Abstract

We present a method for determination of the random-orientation polarimetric scattering properties of an arbitrary, nonsymmetric cluster of spheres. The method is based on calculation of the cluster T matrix, from which the orientation-averaged scattering matrix and total cross sections can be analytically obtained. An efficient numerical method is developed for the T-matrix calculation, which is faster and requires less computer memory than the alternative approach based on matrix inversion. The method also allows calculation of the random orientation scattering properties of a cluster in a fraction of the time required for numerical quadrature. Numerical results for the random orientation scattering matrix are presented for sphere ensembles in the form of densely packed clusters and linear chains.

© 1996 Optical Society of America

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    [CrossRef]
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  27. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
    [CrossRef]

1995 (5)

1994 (2)

1992 (1)

1991 (3)

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc.R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 33, 4716–4731 (1991).
[CrossRef]

1990 (1)

1989 (1)

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

1988 (3)

1986 (3)

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

1984 (2)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

1975 (1)

1974 (1)

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

1971 (2)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Barber, P. W.

Bohren, C. F.

Borghese, F.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Brunning, J. H.

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Denti, P.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Domke, H.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

Draine, B. T.

Edmunds, A. R.

A. R. Edmunds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957), Chap. 4.

Flatau, P. J.

Fuller, K. A.

Ghoul, W. A.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Hovenier, J. W.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.

Kattawar, G. W.

Khlebtsov, N. G.

Lo, Y. T.

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Mackowski, D. W.

McClain, W. M.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Mishchenko, M. I.

Saija, R.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Salzman, G. C.

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Sindoni, O. I.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Singham, M. K.

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Singham, S. B.

S. B. Singham, C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled dipole method,” J. Opt. Soc. Am. A 11, 1867–1872 (1988).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

Stephens, G. L.

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

Travis, L. D.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Xu, Y.-L.

Yeh, C.

Aerosol Sci. Technol. (1)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Appl. Opt. (6)

Astrophys. J. (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Astrophys. Space Sci. (1)

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. Aerosol Sci. (1)

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from nonspherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

J. Chem. Phys. (3)

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Lett. (2)

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Proc.R. Soc. London Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc.R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Other (2)

A. R. Edmunds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957), Chap. 4.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.

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Figures (5)

Fig. 1
Fig. 1

Orientation-averaged scattering matrix elements for a linear chain of spheres.

Fig. 2
Fig. 2

Orientation-averaged scattering matrix elements for a packed cluster of spheres.

Fig. 3
Fig. 3

(a) Asymmetric and (b) symmetric tetrahedral-lattice clusters of eight spheres.

Fig. 4
Fig. 4

Orientation-averaged scattering matrix elements for the symmetric and asymmetric clusters.

Fig. 5
Fig. 5

Variation in fixed-orientation matrix elements for the eight-sphere symmetric cluster. The dotted area represents 1 standard deviation.

Equations (84)

