Abstract

In the framework of image theory the full scattering pattern from model anisotropic particles on a perfectly reflecting surface is calculated for an arbitrary direction of the incident field. The particles on the surface are modeled either as clusters of spherical scatterers or as clusters of hemispheres whose flat faces lie on the reflecting surface. Our approach is based on the expansion of all the fields in terms of spherical multipoles whose transformation properties are used to yield a compact expression for the scattered intensity both from a single particle as a function of its orientation and from a dispersion of randomly oriented particles. The patterns that were calculated for several model scatterers show some features that may provide useful information about the possible anisotropy of actual particles on a reflecting surface.

© 1995 Optical Society of America

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References

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  1. T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
    [CrossRef]
  2. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).
  3. P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
    [CrossRef]
  4. G. Bosi, “Retarded treatment of substrate-related effects on granular films,” Physica A 190, 375–392 (1992).
    [CrossRef]
  5. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991); errata, J. Opt. Soc. Am. A9, 844–845 (1992).
    [CrossRef]
  6. I. V. Lindell, A. H. Sihvola, K. O. Muimonen, P. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
    [CrossRef]
  7. I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem. Part III. General formulation,” IEEE Trans. Antennas Propag. AP-32, 1027–1032 (1984).
    [CrossRef]
  8. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
    [CrossRef]
  9. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially embedded in a ground plane. 1. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [CrossRef]
  10. 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]
  11. E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 10–24.
  12. E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, p. 50; Chap. 5, pp. 76 and 98–106.
  13. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28; Chap. 2, pp. 30–52.
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 453–459; Chap. 16, pp. 739–775.
  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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. B 81, 29–50 (1984).
    [CrossRef]
  18. M. Hammermesh, Group Theory (Addison-Wesley, Reading, Mass., 1962), Chap. 9, p. 322.
  19. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  20. 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]
  21. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of the electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
    [CrossRef]
  22. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975), Chap. 8.1, pp. 257–259.
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 31–32; Chap. 9, p. 114; Chap. 15, p. 297; Chap. 16, p. 329.
  24. 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]
  25. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
    [CrossRef]

1992 (2)

G. Bosi, “Retarded treatment of substrate-related effects on granular films,” Physica A 190, 375–392 (1992).
[CrossRef]

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
[CrossRef]

1991 (2)

1989 (2)

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially embedded in a ground plane. 1. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
[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]

1987 (2)

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (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]

1986 (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

1984 (5)

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

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem. Part III. General formulation,” IEEE Trans. Antennas Propag. AP-32, 1027–1032 (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]

1971 (1)

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

Alanen, E.

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem. Part III. General formulation,” IEEE Trans. Antennas Propag. AP-32, 1027–1032 (1984).
[CrossRef]

Balescu, R.

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975), Chap. 8.1, pp. 257–259.

Barakat, R.

Barber, P.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

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, “Use of group theory for the description of the 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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. B 81, 29–50 (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, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Bosi, G.

G. Bosi, “Retarded treatment of substrate-related effects on granular films,” Physica A 190, 375–392 (1992).
[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, “Use of group theory for the description of the 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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. B 81, 29–50 (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, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Greef, R.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

Greenberg, J. M.

Hammermesh, M.

M. Hammermesh, Group Theory (Addison-Wesley, Reading, Mass., 1962), Chap. 9, p. 322.

Inoue, M.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 453–459; Chap. 16, pp. 739–775.

Johnson, B. R.

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, K. O. Muimonen, P. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem. Part III. General formulation,” IEEE Trans. Antennas Propag. AP-32, 1027–1032 (1984).
[CrossRef]

Muimonen, K. O.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28; Chap. 2, pp. 30–52.

Ohtaka, K.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Rao, T. C.

Rose, E. M.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 10–24.

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, p. 50; Chap. 5, pp. 76 and 98–106.

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, “Use of group theory for the description of the 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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. 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]

Schuerman, D. W.

Sihvola, A. H.

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, “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, “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, “Use of group theory for the description of the 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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. B 81, 29–50 (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]

Takemori, T.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[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, “Use of group theory for the description of the 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, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cim. 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, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 31–32; Chap. 9, p. 114; Chap. 15, p. 297; Chap. 16, p. 329.

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Wang, R. T.

Waterman, P. C.

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

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]

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

IEEE Trans. Antennas Propag. (1)

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem. Part III. General formulation,” IEEE Trans. Antennas Propag. AP-32, 1027–1032 (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. A (5)

J. Phys. Soc. Jpn. (1)

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Nuovo Cim. 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 Cim. B 81, 29–50 (1984).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. D (1)

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

Physica (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Physica A (2)

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica A 137, 243–257 (1986).
[CrossRef]

G. Bosi, “Retarded treatment of substrate-related effects on granular films,” Physica A 190, 375–392 (1992).
[CrossRef]

Other (7)

M. Hammermesh, Group Theory (Addison-Wesley, Reading, Mass., 1962), Chap. 9, p. 322.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 10–24.

E. M. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4, p. 50; Chap. 5, pp. 76 and 98–106.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, pp. 24–28; Chap. 2, pp. 30–52.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 453–459; Chap. 16, pp. 739–775.

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975), Chap. 8.1, pp. 257–259.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 31–32; Chap. 9, p. 114; Chap. 15, p. 297; Chap. 16, p. 329.

