Abstract

We present a comprehensive solution to the classical problem of electromagnetic scattering by aggregates of an arbitrary number of arbitrarily configured spheres that are isotropic and homogeneous but may be of different size and composition. The profile of incident electromagnetic waves is arbitrary. The analysis is based on the framework of the Mie theory for a single sphere and the existing addition theorems for spherical vector wave functions. The classic Mie theory is generalized. Applying the extended Mie theory to all the spherical constituents in an aggregate simultaneously leads to a set of coupled linear equations in the unknown interactive coefficients. We propose an asymptotic iteration technique to solve for these coefficients. The total scattered field of the entire ensemble is constructed with the interactive scattering coefficients by the use of the translational addition theorem a second time. Rigorous analytical expressions are derived for the cross sections in a general case and for all the elements of the amplitude-scattering matrix in a special case of a plane-incident wave propagating along the z axis. As an illustration, we present some of our preliminary numerical results and compare them with previously published laboratory scattering measurements.

© 1995 Optical Society of America

Full Article  |  PDF Article

Errata

Yu-lin Xu, "Electromagnetic scattering by an aggregate of spheres: errata," Appl. Opt. 40, 5508-5508 (2001)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-40-30-5508

References

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  1. L. V. Lorenx, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, H. Valentiner, revues et annotées (Librairie Lehman et Stage, Copenhagen, 1898), pp. 405–529.
  2. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  3. Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81 (1981).
  4. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  5. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  6. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  7. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  8. V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 407–414 (1952).
    [CrossRef]
  9. V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 1099–1118 (1952).
    [CrossRef]
  10. B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23(1954).
  11. A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  12. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  13. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  14. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part I—multiple expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [CrossRef]
  15. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part II—numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
    [CrossRef]
  16. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by ensembles of spheres. I. Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [CrossRef] [PubMed]
  17. K. A. Fuller, G. W. Kattawar, Consummate solution to the problem of classical electromagnetic scattering by ensembles of spheres. II. Clusters of arbitrary configurations,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef] [PubMed]
  18. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  19. K. W. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
    [CrossRef] [PubMed]
  20. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18, 116–120 (1979).
    [CrossRef] [PubMed]
  21. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 4, 227–235 (1984).
  22. 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]
  23. Y. M. Wang, W. C. Chew, A recursive T-matrix approach for the solution of electromagnetic scattering by many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1983).
    [CrossRef]
  24. J. M. Gérardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
    [CrossRef]
  25. K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
    [CrossRef] [PubMed]
  26. G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
    [CrossRef]
  27. G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  28. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  29. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  30. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  31. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  32. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  33. R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. Kohl, eds. (U.S. Army Aberdeen, Md., 1984), pp. 237–247.
  34. G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, Orlando, Fla., 1985), pp. 698–700.
  35. A. Messiah, Quantum Mechanics, Volume II (North-Holland, Amsterdam, 1962), p. 1057.

1993

1991

K. W. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
[CrossRef] [PubMed]

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

1988

1986

1985

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[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. 4, 227–235 (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]

1983

Y. M. Wang, W. C. Chew, A recursive T-matrix approach for the solution of electromagnetic scattering by many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1983).
[CrossRef]

1982

J. M. Gérardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

1981

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81 (1981).

1979

1975

1971

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

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part II—numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

1967

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1962

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1955

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

1954

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23(1954).

1952

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 407–414 (1952).
[CrossRef]

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 1099–1118 (1952).
[CrossRef]

1951

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1908

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, Orlando, Fla., 1985), pp. 698–700.

Asano, S.

Ausloos, M.

J. M. Gérardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bohren, C. F.

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

Borghese, F.

Bruning, J. H.

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

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part II—numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

Chew, W. C.

Y. M. Wang, W. C. Chew, A recursive T-matrix approach for the solution of electromagnetic scattering by many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1983).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Denti, P.

Friedman, B.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23(1954).

Fuller, K. A.

Fuller, K. W.

Gérardy, J. M.

J. M. Gérardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Gouesbet, G.

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

Grehan, G.

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

Gustafson, B. Å. S.

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. Kohl, eds. (U.S. Army Aberdeen, Md., 1984), pp. 237–247.

Huffman, D. R.

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

Kattawar, G. W.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

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

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lo, Y. T.

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

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part II—numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lock, J. A.

Lorenx, L. V.

L. V. Lorenx, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, H. Valentiner, revues et annotées (Librairie Lehman et Stage, Copenhagen, 1898), pp. 405–529.

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]

Maheu, B.

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics, Volume II (North-Holland, Amsterdam, 1962), p. 1057.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81 (1981).

Russek, J.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23(1954).

