Abstract

We apply the recent exact theory of multiple electromagnetic scattering by sphere aggregates to statistically isotropic finite fractal clusters of identical spheres. In the mean-field approximation the usual Mie expansion of the scattered wave is shown to be still valid, with renormalized Mie coefficients as the multipolar terms. We give an efficient method of computing these coefficients, and we compare this mean-field approach with exact results for silica aggregates of fractal dimension 2.

© 1997 Optical Society of America

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  1. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  2. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef] [PubMed]
  3. A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
    [CrossRef]
  4. J. C. Ravey, “Light scattering by aggregates of small dielectric or absorbing spheres,” J. Colloid Interface Sci. 46, 139–146 (1974).
    [CrossRef]
  5. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  6. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).
  7. T. A. Witten, L. M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
    [CrossRef]
  8. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  9. R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
    [CrossRef]
  10. C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. S Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961); O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  12. Y.-L. Xu, “Calculation of the addition coefficients in electro- magnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
    [CrossRef]
  13. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  14. See, e.g., P. R. Weiss, “The application of the Bethe–Peierls method of ferromagnetism,” Phys. Rev. 74, 1493–1504 (1948).
    [CrossRef]
  15. These coefficients appear naturally in the functions τm,n(cos θ) = dPnm(cos θ)/dθ and πm,n(cos θ) = mPnm(cos θ)/sin θ, since τ‖m‖,n = σm,nτm,n and π‖m‖,n = σm,n′πm,n.
  16. V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
    [CrossRef]
  17. W. D. Brown, R. C. Ball, “Computer simulations of chemically limited aggregation,” J. Phys. A 18, 517–521 (1985).
    [CrossRef]

1996

Y.-L. Xu, “Calculation of the addition coefficients in electro- magnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[CrossRef]

1995

R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef] [PubMed]

1994

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

1991

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

1986

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

1985

W. D. Brown, R. C. Ball, “Computer simulations of chemically limited aggregation,” J. Phys. A 18, 517–521 (1985).
[CrossRef]

1981

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

1979

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

1974

J. C. Ravey, “Light scattering by aggregates of small dielectric or absorbing spheres,” J. Colloid Interface Sci. 46, 139–146 (1974).
[CrossRef]

1961

S Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961); O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1948

See, e.g., P. R. Weiss, “The application of the Bethe–Peierls method of ferromagnetism,” Phys. Rev. 74, 1493–1504 (1948).
[CrossRef]

Ball, R. C.

W. D. Brown, R. C. Ball, “Computer simulations of chemically limited aggregation,” J. Phys. A 18, 517–521 (1985).
[CrossRef]

Berry, M. V.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Bohren, C.

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

Botet, R.

R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
[CrossRef]

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Brown, W. D.

W. D. Brown, R. C. Ball, “Computer simulations of chemically limited aggregation,” J. Phys. A 18, 517–521 (1985).
[CrossRef]

Cabane, M.

R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
[CrossRef]

Huffman, D.

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

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Jullien, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Kovacs, J.

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

Mackowski, D. W.

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

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).

Moskovits, M.

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

Percival, I. C.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Rannou, P.

R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
[CrossRef]

Ravey, J. C.

J. C. Ravey, “Light scattering by aggregates of small dielectric or absorbing spheres,” J. Colloid Interface Sci. 46, 139–146 (1974).
[CrossRef]

Sander, L. M.

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

Stein, S

S Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961); O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Tsai, D. P.

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

Weiss, P. R.

See, e.g., P. R. Weiss, “The application of the Bethe–Peierls method of ferromagnetism,” Phys. Rev. 74, 1493–1504 (1948).
[CrossRef]

Witten, T. A.

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Xu, Y.-L.

Y.-L. Xu, “Calculation of the addition coefficients in electro- magnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef] [PubMed]

Appl. Opt.

J. Colloid Interface Sci.

J. C. Ravey, “Light scattering by aggregates of small dielectric or absorbing spheres,” J. Colloid Interface Sci. 46, 139–146 (1974).
[CrossRef]

J. Comput. Phys.

Y.-L. Xu, “Calculation of the addition coefficients in electro- magnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996).
[CrossRef]

J. Phys. A

W. D. Brown, R. C. Ball, “Computer simulations of chemically limited aggregation,” J. Phys. A 18, 517–521 (1985).
[CrossRef]

R. Botet, P. Rannou, M. Cabane, “Sensitivity of some optical properties of fractals to the cut-off functions,” J. Phys. A 28, 297–316 (1995).
[CrossRef]

Opt. Acta

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Phys. Rev.