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E s = i = 1 N S E i , s ,
E s , i = n = 1 m = n n p = 1 2 a mnp i h mnp ( r i ) .
a mnp i + a ¯ np i j = 1 j i N S l = 1 N O , i k = l 1 q = 1 2 H mnp klq ij a klq j = a ¯ np i p mnp i .
a mnp i = j = 1 N S l = 1 N O , i k = l l q = 1 2 T mnp klq ij p klq j .
T nl = i = 1 N S j = 1 N S n = 1 N O , i l = 1 N O , i J n n 0 i T n l ij J l l j 0 .
E θ s = i kr exp ( ikr ) n = 1 N O m = n n p = 1 2 × ( i ) n + 1 a mnp τ mnp ( θ ) exp ( im ϕ ) ,
E ϕ s = 1 kr exp ( ikr ) n = 1 N O m = n n p = 1 2 × ( i ) n + 1 a mnp τ mn 3 p ( θ ) exp ( im ϕ ) .
a mnp = l = 1 N O k = l l q = 1 2 T mnpklq p klq ,
τ mn 1 ( θ ) = d d θ P n m ( cos θ ) ,
τ mn 2 ( θ ) = m sin θ P n m ( cos θ ) .
τ mnp ( θ ) exp ( im ϕ ) = exp ( im α ) k = n n D kn m ( β ) × exp ( ik γ ) τ knp ( θ ) exp ( ik ϕ ) ,
a mnp = exp ( im γ ) k = n n D mn k ( β ) exp ( ik α ) a knp .
p mnp ( α , β , γ ) = exp ( im α ) k = n n D mn k ( β ) × exp [ ik ( γ + Φ ) ] p knp ( 0 , 0 , 0 ) ,
p 1 n 1 = i n + 1 2 2 n + 1 n ( n + 1 ) , p 1 n 1 = i n + 1 2 ( 2 n + 1 ) , p 1 n 2 = p 1 n 1 , p 1 n 2 = p 1 n 1 , p mnp = q mnp = 0 , | m | 1 .
D mn k ( β ) = ( 1 ) k + m D kn m ( β )
a mnp = ( 1 ) k + s exp [ i ( m k ) γ ] exp [ i ( t s ) α ] × D mn t ( β ) T tnpslq D kl s ( β ) exp ( ik Φ ) p klq .
S 1 = τ mn 3 p ( θ ) ( i ) n a mnp 2 ,
S 2 = τ mnp ( θ ) ( i ) n + 1 a mnp 1 ,
S 3 = τ mnp ( θ ) ( i ) n + 1 a mnp 2 ,
S 4 = τ mn 3 p ( θ ) ( i ) n a mnp 1 ,
D mn t ( β ) D kl s ( β ) = w = | n l | n + l ( 1 ) n + l + w × Ĉ tn , sl w Ĉ mn , kl w D m kw t s ( β ) .
1 8 π 2 0 2 π 0 2 π 0 π exp [ i ( u u ) γ ] × exp [ i ( υ υ ) α ] D uw υ ( β ) D u w υ ( β ) sin β d β d α d γ = 1 2 w + 1 f υ w f uw δ u u δ υ υ δ w w ,
f υ w = ( w + υ ) ! ( w υ ) ! ,
i n n ( a u + knp 1 a u + k n p 1 * + a u + knp 2 a u + k n p 2 * ) = 2 D knpk n p u δ u u δ k k ,
i n n ( a u + knp 1 a u + k n p 1 * a u + knp 2 a u + k n p 2 * ) = 2 D knp k n p u δ u u δ k k ,
i n n a u + knp 2 a u + k n p 1 * = i k D knp k n p u δ u u .
D knp k n p u = w = | n n | n + n 1 2 w + 1 × υ = w w f υ w f uw B knp u υ w B k n p u υ w * ,
B knp u υ w = l = L 1 L 2 ( 1 ) n + l A kl np υ w Ĉ u kn , kl w ,
A kl np υ w = i n t = T 1 T 2 q = 1 2 ( 1 ) t Ĉ tn , υ tl w T tnpt υ lq p klq .
