Light scattering by a finite cylinder containing a spherical cavity using Sh-matrices

Optics Communications 282 (2009) 156–166

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Optics Communications

journal homepage: www.elsevier .com/locate/optcom

Light scattering by a finite cylinder containing a spherical cavity using Sh-matrices

Dmitry Petrov a,1, Yuriy Shkuratov a, Gorden Videen b,*

a Astronomical Institute of Kharkov V.N. Karazin National University, 35 Sumskaya Sreet, Kharkov 61022, Ukraineb Army Research Laboratory, AMSRD-ARL-CI-EM, 2800 Powder Mill Road, Adelphi, MD 20783, USA

a r t i c l e i n f o

Article history:Received 18 March 2008Received in revised form 27 August 2008Accepted 20 September 2008

0030-4018/$ - see front matter Published by Elsevierdoi:10.1016/j.optcom.2008.09.060

* Corresponding author. Tel.: +1 301 394 1871; faxE-mail addresses: [email protected] (D. Pe

1 Tel.: +38 057 707 50 63.

a b s t r a c t

We use the Sh-matrix formalism that contains the shape-dependent parameters of the T-matrix to derivean analytical solution for the light scattering from a finite cylinder containing a spherical cavity. The inte-gral expressions for the Sh-matrix elements are simpler than those of the T-matrix elements and the caseof a sphere embedded in a finite cylinder these integrals can be solved analytically.

Published by Elsevier B.V.

1. Introduction

The extended boundary condition method (EBCM) using the T-matrix formalism is widely used to calculate scattering properties of non-spherical particles [1]. The incident and scattered electric fields are expanded in series of vector spherical wave functions, and then a rela-tion between the expansion coefficients of these fields is established by means of a transition matrix, or T-matrix. The elements of theT-matrix depend on the optical and geometrical parameters of the particle system and contain all the scattering information from any illu-mination/observation geometry, including orientation-averaged properties.

Recently, it has been discovered that the shape-dependent factors can be separated from the size- and refractive-index-dependent fac-tors and these are contained in the shape matrix, or Sh-matrix [2–5]. The elements of the Sh-matrices are expressed in terms of surfaceintegrals. Once calculated, the T-matrix of particles having other sizes and refractive indices can be calculated using analytic expressions.In many cases the surface integrals themselves can be calculated analytically, e.g., for Chebyshev particles [4], bi-spheres and capsules [6],merging spheres [7], finite cylinders [8], and corrugated cylinders and capsules [9].

Particles containing cavities represent a class of heterogeneous particles that are of interest in atmospheric sciences. Spherical dropletscontaining cavities approximate water droplets containing cavities that are relevant to radiative-transfer calculations and calculating theradiation budget [10,11]. They also can represent simple biological cells [12]. In this manuscript we consider a system composed of a finitecylinder containing a spherical cavity. In the limit of large cylinder aspect ratio, this represents two fundamental particle systems, a fun-damental two-dimensional cylinder containing a fundamental three-dimensional sphere. The motivation for solving this particular systemis the recent advances in scanning-flow cytometry, in which the light scattering measured from particles contained within micro-capillar-ies is used to characterize their morphology.

2. Theory

In the EBCM of the T-matrix formalism, double integrals over the particle surface must be calculated to satisfy the boundary conditions.This is true also within the Sh-matrix formalism. Due to the axially symmetric morphology, these double integrals reduce to single integralsthat are of relatively simple form that can be solved analytically.

In this manuscript we consider a finite cylinder whose rotation axis is oriented along the polar axis of the coordinate system. The heightof this cylinder is 2h, and the base diameter is 2a. Centered within this cylinder is a spherical cavity whose diameter is 2b. We note, thata P b and h P b. In this coordinate system the shape of the cylinder is described by the following equation:

Rcyl0 hð Þ ¼

hcos h ; 0 6 h < arctan a

h

� �a

sin h ; arctan ah

� �6 h < p� arctan a

h

� �h

cos h ; p� arctan ah

� �6 h < p

8><>: ; ð1Þ

B.V.

: +1 301 394 4797.trov), [email protected] (G. Videen).

mailto:[email protected]

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00304018

http://www.elsevier.com/locate/optcom

D. Petrov et al. / Optics Communications 282 (2009) 156–166 157

And the shape of sphere is described as follows:

Rsph0 ðhÞ ¼ b; ð2Þ

For convenience we designate Xsph = 2pb/k and Xcyl = 2pa/k. General equations relating the Sh-matrix elements to the T-matrix to the lightscattering properties can be found in previous references [4]. We note that a cylinder with a cavity can formally be considered as a layeredparticle. The equations relating the Sh-matrix elements to the T-matrix for layered particles are given in [5]. While the surface integrals re-quired to find the Sh-matrix elements can be solved analytically, their solution is by no means trivial or modest. We present only the finalsolution in Appendix A, which even in compact form requires several pages.

3. Results

Fig. 1 shows examples of cross-sections of the scattering system under consideration. In this case the aspect ratio of the finite cylinder is1. The different panels illustrate the morphologies for different-size spherical cavities.

Fig. 2 shows examples of the two-dimensional light-scattering total intensities (the first Stokes parameter I whose explicit definition isgiven in [13]) in the forward directions (Fig. 2a) and the backward directions (Fig. 2b) for constant-size finite cylinders containing different-size spherical cavities. Fig. 3 shows examples of the polarization images (the two-dimensional distribution of the degree of linear polari-zation of the scattered light) in the forward directions (Fig. 3a) and the backward directions (Fig. 3b) for the same particles. The rows in allpanels show patterns for constant cavity size and the columns show patterns for constant incident angle. In the first columns, the system isilluminated end-on (u = 0), and the light-scattering patterns reflect this symmetry by displaying concentric circles. In the second columns,the system is illuminated at u = 45� incidence and the symmetry is completely broken. In the third column, the system is illuminatedbroadside (u = 90�) and two axes of symmetry exist. In this column a vertical diffraction-band structure appears that is perpendicularto the axis of the cylinder. This band structure is present in both the backward- and forward-scattering intensities and is not as obviousin the polarization images.