See, e.g., P. R. Weiss, “The application of the Bethe–Peierls method of ferromagnetism,” Phys. Rev. 74, 1493–1504 (1948).
[CrossRef]

Phys. Rev. Lett.

T. A. Witten, L. M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Physica A

V. M. Shalaev, R. Botet, D. P. Tsai, J. Kovacs, M. Moskovits, “Localization of dipole excitations and giant optical polarizabilities,” Physica A 207, 197–207 (1994).
[CrossRef]

Proc. R. Soc. London Ser. A

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

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Quart. Appl. Math.

S Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961); O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Other

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

These coefficients appear naturally in the functions τm,n(cos θ) = dPnm(cos θ)/dθ and πm,n(cos θ) = mPnm(cos θ)/sin θ, since τ‖m‖,n = σm,nτm,n and π‖m‖,n = σm,n′πm,n.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).

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

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Figures (1)

Fig. 1
Fig. 1

(a) Real and imaginary parts of the exact coefficients d 1 , 1 ( 1 ) ( j ) [defined in Eqs. (6)] for all the monomers of 64-particle random fractal clusters of fractal dimension 2: radius of the particles, 0.5 μm; wavelength, 0.8 μm; complex refractive index, 1.4 + 0.0001i. There are two orthogonal polarizations of the incident wave per cluster and nine different clusters. The mean-field value (0.996, 0.041) is at the intersection of the two lines. (b) The same except for coefficients d 2 , 2 ( 2 ) ( j ) with a mean-field value of (0., 0.).

Equations (36)

