Abstract

We develop a modification of the T-matrix method that allows for fast calculations of scattering properties of particles with irregular shapes. This modification uses the so-called Sh matrices, the elements of which depend on the shape of particles and do not depend on the particle size or optical constants; i.e., the introduction of Sh matrices makes possible the separation of these parameters within the T-matrix algorithm. For a given shape of a scattering object we calculate the Sh matrices only once and then can quickly calculate the T-matrix elements for a number of sizes and refractive indices. This, in particular, can provide rapid particle-size and refractive index averaging in a particle ensemble. This separation is useful for the derivation of an analytical light-scattering solution for Chebyshev particles.

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  1. L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  2. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
    [CrossRef]
  3. D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
    [CrossRef]
  4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  5. F. M. Kahnert, J. J. Stamnes, and K. Stamnes, "Application of the extended boundary condition method to homogeneous particles with paint-group symmetries," Appl. Opt. 40, 3110-3123 (2001).
    [CrossRef]
  6. M. Kahnert, "Irreducible representations of finite groups in the T -matrix formulation of the electromagnetic scattering problem," J. Opt. Soc. Am. A 22, 1187-1199 (2005).
    [CrossRef]
  7. S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
    [CrossRef]
  8. H. Laitinen and K. Lumme, "T-matrix method for general star-shaped particles: first results," J. Quant. Spectrosc. Radiat. Transf. 60, 325-334 (1998).
    [CrossRef]
  9. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

2006 (1)

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

2005 (1)

2003 (1)

S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
[CrossRef]

2002 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

2001 (1)

1998 (1)

H. Laitinen and K. Lumme, "T-matrix method for general star-shaped particles: first results," J. Quant. Spectrosc. Radiat. Transf. 60, 325-334 (1998).
[CrossRef]

1996 (1)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

1985 (1)

L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

1964 (1)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Baran, A.

S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
[CrossRef]

Edwards, J.

S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
[CrossRef]

Havemann, S.

S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
[CrossRef]

Kahnert, F. M.

Kahnert, M.

Kong, J.

L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Laitinen, H.

H. Laitinen and K. Lumme, "T-matrix method for general star-shaped particles: first results," J. Quant. Spectrosc. Radiat. Transf. 60, 325-334 (1998).
[CrossRef]

Lumme, K.

H. Laitinen and K. Lumme, "T-matrix method for general star-shaped particles: first results," J. Quant. Spectrosc. Radiat. Transf. 60, 325-334 (1998).
[CrossRef]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

Petrov, D.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

Shin, R.

L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Shkuratov, Yu.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

Stamnes, J. J.

Stamnes, K.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Synelnyk, E.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

Tsang, L.

L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Videen, G.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (4)

S. Havemann, A. Baran, and J. Edwards, "Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods," J. Quant. Spectrosc. Radiat. Transf. 79/80, 707-720 (2003).
[CrossRef]

H. Laitinen and K. Lumme, "T-matrix method for general star-shaped particles: first results," J. Quant. Spectrosc. Radiat. Transf. 60, 325-334 (1998).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, "The T-matrix technique for calculations of scattering properties of ensembles of randomly oriented particles with different size," J. Quant. Spectrosc. Radiat. Transf. 102, 85-110 (2006).
[CrossRef]

Other (3)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

L. Tsang, J. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

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Figures (5)

Fig. 1
Fig. 1

Degree of linear polarization of an ensemble of cubes with size averaging over the interval X = 5.0 , , 3.0 and with refractive-index averaging over the interval m 0 = 1.5 , , 2.0 (solid curve). For a comparison we show a curve for a cube with X = 4.0 ; m 0 = 1.75 (line with circles).

Fig. 2
Fig. 2

Normalized intensity and degree of linear polarization of spheres with X = 3.0 , m 0 = 1.2 and 2.0 (dashed and dotted-dashed curves), and with X = 3.0 and homogeneous refractive-index averaging m 0 = 1.2 , , 2.0 (dotted and solid curves represent T-matrix and Mie theory results).

Fig. 3
Fig. 3

Examples of an ensemble of Chebyshev particles: (a) p θ = p φ = 10 , ζ = ξ = 0.005 , (b) ζ = ξ = 0.01 , (c) ζ = ξ = 0.05 , (d) ζ = ξ = 0.1 .

