Abstract

The formalism that was previously devised [J. Opt. Soc. Am. A 9, 1327 (1992)] to deal with the optical properties of homogeneous spheres containing an eccentric spherical inclusion is extended to the case of several inclusions. The extinction efficiency of dielectric spheres containing two identical metallic inclusions is calculated for a few significant geometries. Extinction by a low-density dispersion of the anisotropic scatterers mentioned above is also evaluated. Our results show that the subdivision of the included material has quite visible effects that strongly depend on both the polarization of the incident light and the geometric arrangement of the inclusions.

© 1994 Optical Society of America

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Corrections

F. Borghese, P. Denti, and R. Saija, "Optical properties of spheres containing several spherical inclusions: errata," Appl. Opt. 34, 5556-5556 (1995)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-34-24-5556

References

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  1. M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 3, pp. 42–50; Chap. 5, pp. 232–235.
  2. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a single eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [Crossref]
  3. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [Crossref]
  4. E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 23–24; Chap. 3, pp. 36–38.
  5. 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]
  6. 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]
  7. R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28.
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16, pp. 744–747.
  9. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
    [Crossref]
  10. 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]
  11. O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
    [Crossref]
  12. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [Crossref] [PubMed]
  13. E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, pp. 48–62.
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, pp. 46–55.
  15. R. T. Wang, J. M. Greenberg, D. W. Schuerman, “Experimental results of the dependent light scattering by two spheres,” Opt. Lett. 11, 543–545 (1981).
    [Crossref]
  16. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Reliability of the theoretical description of electromagnetic scattering from non-spherical particles,” J. Aerosol Sci. 20, 1079–1081 (1989).
    [Crossref]
  17. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
    [Crossref]
  18. B. V. Bronk, M. J. Smith, S. Arnold, “Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet,” Opt. Lett. 18, 93–95 (1993).
    [Crossref] [PubMed]
  19. R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, R. G. Pinnick, “Effect of submicrometer-sized particles on micro-droplet lasing,” Opt. Lett. 18, 119–121 (1993).
    [Crossref] [PubMed]

1993 (2)

1992 (1)

1989 (1)

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

1988 (1)

1987 (1)

1984 (4)

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (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]

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 (1)

1980 (1)

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

1979 (1)

Armstrong, R. L.

Arnold, S.

Borghese, F.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a single eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
[Crossref]

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

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[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]

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, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[Crossref]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

Bronk, B. V.

Denti, P.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a single eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
[Crossref]

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

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[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]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (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]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

Fikioris, J. G.

Fuller, K. A.

Greenberg, J. M.

Gu, J.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16, pp. 744–747.

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 3, pp. 42–50; Chap. 5, pp. 232–235.

Newton, R.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28.

Pinnick, R. G.

Rose, E. M.

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, pp. 48–62.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 23–24; Chap. 3, pp. 36–38.

Ruekgauer, T. E.

Saija, R.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a single eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
[Crossref]

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

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[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]

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, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[Crossref]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

Schuerman, D. W.

Sindoni, O. I.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a single eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
[Crossref]

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

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[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]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (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]

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, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

Smith, M. J.

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[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]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (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]

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, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

Uzunoglu, N. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, pp. 46–55.

Wang, R. T.

Xie, J.-G.

Aerosol Sci. Technol. (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]

O. I. Sindoni, F. Borghese, P. Denti, R. Saija, G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984).
[Crossref]

J. Aerosol Sci. (1)

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

J. Math. Phys. (1)

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Nuovo Cimento B (1)

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. (4)

Other (6)

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, pp. 48–62.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, pp. 46–55.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16, pp. 744–747.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 3, pp. 42–50; Chap. 5, pp. 232–235.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 23–24; Chap. 3, pp. 36–38.

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Figures (5)

Fig. 1
Fig. 1

Three regions into which the space is partitioned. In our calculations the center of the external sphere coincides with the origin.