Saija, R.

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. 4, 227–235 (1984).

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Sindoni, O. I.

Stein, A.

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Toscano, G.

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 1099–1118 (1952).
[CrossRef]

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 407–414 (1952).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wang, R. T.

K. A. Fuller, G. W. Kattawar, R. T. Wang, “Electromagnetic scattering from two dielectric spheres: further comparisons between theory and experiment,” Appl. Opt. 25, 2521–2529 (1986).
[CrossRef] [PubMed]

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. Kohl, eds. (U.S. Army Aberdeen, Md., 1984), pp. 237–247.

Wang, Y. M.

Y. M. Wang, W. C. Chew, A recursive T-matrix approach for the solution of electromagnetic scattering by many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1983).
[CrossRef]

Yamamoto, G.

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. 4, 227–235 (1984).

Ann. Phys.

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. M. Wang, W. C. Chew, A recursive T-matrix approach for the solution of electromagnetic scattering by many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1983).
[CrossRef]

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

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres, part II—numerical and experimental results,” IEEE Trans. Antennas Propag. AP-19, 391–400 (1971).
[CrossRef]

J. Acoust. Soc. Am.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. Appl. Phys.

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 407–414 (1952).
[CrossRef]

V. Twersky, “Multiple scattering by an arbitrary planar array of parallel cylinders and by two parallel cylinders,” J. Appl. Phys. 23, 1099–1118 (1952).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Opt. (Paris)

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Philos. Mag.

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81 (1981).

Phys. Rev. B

J. M. Gérardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[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]

Q. Appl. Math.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23(1954).

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Radio Sci.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other

L. V. Lorenx, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, H. Valentiner, revues et annotées (Librairie Lehman et Stage, Copenhagen, 1898), pp. 405–529.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. Kohl, eds. (U.S. Army Aberdeen, Md., 1984), pp. 237–247.

G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, Orlando, Fla., 1985), pp. 698–700.

A. Messiah, Quantum Mechanics, Volume II (North-Holland, Amsterdam, 1962), p. 1057.

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

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Figures (3)

Fig. 1
Fig. 1

Geometry of the multisphere scattering problem.

Fig. 2
Fig. 2

When the direction of propagation of a plane-incident wave is parallel to the z axis, the components of the scattered field (E s , E s ) are rather simply related to the components of the incident wave (E i , E i ). In this case E s = E θ s , E s = −E ϕ s and the incident electric vector is in the xy plane.

Fig. 3
Fig. 3

Angular distributions of six sphere chains. The parameters of these sphere chains are listed in Table 1. (1)–(6), identification numbers of the sphere chains involved. The dotted curve in each panel is the theoretical prediction for i 11, and the solid curve is for i 22. The open circles in each panel are the laboratory scattering measurements for i 11, and the filled circles are for i 22.

Tables (1)

Tables Icon

Table 1 Sphere-System Parameters

Equations (98)