Fig. 4
Fig. 4

Comparison of calculation times for an ensemble of Chebyshev particles at p θ = p φ = 2 using the numerical integration and the analytical solution as a function of the number of expansion terms N max in Eqs. (1, 2). Circles and squares correspond to Chebyshev particles at ζ = ξ = 0.05 and ζ = ξ = 0.5 , respectively. Line with plus signs corresponds to analytical solution time.

Fig. 5
Fig. 5

Normalized intensity and degree of linear polarization of an ensemble of Chebyshev particles with X = 5.0 ; m 0 = 1.313 and p θ = p φ = 10 ; ζ = ξ = 0.001 , , 0.009 .

Equations (124)

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E i n c ( ρ , γ , ϕ ) = n = 1 m = n n [ a m n Rg M m n ( ρ , γ , ϕ ) + b m n Rg N m n ( ρ , γ , ϕ ) ] ,
E s c a ( ρ , γ , ϕ ) = n = 1 m = n n [ p m n M m n ( ρ , γ , ϕ ) + q m n N m n ( ρ , γ , ϕ ) ] ,
p m n = n = 1 m = n n [ T m n m n 11 a m n + T m n m n 12 b m n ] ,
q m n = n = 1 m = n n [ T m n m n 21 a m n + T m n m n 22 b m n ] ,
T m n m n = [ T m n m n 11 T m n m n 12 T m n m n 21 T m n m n 22 ] .
T m n m n = ( Rg Q m n m n ) ( Q m n m n ) 1 .
Rg Q m n m n = [ Rg Q m n m n 11 Rg Q m n m n 12 Rg Q m n m n 21 Rg Q m n m n 22 ] ,
Q m n m n = [ Q m n m n 11 Q m n m n 12 Q m n m n 21 Q m n m n 22 ] ,
Q m n m n 11 = i ( m 0 J m n m n 21 + J m n m n 12 ) ,
Q m n m n 12 = i ( m 0 J m n m n 11 + J m n m n 22 ) ,
Q m n m n 21 = i ( m 0 J m n m n 22 + J m n m n 11 ) ,
Q m n m n 22 = i ( m 0 J m n m n 12 + J m n m n 21 ) ,
Rg Q m n m n 11 = i ( m 0 Rg J m n m n 21 + Rg J m n m n 12 ) ,
Rg Q m n m n 12 = i ( m 0 Rg J m n m n 11 + Rg J m n m n 22 ) ,
Rg Q m n m n 21 = i ( m 0 Rg J m n m n 22 + Rg J m n m n 11 ) ,
Rg Q m n m n 22 = i ( m 0 Rg J m n m n 12 + Rg J m n m n 21 ) .
j ν ( η i z ) = η i ν k = 0 ( ( 1 ) k ( η i 2 1 ) k ( z 2 ) k k ! ) j ν + k ( z ) .
Rg J m n m n 11 ( X , m 0 ) = X n + n + 2 ( m 0 ) n k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( n + k 2 + 3 2 ) Rg S h m n m n , k 1 + k 2 11 ,
Rg J m n m n 12 ( X , m 0 ) = X n + n + 1 ( m 0 ) n k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( n + k 2 + 3 2 ) ( Rg S h m n m n , k 1 + k 2 121 + X 2 n + k 2 + 3 2 Rg S h m n m n , k 1 + k 2 122 ) ,
Rg J m n m n 21 ( X , m 0 ) = X n + n + 1 ( m 0 ) n 1 k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( n + k 2 + 3 2 ) ( Rg S h m n m n , k 1 + k 2 211 + ( X m 0 ) 2 n + k 1 + 3 2 Rg S h m n m n , k 1 + k 2 212 ) ,
Rg J m n m n 22 ( X , m 0 ) = X n + n ( m 0 ) n 1 k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( n + k 2 + 3 2 ) × ( Rg S h m n m n , k 1 + k 2 221 + ( X m 0 ) 2 n + k 1 + 3 2 Rg S h m n m n , k 1 + k 2 222 + X 2 n + k 2 + 3 2 Rg S h m n m n , k 1 + k 2 223 + X 2 ( X m 0 ) 2 ( n + k 1 + 3 2 ) ( n + k 2 + 3 2 ) Rg S h m n m n , k 1 + k 2 224 ) ,
J m n m n 11 ( X , m 0 ) = Rg J m n m n 11 ( X , m 0 ) + X n n + 1 ( m 0 ) n k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( k 2 n + 1 2 ) S h m n m n , k 1 + k 2 11 ,
