Abstract

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.

© 1994 Optical Society of America

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References

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  1. R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A (to be published).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  25. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  26. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [CrossRef]
  27. B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,”J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
    [CrossRef]
  28. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,”Q. Appl. Math. 20, 33–39 (1962).

1992

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,”J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

1991

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

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

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

1989

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

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

1987

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,”J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

1984

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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (1984).
[CrossRef]

1981

1980

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[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]

1975

1971

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. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

1962

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,”Q. Appl. Math. 20, 33–39 (1962).

Ausloos, M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Barber, P. W.

Bohren, C. F.

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

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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (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]

Botet, R.

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

Brunning, J. H.

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

Chen, H. Y.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,”Q. Appl. Math. 20, 33–39 (1962).

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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (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]

Dobbins, R. A.

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Draine, B. T.

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

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,”J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Edmunds, A. R.

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

Fuller, K. A.

Gerardy, J. M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Greenberg, J. M.

Huffman, D. R.

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

Iskander, M. F.

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).

Kattawar, G. W.

Khlebtsov, N. G.

Ku, J. C.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,”J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Lo, Y. T.

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

Megaridis, C. M.

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Mishchenko, M. I.

Mountain, R. D.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Mulholland, G. W.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Penner, J. E.

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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (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]

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” Department of Meteorology Sci. Rep. 9 (University of California at Los Angeles, Los Angeles, Calif., 1955).

Schuerman, D. W.

Shim, K.-H.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,”J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (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]

Tien, C. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,”J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (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]

Wang, R. T.

Waterman, P. C.

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

Wiscombe, W. J.

Yeh, C.

Aerosol Sci. Technol.

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.

Astrophys. J.

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

IEEE Trans. Antennas Propag.

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

J. Aerosol Sci.

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. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,”J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,”J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Langmuir

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Nuovo Cimento B

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento B 81, 29–50 (1984).
[CrossRef]

Opt. Lett.

Phys. Rev. B

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Phys. Rev. D

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

Q. Appl. Math.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,”Q. Appl. Math. 20, 33–39 (1962).

Other

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

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

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

D. S. Saxon, “Lectures on the scattering of light,” Department of Meteorology Sci. Rep. 9 (University of California at Los Angeles, Los Angeles, Calif., 1955).

K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A (to be published).

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Figures (5)

Fig. 1
Fig. 1

Comparison of theoretical predictions of complex forward-scattering amplitude for a two-sphere cluster with the experimental measurements of Wang et al.22

Fig. 2
Fig. 2

Random-orientation absorption cross section of a two-sphere cluster, divided by that for independent spheres, versus the number of orders in the scattered-field expansion.

Fig. 3
Fig. 3

40-sphere fractal cluster generated with the sequential algorithm.

Fig. 4
Fig. 4

Ratio of cluster efficiency and volume equivalent size parameter xV versus size parameter for exact (Nt = 2), electric dipole, and volume equivalent-sphere predictions. The refractive index is m = 1.6 + 0.6i.

Fig. 5
Fig. 5

Same as Fig. 4 but with m = 2 + 1i.

Tables (1)

Tables Icon

Table 1 Results for a Cluster of Two Identical Touching Spheresa

Equations (72)

Equations on this page are rendered with MathJax. Learn more.