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

× × E - k 2 E = 0 ,             × × H - k 2 H = 0 ,
M m n ( J ) = [ i θ i π m n ( cos θ ) - i ϕ τ m n ( cos θ ) ] z n ( J ) ( k r ) exp ( i m ϕ ) , N m n ( J ) = i r n ( n + 1 ) P n m ( cos θ ) z n ( J ) ( k r ) k r exp ( i m ϕ ) + [ i θ τ m n ( cos θ ) + i ϕ i π m n ( cos θ ) ] × 1 k r d d r [ r z n ( J ) ( k r ) ] exp ( i m ϕ ) ,
π m n ( cos θ ) = m sin θ P n m ( cos θ ) , τ m n ( cos θ ) = d d θ P n m ( cos θ ) .
E s ( j ) = n = 1 m = - n n i E m n [ a m n j N m n ( 3 ) + b m n j M m n ( 3 ) ] , H s ( j ) = k ω μ n = 1 m = - n n E m n [ b m n j N m n ( 3 ) + a m n j M m n ( 3 ) ] , E I ( j ) = - n = 1 m = - n n i E m n [ d m n j N m n ( 1 ) + c m n j M m n ( 1 ) ] , H I ( j ) = - k j ω μ j n = 1 m = - n n E m n [ c m n j N m n ( 1 ) + d m n j M m n ( 1 ) ] ,
E m n = E 0 i n ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! .
E 1 n = E n = E 0 i n 2 n + 1 n ( n + 1 ) .
E i ( j ) = - n = 1 m = - n n i E m n [ p m n j N m n ( 1 ) + q m n j M m n ( 1 ) ] , H i ( j ) = - k ω μ n = 1 m = - n n E m n [ q m n j N m n ( 1 ) + p m n j M m n ( 1 ) ] ,
E i ( j ) + E s ( j ) - E I ( j ) ] × i r j = [ H i ( j ) + H s ( j ) - H I ( j ) ] × i r j = 0.
E i θ ( j ) + E s θ ( j ) = E I θ ( j ) ,             E i ϕ ( j ) + E s ϕ ( j ) = E I ϕ ( j ) , H i θ ( j ) + H s θ ( j ) = H I θ ( j ) ,             H i ϕ ( j ) + H s ϕ ( j ) = H I ϕ ( j ) .
j n ( m j x j ) c m n j + h n ( 1 ) ( x j ) b m n j = q m n j j n ( x j ) , μ [ m j x j j n ( m j x j ) ] c m n j + μ j [ x j h n ( 1 ) ( x j ) ] b m n j = q m n j μ j [ x j j n ( x j ) ] , μ m j j n ( m j x j ) d m n j + μ j h n ( 1 ) ( x j ) a m n j = p m n j μ j j n ( x j ) , [ m j x j j n ( m j x j ) ] d m n j + m j [ x j h n ( 1 ) ( x j ) ] a m n j = p m n j m j [ x j j n ( x j ) ] .
x j = k a j = 2 π N 0 a j λ ,             m j = k j k = N j N 0 ,
a m n j = μ ( m j ) 2 j n ( m j x j ) [ x j j n ( x j ) ] - μ j j n ( x j ) [ m j x j j n ( m j x j ) ] μ ( m j ) 2 j n ( m j x j ) [ x j h n ( 1 ) ( x j ) ] - μ j h n ( 1 ) ( x j ) [ m j x j j n ( m j x j ) ] p m n j , b m n j = μ j j n ( m j x j ) [ x j j n ( x j ) ] - μ j n ( x j ) [ m j x j j n ( m j x j ) ] μ j j n ( m j x j ) [ x j h n ( 1 ) ( x j ) ] - μ h n ( 1 ) ( x j ) [ m j x j j n ( m j x j ) ] q m n j ,
c m n j = μ j j n ( x j ) [ x j h n ( 1 ) ( x j ) ] - μ j h n ( 1 ) ( x j ) [ x j j n ( x j ) ] μ j j n ( m j x j ) [ x j h n ( 1 ) ( x j ) ] - μ h n ( 1 ) ( x j ) [ m j x j j n ( m j x j ) ] q m n j , d m n j = μ j m j j n ( x j ) [ x j h n ( 1 ) ( x j ) ] - μ j m j h n ( 1 ) ( x j ) [ x j j n ( x j ) ] μ ( m j ) 2 j n ( m j x j ) [ x j h n ( 1 ) ( x j ) ] - μ j h n ( 1 ) ( x j ) [ m j x j j n ( m j x j ) ] p m n j .