J m n m n 12 ( X , m 0 ) = Rg J m n m n 12 ( X , m 0 ) + X n n ( m 0 ) n k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( k 2 n + 1 2 ) ( S h m n m n , k 1 + k 2 121 + X 2 k 2 n + 1 2 S h m n m n , k 1 + k 2 122 ) ,
J m n m n 21 ( X , m 0 ) = Rg J m n m n 21 ( X , m 0 ) + X n n ( m 0 ) n 1 k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( k 2 n + 1 2 ) ( S h m n m n , k 1 + k 2 211 + ( X m 0 ) 2 n + k 1 + 3 2 S h m n m n , k 1 + k 2 212 ) ,
J m n m n 22 ( X , m 0 ) = Rg J m n m n 22 ( X , m 0 ) + X n n 1 ( m 0 ) n 1 k 1 = 0 ( X m 0 ) 2 k 1 k 1 ! Γ ( n + k 1 + 3 2 ) k 2 = 0 ( X ) 2 k 2 k 2 ! Γ ( k 2 n + 1 2 ) × ( S h m n m n , k 1 + k 2 221 + ( X m 0 ) 2 n + k 1 + 3 2 S h m n m n , k 1 + k 2 222 + X 2 k 2 n + 1 2 S h m n m n , k 1 + k 2 223 + X 2 ( X m 0 ) 2 ( n + k 1 + 3 2 ) ( k 2 n + 1 2 ) S h m n m n k 1 + k 2 224 ) ,
X i = X min + ( X max X min ) i N , i = 0 , , N .
m 0 j = m 0 min + ( m 0 max m 0 min ) j N , j = 0 , , N .
C s c a = 1 N N i = 0 N f ( X i ) j = 0 N g ( m 0 j ) C s c a ( X i , m 0 j ) ,
α p s = 1 N N C s c a i = 0 N f ( X i ) j = 0 N g ( m 0 j ) C s c a ( X i , m 0 j ) α p s ( X i , m 0 j ) , p = 1 , , 4 ,
β p s = 1 N N C s c a i = 0 N f ( X i ) j = 0 N g ( m 0 j ) C s c a ( X i , m 0 j ) β p s ( X i , m 0 j ) , p = 1 , 2 ,
R 0 ( θ , φ ) = ( 1 + ζ T ̃ p θ ( cos θ ) ) ( 1 + ξ T ̃ p φ ( cos φ ) ) ,
T ̃ n ( z ) = n 2 m = 0 [ N 2 ] ( 1 ) m ( n m 1 ) ! m ! ( n 2 m ) ! ( 2 z ) n 2 m .
Rg J m n m n 11 = i ( 1 ) m m A n n 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R 2 } ,
Rg J m n m n 12 = ( 1 ) m m A n n [ 0 π d θ { sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R 2 } + 0 π d θ { sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R θ } + i 0 π d θ { d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R φ } ] ,
Rg J m n m n 21 = ( 1 ) m m 1 A n n [ 0 π d θ { sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R 2 } + 1 m 0 0 π d θ { sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R θ } i m 0 0 π d θ { d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R φ } ] ,
Rg J m n m n 22 = i ( 1 ) m m A n n [ 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R 2 } + 0 π d θ { sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R θ } + i 0 π d θ { d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R φ } + 1 m 0 0 π d θ { sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R θ } i m 0 0 π d θ { d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) j n ( R ) R φ } ] ,
J m n m n 11 = i ( 1 ) m m A n n 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R 2 } ,
J m n m n 12 = ( 1 ) m m A n n [ 0 π d θ { sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R 2 } + 0 π d θ { sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R θ } + i 0 π d θ { d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R φ } ] ,
J m n m n 21 = ( 1 ) m m A n n [ 0 π d θ { sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R 2 } + 1 m 0 0 π d θ { sin θ d m n ( θ ) τ m n ( θ ) j = 0 q 1 exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R θ } i m 0 0 π d θ { d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R φ } ] ,