Qualitatively, we see very little effect when the cavity is small. The patterns for Xsph = 2.0 appear much the same as those for Xsph = 0, forthe homogeneous cylinder. There are, however, quantitative differences, especially in the backward scattering and in the polarizations. Asthe cavity size increases, the patterns change significantly. For large cavity sizes, we can see not only changes in amplitude, but shifts in thepositions of maxima and minima that could be used to characterize particles, especially through Fourier analysis and diffraction tech-niques. Additional interference minima appear in the azimuthal directions that are not present for smaller cavities.

To verify the formulae presented in Appendix A we have compared Sh-matrix calculations with those of the discrete dipole approxima-tion (DDA). We use the DDA code developed by Zubko [14]. Figs. 4 and 5 show the intensity and degree of linear polarization of cylinders ofdifferent lengths containing a spherical cavity. As can be seen in all cases very good coincidence is observed.

Fig. 1. Examples showing cross-sections of cylinder, having a/h = 1.0 containing a spherical cavity.

Fig. 2. Intensity maps (the two-dimensional distribution of intensity of scattered light) in the (a) forward hemisphere and (b) backward hemisphere of light scattered byXcyl = 10.0 cylinders having a/h = 1.0 containing a spherical cavity at different cavity size: Xsph = 0 (first row), Xsph = 2.0 (second row), Xsph = 4.0 (third row), Xsph = 6.0 (fourthrow), Xsph = 8.0 (fifth row) and Xsph = 10.0 (sixth row). The cylinder is in a fixed orientation for different angles of light incidence: / = 0� (light strikes the particle end-on)corresponds to the first column, / = 45� to the second column, and / = 90� (light strikes the particle broadside) to the third column. The refractive index of the cylinder ism0 = 1.5 + 0i.

158 D. Petrov et al. / Optics Communications 282 (2009) 156–166

4. Discussion

In this manuscript we present analytical expressions for the shape matrix. These elements depend only on the particle morphology, sothey can be used to calculate rapidly the T-matrices of a polydispersion of particles having identical shape, but of different size and/orrefractive index. The T-matrix also can be used to calculate the scattering from the particle in any orientation or the scattering propertiesaveraged over orientation. We present sample images of scattering patterns from finite cylinders containing spherical cavities of differentsize. These images retain many of the light-scattering features of the cylinder until the dimension of the spherical cavity is greater than 20%of that of the cylinder.

One of the goals of this study was to consider the geometry of particles illuminated in a flow cytometer, i.e., in a capillary. For such ageometry the aspect ratio of the cylinder should be made quite large and the incident beam should be restricted in size, preferably Gauss-ian in nature, so that the edges of the cylinder would not affect the scattering pattern significantly. In scanning-flow cytometry, theGaussian beam is coupled into the capillary that is illuminated end-on, and the light that is scattered by the particle out of the cylinderis measured as a function of scattering angle. Interestingly, this is the same geometry that is relevant in fiber-optics applications. Ofprime concern in this field is the energy loss in transmission due to imperfections within the fiber. In this manuscript we have demon-strated that an analytical solution to the scattering from such a fiber containing a spherical cavity is obtainable. While the finite-differ-ence time-domain (FDTD) is the obvious method for considering such geometries, we suggest the Sh-matrix as a possible alternativemethodology that has some advantages for certain applications. Primarily, an analytical solution may be obtained that is independentof refractive index and beam shape or mode. We consider such studies requiring different beam shapes to be beyond the scope of thepresent manuscript.

Appendix A

In this appendix we present the analytical equations for the Sh-matrix elements for a finite cylinder containing a spherical cavity. Notethat equations in Appendix for a cylinder with a spherical cavity are almost the same as for a homogeneous cylinder, while the presence ofa cavity introduces only a few terms, depending on cavity radius b


RgSh11mnm0n0k ¼ �i

�1ð Þm0�mþk

22kþn0þnþ1 Ann0p2dmm0 �

h2kþnþn0þ2 Ið00Þmnm0n0 2kþ nþ n0 þ 2;Hð Þ � Ið00Þ

mnm0n0 2kþ nþ n0 þ 2; 0ð Þh i

þa2kþnþn0þ2 Ið01Þmnm0n0 2kþ nþ n0 þ 2;p�Hð Þ � Ið01Þ

mnm0n0 2kþ nþ n0 þ 2;Hð Þh i

þh2kþnþn0þ2 Ið00Þmnm0n0 2kþ nþ n0 þ 2;pð Þ � Ið00Þ

mnm0n0 2kþ nþ n0 þ 2;p�Hð Þh i

266664

377775; ðA:1Þ

RgSh122mnm0n0k ¼ p2 �1ð Þk

22kþ2nþ2 Ann0dmm0b2kþ2nþ3 2n nþ 1ð Þ

2nþ 1dnn0 � p2 �1ð Þm

0�mþk

22kþn0þnþ2 dmm0Ann0

�

h2kþnþn0þ3 Ið10Þmnm0n0 2kþ nþ n0 þ 3;Hð Þ � Ið10Þ

mnm0n0 2kþ nþ n0 þ 3;0ð Þh in




mnm0n0 2kþ nþ n0 þ 3;p�Hð Þh io

2666664

3777775; ðA:2Þ

RgSh121mnm0n0k¼�p2 �1ð Þk

22kþ2nþ1 Ann0dmm0b2kþ2nþ1 2n nþ1ð Þ2

2nþ1dnn0 þp2 �1ð Þk


�

nþ1ð Þ h2kþnþn0þ1 Ið10Þmnm0n0 2kþnþn0 þ1;Hð Þ� Ið10Þ

mnm0n0 2kþnþn0 þ1;0ð Þh in

þa2kþnþn0þ1 Ið11Þmnm0n0 2kþnþn0 þ1;p�Hð Þ� Ið11Þ

mnm0n0 2kþnþn0 þ1;Hð Þh i

þh2kþnþn0þ1 Ið10Þmnm0n0 2kþnþn0 þ1;pð Þ� Ið10Þ

mnm0n0 2kþnþn0 þ1;p�Hð Þh io

þN mð Þ h2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2kþnþn0 þ2;Hð Þ

��h2kþnþn0þ1 n0þ1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðCÞmnm0n0�1 2kþnþn0 þ2;Hð Þ