a m n j = a n j p m n j ,             b m n j = b n j q m n j ,
c m n j = c n j q m n j ,             d m n j = d n j p m n j ,
E i ( j ) = E 0 ( j ) + l j E s ( l , j ) , H i ( j ) = H 0 ( j ) + l j H s ( l , j ) .
E 0 ( j ) = - n = 1 m = - n n i E m n [ p m n j , j N m n ( 1 ) + q m n j , j M m n ( 1 ) ] , H 0 ( j ) = - k ω μ n = 1 m = - n n E m n [ q m n j , j N m n ( 1 ) + p m n j , j M m n ( 1 ) ] .
p m n 0 = p m n j 0 , j 0 ,             q m n 0 = q m n j 0 , j 0 ,
k = k ( i x sin α cos β + i y sin α sin β + i z cos α ) ,
p m n j , j = exp ( i k · d j 0 , j ) p m n 0 ,             q m n j , j = exp ( i k · d j 0 , j ) q m n 0 ,
p m n 0 = 1 n ( n + 1 ) [ τ m n ( cos α ) cos α - i π m n ( cos α ) sin β ] , q m n 0 = 1 n ( n + 1 ) [ τ m n ( cos α ) cos β - i π m n ( cos α ) sin β ] .
M m n = ν = 0 μ = - ν ν ( A 0 μ ν m n M μ ν + B 0 μ ν m n N μ ν ) , N m n = ν = 0 μ = - ν ν ( B 0 μ ν m n M μ ν + A 0 μ ν m n N μ ν ) ,
M m n ( 3 ) ( l ) = ν = 0 μ = - ν ν [ A 0 μ ν m n ( l , j ) M μ ν ( 1 ) ( j ) + B 0 μ ν m n ( l , j ) N μ ν ( 1 ) ( j ) ] , N m n ( 3 ) ( l ) = ν = 0 μ = - ν ν [ B 0 μ ν m n ( l , j ) M μ ν ( 1 ) ( j ) + A 0 μ ν m n ( l , j ) N μ ν ( 1 ) ( j ) ] .
E s ( l , j ) = n = 1 m = - n n i E m n ( a m n l { ν = 1 μ = - ν ν [ A 0 μ ν m n ( l , j ) N μ ν ( 1 ) + B 0 μ ν m n ( l , j ) M μ ν ( 1 ) ] } + b m n l { ν = 1 μ = - ν ν [ B 0 μ ν m n ( l , j ) N μ ν ( 1 ) + A 0 μ ν m n ( l , j ) M μ ν ( 1 ) ] } ) , H s ( l , j ) = k ω μ n = 1 m = - n n E m n ( b m n l { ν = 1 μ = - ν ν [ A 0 μ ν m n ( l , j ) N μ ν ( 1 ) + B 0 μ ν m n ( l , j ) M μ ν ( 1 ) ] } + a m n l { ν = 1 μ = - ν ν [ B 0 μ ν m n ( l , j ) N μ ν ( 1 ) + A 0 μ ν m n ( l , j ) M μ ν ( 1 ) ] } ) .
E s ( l , j ) = - n = 1 m = - n n i E m n [ p m n l , j N m n ( 1 ) + q m n l , j M m n ( 1 ) ] , H s ( l , j ) = - k ω μ n = 1 m = - n n E m n [ q m n l , j N m n ( 1 ) + p m n l , j M m n ( 1 ) ] ,
p m n 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 ) ]             ( l j ) , q m n l , j = - ν = 1 μ = - ν ν [ a μ ν l B m n μ ν ( l , j ) + b μ ν l A m n μ ν ( l , j ) ]             ( l j ) .
A m n μ ν = E μ ν E m n A 0 m n μ ν = i ν - n ( 2 ν + 1 ) ( n + m ) ! ( ν - μ ) ! ( 2 n + 1 ) ( n - m ) ! ( ν + μ ) ! A 0 m n μ ν , B m n μ ν = E μ ν E m n B 0 m n μ ν = i ν - n ( 2 ν + 1 ) ( n + m ) ! ( ν - μ ) ! ( 2 n + 1 ) ( n - m ) ! ( ν + μ ) ! B 0 m n μ ν .
p m n j = l = 1 L p m n l , j ,             q m n j = l = 1 L q m n l . j
p m n j = p m n j , j - l j ( 1 , L ) 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 ) ] , q m n j = q m n j , j - l j ( 1 , L ) ν = 1 μ = - ν ν [ a μ ν l B m n μ ν ( l , j ) + b μ ν l A m n μ ν ( l , j ) ] ,