E s = i N s E s i ,
E s i = n = 1 m = - n n [ a m n i N m n ( 3 ) ( K r i , θ i , ϕ i ) + b m n i M m n ( 3 ) ( k r i , θ i , ϕ i ) ] ,
E 1 i = n = 1 m = - n n [ d m n i N m n ( 1 ) ( m i k r i , θ i , ϕ i ) + c m n i M m n ( 1 ) ( m i k r i , θ i , ϕ i ) ] ,
E 0 i = - n = 1 m = - n n [ p m n i N m n ( 1 ) ( k r i , θ i , ϕ i ) + q m n i M m n ( 1 ) ( k r i , θ i , ϕ i ) ] ,
M m n ( 3 ) ( k r j , θ j , ϕ j ) = l = 1 k = - l l [ A m n k l ( 3 ) ( k R i j , Θ i j , Φ i j ) M k l ( 1 ) ( k r i , θ i , ϕ i ) + B m n k l ( 3 ) ( k R i j , Θ i j , Φ i j ) N k l ( 1 ) ( k r i , θ i , ϕ i ) ] ,
N m n ( 3 ) ( k r j , θ j , ϕ j ) = l = 1 k = - l l [ A m n k l ( 3 ) ( k R i j , Θ i j , Φ i j ) N k l ( 1 ) ( k r i , θ i , ϕ i ) + B m n k l ( 3 ) ( k R i j , Θ i j , Φ i j ) M k l ( 1 ) ( k r i , θ i , ϕ i ) ] .
M m n ( 3 ) ( k r j , θ j , ϕ j ) = l = 1 k = - l l [ A m n k l ( 1 ) ( k R i j , Θ i j , Φ i j ) M k l ( 3 ) ( k r i , θ i , ϕ i ) + B m n k l ( 1 ) ( k R i j , Θ i j , Φ i j ) N k l ( 3 ) ( k r i , θ i , ϕ i ) ] ,
N m n ( 3 ) ( k r i , θ j , ϕ j ) = l = 1 k = - l l [ A m n k l ( 1 ) ( k R i j , Θ i j , Φ i j ) N k l ( 3 ) ( k r i , θ i , ϕ i ) + B m n k l ( 1 ) ( k R i j , Θ i j , Φ i j ) M k l ( 3 ) ( k r i , θ i , ϕ i ) ] ,
a m n i = a ¯ n i { p m n i - j = 1 j 1 N s l = 1 N t j k = - l l [ A k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) a k l j + B k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) b k l j ] } ,
b m n i = b ¯ n i { q m n i - j = 1 j 1 N s l = 1 N t j k = - l l [ A k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) b k l j + B k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) a k l j ] } .
a ¯ n i = m i ψ n ( x i ) ψ n ( m i x i ) - ψ n ( x i ) ψ n ( m i x i ) m i ξ n ( x i ) ψ n ( m i x i ) - ξ n ( x i ) ψ n ( m i x i ) ,
b ¯ n i = ψ n ( x ) ψ n ( m i x i ) - m i ψ n ( x i ) ψ n ( m i x i ) ξ n ( x i ) ψ n ( m i x i ) - m i ξ n ( x i ) ψ n ( m i x i ) ,
a m n p i + a ¯ n p i j = 1 j 1 N s l = 1 N t j k = - l l q = 1 2 H m n p k l q i j a k l q j = a ¯ n p i p m n p i .
H m n 1 k 1 l 1 i j = H m n 2 k l 2 i j = A k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) , H m n 1 k l 2 i j = H m n 2 k l 1 i j = B k l m n ( 3 ) ( k R i j , Θ i j , Φ i j ) .
M i = 2 N t i ( N t i + 2 ) .
a m n p i = j = 1 N s l = 1 N t j k = - 1 l q = 1 2 T m n p k l q i j p k l q j .
p m n p i = J m n p k l q i 0 p k l q 0 ,
a m n p 0 = i = 1 N s J m n p k l q 0 i a k l q i .
a m n p 0 = i , j J m n p m n p 0 i T m n p k l q i j J k l q k l q j 0 p k l q 0 = T m n p k l q 0 p k l q 0 .