�h2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2kþnþn0 þ2;0ð Þþh2kþnþn0þ1 n0þ1ð Þ


2n0þ1 IðCÞmnm0n0�1 2kþnþn0 þ2;0ð Þ

þa2kþnþn0þ1 n0þ1ð Þffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðSÞmnm0n0�1 2kþnþn0 þ2;p�Hð Þ�a2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðSÞmnm0n0þ1 2kþnþn0 þ2;p�Hð Þ

þa2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðSÞmnm0n0þ1 2kþnþn0 þ2;Hð Þ�a2kþnþn0þ1 n0þ1ð Þ


2n0þ1 IðSÞmnm0n0�1 2kþnþn0 þ2;Hð Þ

þh2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2kþnþn0 þ2;pð Þ�h2kþnþn0þ1 n0þ1ð Þ


2n0þ1 IðCÞmnm0n0�1 2kþnþn0 þ2;pð Þ

þh2kþnþn0þ1 n0þ1ð Þffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðCÞmnm0n0�1 2kþnþn0 þ2;p�Hð Þ�h2kþnþn0þ1 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2kþnþn0 þ2;p�Hð Þ

�

266666666666666666666666666664

377777777777777777777777777775

;

ðA:3Þ

RgSh212mnm0n0k ¼ �p2 �1ð Þ2k

22kþ2nþ2 Ann0dmm0b2kþ2nþ3 2n nþ 1ð Þ

2nþ 1dnn0 þ p2 �1ð Þm

0�mþk


�

h2kþnþn0þ3 Ið10Þmnm0n0 m; n;m0;n0;2kþ nþ n0 þ 3;Hð Þ � Ið10Þ

mnm0n0 m; n;m0;n0;2kþ nþ n0 þ 3;0ð Þh in

þa2kþnþn0þ3 Ið11Þmnm0n0 m;n;m0; n0;2kþ nþ n0 þ 3;p�Hð Þ � Ið11Þ

mnm0n0 m;n;m0;n0;2kþ nþ n0 þ 3;Hð Þh i

þh2kþnþn0þ3 Ið10Þmnm0n0 m;n;m0; n0;2kþ nþ n0 þ 3;pð Þ � Ið10Þ

mnm0n0 m;n;m0;n0;2kþ nþ n0 þ 3;p�Hð Þh io

2666664

3777775; ðA:4Þ

RgSh211mnm0n0k¼p2 �1ð Þk

22kþ2nþ1 Ann0dmm0b2kþ2nþ1 2n nþ1ð Þ2

2nþ1dnn0 �p2 �1ð Þk


�

n0 þ1ð Þ h2kþnþn0þ1 Ið10Þmnm0n0 2kþnþn0 þ1;Hð Þ� Ið10Þ

mnm0n0 2kþnþn0 þ1;0ð Þh in

þa2kþnþn0þ1 Ið11Þmnm0n0 2kþnþn0 þ1;p�Hð Þ� Ið11Þ

mnm0n0 2kþnþn0 þ1;Hð Þh i

þh2kþnþn0þ1 Ið10Þmnm0n0 2kþnþn0 þ1;pð Þ� Ið10Þ

mnm0n0 m;n;m0;n0;2kþnþn0 þ1;p�Hð Þh io

þN m0ð Þ h2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2kþnþn0 þ2;Hð Þ

��h2kþnþn0þ1 nþ1ð Þ

ffiffiffiffiffiffiffiffiffiffiffin2�m2p

2nþ1 IðCÞmn�1m0n0 2kþnþn0 þ2;Hð Þ

�h2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2kþnþn0 þ2;0ð Þþh2kþnþn0þ1 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2kþnþn0 þ2;0ð Þ

þa2kþnþn0þ1 nþ1ð Þffiffiffiffiffiffiffiffiffiffiffin2�m2p

2nþ1 IðSÞmn�1m0n0 2kþnþn0 þ2;p�Hð Þ�a2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðSÞmnþ1m0n0 2kþnþn0 þ2;p�Hð Þ

þa2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðSÞmnþ1m0n0 2kþnþn0 þ2;Hð Þ�a2kþnþn0þ1 nþ1ð Þ


2nþ1 IðSÞmn�1m0n0 2kþnþn0 þ2;Hð Þ

þh2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2kþnþn0 þ2;pð Þ�h2kþnþn0þ1 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2kþnþn0 þ2;pð Þ

�h2kþnþn0þ1 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2kþnþn0 þ2;p�Hð Þþh2kþnþn0þ1 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2kþnþn0 þ2;p�Hð Þo

266666666666666666666666666666664

377777777777777777777777777777775

; ðA:5Þ

Fig. 3. Maps of polarization degree in the (a) forward hemisphere and (b) backward hemisphere of light scattered by Xcyl = 10.0 cylinders having a/h = 1.0 containing aspherical cavity at different cavity size: Xsph = 0 (first row), Xsph = 2.0 (second row), Xsph = 4.0 (third row), Xsph = 6.0 (fourth row), Xsph = 8.0 (fifth row) and Xsph = 10.0 (sixthrow). The cylinder is in a fixed orientation for different angles of light incidence: / = 0� (light strikes the particle end-on) corresponds to the first column, / = 45� to the secondcolumn, and / = 90� (light strikes the particle broadside) to the third column. The refractive index of the cylinder is m0 = 1.5 + 0i.


