Abstract

The formalism presented a few years ago by Fikioris and Uzunoglu [ J. Opt. Soc. Am. 69, 1359 ( 1979)] to describe the electromagnetic scattering by homogeneous spheres containing an eccentric spherical inclusion is reformulated. The resulting approach is an extension of our previous formalism [ Aerosol Sci. Technol. 3, 27 ( 1984)] designed to deal with the dependent scattering by aggregated spheres and is put in a form readily extensible to the case of spheres containing more than one inclusion. A comparison of our results with those of Fikioris and Uzunoglu is made, and the differences are explained in terms of the approximations that they used. Specific results for dielectric spheres containing either a metallic inclusion or a dielectric inclusion with parameters quite incompatible with the approximation scheme of Fikioris and Uzunoglu are also presented. These scatterers have a response that depends on the direction of incidence and, in general, also on the polarization, thus making them distinguishable from spheres with a centered inclusion as well as from homogeneous spheres.

© 1992 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  3. R. Ruppin, “Optical properties of metal spheres with a diffuse surface,” J. Opt. Soc. Am. 66, 449–453 (1976).
    [CrossRef]
  4. 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]
  5. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [CrossRef]
  6. 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]
  7. E. M. Rose, Multipole Fields (Wiley, New York, 1955).
  8. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
    [CrossRef]
  9. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81B, 29–50 (1984).
  10. 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]
  11. 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]
  12. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  13. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).
  18. E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).
  19. See, for instance, Ref. 4 and references therein.
  20. K. A. Fuller, “Recent progress in the study of the optical resonances of two-sphere systems,” in Proceedings of the Second International Conference on Optical Particle Sizing, E. D. Hirleman, ed. (University of Arizona, Tempe, Ariz., 1990), pp. 65–68.
  21. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).
  22. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  23. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
    [CrossRef]

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 (2)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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]

1984 (4)

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 81B, 29–50 (1984).

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]

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)

1976 (1)

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

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 non-spherical particles,”J. Aerosol Sci. 20, 1079–1081 (1989).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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 81B, 29–50 (1984).

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]

Denti, P.

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, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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 81B, 29–50 (1984).

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]

Fikioris, J. G.

Fuller, K. A.

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]

K. A. Fuller, “Recent progress in the study of the optical resonances of two-sphere systems,” in Proceedings of the Second International Conference on Optical Particle Sizing, E. D. Hirleman, ed. (University of Arizona, Tempe, Ariz., 1990), pp. 65–68.

Greenberg, J. M.

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969).

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).

Newton, R.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Rose, E. M.

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

E. M. Rose, Multipole Fields (Wiley, New York, 1955).

Ruppin, R.

Saija, R.

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, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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 81B, 29–50 (1984).

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]

Schuerman, D. W.

Sindoni, O. I.

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, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (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 81B, 29–50 (1984).

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

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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 81B, 29–50 (1984).

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]

Uzunoglu, N. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wang, R. T.

Aerosol Sci. Technol. (3)

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, “Optical absorption coefficient of a dispersion of clusters composed of a large number of spheres,” Aerosol Sci. Technol. 6, 173–181 (1987).
[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. (2)

J. Opt. Soc. Am. A (2)

Nuovo Cimento (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 81B, 29–50 (1984).

Opt. Lett. (2)

Other (11)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

E. M. Rose, Multipole Fields (Wiley, New York, 1955).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light (Academic, New York, 1969).

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

See, for instance, Ref. 4 and references therein.

K. A. Fuller, “Recent progress in the study of the optical resonances of two-sphere systems,” in Proceedings of the Second International Conference on Optical Particle Sizing, E. D. Hirleman, ed. (University of Arizona, Tempe, Ariz., 1990), pp. 65–68.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).

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

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Sketch of the three regions into which the space is partitioned. In actual calculations the center of the external sphere coincides with the origin, and the center of the inclusion lies on the z axis.

Fig. 2
Fig. 2

Backscattering efficiency σB/A= (4n2/ρ02)|f(ϑ = π)|2, where A = πρ02 and ϑ is the angle of scattering, as a function of xE. The dotted curve shows the approximate results of Fikioris and Uzunoglu (Fig. 2 of Ref. 5), and the solid curve shows our results for the same scatterer.

Fig. 3
Fig. 3

(a) (0) and (b) Q(0) as a function of xE for a dielectric sphere containing a metallic inclusion (solid curves). The curves are labeled by the values of ϑinc, and when necessary it is also indicated whether the polarization vector is parallel (l) or perpendicular (r) to the scattering plane. The figures also show (a) P ¯ and (b) Q ¯ as a function of xE (dotted curves).