J m n m n 22 = i ( 1 ) m m A n n [ 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R 2 } + 0 π d θ { sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R θ } + i 0 π d θ { d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R φ } + 1 m 0 0 π d θ { sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R θ } i m 0 0 π d θ { d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] j n ( m 0 R ) h n ( R ) R φ } ] ,
π m n ( θ ) = m sin θ d 0 m n ( θ ) ,
τ m n ( θ ) = d d θ d 0 m n ( θ ) ,
d m n s ( θ ) = ( s + m ) ! ( s m ) ! ( s + n ) ! ( s n ) ! k = 0 min ( s + m , s n ) ( 1 ) k ( cos θ 2 ) 2 s 2 k + m n ( sin θ 2 ) 2 k m + n k ! ( s + m k ) ! ( s n k ) ! ( n m k ) ! ,
A n n = ( 2 n + 1 ) 4 π n ( n + 1 ) ( 2 n + 1 ) 4 π n ( n + 1 ) ,
Rg S h m n m n k 11 = i π ( 1 ) m m + k 2 2 k + n + n + 2 A n n 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 2 } ,
Rg S h m n m n k 121 = π ( 1 ) m m + k 2 2 k + n + n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n R 0 θ + i 0 π d θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n R 0 φ ] ,
Rg S h m n m n k 122 = π ( 1 ) m m + k 2 2 k + n + n + 3 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 3 ] ,
Rg S h m n m n k 211 = π ( 1 ) m m + k 2 2 k + n + n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n R 0 θ i 0 π d θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n R 0 φ ] ,
Rg S h m n m n k 212 = Rg S h m n m n k 122 ,
Rg S h m n m n k 221 = i π ( 1 ) m m + k 2 2 k + n + n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n ( n + 1 ) ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n 1 R 0 θ ( n + 1 ) + i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n 1 R 0 φ ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n 1 R 0 θ ( n + 1 ) i 0 π d ( θ ) d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n 1 R 0 φ ( n + 1 ) ] ,
Rg S h m n m n k 222 = i π ( 1 ) m m + k 2 2 k + n + n + 3 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 2 ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 R 0 θ i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 R 0 φ ] ,
Rg S h m n m n k 223 = i π ( 1 ) m m + k 2 2 k + n + n + 3 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 2 ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 R 0 θ i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 1 R 0 φ ] ,
Rg S h m n m n k 224 = i π ( 1 ) m m + k 2 2 k + n + n + 4 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k + n + n + 4 ] ,
S h m n m n k 11 = Rg S h m n m n k 11 + π ( 1 ) m m + n 1 + k 2 2 k + n n + 1 A n n 0 π d θ { sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n + 1 } ,
S h m n m n k 121 = Rg S h m n m n k 121 + i π ( 1 ) m m + n 1 + k 2 2 k + n n + 1 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n n + 0 π d θ sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 1 R 0 θ + i 0 π d θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 1 R 0 φ ] ,