RgSh221mnm0n0k ¼ �ip2 �1ð Þk

22kþn0þnþ1 dmm0Ann0 �

n0 þ 1ð Þ nþ 1ð Þ h2kþnþn0 Ið00Þmnm0n0 2kþ nþ n0;Hð Þ � Ið00Þ

mnm0n0 2kþ nþ n0;0ð Þh in

þa2kþnþn0 Ið01Þmnm0n0 2kþ nþ n0;p�Hð Þ � Ið01Þ

mnm0n0 2kþ nþ n0;Hð Þh i

þh2kþnþn0 Ið00Þmnm0n0 2kþ nþ n0;pð Þ � Ið00Þ

mnm0n0 2kþ nþ n0;p�Hð Þh ioþ N mð Þ nþ 1ð Þ mj j þ N m0ð Þ n0 þ 1ð Þ m0j j½ ��

h2kþnþn0 IðCÞmnm0n0 2kþ nþ n0 þ 1;Hð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 1;0ð Þh in

þa2kþnþn0 IðSÞmnm0n0 2kþ nþ n0 þ 1;p�Hð Þ � IðSÞmnm0n0 2kþ nþ n0 þ 1;Hð Þh i

þh2kþnþn0 IðCÞmnm0n0 2kþ nþ n0 þ 1;pð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 1;p�Hð Þh io

266666666666666666666666664

377777777777777777777777775

; ðA:6Þ

RgSh222mnm0n0k ¼ ip2 �1ð Þk


nþ 1ð Þ h2kþnþn0þ2 Ið00Þmnm0n0 2kþ nþ n0 þ 2;Hð Þ � Ið00Þ






þN m0ð Þ m0j j h2kþnþn0þ2 IðCÞmnm0n0 2kþ nþ n0 þ 3;Hð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 3;0ð Þh in

þa2kþnþn0þ2 IðSÞmnm0n0 2kþ nþ n0 þ 3;p�Hð Þ � IðSÞmnm0n0 2kþ nþ n0 þ 3;Hð Þ� �

þh2kþnþn0þ2 IðCÞmnm0n0 2kþ nþ n0 þ 3;pð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 3;p�Hð Þ� �o

26666666666666664

37777777777777775

; ðA:7Þ

1e-005

0.0001

0.001

0.01

0.1

1

I

0 60 120 180ϑ°

-80

-40

0

40

80P,%

T-matrixDDA

Xcyl = 10.0

a/h = 0.5

Xsph = 3.0

m0 =1.313+0ifixed orientations

ϕ =0°

Fig. 4. The intensity and degree of linear polarization of Xcyl = 10.0 cylinders having a/h = 0.5 containing a spherical cavity Xsph = 3.0. The refractive index is m0 = 1.313 + 0i and/ = 0� (light strikes the particle end-on).


RgSh223mnm0n0k ¼ ip2 �1ð Þk


n0 þ 1ð Þ h2kþnþn0þ2 Ið00Þmnm0n0 2kþ nþ n0 þ 2;Hð Þ � Ið00Þ






þN mð Þ mj j h2kþnþn0þ2 IðCÞmnm0n0 2kþ nþ n0 þ 3;Hð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 3;0ð Þh in

þa2kþnþn0þ2 IðSÞmnm0n0 2kþ nþ n0 þ 3;p�Hð Þ � IðSÞmnm0n0 2kþ nþ n0 þ 3;Hð Þ� �

þh2kþnþn0þ2 IðCÞmnm0n0 2kþ nþ n0 þ 3;pð Þ � IðCÞmnm0n0 2kþ nþ n0 þ 3;p�Hð Þ� �o

26666666666666664

37777777777777775

; ðA:8Þ

RgSh224mnm0n0k ¼ �ip2 �1ð Þk


�

h2kþnþn0þ4 Ið00Þmnm0n0 m; n;m0;n0;2kþ nþ n0 þ 4;Hð Þ � Ið00Þ

mnm0n0 m; n;m0;n0;2kþ nþ n0 þ 4;0ð Þh i

þa2kþnþn0þ4 Ið01Þmnm0n0 m;n;m0;n0;2kþ nþ n0 þ 4;p�Hð Þ � Ið01Þ

mnm0n0 m; n;m0;n0;2kþ nþ n0 þ 4;Hð Þh iþh2kþnþn0þ4 Ið00Þ

mnm0n0 2kþ nþ n0 þ 4;pð Þ � Ið00Þmnm0n0 2kþ nþ n0 þ 4;p�Hð Þ

h io

266664

377775; ðA:9Þ

Sh11mnm0n0k ¼

�1ð Þk

22kþn0�nAnn0p2dmm0

�

h2k�nþn0þ1 Ið00Þmnm0n0 m; n;m0;n0;2k� nþ n0 þ 1;Hð Þ � Ið00Þ

mnm0n0 m; n;m0;n0;2k� nþ n0 þ 1;0ð Þh i

þa2k�nþn0þ1 Ið01Þmnm0n0 m;n;m0;n0;2k� nþ n0 þ 1;p�Hð Þ � Ið01Þ

mnm0n0 m; n;m0;n0;2k� nþ n0 þ 1;Hð Þh i

þh2k�nþn0þ1 Ið00Þmnm0n0 m;n;m0;n0;2k� nþ n0 þ 1;pð Þ � Ið00Þ

mnm0n0 m;n;m0; n0;2k� nþ n0 þ 1;p�Hð Þh i

266664

377775; ðA:10Þ

1e-005

0.0001

0.001

0.01

0.1

1

I

0 60 120 180ϑ°

-80

-40

0

40

80P,%

T-matrixDDA

Xcyl = 20.0

a/h = 0.1

Xsph = 1.0

m0 =1.313+0ifixed orientations

ϕ =0°

Fig. 5. The intensity and degree of linear polarization of Xcyl = 20.0 cylinders having a/h = 0.1 containing a spherical cavity Xsph = 1.0. The refractive index is m0 = 1.313 + 0i and/ = 0� (light strikes the particle end-on).