( p m n i , q m n i ) = ( p m n , q m n ) exp i [ Z i cos β + ( sin β ) × ( X i cos α + Y i sin α ) ] , p m n = - i n + 1 1 E m n [ τ m n ( β ) cos γ - i π m n ( β ) sin γ ] × exp ( - i m α ) , q m n = - i n 1 E m n [ τ m n ( β ) sin γ + i π m n ( β ) cos γ ] × exp ( - i m α ) ,
E m n = n ( n + 1 ) 2 n + 1 ( n + m ) ! ( n - m ) ! ,
τ m n ( β ) = d d β P n m ( cos β ) ,             π m n ( β ) = m sin β P n m ( cos β ) .
C ext i = 4 π k 2 Re ( E m n P m n p i * a m n p i ) ,
C ext = i = 1 N s C ext i .
C ext = 4 π k 2 Re ( E m n P m n p 0 * a m n p 0 ) .
C ext = 4 π k 2 Re ( E m n p m n p 0 * T m n p k l q 0 p k l q 0 ) .
1 8 π 2 0 2 π 0 π 0 2 π p m n p * p k l q d γ cos β d β d α = 1 2 E m n δ k m δ l n δ p q ,
C ¯ ext = 2 π k 2 Re T m n p m n p 0 = 2 π k 2 Re ( Tr T 0 ) .
C ¯ ext = 2 π k 2 Re ( J m n p m n p 0 i T m n p k l q i j J k l q m n p j 0 ) ,
E m n J m n p k l q i j * = E k l J k l q m n p j i .
n = 1 m = - n n p = 1 2 J m n p m n p j j J m n p k l q j i = J m n p k l q j i .
C ¯ ext = 2 π k 2 Re ( J k l q m n p j i T m n p k l q i j ) .
C ¯ ext = 2 π k 2 Re [ T m n p m n p i i + ( J m n p k l q i j T k l q m n p j i + E k l E m n J k l q m n p i j * T k l q m n p i j ) ] ,             j > i .
C sca = 4 π k 2 E m n a m n p 0 * a m n p 0 .
C sca = 4 π k 2 Re ( E k l a k l q i * J k j q m n p i j a m n p j ) .
C sca = 4 π k 2 E m n [ a m n p i * a m n p i + 2 Re ( a m n p i * J m n p k j q i j a k l q j ) ] ,             j > i .
C ¯ sca = 2 π k 2 Re { E m n E k l [ J m n p u v w i i T u v w k l q i j × ( T m n p u v w i j J u v w k l q j j ) * ] } .
C ¯ sca = 2 π k 2 E m n E k l ( T m n p k l q 0 T m n p k l q 0 * ) ,
C abs i = a i 2 I 0 0 2 π 0 π I 1 ( r = a i ) sin θ d θ d θ ϕ = a i 2 2 I 0 Re [ 0 2 π 0 π ( E 1 θ H 1 ϕ * - E 1 ϕ H 1 θ * ) r = a i × sin θ d θ d ϕ ] ,
C abs i = 4 π k 2 n = 1 N t i m = - n n E m n ( d ¯ n i a m n i 2 + c ¯ n i b m n i 2 ) ,
d ¯ n i = Re [ i ψ n ( m i x i ) ψ n * ( m i x i ) m i * ] m i ψ n ( m i x i ) ψ n ( x i ) - ψ n ( x i ) ψ n ( m i x i ) 2 ,
c ¯ n i = Re [ i ψ n ( m i x i ) ψ n * ( m i x i ) m i ] ψ n ( m i x i ) ψ n ( x i ) - m i ψ n ( x i ) ψ n ( m i x i ) 2 .
C abs = i = 1 N s C abs i .
C abs = 4 π k 2 E m n d ¯ n p i a m n p i a m n p i * ,
C ¯ abs = 2 π k 2 d ¯ n p i E m n E k l Re ( T m n p k l q i j J k l q k l q j j T m n p k l q i j * ) ,
C ¯ abs = 2 π k 2 d ¯ n p i E m n E k i [ T m n p k l q i j T m n p k l q i j * + 2 Re ( T m n p k l q i j J k l q k l q j j T m n p k l q i j * ) ] ,             j > j .
S ( 0 ) = 4 π k 2 G i = 1 N s E m n p m n p i * a m n p i ,
N t i x i + 4 ( x i ) 1 / 3 + 2.
N s = k f ( R g d p ) D f ,
R g 2 = 1 N s i = 1 N s r i 2 ,
( I - a ¯ n p 2 H m n p m n p 21 a ¯ n p 1 H m n p m l q 12 ) a m l q 2 = a ¯ n p 2 ( p m n 2 - H m n p m n p 21 a ¯ n p 1 p m n 1 ) .
T 22 = ( I - a ¯ 2 H 21 a ¯ 1 H 12 ) - 1 a ¯ 2 ,