Fig. 4
Fig. 4

(ϑ) versus Q(ϑ) as a function of ϑ for (a) ϑinc = 0, (b) ϑinc = π/4, and (c) ϑinc = π/2 for a dielectric sphere containing a metallic inclusion. The open circles mark a 30° increment of ϑ; the forward scattering side is marked F. The solid curves are for xE = −2 (B) and xE = 2 (T). The dotted curves refer to xE = 0 (centered inclusion). For the sake of comparison we also show h(ϑ) versus Q h(ϑ) (dashed curves).

Fig. 5
Fig. 5

(a) (0) and (b) Q(0) as a function of xE for a dielectric sphere containing an empty cavity (solid curves). The curves are labeled by the values of ϑinc, and when necessary it is also indicated whether the polarization vector is parallel (l) or perpendicular (r) to the scattering plane. The figures also show (a) P ¯ and (b) Q ¯ as a function of xE (dotted curves). The maximum value of the eccentricity is xE = 2.1705.

Equations (42)

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K = k n ,             K 0 = k n 0 ,             K 1 = k n 1
x 0 = k ρ 0 ,             x 1 = k ρ 1 ,
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 ) + P 1 l m ( p ) H l m ( p ) ( K 0 , r 1 ) ] ,
E = E 0 p l m C 1 l m ( p ) J l m ( p ) ( K 1 , r 1 ) .
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 ) - 1 I 1 0 R W I 0 1 ( R 0 ) - 1 ] [ P 1 P 0 ] = [ 0 W ]
M P = W ,
M - 1 = [ Z 1 Z 10 Z 01 Z 0 ] ,
Z = [ Z 10 Z 0 ] ,
P = Z W .
A = T P = SW ,
T = [ M W I 0 1             M 0 ] .
N η η = n [ δ η η + 2 π K 2 d ( Θ ) f η η ( Θ ) d Θ ] ,
f η η ( Θ ) = f η ( Θ ) · e ^ n *
n η = Re [ N η η ] ,             γ η = 2 k Im [ N η 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 ) .
ϑ sca = ϑ inc + ϑ , φ sca = 0 for ϑ inc + ϑ < π , ϑ sca = 2 π - ( ϑ inc + ϑ ) , φ sca = π for ϑ inc + ϑ < π .
P η ( ϑ ) = 4 K ρ 0 2 Re [ f η η ( ϑ ) ] ,             Q η ( ϑ ) = 4 K ρ 0 2 Im [ f η η ( ϑ ) ] ,
M = 1 - 1 ν ( ν + i γ ) ,
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 ( k ^ inc × e ^ ) · X l m * ( k ^ inc ) .
n ¯ 1 = n 0 n 1 - 1 ,             n ¯ 0 = n n 0 - 1
u l ( x ) = x j l ( x ) ,             w l ( x ) = x h l ( x )
R 1 l m , l m ( p , p ) = δ p p δ l l δ m m R 1 l ( p ) , R 0 l m , l m ( p , p ) = δ p p δ l l δ m m R 0 l ( p ) , R W l m , l m ( p , p ) = δ p p δ l l δ m m R W l ( p ) ,
R 1 l ( p ) = ( 1 + n ¯ 1 δ p 1 ) u l ( K 1 ρ 1 ) u l ( K 0 ρ 1 ) - ( 1 + n ¯ 1 δ p 2 ) u l ( K 1 ρ 1 ) u l ( K 0 ρ 1 ) ( 1 + n ¯ 1 δ p 1 ) u l ( K 1 ρ 1 ) w l ( K 0 ρ 1 ) - ( 1 + n ¯ 1 δ p 2 ) u l ( K 1 ρ 1 ) w l ( K 0 ρ 1 ) , 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 , 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 m , l m ( p , p ) = δ p p δ l l δ m m M W l ( p ) , M 0 l m , l m ( p , p ) = δ p p δ l l δ m m M 0 l ( p ) ,
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 ) ] , 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 ) ] ,
I 1 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 10 ) C ( 1 , l , l ; - μ , m + μ ) ,
G l m , l m ( p , p ) ( K , R ) = 4 π λ i l - l - λ λ ( l , m ; l , m ) × j λ ( K R ) Y λ , m - m * ( R ^ ) .
λ ( l , m ; l , m ) = Y l m Y l m * Y λ , m - m d Ω .
ϕ ( + ) = i [ n ( + ) k · r - ω t ] ,
n ( + ) = n + i n .
E sca ( + ) = E 0 exp [ i n ( + ) k r ] r f ,
f = i n ( + ) k S B H = 1 n ( + ) k S B J .
ϕ ( - ) = - i [ n ( - ) k · r - ω t ] ,
n ( - ) = n - i n .
n ( - ) * = n ( + ) , E inc ( - ) * = E inc ( + ) ,             E sca ( - ) * = E sca ( + ) ,
S ( - ) * = - i n ( + ) k f .

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