Sh221mnm0n0k ¼ p2 �1ð Þk

22kþn0�ndmm0Ann0

�

�n n0 þ 1ð Þ h2kþnþn0 Ið00Þmnm0n0 2k� nþ n0 � 1;Hð Þ � Ið00Þ

mnm0n0 2k� nþ n0 � 1; 0ð Þh in

þa2k�nþn0�1 Ið01Þmnm0n0 2k� nþ n0 � 1;p�Hð Þ � Ið01Þ

mnm0n0 2k� nþ n0 � 1;Hð Þh i

þh2k�nþn0�1 Ið00Þmnm0n0 2k� nþ n0 � 1;pð Þ � Ið00Þ

mnm0n0 2k� nþ n0 � 1;p�Hð Þh io

þ N m0ð Þ n0 þ 1ð Þ m0j j � nN mð Þ mj j½ � h2k�nþn0�1 IðCÞmnm0n0 2k� nþ n0;Hð Þ � IðCÞmnm0n0 2k� nþ n0;0ð Þh in

þa2k�nþn0�1 IðSÞmnm0n0 2k� nþ n0;p�Hð Þ � IðSÞmnm0n0 2k� nþ n0;Hð Þ� �

þh2k�nþn0�1 IðCÞmnm0n0 2k� nþ n0;pð Þ � IðCÞmnm0n0 2k� nþ n0;p�Hð Þ� �o

266666666666666666666664

377777777777777777777775

; ðA:11Þ

Sh122mnm0n0k ¼ �ip2 �1ð Þnþk

22kþ1 Ann0dmm0b2kþ2 2n nþ 1ð Þ

2nþ 1dnn0 þ ip2 �1ð Þk

22k�n0þnþ1 dmm0Ann0

�

h2k�nþn0þ2 Ið10Þmnm0n0 2k� nþ n0 þ 2;Hð Þ � Ið10Þ

mnm0n0 2k� nþ n0 þ 2; 0ð Þh i

þa2k�nþn0þ2 Ið11Þmnm0n0 2k� nþ n0 þ 2;p�Hð Þ � Ið11Þ

mnm0n0 2k� nþ n0 þ 2;Hð Þh i

þh2k�nþn0þ2 Ið10Þmnm0n0 2k� nþ n0 þ 2;pð Þ � Ið10Þ

mnm0n0 2k� nþ n0 þ 2;p�Hð Þh i

266664

377775; ðA:12Þ


Sh212mnm0n0k ¼ ip2 �1ð Þnþk

22kAnn0dmm0b

2kþ2 2n nþ 1ð Þ2nþ 1

dnn0 � ip2 �1ð Þk

22kþn0�ndmm0Ann0

�


mnm0n0 2k� nþ n0 þ 2; 0ð Þh i




mnm0n0 2k� nþ n0 þ 2;p�Hð Þh i

266664

377775; ðA:13Þ

Sh224mnm0n0k ¼ p2 �1ð Þm

0�mþk

22kþn0�nþ2 dmm0Ann0 �


mnm0n0 2k� nþ n0 þ 3;0ð Þh i




mnm0n0 2k� nþ n0 þ 3;p�Hð Þh io

266664

377775; ðA:14Þ

Sh121mnm0n0k¼�ip2 �1ð Þnþk

22kAnn0dmm0b

2k 2n2 nþ1ð Þ2nþ1

dnn0 þ ip2 �1ð Þk

22kþn0�ndmm0Ann0

�

�n h2k�nþn0 Ið10Þmnm0n0 m;n;m0;n0;2k�nþn0;Hð Þ� Ið10Þ

mnm0n0 m;n;m0;n0;2k�nþn0;0ð Þh in

þa2k�nþn0 Ið11Þmnm0n0 m;n;m0;n0;2k�nþn0;p�Hð Þ� Ið11Þ

mnm0n0 m;n;m0;n0;2k�nþn0;Hð Þh i

þh2k�nþn0 Ið10Þmnm0n0 m;n;m0;n0;2k�nþn0 þ1;pð Þ� Ið10Þ

mnm0n0 m;n;m0;n0;2k�nþn0;p�Hð Þh io

þN mð Þ h2k�nþn0 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2k�nþn0 þ1;Hð Þ

��a2k�nþn0 n0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðSÞmnm0n0þ1 2k�nþn0 þ1;p�Hð Þ

�h2k�nþn0 n0þ1ð Þffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðCÞmnm0n0�1 2k�nþn0 þ1;Hð Þ�h2k�nþn0 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2k�nþn0 þ1;0ð Þ

þh2k�nþn0 n0þ1ð Þffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðCÞmnm0n0�1 2k�nþn0 þ1;0ð Þþa2k�nþn0 n0þ1ð Þffiffiffiffiffiffiffiffiffiffiffiffin02�m02p

2n0þ1 IðSÞmnm0n0�1 2k�nþn0 þ1;p�Hð Þ

þa2k�nþn0 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðSÞmnm0n0þ1 2k�nþn0 þ1;Hð Þ�a2k�nþn0 n0þ1ð Þ


2n0þ1 IðSÞmnm0n0�1 2k�nþn0 þ1;Hð Þ

þh2k�nþn0 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2k�nþn0 þ1;pð Þ�h2k�nþn0 n0þ1ð Þ