a m n j = a n j { p m n j , j - l j ( 1 , L ) 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 j { q m n j , j - l j ( 1 , L ) ν = 1 μ = - ν ν [ a μ ν l B m n μ ν ( l , j ) + b μ ν l A m n μ ν ( l , j ) ] } .
d n j a m n j - a n j d m n j = 0 ,             c n j b m n j - b n j c m n j = 0.
p 1 m n i = p m n j , j ,             q 1 m n j = q m n j , j ,
a 1 μ ν l = a ν l p 1 μ ν 1 ,             b 1 μ ν l = b ν l q 1 μ ν l .
p i m n j = p m n j , j - l j ( 1 , L ) ν = 1 μ = - ν ν [ a i - 1 μ ν l A m n μ ν ( l , j ) + b i - 1 μ ν l B m n μ ν ( l , j ) ] , q i m n j = q m n j , j - l j ( 1 , L ) ν = 1 μ = - ν ν [ a i - 1 μ ν l B m n μ ν ( l , j ) + b i - 1 μ ν l A m n μ ν ( l , j ) ] ,
a i μ ν l = ( 1 - f ) a i - 1 μ ν l + f a ν l p i μ ν l , b i μ ν l = ( 1 - f ) b i - 1 μ ν l + f b ν l q i μ ν l ,
E s = n = 1 m = - n n i E m n [ a m n N m n ( 3 ) + b m n M m n ( 3 ) ] , H s = k ω μ n = 1 m = - n n E m n [ b m n N m n ( 3 ) + a m n M m n ( 3 ) ] ,
a m n = l = 1 L ν = 1 μ = - ν ν [ a μ ν l A m n μ ν ( l , j 0 ) + b μ ν l B m n μ ν ( l , j 0 ) ] , b m n = l = 1 L ν = 1 μ = - ν ν [ a μ ν l B m n μ ν ( l , j 0 ) + b μ ν l A m n μ ν ( l , j 0 ) ] .
A m n μ ν ( j 0 , j 0 ) = δ μ m δ ν n ,
B m n μ ν ( j 0 , j 0 ) = 0 ,
E I ( j , j 0 ) = - n = 1 m = - n n i E m n [ d m n j , j 0 N m n ( 1 ) + c m n j , j 0 M m n ( 1 ) ] , H I ( j , j 0 ) = - k j ω μ j n = 1 m = - n n E m n [ c m n j , j 0 N m n ( 1 ) + d m n j , j 0 M m n ( 1 ) ] ,
c m n j , j 0 = ν = 1 μ = - ν ν [ c μ ν j A m n μ ν ( j , j 0 ) + d μ ν j B m n μ ν ( j , j 0 ) ] , d m n j , j 0 = ν = 1 μ = - ν ν [ d μ ν j B m n μ ν ( j , j 0 ) + c μ ν j A m n μ ν ( j , j 0 ) ] .
S = S i + S s + S ext ,
S i = ½ Re ( E i × H i * ) ,             S s = ½ Re ( E s × H s * ) , S ext = ½ Re ( E i × H s * + E s × H i * ) .
W a = W ext - W s ,
W ext = - A S ext · i r d A ,             W s = - A S s · i r d A ,
W ext = 1 2 Re 0 2 π 0 π ( E i ϕ H s θ * - E i θ H s ϕ * - E s θ H i ϕ * + E s ϕ H i θ * ) r 2 sin θ d θ d ϕ , W s = 1 2 Re 0 2 π 0 π ( E s θ H s ϕ * - E s ϕ H s θ * ) r 2 sin θ d θ d ϕ ,
E i θ = n = 1 m = - n n E m n ( - i p m n 0 ψ n τ m n + q m n 0 ψ n π m n ) exp ( i m ϕ ) k r , E i ϕ = n = 1 m = - n n E m n ( i q m n 0 ψ n τ m n + p m n 0 ψ n π m n ) exp ( i m ϕ ) k r , H i θ = k ω μ 0 n = 1 m = - n n E m n × ( i p m n 0 ψ n τ m n - q m n 0 ψ n π m n ) exp ( i m ϕ ) k r , H i ϕ = k ω μ 0 n = 1 m = - n n E m n × ( i q m n 0 ψ n π m n + p m n 0 ψ n τ m n ) exp ( i m ϕ ) k r ,
E s θ = n = 1 m = - n n E m n ( - i a m n ξ n τ m n - b m n ξ n π m n ) exp ( i m ϕ ) k r , E s ϕ = n = 1 m = - n n E m n ( - i b m n ξ n τ m n - a m n ξ n π m n ) exp ( i m ϕ ) k r , H s θ = k ω μ 0 n = 1 m = - n n E m n × ( i a m n ξ n π m n + b m n ξ n τ m n ) exp ( i m ϕ ) k r , H s ϕ = k ω μ 0 n = 1 m = - n n E m n × ( i b m n ξ n π m n - a m n ξ n τ m n ) exp ( i m ϕ ) k r ,