S h m n m n k 122 = Rg S h m n m n k 122 i π ( 1 ) m m n + 1 + k 2 2 k + n n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n + 2 ] ,
S h m n m n k 211 = Rg S h m n m n k 211 i π ( 1 ) m m n 1 + k 2 2 k + n n + 1 A n n [ 0 π d θ sin θ [ π m n ( θ ) π m n ( θ ) + τ m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 1 R 0 θ i 0 π d θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 1 R 0 φ ] ,
S h m n m n k 212 = S h m n m n k 122 ,
S h m n m n k 221 = Rg S h m n m n k 221 + π ( 1 ) m m n 1 + k 2 2 k + n n + 1 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 1 ( n + 1 ) ( n ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 2 R 0 θ ( n + 1 ) + i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 2 R 0 φ ( n + 1 ) 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 2 R 0 θ n + i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n 2 R 0 φ n ] ,
S h m n m n k 222 = Rg S h m n m n k 222 π ( 1 ) m m n 1 + k 2 2 k + n n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n + 1 n + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n R 0 θ + i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n R 0 φ ] ,
S h m n m n k 223 = Rg S h m n m n k 223 π ( 1 ) m m n 1 + k 2 2 k + n n + 2 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n + 1 ( n + 1 ) + 0 π d θ sin θ d m n ( θ ) π m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n R 0 θ i 0 π d θ d m n ( θ ) τ m n ( θ ) 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n R 0 φ ] ,
S h m n m n k 224 = Rg S h m n m n k 224 + π ( 1 ) m m n + 1 + k 2 2 k + n n + 3 A n n [ 0 π d θ sin θ [ π m n ( θ ) τ m n ( θ ) + π m n ( θ ) τ m n ( θ ) ] 0 2 π d φ exp [ i φ ( m m ) ] ( R 0 ) 2 k n + n + 3 ] ,
Rg S h m n m n , k 11 = i A n n π ( 1 ) m m + k 2 2 k + n + n + 2 I m n m n ( 1 ) ( p θ , 2 k + n + n + 2 ) Φ 2 k + n + n + 2 , m m ( 0 ) ,
Rg S h m n m n , k 121 = A n n π ( 1 ) m m + k 2 2 k + n + n + 2 ( n + 1 ) [ I m n m n ( 2 ) ( p θ , 2 k + n + n + 1 ) Φ 2 k + n + n + 1 , m m ( 0 ) L m n m n ( 01 ) ( 2 k + n + n ) Φ 2 k + n + n , m m ( 0 ) + i m n I m n m n ( θ ) ( 2 k + n + n + 1 ) Φ 2 k + n + n , m m ( 2 ) ] ,
Rg S h m n m n , k 122 = A n n π ( 1 ) m m + k 2 2 k + n + n + 3 [ I m n m n ( 2 ) ( p θ , 2 k + n + n + 3 ) Φ 2 k + n + n + 3 , m m ( 0 ) ] ,
Rg S h m n m n , k 211 = A n n π ( 1 ) m m + k 2 2 k + n + n + 2 ( n + 1 ) [ I m n m n ( 2 ) ( p θ , 2 k + n + n + 1 ) Φ 2 k + n + n + 1 , m m ( 0 ) L m n m n ( 00 ) ( 2 k + n + n ) Φ 2 k + n + n , m m ( 0 ) + i m n I m n m n ( θ ) ( 2 k + n + n + 1 ) Φ 2 k + n + n , m m ( 2 ) ] ,
Rg S h m n m n , k 212 = Rg S h m n m n , k 122 ,
Rg S h m n m n , k 221 = i A n n π ( 1 ) m m + k 2 2 k + n + n + 2 ( n + 1 ) ( n + 1 ) [ I m n m n ( 1 ) ( p θ , 2 k + n + n ) Φ 2 k + n + n , m m ( 0 ) + m n I m n m n ( θ ) ( 2 k + n + n 1 ) Φ 2 k + n + n , m m ( 2 ) i m n L m n m n ( 11 ) ( 2 k + n + n ) Φ 2 k + n + n 1 , m m ( 2 ) + m n I m n m n ( θ ) ( 2 k + n + n 1 ) Φ 2 k + n + n , m m ( 0 ) + i L m n m n ( 10 ) ( 2 k + n + n ) Φ 2 k + n + n 1 , m m ( 2 ) ] ,