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E S ( j ) = i E 0 n = 1 m = - n n i n ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × [ a m , n j N m , n ( 3 ) ( j ) + b m , n j M m , n ( 3 ) ( j ) ] , H S ( j ) = k ω μ m E 0 n = 1 m = - n n i n ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × [ a m , n j M m , n ( 3 ) ( j ) + b m , n j N m , n ( 3 ) ( j ) ] ,
a m , n j = a n { p m , n 0 exp ( i k inc r j o , j ) - l j ν = 1 μ = - ν ν [ a μ , ν l A m , n μ , ν ( l , j ) + b μ , ν l B m , n μ , ν ( l , j ) ] } , b m , n j = b n { q m , n 0 exp ( i k inc r j o , j ) - l j ν = 1 μ = - ν ν [ a μ , ν l B m , n μ , ν ( l , j ) + b μ , ν l B m , n μ , ν ( l , j ) ] } ,
p 1 , n 0 = q 1 , n 0 = ( 1 / 2 ) exp ( - i β ) , p - 1 , n 0 = - q - 1 , n 0 = - 1 2 n ( n + 1 ) exp ( i β ) , p m , n 0 = q m , n 0 = 0             if             m ± 1 ,
A m , n μ , ν ( l , j ) = ( - 1 ) m 2 ν + 1 2 n ( n + 1 ) p = n - ν n + ν i p [ n ( n + 1 ) + ν ( ν + 1 ) - p ( p + 1 ) ( 2 p + 1 ) 2 × [ - 1 1 P ν - μ ( x ) P n m ( x ) P p m - μ ( x ) d x ] × h p ( 1 ) ( k r l , j ) P p μ - m ( cos θ l , j ) exp [ i ( μ - m ) ϕ l , j ] , B m , n μ , ν ( l , j ) = ( - 1 ) m 2 ν + 1 2 n ( n + 1 ) p = n - ν n + ν i p ( 2 p + 1 ) × { - 1 1 P ν - μ ( x ) [ m P n m ( x ) d P p m - μ ( x ) d x + ( m - μ ) P p m - μ ( x ) d P n m d x ] d x } × h p ( 1 ) ( k r l , j ) P p μ - m ( cos θ l , j ) exp [ i ( μ - m ) ϕ l , j ] .
a m , n j exp ( - i k inc r j o , j ) = a n p m , n 0 , b m , n j exp ( - i k inc r j o , j ) = b n q m , n 0 ,
d m , n ( 1 ) ( j ) = 2 ς m , n a m , n j exp ( - i k inc r j o , j ) exp ( i m β ) , d m , n ( 2 ) ( j ) = 2 ς m , n b m , n j exp ( - i k inc r j o , j ) exp ( i m β ) ,
ς m , n = ς m , n = 1 , ς - m , n = - ς - m , n = ( - 1 ) m ( n + m ) ! / ( n - m ) !             for             m 1 ,
d ¯ m , n ( 1 ) = a n { δ ( m ± 1 ) - ν = 1 μ = - ν ν l j × [ d ¯ μ , ν ( 1 ) A m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν + d ¯ μ , ν ( 2 ) B m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν ] } , d ¯ m , n ( 2 ) = b n { δ ( m ± 1 ) - ν = 1 μ = - ν ν l j × [ d ¯ μ , ν ( 1 ) B m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν + d ¯ μ , ν ( 2 ) A m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν ] } .
l j A m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν ( N - 1 ) A m , n μ , ν ( l , j ) exp ( i k inc r j , l ) ς m , n ς μ , ν , ,
P ( r ) c D 4 π R g 3 ( r R g ) D - 3 exp [ - c ( r / R g ) D ]
A m , n m , ν ( l , j ) ς m , n ς m , v = A - m , n - m , ν ( l , j ) ς - m , n ς - m , ν = A m , n m , ν ( l , j ) ς m , n ς m , ν = A - m , n - m , ν ( l , j ) ς - m , n ς - m , ν , B m , n m , ν ( l , j ) ς m , n ς m , v = B - m , n - m , ν ( l , j ) ς - m , n ς - m , ν = B m , n m , ν ( l , j ) ς m , n ς m , ν = B - m , n - m , ν ( l , j ) ς - m , n ς - m , ν .
d ¯ 1 , n ( 1 ) = a n { 1 - ( N - 1 ) ν = 1 [ A ¯ 1 , n 1 , ν d ¯ 1 , ν ( 1 ) + B ¯ 1 , n 1 , ν d ¯ 1 , ν ( 2 ) ] } , d ¯ 1 , n ( 2 ) = b n { 1 - ( N - 1 ) ν = 1 [ B ¯ 1 , n 1 , ν d ¯ 1 , ν ( 1 ) + A ¯ 1 , n 1 , ν d ¯ 1 , ν ( 2 ) ] } ,