2n0þ1 IðCÞmnm0n0�1 2k�nþn0 þ1;pð Þ

�h2k�nþn0 n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0þ1ð Þ2�m02

p2n0þ1 IðCÞmnm0n0þ1 2k�nþn0 þ1;p�Hð Þþh2k�nþn0 n0þ1ð Þ


2n0þ1 IðCÞmnm0n0�1 2k�nþn0 þ1;p�Hð Þo

26666666666666666666666666664

37777777777777777777777777775

; ðA:15Þ

Sh222mnm0n0k ¼ �p2 �1ð Þk

22kþn0�nþ1 dmm0Ann0

�

�n h2k�nþn0þ1 Ið00Þmnm0n0 2k� nþ n0 þ 1;Hð Þ � Ið00Þ

mnm0n0 2k� nþ n0 þ 1; 0ð Þh in





þN m0ð Þ m0j j h2k�nþn0þ1 IðCÞmnm0n0 2k� nþ n0 þ 2;Hð Þ � IðCÞmnm0n0 2k� nþ n0 þ 2; 0ð Þh in

þa2k�nþn0þ1 IðSÞmnm0n0 2k� nþ n0 þ 2;p�Hð Þ � IðSÞmnm0n0 2k� nþ n0 þ 2;Hð Þh i

þh2k�nþn0þ1 IðCÞmnm0n0 m;n;m0; n0;2k� nþ n0 þ 2;pð Þ � IðCÞmnm0n0 m; n;m0;n0;2k� nþ n0 þ 2;p�Hð Þh io

26666666666666664

37777777777777775

; ðA:16Þ

Sh211mnm0n0k¼�ip2 �1ð Þnþk

22kAnn0dmm0b

2k0þ2 2n nþ1ð Þ2

2nþ1dnn0 þ ip2 �1ð Þk

22kþn0�ndmm0Ann0

�

n0 þ1ð Þ h2k�nþn0 Ið10Þmnm0n0 2k�nþn0;Hð Þ� Ið10Þ

mnm0n0 2k�nþn0;0ð Þh in

þa2k�nþn0 Ið11Þmnm0n0 2k�nþn0;p�Hð Þ� Ið11Þ

mnm0n0 m;n;m0;n0;2k�nþn0;Hð Þh i

þh2k�nþn0 Ið10Þmnm0n0 2k�nþn0;pð Þ� Ið10Þ

mnm0n0 2k�nþn0;p�Hð Þh io

þN m0ð Þ h2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2k�nþn0 þ1;Hð Þ

��h2k�nþn0 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2k�nþn0 þ1;Hð Þ

þh2k�nþn0 nþ1ð Þffiffiffiffiffiffiffiffiffiffiffin2�m2p

2nþ1 IðCÞmn�1m0n0 2k�nþn0 þ1;0ð Þ�h2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2k�nþn0 þ1;0ð Þ

þa2k�nþn0 nþ1ð Þffiffiffiffiffiffiffiffiffiffiffin2�m2p

2nþ1 IðSÞmn�1m0n0 2k�nþn0 þ1;p�Hð Þ�a2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðSÞmnþ1m0n0 2k�nþn0 þ1;p�Hð Þ

þa2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðSÞmnþ1m0n0 2k�nþn0 þ1;Hð Þ�a2k�nþn0 nþ1ð Þ


2nþ1 IðSÞmn�1m0n0 2k�nþn0 þ1;Hð Þ

þh2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2k�nþn0 þ1;pð Þ�h2k�nþn0 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2k�nþn0 þ1;pð Þ

�h2k�nþn0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ1ð Þ2�m2

p2nþ1 IðCÞmnþ1m0n0 2k�nþn0 þ1;p�Hð Þþh2k�nþn0 nþ1ð Þ


2nþ1 IðCÞmn�1m0n0 2k�nþn0 þ1;p�Hð Þo

26666666666666666666666666664

37777777777777777777777777775

; ðA:17Þ


Sh223mnm0n0k ¼ �p2 �1ð Þk

22kþn0�nþ1 dmm0Ann0 �

n0 þ 1ð Þ h2k�nþn0þ1 Ið00Þmnm0n0 2k� nþ n0 þ 1;Hð Þ � Ið00Þ

mnm0n0 2k� nþ n0 þ 1;0ð Þh in





þN mð Þ mj j h2k�nþn0þ1 IðCÞmnm0n0 2k� nþ n0 þ 2;Hð Þ � IðCÞmnm0n0 2k� nþ n0 þ 2; 0ð Þh in

þa2k�nþn0þ1 IðSÞmnm0n0 2k� nþ n0 þ 2;p�Hð Þ � IðSÞmnm0n0 2k� nþ n0 þ 2;Hð Þ� �

þh2k�nþn0þ1 IðCÞmnm0n0 2k� nþ n0 þ 2;pð Þ � IðCÞmnm0n0 2k� nþ n0 þ 2;p�Hð Þ� �o

26666666666666664

37777777777777775

; ðA:18Þ

where

Ið00Þmnm0n0 z; hð Þ ¼ Nm0Nmþ1m

n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ2 �m02

q2n0 þ 1

I0mnm0n0þ1 z; hð Þ � n0 þ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1I0mnm0n0�1 z; hð Þ

24

35

þ Nm0þ1Nmm0n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ 1ð Þ2 �m2

q2nþ 1

I0mnþ1m0n0 z; hð Þ � nþ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I0mn�1m0n0 z; hð Þ

24

35; ðA:19Þ

Ið01Þmnm0n0 z; hð Þ ¼ Nm0Nmþ1m


q2n0 þ 1

I00mnm0n0þ1 z� 1; hð Þ � n0 þ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1I00mnm0n0�1 z� 1; hð Þ

24

35

þ Nm0þ1Nmm0n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ 1ð Þ2 �m2

q2nþ 1

I00mnþ1m0n0 z� 1; hð Þ � nþ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I00mn�1m0n0 z� 1; hð Þ