Rg S h m n m n , k 222 = i A n n π ( 1 ) m m + k 2 2 k + n + n + 3 ( n + 1 ) [ I m n m n ( 1 ) ( p θ , 2 k + n + n + 2 ) Φ 2 k + n + n + 2 , m m ( 0 ) + m n I m n m n ( θ ) ( 2 k + n + n + 1 ) Φ 2 k + n + n + 2 , m m ( 2 ) + i L m n m n ( 11 ) ( 2 k + n + n + 2 ) Φ 2 k + n + n + 1 , m m ( 2 ) ] ,
Rg S h m n m n , k 223 = i A n n π ( 1 ) m m + k 2 2 k + n + n + 3 ( n + 1 ) [ I m n m n ( 1 ) ( p θ , 2 k + n + n + 2 ) Φ 2 k + n + n + 2 , m m ( 0 ) + m n I m n m n ( θ ) ( 2 k + n + n + 1 ) Φ 2 k + n + n + 2 , m m ( 2 ) + i L m n m n ( 10 ) ( 2 k + n + n + 2 ) Φ 2 k + n + n + 1 , m m ( 2 ) ] ,
Rg S h m n m n , k 224 = i A n n π ( 1 ) m m + k 2 2 k + n + n + 4 [ I m n m n ( 1 ) ( p θ , 2 k + n + n + 4 ) Φ 2 k + n + n + 4 , m m ( 0 ) ] ,
S h m n m n , k 11 = Rg S h m n m n , k 11 A n n π ( 1 ) m m + n + k 2 2 k n + n + 1 I m n m n ( 1 ) ( p θ , 2 k n + n + 1 ) Φ 2 k n + n + 1 , m m ( 0 ) ,
S h m n m n , k 121 = Rg S h m n m n , k 121 + i A n n π ( 1 ) m m + n + k 2 2 k n + n + 1 [ n I m n m n ( 2 ) ( p θ , 2 k n + n ) Φ 2 k n + n , m m ( 0 ) + ( n + 1 ) L m n m n ( 01 ) ( 2 k n + n 1 ) Φ 2 k n + n , m m ( 0 ) i m ( n + 1 ) n I m n m n ( θ ) ( 2 k n + n ) Φ 2 k n + n 1 , m m ( 2 ) ] ,
S h m n m n , k 122 = Rg S h m n m n , k 122 + i A n n π ( 1 ) m m + n + k 2 2 k n + n + 2 [ I m n m n ( 2 ) ( p θ , 2 k n + n + 2 ) Φ 2 k n + n + 2 , m m ( 0 ) ] ,
S h m n m n , k 211 = Rg S h m n m n , k 211 + i A n n π ( 1 ) m m + n + k 2 2 k n + n + 1 ( n + 1 ) [ I m n m n ( 2 ) ( p θ , 2 k n + n ) Φ 2 k n + n , m m ( 0 ) L m n m n ( 00 ) ( 2 k n + n 1 ) Φ 2 k n + n , m m ( 0 ) i m n I m n m n ( θ ) ( 2 k n + n ) Φ 2 k n + n 1 , m m ( 2 ) ] ,
S h m n m n , k 212 = S h m n m n , k 122 ,
S h m n m n , k 221 = Rg S h m n m n , k 221 + A n n π ( 1 ) m m + n + k 2 2 k n + n + 1 ( n + 1 ) [ I m n m n ( 1 ) ( p θ , 2 k n + n 1 ) Φ 2 k n + n 1 , m m ( 0 ) + n ( n m ( n + 1 ) m ) I m n m n ( θ ) ( 2 k n + n 2 ) Φ 2 k n + n 1 , m m ( 2 ) + i ( n + 1 ) L m n m n ( 11 ) ( 2 k n + n 1 ) Φ 2 k n + n 2 , m m ( 2 ) + i n T m n m n ( 10 ) ( 2 k n + n 1 ) Φ 2 k n + n 2 , m m ( 2 ) ] ,
S h m n m n , k 222 = Rg S h m n m n , k 222 A n n π ( 1 ) m m + n + k 2 2 k n + n + 2 [ n I m n m n ( 1 ) ( p θ , 2 k n + n + 1 ) Φ 2 k n + n + 1 , m m ( 0 ) m n ( n + 1 ) I m n m n ( θ ) ( 2 k n + n ) Φ 2 k n + n + 1 , m m ( 2 ) + i ( n + 1 ) L m n m n ( 11 ) ( 2 k n + n + 1 ) Φ 2 k n + n , m m ( 2 ) ] ,
S h m n m n , k 223 = Rg S h m n m n , k 223 + A n n π ( 1 ) m m + n + k 2 2 k n + n + 2 ( n + 1 ) [ I m n m n ( 1 ) ( p θ , 2 k n + n + 1 ) Φ 2 k n + n + 1 , m m ( 0 ) + m n I m n m n ( θ ) ( 2 k n + n ) Φ 2 k n + n + 1 , m m ( 2 ) + i L m n m n ( 10 ) ( 2 k n + n + 1 ) Φ 2 k n + n , m m ( 2 ) ] ,
S h m n m n , k 224 = Rg S h m n m n , k 224 A n n π ( 1 ) m m + n + k 2 2 k n + n + 3 [ I m n m n ( 1 ) ( p θ , 2 k n + n + 3 )