ψ n ( ρ ) = ρ j n ( ρ ) ,             ξ n ( ρ ) = ρ h n ( 1 ) ( ρ ) .
W s = 2 π E 0 2 k ω μ 0 n = 1 m = - n n n ( n + 1 ) ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × Re ( - i ξ n * ξ n a m n a m n * + i ξ n ξ n * b m n b m n * ) ,
E m n E m n * = E 0 2 ( 2 n + 1 ) 2 [ ( n - m ) ! ( n + m ) ! ] 2 ,
0 2 π exp ( i m ϕ ) [ exp ( i m ϕ ) ] * d ϕ = 2 π δ m m ,
0 π ( π m n π m ν + τ m n τ m ν ) sin θ d θ = δ ν n 2 n ( n + 1 ) 2 n + 1 ( n + m ) ! ( n - m ) ! .
h n ( 1 ) ( k r ) ~ ( - i ) n exp ( i k r ) i k r ,             k r n 2 ,
ξ n ~ ( - i ) n + 1 exp ( i k r ) ,             ξ n ~ ( - i ) n exp ( i k r ) ,
i ξ n ξ n * = - i ξ n * ξ n = 1.
C sca = W s I i = 4 π k 2 n = 1 m = - n n n ( n + 1 ) ( 2 n + 1 ) × ( n - m ) ! ( n + m ) ! ( a m n 2 + b m n 2 ) ,
C ext = W ext I i = 4 π k 2 n = 1 m = - n n n ( n + 1 ) ( 2 n + 1 ) × ( n - m ) ! ( n + m ) ! Re ( p m n 0 * a m n + q m n 0 * b m n ) .
Re ( p m n 0 a m n * ) = Re ( p m n 0 * a m n ) , Re ( q m n 0 b m n * ) = Re ( q m n 0 * b m n ) .
ψ n ( ρ ) ~ cos [ ρ - n ( n + 1 ) π 2 ] , ψ n ( ρ ) ~ - sin [ ρ - n ( n + 1 ) π 2 ] , - i ( ψ n ξ n * + ψ n * ξ n ) = i ( ψ n ξ n * + ψ n * ξ n ) = 1.
C abs = C ext - C sca .
E s θ ~ E 0 exp ( i k r ) - i k r n = 1 m = - n n ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × ( a m n τ m n + b m n π m n ) exp ( i m ϕ ) , E s ϕ ~ E 0 exp ( i k r ) - i k r n = 1 m = - n n ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × ( a m n π m n + b m n τ m n ) i exp ( i m ϕ ) .
( E s E s ) = exp [ i k ( r - z ) ] - i k r [ S 2 S 3 S 4 S 1 ] ( E i E i ) ,
E i = E 0 ( cos ϕ cos β + sin ϕ sin β ) , E i = E 0 ( sin ϕ cos β - cos ϕ sin β ) ,
( E s E s ) = ( E s θ - E s ϕ ) = exp [ i k ( r - z ) ] - i k r [ S 2 S 3 S 4 S 1 ] [ E 0 cos ( ϕ - β ) E 0 sin ( ϕ - β ) ] .
π - m , n = ( - 1 ) m + 1 ( n - m ) ! ( n + m ) ! π m n , τ - m , n = ( - 1 ) m ( n - m ) ! ( n + m ) ! τ m n ,
S 2 cos ( ϕ - β ) + S 3 sin ( ϕ - β ) = n - 1 m = 0 n ( Ψ m n cos m ϕ + Φ m n i sin m ϕ ) , S 4 cos ( ϕ - β ) + S 1 sin ( ϕ - β ) = i n - 1 m = 0 n ( Θ m n cos m ϕ + Ξ m n i sin m ϕ ) ,
Ψ m n = 2 n + 1 1 + δ 0 m [ ( n - m ) ! ( n + m ) ! ( a m n τ m n + b m n π m n ) + ( - 1 ) m ( a - m n τ m n - b - m n π m n ) ] , Φ m n = 2 n + 1 1 + δ 0 m [ ( n - m ) ! ( n + m ) ! ( a m n τ m n + b m n π m n ) - ( - 1 ) m ( a - m n τ m n - b - m n π m n ) ] , Θ m n = 2 n + 1 1 + δ 0 m [ ( n - m ) ! ( n + m ) ! ( a m n π m n + b m n τ m n ) - ( - 1 ) m ( a - m n π m n - b - m n τ m n ) ] , Ξ m n = 2 n + 1 1 + δ 0 m [ ( n - m ) ! ( n + m ) ! ( a m n π m n + b m n τ m n ) + ( - 1 ) m ( a - m n π m n - b - m n τ m n ) ] .