L 1 = max ( 1 , | w n | ) , L 2 = min ( N O , w + n ) , T 1 = max ( n , l + υ ) , T 2 = min ( n , l + υ ) .
τ mnp = 1 2 [ n ( n + 1 ) D 1 n m + ( 1 ) p D 1 n m ] .
S 11 = Re [ w = 0 2 N O ( a 0 , 1 , w + a 0 , 1 , w ) D 0 w 0 ( θ ) ] ,
S 14 = Re [ w = 0 2 N O ( a 0 , 1 , w a 0 , 1 , w ) D 0 w 0 ( θ ) ] ,
S 44 = Re [ w = 0 2 N O ( b 0 , 1 , w b 0 , 1 , w ) D 0 w 0 ( θ ) ] ,
S 42 = Re [ w = 0 2 N O ( b 0 , 1 , w + b 0 , 1 , w ) D 0 w 0 ( θ ) ] ,
S 12 = Re [ w = 2 2 N O ( a 2 , 1 , w + a 2 , 1 , w ) D 0 w 2 ( θ ) ] ,
S 24 = Re [ w = 2 2 N O ( a 2 , 1 , w a 2 , 1 , w ) D 0 w 2 ( θ ) ] ,
S 34 = Im [ w = 2 2 N O ( b 2 , 1 , w b 2 , 1 , w ) D 0 w 2 ( θ ) ] ,
S 31 = Im [ w = 2 2 N O ( b 2 , 1 , w + b 2 , 1 , w ) D 0 w 2 ( θ ) ] ,
S 13 = 2 Im [ w = 2 2 N O a 2 , 0 , w D 0 w 2 ( θ ) ] ,
S 41 = 2 Re [ w = 2 2 N O b 2 , 0 , w D 0 w 2 ( θ ) ] ,
S 22 = Re { w = 2 2 N O [ c 20 w D 2 w 2 ( θ ) + c 20 w D 2 w 2 ( θ ) ] } ,
S 23 = Im { w = 2 2 N O [ c 20 w D 2 w 2 ( θ ) + c 20 w D 2 w 2 ( θ ) ] } ,
S 33 = Re { w = 2 2 N O [ c 20 w D 2 w 2 ( θ ) c 20 w D 2 w 2 ( θ ) ] } ,
S 32 = Im { w = 2 2 N O [ c 20 w D 2 w 2 ( θ ) c 20 w D 2 w 2 ( θ ) ] } .
a 0 , k , w = n , p , n f 1 n Ĉ 1 n , 1 n w F knp k n p w ,
b 0 , k , w = n , p , n f 1 n Ĉ 1 n , 1 n w F knp k n 3 p w ,
a 2 , k , w = n , p , n ( 1 ) p Ĉ 1 n , 1 n w F knp k n p w ,
b 2 , k , w = n , p , n ( 1 ) 3 p Ĉ 1 n , 1 n w F knp k n 3 p w ,
a 2 , 0 , w = n , p , n f 1 n Ĉ 1 n , 1 n w F 1 np 1 n p w ,
b 2 , 0 , w = n , p , n f 1 n Ĉ 1 n , 1 n w F 1 np 1 n 3 p w ,
c 2 , 0 , w = f 2 w n , p , n , p ( 1 ) p Ĉ 1 n , 1 n w F 1 np 1 n p w ,
c 2 , 0 , w = n , p , n , p ( 1 ) p + n + n + w Ĉ 1 n , 1 n w F 1 np 1 n p w ,
F knp k n p w = 2 u ( 1 ) u f u + k n × Ĉ u + kn , u k n w D knp k n p u .
C sca = 2 π k 2 Re ( a 0 , 1 , 0 + a 0 , 1 , 0 ) .
C sca cos θ = 2 π 3 k 2 Re ( a 0 , 1 , 1 + a 0 , 1 , 1 ) .
C ext = 2 π k 2 Re ( n , m , p T mnp mnp ) .
C sca = 2 π k 2 n , m , p k , l , q n ( n + 1 ) ( 2 l + 1 ) f mn l ( l + 1 ) ( 2 n + 1 ) f kl | T mnp klq | 2 .
[ I ā 1 H 12 ā 1 H 13 . . . ā 2 H 21 I ā 2 H 23 . . . ā 3 H 31 ā 3 H 32 I . . . ] [ T 11 T 12 T 13 . . . T 21 T 22 T 23 . . . T 31 T 32 T 33 . . . ] = [ a 1 I 0 0 . . . 0 ā 2 I 0 . . . 0 0 ā 3 I . . . ] ,