A ¯ 1 , n 1 , ν = 2 ν + 1 n ( n + 1 ) ν ( ν + 1 ) p = n - ν n + ν [ n ( n + 1 ) + ν ( ν + 1 ) - p ( p + 1 ) ] a ( ν , n ; p ) S p ( k R g ) , B ¯ 1 , n 1 , ν = 2 2 ν + 1 n ( n + 1 ) ν ( ν + 1 ) p = n - ν n + ν b ( ν , n ; p ) S p ( k R g ) ,
a ( ν , n ; p ) = 2 p + 1 2 - 1 1 P ν 1 ( x ) P n 1 ( x ) P p ( x ) d x ,
b ( ν , n ; p ) = 2 p + 1 2 - 1 1 P ν 1 ( x ) P n 1 ( x ) d P p ( x ) d x d x ,
S p ( k R g ) = π 2 k 3 0 u J p + ( 1 / 2 ) ( u ) H p + ( 1 / 2 ) ( 1 ) P ( u / k ) d u .
( N - 1 ) S p ( k R g ) = π c D ( N - 1 ) 4 ( k R g ) D 0 J p + ( 1 / 2 ) ( u ) × H p + ( 1 / 2 ) ( 1 ) ( u ) u D - 2 × exp [ - c ( u / k R g ) D ] d u .
( N - 1 ) S p ( k R g ) ~ c 2 / D Γ ( 1 - 2 / D ) 4 N ( k R g ) 2 .
d ¯ 1 , n ( 1 ) = a n 1 + 2 L ν = 1 ( 2 ν + 1 ) a v ,
d ¯ 1 , n ( 2 ) = b n 1 + 2 L ν = 1 ( 2 ν + 1 ) b ν ,
2 ν + 1 n ( n + 1 ) ν ( ν + 1 ) p = n - ν n + ν [ n ( n + 1 ) + ν ( ν + 1 ) ] - p ( p + 1 ) a ( ν , n ; p ) = 2 ( ν + 1 ) , 2 ν + 1 n ( n + 1 ) ν ( ν + 1 ) p = n - ν n + ν b ( ν , n ; p ) = 0.
( N - 1 ) S p ( k R g ) ~ c N ln [ N ( k a ) 2 ] 4 ( k R g ) 2 ,
( N - 1 ) S p ( k R g ) ~ π c D 4 ( k a ) D N - 1 ( R g / a ) D 0 J p + ( 1 / 2 ) ( u ) × H p + ( 1 / 2 ) ( 1 ) ( u ) u D - 2 d u ,
E S ( j ) = E 0 exp ( i k inc r j o , j ) n = 1 i n 2 n + 1 n ( n + 1 ) × [ i d ¯ 1 , n ( 1 ) N e 1 n ( 3 ) - d ¯ 1 , n ( 2 ) M o 1 n ( 3 ) ] , H S ( j ) = k ω μ m E 0 exp ( i k inc r j o , j ) n = 1 i n 2 n + 1 n ( n + 1 ) × [ i d ¯ 1 , n ( 2 ) N o 1 n ( 3 ) - d ¯ 1 , n ( 1 ) M e 1 n ( 3 ) ] ,
S α , β ( θ ) = S α , β 0 ( θ ) j , l = 1 N exp [ i ( k sca - k inc ) r j , l ]
S α , β ( θ ) = N S α , β 0 ( θ ) { 1 + 4 π ( N - 1 ) × 0 r 2 P ( r ) sin [ 2 k r     sin ( θ / 2 ) ] 2 k r     sin ( θ / 2 )     d r } ,
σ ext = 2 π N k 2 n = 1 ( 2 n + 1 ) Re [ d ¯ 1 , n ( 1 ) + d ¯ 1 , n ( 2 ) ] .
a ( ν , n ; p ) = ( - 1 ) ( ν + p - n ) / 2 2 p + 1 2 p p ! n + ν - p + 1 n ( n + 1 ) [ ( p + ν - n ) / 2 ] ! [ ( p + n - ν ) / 2 ] ! ( n + ν - p + 1 ) ! ! ( n + ν + p + 1 ) ! ! × j = max ( 0 , p - n - 1 ) min ( p , ν - 1 ) ( - 1 ) j ( j + ν + 1 ) ! ( p + n - 1 - j ) ! j ! ( p - j ) ! ( ν - 1 - j ) ! ( j + n - p + 1 ) ! ,
a ( ν , n ; p ) = 2 ν - 1 ν - 1 n 2 n + 1 a ( ν - 1 , n + 1 ; p ) + 2 ν - 1 ν - 1 n + 1 2 n + 1 a ( ν - 1 , n - 1 ; p ) - ν ν - 1 a ( ν - 2 , n ; p ) ,
a ( 0 , n ; p ) = 0 ,
a ( 1 , n ; p ) = 0             if             p n ± 1 ,
a ( 1 , n ; p ) = n ( n + 1 ) ( n - p ) 2 n + 1             if             p = n ± 1 ,
a ( ν , n ; p ) = a ( n , ν ; p ) .
b ( ν , n ; p ) = ( 2 p + 1 ) l = 0 [ ( p - 1 - n - ν ) / 2 ] a ( ν , n ; p - 2 l - 1 ) ,
Re [ S p ( k R g ) ] = 1 2 l = 0 ( - 1 ) l ( l + p ) ! 2 Γ { [ ( 2 l + 2 p ) / D ] + 1 } l ! ( 2 l + 2 p + 1 ) ! ( l + 2 p + 1 ) ! × ( 2 k R g c 1 / D ) 2 l + 2 p ,
Im [ S p ( k R g ) ] = l = 0 ( - 1 ) l + 1 l ! ( 2 l + 2 p ) ! ( l + 2 p ) ! Γ { [ ( 2 l + 2 p - 1 ) / D ] + 1 } ( 2 l ) ! ( l + p ) ! 2 ( 2 l + 4 p + 1 ) ! ( 2 k R g c 1 / D ) 2 l + 2 p - 1 - l = 0 p - 1 ( 2 l ) ! ( l + p ) ! ( 2 p - 2 l ) ! Γ { [ ( 2 l - 1 ) / D ] + 1 } l ! 2 ( 2 l + 2 p + 1 ) ! ( p - l ) ! ( 2 k R g c 1 / D ) 2 l - 1 ,

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