24

35; ðA:20Þ

Ið10Þmnm0n0 z; hð Þ ¼ Nm0þ1Nmþ1mm0I0mnm0n0 z; hð Þ

þ Nm0Nm


q2n0 þ 1

nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ 1ð Þ2 �m2

q2nþ 1

I0mnþ1m0n0þ1 z; hð Þþ

8<: n0 þ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1nþ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I0mn�1m0n0�1 z; hð Þ

� n0 þ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1


q2nþ 1

I0mnþ1m0n0�1 z; hð Þ �n0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ2 �m02

q2n0 þ 1

nþ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I0mn�1m0n0þ1 z; hð Þ

9=;; ðA:21Þ

Nm ¼1; m P 0�1ð Þm; m < 0

�; ; ðA:22Þ

Ið11Þmnm0n0 z; hð Þ ¼ Nm0þ1Nmþ1mm0I00mnm0n0 z� 1; hð Þ

þ Nm0Nm


q2n0 þ 1


q2nþ 1

I00mnþ1m0n0þ1 z� 1; hð Þ

8<:

þ n0 þ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1nþ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I0mn�1m0n0�1 z� 1; hð Þ� n0 þ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1


q2nþ 1

I00mnþ1m0n0�1 z� 1; hð Þ

�n0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ2 �m02

q2n0 þ 1

nþ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1I00mn�1m0n0þ1 z� 1; hð Þ

9=;; ðA:23Þ

where

I0mnm0n0 z;hð Þ¼n!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� mj jð Þ! nþ mj jð Þ!

pn0!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 � m0j jð Þ! n0 þ m0j jð Þ!

p Xn� mj j

k¼0

�1ð Þk

k! n�kð Þ! n� mj j�kð Þ! mj jþkð Þ!

�Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 �k0� �

! n0 � m0j j�k0� �

! m0j jþk0� �

!

�

2 �1ð ÞzP1

k00¼0

C 0 z;k00� �

�2ð Þk00X 2k00 þ2n�2k� mj jþ2n0 �2k0 � m0j j�1;2kþ mj jþ2k0 þ m0j j�1; h2� �

; z>0; h> p2

2P1

k00¼0

C 0 z;k00� �

�2ð Þk00X 2n�2k� mj jþ2n0 �2k0 � m0j j�1;2k00 þ2kþ mj jþ2k0 þ m0j j�1; h2� �

; z>0; h6 p2

�1ð ÞzPz

k00¼0

Ck00

z �2ð Þk00X 2k00 þ2n�2k� mj jþ2n0 �2k0 � m0j j�1;2kþ mj jþ2k0 þ m0j j�1; h2� �

; z60

8>>>>>>>><>>>>>>>>:

; ðA:24Þ


I00mnm0n0 z;hð Þ¼n!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� mj jð Þ! nþ mj jð Þ!

pn0!


p Xn� mj j

k¼0

�1ð Þk


�umn0� m0j jk0¼0

�1ð Þk02�zþ2

k0! n0 �k0� �

! n0 � m0j j�k0� �

! m0j jþk0� �

!X 2n�2k� mj jþ2n0 �2k0 � m0j j�z;2kþ mj jþ2k0 þ m0j j�z;

h2

; ðA:25Þ

IðCÞmnm0n0 z;hð Þ¼4n!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� mj jð Þ! nþ mj jð Þ!

pn0!


p Xn� mj j

k¼0

�1ð Þk


�Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 �k0� �

! n0 � m0j j�k0� �

! m0j jþk0� �

!

�

f2 �1ð ÞzP1

k00¼0

C 0 z;k00� �

�2ð Þk00X 2k00 þ2n�2k� mj jþ2n0 �2k0 � m0j jþ1;2kþ mj jþ2k0 þ m0j jþ1; h2� �

; z>0; h> p2

2P1

k00¼0

C 0 z;k00� �

�2ð Þk00X 2n�2k� mj jþ2n0 �2k0 � m0j jþ1;2k00 þ2kþ mj jþ2k0 þ m0j jþ1; h2� �

; z>0; h6 p2

�1ð ÞzPz

k00¼0

Ck00

z �2ð Þk00X 2k00 þ2n�2k� mj jþ2n0 �2k0 � m0j jþ1;2kþ mj jþ2k0 þ m0j jþ1; h2� �

; z60

8>>>>>>>><>>>>>>>>:

; ðA:26Þ

IðSÞmnm0n0 z; hð Þ ¼ n!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� mj jð Þ! nþ mj jð Þ!

pn0!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 � m0j jð Þ! n0 þ m0j jð Þ!

p�

�Xn� mj j

k¼0

�1ð Þk

k! n� kð Þ! n� mj j � kð Þ! mj j þ kð Þ!Xn0� m0j j

k0¼0

�1ð Þk02�zþ1

k0! n0 � k0� �

! n0 � m0j j � k0� �

! m0j j þ k0� �

!

�X 2n� 2k� mj j þ 2n0 � 2k0 � m0j j � zþ 1;2kþ mj j þ 2k0 þ m0j j � z� 1; h

2

� ��X 2n� 2k� mj j þ 2n0 � 2k0 � m0j j � z� 1;2kþ mj j þ 2k0 þ m0j j � zþ 1; h

2

� �" #

; ðA:27Þ

X l; m; hð Þ ¼

� ðcos hÞlþ1

lþm ðsin hÞm�1 þPn�1

k¼1

m�1ð Þ!! lþm�2k�2ð Þ!!m�2k�1ð Þ!! lþm�2ð Þ!! ðsin hÞm�2k�1

þ m�1ð Þ!!l!!

lþmð Þ!! ZðCÞ l; hð Þ; m ¼ 2n; l–� 2;�4; . . .

