetric parts of the TE mode and for both parts of the TM mode are constructed analogously. The expressions for the averaged cross-sections (10) and (11) hold here as well if one replaces ðmÞ the associated Legendre functions by the angular spheroidal functions in all integrals zðmÞ ln , oln , ðmÞ ðmÞ ðmÞ kln , tln , and Oln as follows: Z 1
1
1 zðmÞ ! N ðc ÞN ðc Þ S ml ðc1 ; ZÞ S0mn ðc1 ; ZÞ dZ ml 1 mn 1 ln
1
1 ¼ N 1 ml ðc1 ÞN mn ðc1 Þ
1 0X 1 0 X
ðmÞ mn d ml i ðci Þ d j ðcj Þzij ;
(20)
i¼0;1 j¼0;1
where zðmÞ ij is the integral (12) and the expansion (19) is used.
3. Discussion and numerical results 3.1. Comparison with earlier works As far as we know analytical averaging of the scattered field characteristics for randomly oriented particles was performed so far only within the standard version of the TMM (see [8] and references therein). The orientation-averaged extinction and scattering cross-sections were found to be equal to sums of the traces and the norms of the T-matrices, respectively. It is interesting to compare these well-known formulae with those obtained in Section 2. In the axisymmetric parts represented by the first terms in Eqs. (10) and (11) we also get the trace and practically the norm of the matrix T A . The analogy can be explained by the fact that in this case we actually solve a scalar scattering problem using an orthogonal basis. In the nonaxisymmetric parts (see the terms including the matrices T N;a ; T N;b ) our expressions are more complicated than the trace and the norm. It is a result of our solution of the vector scattering problem using a nonorthogonal basis (see Eq. (3)). If the basis had been orthogonal, we would have had the diagonal matrices zðmÞ ¼ ði=2ÞE, kðmÞ ¼ 0, oðmÞ ¼ tðmÞ ¼ vE, and O ¼ v 1 E, where v is a constant and E the unit matrix. It is worth noting that for nonpolarized incident radiation, the expression (10) simplifies due to ðmÞ symmetry properties of the matrices T N;b m and z " # 1 1 X 1 X X 4p A N;a ð21Þ hC ext i ¼ 2 Re T~ ll þ 2 T~ m;ll ; k1 m¼1 l¼m l¼1 where the matrices T~ ¼ 12ðT TE þ T TM Þ. Now only the traces of the T-matrices are present, and moreover because of the symmetry information on the scatterer involved in the matrices T N;b m
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becomes unimportant. For the scattering cross-section hC sca i, a similar simplification does not i given by Eq. (11). occur and one should use hC TE;TM sca 3.2. Applicability of the approaches The applicability of both TMM modifications used in Section 2 has been analytically and numerically investigated for single perfectly conducting particles in [11]. The conclusions made in that paper are also valid for dielectric homogeneous particles and for any version of the TMM involving surface integral equations and monopole field/potential expansions in terms of wave functions. The results of [11] can be formulated as follows. The use of the TMM in the near-field zone (up to the particle surface) and the far-field zone is mathematically correct under different conditions. In the first case the Rayleigh hypothesis about the convergence of the field (potential) expansions everywhere up to the scatterer border must be satisfied. In the second case the condition is essentially weaker (see the condition 3 and its discussion in [3]). Forpexample, the Rayleigh hypothesis is valid for spheroids if and only if the aspect ratio ffiffiffi a=bo 2, while the second condition is satisfied always. In other words, one can use the TMM to calculate such characteristics of the scattered radiation in the far-field zone as cross-sections, scattering matrix elements, etc. for a homogeneous spheroid with any aspect ratio (however, such calculations can require very large computational efforts). For scatterers of other shapes, e.g. the Chebyshev particles, the situation is less simple, and in the far-field zone the method is mathematically correct only in a certain region of the particle shape parameters. These results are generalized for layered scatterers in [3]. It is found that while the applicability in the whole near-field zone still needs the Rayleigh hypothesis to be valid, the condition for the farfield zone becomes more complex and involves information on all layer boundaries. As a result one can utilize the TMM to calculate, e.g., cross-sections off a particle with spheroidal boundaries of the layers only if these surfaces are confocal. Otherwise there appears a certain region of the semiaxes of the boundaries where the TMM fails to be mathematically correct and hence one will encounter divergence of the solution. Note that the conditions formulated in [3] allows one to determine the applicability range for scatterers not only with spheroidal, but also with more complex layer boundaries. Obviously, the theoretical applicability range of the analytical averaging procedure is generally equal to that derived for single layered scatterers. However, important differences appear when we consider numerical problems which make the real range narrower. First, we find that the modified TMM in spherical coordinates (and probably any similar approach) generally gives the values of the orientation-averaged cross-sections with the accuracy of 4–5 orders higher than the accuracy of the cross-sections for the particle in a fixed orientation. As a result, the numerical instability limiting the applicability of the TMM (in particular for spheroids of large aspect ratios, see, e.g., [8]) arises in the averaging procedure at much larger ratios (at least in the modification of the TMM under consideration). Second, in order to derive the orientation-averaged cross-sections within the modified TMM in spheroidal coordinates, one does not need to compute the angular spheroidal functions but only (see Eqs. (16)–(19)). This circumstance extends the coefficients of their expansions d mn k significantly the practical applicability range of this TMM modification (and the equivalent separation of variables method for spheroids) because this range is mainly limited by the problems of numerical calculations of the spheroidal functions.