S 2 ( θ , ϕ ) = n = 1 m = 0 n { Ψ m n   cos [ ( m - 1 ) ϕ + β ] + i Φ m n sin [ ( m - 1 ) ϕ + β ] } , S 3 ( θ , ϕ ) = - n = 1 m = 0 n { Ψ m n   sin [ ( m - 1 ) ϕ + β ] - i Φ m n cos [ ( m - 1 ) ϕ + β ] } , S 4 ( θ , ϕ ) = - n = 1 m = 0 n { i Θ m n   cos [ ( m - 1 ) ϕ + β ] - Ξ m n sin [ ( m - 1 ) ϕ + β ] } , S 1 ( θ , ϕ ) = n = 1 m = 0 n { i Θ m n   sin [ ( m - 1 ) ϕ + β ] + Ξ m n cos [ ( m - 1 ) ϕ + β ] } .
V = [ S 2 cos ( ϕ - β ) + S 3 sin ( ϕ - β ) ] i θ - [ S 4 cos ( ϕ - β ) + S 1 sin ( ϕ - β ) ] i ϕ .
C sca = 0 2 π 0 π V 2 k 2 sin θ d θ d ϕ ,
C ext = 4 π k 2 Re [ ( V · i V ) θ = 0 ] ,
i V = i x   cos β + i y sin β = sin θ cos ( ϕ - β ) i r + cos θ cos ( ϕ - β ) i θ - sin ( ϕ - β ) i ϕ .
S 2 ( 0 , ϕ ) cos ( ϕ - β ) + S 3 ( 0 , ϕ ) sin ( ϕ - β ) = n = 1 2 n + 1 2 × [ a 1 n + b 1 n - n ( n + 1 ) ( a - 1 n - b - 1 n ) ] cos ϕ + n = 1 2 n + 1 2 × [ a 1 n + b 1 n + n ( n + 1 ) ( a - 1 n - b - 1 n ) ] i sin ϕ , S 4 ( 0 , ϕ ) cos ( ϕ - β ) + S 1 ( 0 , ϕ ) sin ( ϕ - β ) = n = 1 2 n + 1 2 × [ a 1 n + b 1 n - n ( n + 1 ) ( a - 1 n - b - 1 n ) ] i cos ϕ + n = 1 2 n + 1 2 × [ a 1 n + b 1 n - n ( n + 1 ) ( a - 1 n - b - 1 n ) ] sin ϕ ,
π m n ( 1 ) = { 1 2 m = - 1 n ( n + 1 ) 2 m = 1 0 otherwise , π m n ( 1 ) = { - 1 2 m = - 1 n ( n + 1 ) 2 m = 1 0 otherwise .
Re [ S ( 0 ) ] = Re [ V · i v ) θ = 0 ] = n = 1 ( 2 n + 1 ) Re [ p 1 n 0 * a 1 n + q 1 n 0 * b 1 n + n 2 ( n + 1 ) 2 ( p - 1 n 0 * a - 1 n + q - 1 n 0 * b - 1 n ) ] ,
Re [ ( a - 1 n + b - 1 n ) exp ( - i ϕ ) ] = Re [ ( a - 1 n * + b - 1 n * ) exp ( i ϕ ) ] , Re ( p - 1 n 0 a - 1 n * ) = Re ( p - 1 n 0 * a - 1 n ) , Re ( q - 1 n 0 b - 1 n * ) = Re ( q - 1 n 0 * b - 1 n ) ,
p 1 n 0 = q 1 n 0 = exp ( - i β ) 2 , p - 1 n 0 = - exp ( - i β ) 2 n ( n + 1 ) ,             q - 1 n 0 = exp ( - i β ) 2 n ( n + 1 ) .
C ext = 4 π k 2 Re [ S ( 0 ) ] ,
C ext = 4 π k 2 n = 1 ( 2 n + 1 ) Re [ p 1 n 0 * a 1 n + q 1 n 0 * b 1 n + n 2 ( n + 1 ) 2 ( p - 1 n 0 * a - 1 n + q - 1 n 0 * b - 1 n ) ] ,
C ext x = 2 π k 2 n = 1 ( 2 n + 1 ) × Re [ a 1 n + b 1 n - n ( n + 1 ) ( a - 1 n - b - 1 n ) ] , C ext y = 2 π k 2 n = 1 ( 2 n + 1 ) × Re { i [ a 1 n + b 1 n + n ( n + 1 ) ( a - 1 n - b - 1 n ) ] } ,
S 2 x ( θ , ϕ ) = n = 1 m = 0 n × [ Ψ m n cos ( m - 1 ) ϕ + i Φ m n sin ( m - 1 ) ϕ ] , S 3 x ( θ , ϕ ) = - n = 1 m = 0 n × [ Ψ m n sin ( m - 1 ) ϕ - i Φ m n cos ( m - 1 ) ϕ ] , S 4 x ( θ , ϕ ) = - n = 1 m = 0 n × [ i Θ m n cos ( m - 1 ) ϕ - Ξ m n sin ( m - 1 ) ϕ ] , S 1 x ( θ , ϕ ) = n = 1 m = 0 n × [ i Θ m n sin ( m - 1 ) ϕ + Ξ m n cos ( m - 1 ) ϕ ] ,
S 2 y ( θ , ϕ ) = - n = 1 m = 0 n × [ Ψ m n sin ( m - 1 ) ϕ - i Φ m n cos ( m - 1 ) ϕ ] , S 3 y ( θ , ϕ ) = - n = 1 m = 0 n × { Ψ m n cos ( m - 1 ) ϕ + i Φ m n sin ( m - 1 ) ϕ } , S 4 y ( θ , ϕ ) = n = 1 m = 0 n × [ i Θ m n sin ( m - 1 ) ϕ + Ξ m n cos ( m - 1 ) ϕ ] , S 1 y ( θ , ϕ ) = n = 1 m = 0 n × [ i Θ m n cos ( m - 1 ) ϕ - Ξ m n sin ( m - 1 ) ϕ ] ,