Fig. 2
Fig. 2

Q η (curve c) and Q ¯ (curve c ¯ ) for spheres containing two identical spherical inclusions, arranged as in the inset in (a), plotted versus x E 2 in the range from 0.7937 to 2.2063. For the sake of comparison, Q η (curve s) and Q ¯ (curve s ¯ ) are also plotted versus x E for spheres containing the equivalent inclusion. The angles of incidence are φ0 = 0 in all cases and ϑ0 = 0, π/4, and π/2 in (a), (b), and (c), respectively. The subscript l refers to the polarization parallel to the plane of scattering and the subscript r refers to the perpendicular polarization.

Fig. 3
Fig. 3

Q η and Q ¯ , the same as in Fig. 2 but with the geometric arrangement in the inset in (a).

Fig. 4
Fig. 4

Q η and Q ¯ , the same as in Fig. 2 but with the geometric arrangement in the inset in (a).

Fig. 5
Fig. 5

Collection of the curves for Q ¯ versus x E 2 . For the sake of comparison we also report Q ¯ versus x E for the sphere containing the equivalent inclusion (curve 1). The labels 2, 3, and 4 refer to the geometries in Figs. 2, 3, and 4, respectively.

Equations (41)

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K = k n ,             K 0 = k n 0 ,             K α = k n α ,
x 0 = k ρ 0 ,             x α = k ρ α ,
E inc = E 0 e ^ exp ( i K inc · r 0 ) ,
H L M ( 1 ) ( K , r ) = h L ( K r ) X L M ( r ^ ) , H L M ( 2 ) ( K , r ) = 1 K × H L M ( 1 ) ( K , r ) ,
E = E 0 p l m [ A l m ( p ) H l m ( p ) ( K , r 0 ) + W l m ( p ) J l m ( p ) ( K , r 0 ) ] ,
E = E 0 p l m [ P 0 l m ( p ) J l m ( p ) ( K 0 , r 0 ) + α P α l m ( p ) H l m ( p ) ( K 0 , r α ) ] ,
E = E 0 p l m C α l m ( p ) J l m ( p ) ( K α , r α ) ,
f = 1 K l m ( - i ) l + 1 [ A l m ( 1 ) X l m ( k ^ sca ) + i A l m ( 2 ) k ^ sca × X l m ( k ^ sca ) ] ,
| ( R ) - 1 + H I 0 R W I 0 ( R 0 ) - 1 | | P P 0 | = | 0 W | ,
M P = W ,
P = M - 1 W ,
M - 1 = | Z I Z I 0 Z 0 I Z 0 | ,
L = | Z I 0 Z 0 | ,
P = L W .
A = J P = SW ,
J = M W l 0 M 0 ,
n η = Re [ N η η ] ,             γ η = 2 k Im [ N η η ] ,
N η η = n [ δ η η + 2 π K 2 ρ ( Θ ) f η η ( Θ ) d Θ ] .
f η η ( Θ ) = f η ( Θ ) · e ^ η *
N η η = n [ δ η η + 2 π ρ ¯ K 2 f ¯ η η ] ,
f ¯ η η = f ¯ η · e ^ η * ;
A l m ( p ) = 1 2 l + 1 p m S l m , l m ( p , p ) ( Θ 0 ) W l m ( p ) .
Q η = 4 k ρ 0 2 Im [ f η η ] ,