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It is also worth mentioning the complementary nature of the approaches described in Section 2. The modified TMM in spherical coordinates can be applied to randomly oriented particles with quite different shapes of the layer boundaries but has relatively narrow range of the applicability especially for particles of large eccentricity. The modified TMM in spheroidal coordinates can efficiently treat only layered particles with the confocal spheroidal boundaries but their eccentricity can be very large.
(a)
(b) Fig. 1. Extinction cross-sections of randomly oriented prolate particles in the 3 mm (a) and 10=20 mm (b) band regions. The results for three-layered spheroids with the different middle layer material (‘sil’ means silicate, ‘org’ organics, and ‘ice’ water ice) are shown by short-dashed lines and those for homogeneous spheroids with the refractive indices correspondingly derived using the EMT by longdashed lines. The radius of the sphere having the volume equal to that of the spheroids is 1:44 mm, the aspect ratios of all layer boundaries a=b ¼1.4.
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3.3. Some numerical results For the sake of illustration, we have calculated the orientation-averaged extinction crosssections for layered analogs of cosmic dust grains used in the model of Greenberg [1,2]. This physical model assumes that in diffuse interstellar clouds silicate grains are coated with a layer of carbonaceous organic refractory, while in dense molecular clouds these grains are further coated with an outer ice mantle. There are also evidences for a nonspherical shape of the grains.
(a)
(b) Fig. 2. Extinction cross-sections of randomly oriented prolate spheroids with different aspect ratios a=b in the 3 mm (a) and 10=20 mm (b) band regions. The refractive index was derived using the EMT for the equal volume fractions of silicate, organics and ice. The radius of the sphere having the volume equal to that of the spheroids is 1:44 mm.
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In Fig. 1 we present the extinction cross-sections calculated for randomly oriented three-layered prolate spheroidal particles with a silicate core, an organics layer and an ice mantle. The refractive indices for the astronomical silicate, refractory organics and ice were taken from [12]. All layer boundaries have the aspect ratio a=b ¼ 1:4, i.e. are not confocal and hence the method from Section 2.1 was employed. The size of the equivolume spherical particles was selected to be 1:44 mm (when the volume fractions of all layers are equal, the radius of the core is 1 mm, that of the layer 1:26 mmÞ. For particles of the smaller size ðt0:5 mmÞ, small scatterer size approximations (e.g., the special version of the Rayleigh approximation for layered ellipsoids with the nonconfocal boundaries of layers from [13]) are valid [14] and hC ext i are nearly proportional to the particle size, i.e. the band shape is independent of it. For particles as large as 5–10 mm, the 10 and 20 mm silicate bands disappear, as opposed to spectra observed for the interstellar medium. These spectra are often explained using the Effective Medium Theory (EMT) which defines the mean refractive index for an inhomogeneous scatterer and approximates the optical properties of the scatterer by those of a homogeneous particle of the same shape. Therefore in Fig. 1 we also show the results obtained for randomly oriented homogeneous spheroids with the refractive index derived using the standard (Bruggeman) rule of the EMT. Fig. 2 demonstrates structure effects on the 3 mm water ice and 10 and 20 mm silicate bands. To show the role of the middle layer, we give curves for particles in which organics is replaced by silicate or ice. The figure also illustrates the shape effects by considering randomly oriented spheroids of different aspect ratios (those with the large aspect ratios were treated by the method from Section 2.2). One can see that the shape and structure effects are different and both effects can be important in astrophysical applications.
4. Conclusions We have demonstrated that analytical averaging of the extinction and scattering cross-sections for randomly oriented layered scatterers is also possible within the nonstandard versions of the TMM, including the SVM for spheroids. Although the SVM is only efficient for scatterers with the confocal spheroidal boundaries of layers, it is applicable to particles of high eccentricity. The suggested analytical averaging procedure for the SVM-like modification of the TMM from Section 2.2. has an even wider applicability range than the SVM for single particles, since only the radial spheroidal functions need to be calculated in this case. Thus, this procedure is likely to provide a unique way to quickly get exact results for ensembles of randomly oriented spheroids of high eccentricity and, together with the similar procedure for the modified TMM from Section 2.1, they present a set of tools useful for applications of models of layered nonspherical particles.
Acknowledgements The authors thank A. Li for providing the optical constants of astronomical silicate, organics, and ice used in the calculations. The work was partly supported by the Russian Ministry of
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Education (grant E02-11.0-8). M.P. and V.I. acknowledge the support within the Russian federal programs for Astronomy and for prominent scientific schools (grant 1088.2003.2).
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