[ I ā 1 H 12 ā 1 H 13 . . . ā 2 H 21 I ā 2 H 23 . . . ā 3 H 31 ā 3 H 32 I . . . ] [ T 11 T 12 T 13 . . . T 21 T 22 T 23 . . . T 31 T 32 T 33 . . . ] × ( J 10 J 20 J 30 ) = [ ā 1 I 0 0 . . . 0 ā 2 I 0 . . . 0 0 ā 3 I . . . ] ( J 10 J 20 J 30 ) ,
T i = j = 1 N S T ij J j 0 ,
[ I ā 1 H 12 ā 1 H 13 . . . ā 2 H 21 I ā 2 H 23 . . . ā 3 H 31 ā 3 H 32 I . . . ] ( T 1 T 2 T 3 ) = ( ā 1 J 10 ā 2 J 20 ā 3 J 30 ) .
T mnp klq i = ā np i J mnp klq i 0 ā np i H mnp m n p ij T m n p klq j .
T mnp klq = J mnp m n p 0 i T m n p klq i .
T mnp klq = ( 1 ) m + k l ( l + 1 ) ( 2 n + 1 ) n ( n + 1 ) ( 2 l + 1 ) T klq mnp .
Q ext , i = 2 x i 2 Re ( n , m , p l , k , q J klq mnp 0 i T mnp klq i ) ,
Q abs , i = 2 x i 2 n , m , p l , k , q d ¯ np i n ( n + 1 ) ( 2 l + 1 ) f mn l ( l + 1 ) ( 2 n + 1 ) f kl × T mnp klq i T mnp klq i * ,
H ij = H mnp klq ( r ij , θ ij , ϕ ij ) = ( 1 ) m exp [ i ( k m ) ϕ ij ] m ( 1 ) m × D mn m ( θ ij ) H m np m lq ( r ij , 0 , 0 ) D m k ( θ ij ) ,
H mnp mlp ij = z ij [ n + m + 1 ( n + 1 ) ( 2 n + 3 ) C mn + 1 ml ij + n m n ( 2 n 1 ) C mn 1 ml ij ] + C mn ml ij ,
H mnp ml 3 p ij = z ij im n ( m + 1 ) C mn ml ij .
C mn ml ij = ( 1 ) m i n 1 ( 2 n + 1 ) × w i w a ( m , l ; m , n ; w ) h w ( r ij ) ,
P l m ( cos θ ) P n m ( cos θ ) = w a ( m , l ; m , n ; w ) P w ( cos θ ) ,
a ( m , l ; m , n ; w ) = ( 1 ) n + l + w Ĉ ml , mn w Ĉ 0 l , 0 n w .
D kn m = ( 1 ) m + k [ ( n k ) ! ( n + m ) ! ( n + k ) ! ( n m ) ! ] 1 / 2 d km ( n ) .
D kn 0 ( β ) = P n k ( cos β ) ,
D kn m + 1 = cos 2 ( β / 2 ) D k 1 n m ( n k ) ( n + k + 1 ) × sin 2 ( β / 2 ) D k + 1 n m k ( sin β ) D k n m .
Ĉ mn , kl w = ( f mn f kl f m + k w ) 1 / 2 C mn , kl m + k w .
Ĉ mn , kl w = g nlw ( n + w ) ! ( l + k ) ! ( w m k ) ! S mn , kl w ,
g nlw = [ ( 2 w + 1 ) ( n + l w ) ! ( w + n l ) ! ( w + l n ) ! ( n + l + w + 1 ) ! ] 1 / 2 .
S mn , kl w 1 = b w S mn , kl w + c w S mn , kl w + 1 ,
b w = ( 2 w + 1 ) { ( m k ) w ( w + 1 ) ( m + k ) [ n ( n + 1 ) l ( l + 1 ) ] } ( w + 1 ) ( n + l w + 1 ) ( n + l + w + 1 ) ,
c w = w ( w + n l + 1 ) ( w + l n + 1 ) ( w + m + k + 1 ) ( w m k + 1 ) ( w + 1 ) ( n + l w + 1 ) ( n + l + w + 1 ) ,
S mn , kl n + l + 1 = 0 , S mn , kl n + l = 1 ( n m ) ! ( l + k ) ! ( n + m ) ! ( l k ) ! .
S mn , kl w = ( 1 ) w + n + k S m + kw , kl n = ( 1 ) w + n + k S kl , m k w n = ( 1 ) n + m S mn , m k w l = ( 1 ) n + m S m + k w , mn l .

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