i 11 = S 1 y ( θ , 0 ) 2 , i 22 = S 2 x ( θ , 0 ) 2 .
a 1 n = a n 2 ,             b 1 n = b n 2 , a - 1 n = - a n 2 n ( n + 1 ) ,             b - 1 n = b n 2 n ( n + 1 ) ,
C sca x = 2 π k 2 n = 1 ( 2 n + 1 ) ( a n 2 + b n 2 ) , C ext x = 2 π k 2 n = 1 ( 2 n + 1 ) Re ( a n + b n ) , S 2 x ( θ ) = S 2 x ( θ , ϕ ) = n = 1 2 n + 1 ( n + 1 ) ( a n τ n + b n π n ) , S 3 x ( θ ) = S 4 x ( θ ) = S 3 x ( θ , ϕ ) = S 4 x ( θ , ϕ ) = 0 , S 1 x ( θ ) = S 1 x ( θ , ϕ ) = n = 1 2 n + 1 n ( n + 1 ) ( a n π n + b n τ n ) ,
a 1 n = - i a n 2 ,             b 1 n = - i b n 2 , a - 1 n = - i a n 2 n ( n + 1 ) ,             b - 1 n = i b n 2 n ( n + 1 ) ,
π m n = m P n m ( x ) ( 1 - x 2 ) 1 / 2 , τ m n = ( 1 - x 2 ) 1 / 2 d P n m ( x ) d x ,             - 1 x 1 .
π m n + 1 = 2 n + 1 n - m + 1 x π m n - n + m n - m + 1 π m n - 1 , π m + 1 n = 2 ( m + 1 ) x ( 1 - x 2 ) 1 / 2 π m n - ( m + 1 ) ( n + m ) ( n - m + 1 ) m - 1 π m - 1 n ,             m 1 , π n n = ( 1 - x ) 1 / 2 n ( 2 n - 1 ) n - 1 π ( n - 1 ) ( n - 1 ) ,             n 1 ,
τ m n = n - m + 1 m π m n + 1 - n + 1 m x π m n ,             m 0 , τ 0 n + 1 = 2 n + 1 n x τ 0 n - n + 1 n x τ 0 n - 1 ,
π - m n = ( - 1 ) m + 1 ( n - m ) ! ( n + m ) ! π m n , π m n ( - x ) = ( - 1 ) n + m π m n ( x ) ,
τ - m n = ( - 1 ) m ( n - m ) ! ( n + m ) ! τ m n , τ m n ( - x ) = ( - 1 ) n + m + 1 τ m n ( x ) .
π 00 = 0 ,             π 01 = 0 ,             π 10 = 0 ,             π 11 = 1 , τ 00 = 0 ,             τ 0 , 1 = - ( 1 - x 2 ) 1 / 2 ,             τ 10 = 0 ,             τ 11 = x ,
τ m n ( ± 1 ) = { ( ± ) n + 1 ( 1 2 ) m = - 1 ( ± ) n + 1 [ n ( n + 1 ) 2 ] m = 1 0 otherwise , τ m n ( ± 1 ) = { - ( ± ) n ( 1 2 ) m = - 1 ( ± ) n [ n ( n + 1 ) 2 ] m = 1 0 otherwise .
A 0 μ ν m n ( l , j ) = ( - 1 ) μ i ν - n 2 ν + 1 2 ν ( ν + 1 ) p = n - ν n + ν ( - i ) p × [ n ( n + 1 ) + ν ( ν + 1 ) - p ( p + 1 ) ] × a ( m , n , - μ , ν , p ) h p ( 1 ) ( k d l , j ) × P p m - μ ( cos θ l , j ) exp [ i ( m - μ ) ϕ l , j ] , B 0 μ ν m n ( l , j ) = ( - 1 ) μ i ν - n 2 ν + 1 2 ν ( ν + 1 ) × p = n - ν n + ν ( - i ) p b ( m , n , - μ , ν , p , p - 1 ) × h p ( 1 ) ( k d l , j ) P p m - μ ( cos θ l , j ) exp [ i ( m - μ ) ϕ l , j ] ,
b ( m , n , - μ , ν , p , p - 1 ) = 2 p + 1 2 p - 1 [ ( ν - μ ) ( ν + μ + 1 ) a ( m , n , - μ - 1 , ν , p - 1 ) - ( p - m + μ ) ( p - m + μ - 1 ) × a ( m , n , - μ + 1 , ν , p - 1 ) + 2 μ ( p - m + μ ) a ( m , n , - μ , ν , p - 1 ) ] ,
P n m ( cos θ ) P ν μ ( cos θ ) = p = n - ν n + ν a ( m , n , μ , ν , p ) P p m + μ ( cos θ ) ,
a ( m , n , μ , ν , p ) = 2 p + 1 2 ( p - m - μ ) ! ( p + m + μ ) ! × - 1 1 P n m ( x ) P ν μ ( x ) P p m + μ ( x ) d x .

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