ðsin hÞmþ1

�l�1 ðcos hÞlþ1 þPm�1

k¼1

�l�m�2ð Þ �l�m�4ð Þ�...� �l�m�2kð Þ�l�3ð Þ �l�5ð Þ�...� �l�2k�1ð Þ ðcos hÞlþ2kþ1

þ �l�m�2ð Þ �l�m�4ð Þ�...� �mð Þ�l�1ð Þ!! ZðSÞ l; hð Þ;

264

375; m ¼ 2n; �l ¼ 2m

� ðcos hÞlþ1

lþm ðsin hÞm�1 þPnk¼1

2kC nð Þ lþm�2k�2ð Þ!!C n�kð Þ lþm�2ð Þ!! ðsin hÞm�2k�1

; m ¼ 2nþ 1; l–� 1;�3; . . .

12

Pmk ¼ 0k–n

ð�1ÞkCkmðcos hÞ2k�mþ1

n�k þ �1ð Þnþ1Cnm ln cos hj j; n 6 m

12

Pmk¼0

ð�1ÞkCkmðcos hÞ2k�mþ1

n�k ; n > m

26666664

37777775

; m ¼ 2nþ 1; �l ¼ 2mþ 1

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

; ðA:28Þ

ZðCÞ l; hð Þ ¼

Cnl

h2l þ 1

2l�1

Pn�1

k¼1Ck

lsin l�2kð Þh½ �

l�2k ; l ¼ 2n; l P 0

Pnk¼0

Ckn�1ð Þk

2kþ1 sin hð Þ2kþ1; a ¼ 2nþ 1; a > 0

� sin hlþ1 cos hð Þlþ1 þ sumn�1

k¼12kn n�1ð Þ... n�kþ1ð Þ

2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þ cos hð Þlþ2kþ1� �

; �l ¼ 2n; l < 0

� sin hlþ1 cos hð Þlþ1 þ

Pn�2

k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ2kn n�1ð Þ... n�kþ1ð Þ

cos hð Þlþ2kþ1

þ 2n�1ð Þ!!2nn!

ln tan h2

� �� ; �l ¼ 2nþ 1; l < 0

8>>>>>>>>>>>><>>>>>>>>>>>>:

; ðA:29Þ

ZðSÞ m; hð Þ ¼

m�1ð Þ!!h2m2 m

2ð Þ!� cos h

m sin hð Þm�1 þ 1C m

2ð ÞPn�1

k¼1C m

2� kþ 1� � m�1ð Þ m�3ð Þ... m�2kþ1ð Þ sin hð Þm�2k�1

2k

; m ¼ 2n; m > 0

Pnk¼0

Ckn�1ð Þkþ1

2kþ1 cos hð Þ2kþ1; m ¼ 2nþ 1; m > 0

cos hmþ1 sin hð Þmþ1 þ

Pn�1

k¼1

2kn n�1ð Þ... n�kþ1ð Þ2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þ sin hð Þmþ2kþ1

; �m ¼ 2n; m < 0

cos hm�1 sin hð Þmþ1 þ

Pn�2

k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ2kn n�1ð Þ... n�kþ1ð Þ

cos hð Þmþ2kþ1

þ 2n�1ð Þ!!2nn!

ln tan h2

� �� ; �m ¼ 2nþ 1; m 6 0

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

; ðA:30Þ

�
where dn;m ¼
1;n ¼ m0;n–m

is Kronecker’s delta, Cmn ¼ n!

m! n�mð Þ! is the binomial coefficients,� �
H ¼ arctan
ah; ðA:31Þ


and

C0mn ¼�1ð Þn nþm� 1ð Þ!

n! m� 1ð Þ! : ðA:32Þ

References

[1] M.I. Mishchenko, L.D. Travis, D.W. Mackowski, J. Quant. Spectrosc. Rad. Transfer 55 (1996) 535.[2] D. Petrov, E. Synelnyk, Yu. Shkuratov, G. Videen, J. Quant. Spectrosc. Rad. Transfer 102 (2006) 85.[3] D. Petrov, Y. Shkuratov, G. Videen, Opt. Lett. 32 (2007) 1168.[4] D.V. Petrov, Yu. G. Shkuratov, G. Videen, J. Opt. Soc. Am. A 24 (2007) 1103.[5] D. Petrov, Yu. Shkuratov, E. Zubko, G. Videen, J. Quant. Spectrosc. Rad. Transfer 106 (2007) 437.[6] D. Petrov, G. Videen, Yu. Shkuratov, M. Kaydash, J. Quant. Spectrosc. Rad. Transfer 108 (2007) 81, doi:10.1016/j.jqsrt.2007.04.010.[7] D. Petrov, Yu. Shkuratov, G. Videen, Opt. Comm. (2008).[8] D. Petrov, Yu. Shkuratov, G. Videen, J. Quant. Spectrosc. Rad. Transfer 109 (2008) 1474.[9] D. Petrov, Y. Shkuratov, G. Videen, J. Quant. Spectrosc. Radiative Transfer 109 (2008) 650.

[10] P. Chylek, G. Lesins, G. Videen, J. Wong, R.G. Pinnick, D. Ngo, J.D. Klett, J. Geophys. Res. 101 (D18) (1996). 23, 365, 371.[11] G. Videen, D. Ngo, P. Chylek, R.G. Pinnick, J. Opt. Soc. Am. A 12 (1995) 922.[12] G. Videen, D. Ngo, J. Biomed. Opt. 3 (2) (1998) 212.[13] M.I. Mishchenko, L.D. Travis, A.A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press, Cambridge, 2002. 690 p..[14] E. Zubko, Y. Shkuratov, M. Mishchenko, G. Videen, J. Quant. Spectrosc. Rad. Transfer 109 (2008) 2195.

http://dx.doi.org/10.1016/j.jqsrt.2007.04.010

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