Q ¯ = 4 k ρ 0 2 Im [ f ¯ η η ] ,
W l m ( 1 ) ( k ^ inc ) = 4 π i l e ^ · X l m * ( k ^ inc ) , W l m ( 2 ) ( k ^ inc ) = 4 π i l + 1 ( k ^ inc × e ^ ) · X l m * ( k ^ inc ) .
n ¯ α = n 0 n α - 1 ,             n ¯ 0 = n n 0 - 1 ,
u l ( x ) = x j l ( x ) ,             w l ( x ) = x h l ( x ) ,
R α l m , α l m ( p , p ) = δ p p δ l l δ m m δ α α R α l ( p ) ,
R α l ( p ) = ( 1 + n ¯ α δ p 1 ) u l ( K α ρ α ) u l ( K 0 ρ α ) - ( 1 + n ¯ α δ p 2 ) u l ( K α ρ α ) u l ( K 0 ρ α ) ( 1 + n ¯ α δ p 1 ) u l ( K α ρ α ) w l ( K 0 ρ α ) - ( 1 + n ¯ α δ p 2 ) u l ( K α ρ α ) w l ( K 0 ρ α ) .
R α l ( p ) = G α l ( p ) u l ( K α ρ α ) - ( 1 + n ¯ α δ p 2 ) 2 G α l ( p ) u l ( K 0 ρ α ) G α l ( p ) w l ( K α ρ α ) - ( 1 + n ¯ α δ p 2 ) 2 G α l ( p ) w l ( K 0 ρ α ) ,
d 2 G α l ( p ) d ξ 2 - 2 n α d n α d ξ d G α l ( 2 ) d ξ δ p 2 + [ n α 2 - l ( l + 1 ) ξ 2 ] G α l ( p ) = 0.
R 0 l m , l m ( p , p ) = δ p p δ l l δ m m R 0 l ( p ) ,
R 0 l ( p ) = i [ ( 1 + n ¯ 0 δ p 1 ) u l ( K 0 ρ 0 ) w l ( K ρ 0 ) - ( 1 + n ¯ 0 δ p 2 ) u l ( K 0 ρ 0 ) w l ( K ρ 0 ) ] - 1 .
M 0 l m , l m ( p , p ) = δ p p δ l l δ m m M 0 l ( p ) ,
M 0 l ( p ) = i [ ( 1 + n ¯ 0 δ p 1 ) u l ( K 0 ρ 0 ) u l ( K ρ 0 ) - ( 1 + n ¯ 0 δ p 2 ) u l ( K 0 ρ 0 ) u l ( K ρ 0 ) ] .
R W α l m , l m ( p , p ) = δ p p δ l l δ m m R W l ( p ) , M W α l m , l m ( p , p ) = δ p p δ l l δ m m M W l ( p ) ,
R W l ( p ) = - i [ ( 1 + n ¯ 0 δ p 1 ) w l ( K 0 ρ 0 ) w l ( K ρ 0 ) - ( 1 + n ¯ 0 δ p 2 ) w l ( K 0 ρ 0 ) w l ( K ρ 0 ) ] , M W l ( p ) = i [ ( 1 + n ¯ 0 δ p 1 ) w l ( K 0 ρ 0 ) u l ( K ρ 0 ) - ( 1 + n ¯ 0 δ p 2 ) w l ( K 0 ρ 0 ) u l ( K ρ 0 ) ] .
I α l m , 0 l m ( p , p ) = [ δ p p - i ( 2 l + 1 l ) 1 / 2 ( 1 - δ p p ) ] × μ C ( 1 , l + 1 - δ p p , l ; - μ , m + μ ) × G l + 1 - δ p p , m + μ ; l , m + μ ( p , p ) ( K 0 , R α 0 ) × C ( 1 , l , l ; - μ , m + μ ) ,
G l m , l m ( p , p ) ( K , R ) = 4 π λ i l - l - λ J λ ( l , m ; l , m ) j λ ( K R ) Y λ , m - m * ( R ^ ) .
J λ ( l , m ; l , m ) = Y l m Y l m * Y λ , m - m d Ω .
H α l m , α l m ( p , p ) = ( 1 - δ α α ) [ δ p p - i ( 2 l + 1 l ) 1 / 2 ( 1 - δ p p ) ] × μ C ( 1 , l + 1 - δ p p , l ; - μ , m + μ ) × G H l + 1 - δ p p , m + μ ; l , m + μ ( p , p ) ( K 0 , R α α ) × C ( 1 , l , l ; - μ , m + μ ) ,

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