T 21 = - T 22 H 21 a ¯ 1 ,
T 12 = - a ¯ 1 H 12 T 22 ,
T 11 = a ¯ 1 ( I + H 12 T 22 H 21 a ¯ 1 ) .
C ¯ ext 2 = 2 π k 2 Re ( T m n p m n p 22 - T m n p m n p 22 × H m n p m l q 21 a ¯ l q 1 J m l q m n p 12 ) .
C ¯ abs 2 = 2 π k 2 Re ( S m n p m n p 2 + S m n p m l q 2 U m n p m l q 2 ) ,
S m n p m l q 2 = E m n d ¯ n p 2 T m n p m n p 22 T m n p m l q 22 * / E m l ,
U m n p m l q 2 = E m l a ¯ n p 1 H m n p m n p 21 ( H m l q m n p 21 * a ¯ n p 1 * - 2 J m l q m n p 21 * ) / E m n .
C ¯ ext , 2 , d p = 4 π k 2 × Re ( a ¯ 1 { 3 - a ¯ 1 [ 2 g 1 ( f 1 - a ¯ 1 g 1 ) 1 - ( a ¯ 1 g 1 ) 2 + g 2 ( f 2 - a ¯ 1 g 2 ) 1 - ( a ¯ 1 g 2 ) 2 ] } ) ,
C ¯ abs , 2 , d p = 4 π k 2 d ¯ 1 a ¯ 1 2 × Re [ ( 3 - a ¯ 1 { 2 g 1 [ ( a ¯ 1 g 1 ) * ( a ¯ 1 g 1 2 - 1 ) + 2 ( f 1 - a ¯ 1 g 1 ) ] 1 - ( a ¯ 1 g 1 ) 2 2 + g 2 [ a ¯ 1 g 2 * ( a ¯ 1 g 2 2 - 1 ) + 2 ( f 2 - a ¯ 1 g 2 ) ] 1 - ( a ¯ 1 g 2 ) 2 2 } ) ] ,
g 1 = h 0 ( k R 12 ) - ½ h 2 ( k R 12 ) , g 2 = h 0 ( k R 12 ) + h 2 ( k R 12 ) , f 1 = j 0 ( k R 12 ) - ½ j 2 ( k R 12 ) , f 2 = j 0 ( k R 12 ) + j 2 ( k R 12 ) .
A k l m n ( ν ) = 1 2 n ( n + 1 ) [ ( n - m ) ( n + m + 1 ) C k + 1 l , m + 1 n ( ν ) + 2 m k C k l m n ( ν ) + ( l + k ) ( l - k + 1 ) C k - 1 l , m - 1 n ( ν ) ] ,
B k l m n ( ν ) = - i 2 n + 1 2 n ( n + 1 ) ( 2 n - 1 ) [ ( n - m ) ( n - m - 1 ) × C k + 1 l ; m + 1 , n - 1 ( ν ) + 2 k ( n - m ) C k l , m n - 1 ( ν ) - ( l + k ) × ( l - k + 1 ) C k - 1 l ; m - 1 , n - 1 ( ν ) ] .
1 2 l + 1 [ C k l - 1 , m n ( ν ) + C k l + 1 , m n ( ν ) ] = 1 2 n - 1 C k - 1 l ; m - 1 , n - 1 ( ν ) + 1 2 n + 3 C k - 1 l ; m - 1 , n + 1 ( ν ) ,
1 2 l + 1 [ ( l + k ) ( l + k + 1 ) C k l - 1 , m n ( ν ) + ( l - k ) ( l - k + 1 ) C k l + 1 , m n ( ν ) ] = ( n - m ) ( n - m - 1 ) 2 n - 1 C k + 1 l ; m + 1 , n - 1 ( ν ) + ( n + m + 1 ) ( n + m + 2 ) 2 n + 3 C k + 1 l ; m + 1 , n - 1 ( ν ) ,
1 2 l + 1 [ ( l + k ) C k l - 1 , m n ( ν ) - ( l - k + 1 ) C k l + 1 , m n ( ν ) ] = - n - m 2 n - 1 C k l , m n - 1 ( ν ) + n + m + 1 2 n + 3 C k l , m n + 1 ( ν ) .
C 00 m n ( ν ) = { ( - 1 ) n + m ( 2 n + 1 ) j n ( k R i j ) P n - m ( cos Θ i j ) exp ( - i m Φ i j ) ν = 1 ( - 1 ) n + m ( 2 n + 1 ) h n ( k R i j ) P n - m ( cos Θ i j ) exp ( - i m Φ i j ) ν = 3 ,
A - 11 m n = ( - 1 ) m + n 2 n ( n + 1 ) [ ( n + 1 ) ( n - m ) ( n - m - 1 ) u - m - 1 , n - 1 - n ( n + m + 1 ) ( n + m + 2 ) u - m - 1 , n + 1 ] ,
A 01 m n = - ( - 1 ) m + n n ( n + 1 ) [ ( n + 1 ) ( n - m ) u - m , n - 1 - n ( n + m + 1 ) u - m , n + 1 ] ,
A 11 m n = ( - 1 ) m + n n ( n + 1 ) [ ( n + 1 ) u - m + 1 , n - 1 - n u - m + 1 , n + 1 ] ,
u m n = z n ( R i j ) P n m ( cos Θ i j ) exp ( i m Φ i j ) ,

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