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Figures (4)

Fig. 1
Fig. 1

Sketch of the arrangement that we chose to display our results. The reflecting surface coincides with the xz plane, and the y axis points toward the accessible half-space, whose refractive index is n = 1. The polar angles of the direction of observation range in the intervals 0° ≤ ϑS ≤ 180° and 0° ≤ φS ≤ 180°. The direction of incidence k ^ I, indicated by the arrow, has the polar angles ϑI = 90° and φI = 225°. The arrangement of the four-hemisphere linear chain whose scattering pattern is reported in Fig. 2(d) below is also shown.

Fig. 2
Fig. 2

Pattern of the scattered intensity for (a) a single sphere on the reflecting surface, (b) a dispersion of randomly oriented two-sphere clusters, (c) a dispersion of randomly oriented four-hemisphere linear chains, (d) a four-hemisphere linear chain oriented as shown in Fig. 1. The wavelength of the incident wave is λ = 628.3 nm, and the refractive index of all the scatterers is n0 = 3. The radius of the single sphere in (a) is bs = 126.0 nm, whereas the radius of the spheres of the two-sphere cluster in (b) and of the hemispheres of the four-hemisphere linear chain in (c) and (d) is bh = 100 nm. We actually report (in square meters) the quantities r2Iφφ/I0 in (a) and (d) and r2〈Iφφ〉/I0 in (b) and (c) as a function of the angles of observation ϑS and φS. r is the distance of observation; I0, the incident intensity; Iφφ, the observed intensity; 〈Iφφ〉, the orientationally averaged observed intensity.

Fig. 3
Fig. 3

Pattern of the scattered intensity for (a) a single sphere, (b) a dispersion of randomly oriented two-sphere clusters, (c) a dispersion of randomly oriented two-hemisphere clusters, and (d) a dispersion of randomly oriented four-hemisphere linear chains. The radii and the refractive indices are the same as in Fig. 2; the radius of the hemispheres of the two-hemisphere cluster equals the radius of the single sphere. We report (in square meters) r2Iφϑ/I0 in (a) and r2〈Iφϑ 〉/I0 in (b)–(d).

Fig. 4
Fig. 4

Same as Fig. 3, except that r2Iϑφ/I0 and r2〈Iϑφ〉/I0 are considered.

Equations (45)

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E inc = E 0 e ^ I exp ( i k I · r )
E inc = E 0 p l m W I l m ( p ) J l m ( p ) ( n k , r ) ,
J l m ( 1 ) ( k , r ) = j l ( k r ) X l m ( r ^ ) , J l m ( 2 ) ( k , r ) = ( 1 / k ) × J l m ( 1 ) ( k , r ) ,
W I l m ( p ) = W l m ( p ) ( e ^ I , k ^ I ) ,
W l m ( 1 ) ( e ^ , k ^ ) = 4 π i l e ^ · X l m * ( k ^ ) , W l m ( 2 ) ( e ^ , k ^ ) = 4 π i l + 1 ( k ^ × e ^ ) · X l m * ( k ^ ) .
E ref = E 0 p l m W R l m ( p ) J l m ( p ) ( n k , r ) .
( e ^ I + e ^ R ) × n ^ = 0 ,
e R = e I ,             e R = - e I
u ^ I = ( 1 / 2 ) ( u ^ I i u ^ I ) , u ^ R = ( 1 / 2 ) ( u ^ R i u ^ R ) ,
e R - = e I + ,             e R + = e I - .
E sca = E 0 p l m A l m ( p ) H l m ( p ) ( n k , r ) ,
A l m ( p ) = - p l m S l m , l m ( p , p ) W E l m ( p ) ,
W E l m ( p ) = W I l m ( p ) + W R l m ( p ) .
E η η sca = E 0 exp ( i n k r ) r f η η ,
I η η = | E 0 r f η η | 2 ,
f η η = i 4 π n k p l m p l m W S η l m ( p ) * S l m , l m ( p , p ) W E η l m ( p ) ,
W S η l m ( p ) = W l m ( p ) ( e ^ S , k ^ S ) ,
f η η = i 4 π n k p l m p l m μ μ W S η l m ( p ) * D μ m ( l ) * ( Θ ) × S ¯ l μ , l μ ( p , p ) D μ m ( l ) ( Θ ) W E η l m ( p ) ,
I ¯ η η = ( ν E ν η η * ) ( ν E ν η η ) = N E η η ( R , Θ ) 2 + ( N 2 - N ) × E η η * ( R , Θ ) E η η ( R , Θ ) ,
I ¯ η η = N I η η = N I η η ( R , Θ ) w ( R , Θ ) d R d Θ ,
I η η I η η ( R , Θ ) w R ( R ) w Θ ( Θ ) d R d Θ w R ( R ) d R I η η ( 0 , Θ ) w Θ ( Θ ) d Θ = I η η ( 0 , Θ ) w Θ ( Θ ) d Θ
w R ( R ) d R = 1.
D μ m ( l ) ( α , 0 , 0 ) = exp ( - i μ α ) δ μ m ,
f η η = i 4 π n k p l m p l m W S η l m ( p ) * exp ( i m α ) × S ¯ l m , l m ( p , p ) exp ( - i m α ) W E η l m ( p ) .
I ¯ η η = N 16 π 2 k 2 r 2 | E 0 n | 2 × m m m m F η η m m * F η η m m m - m - m + m ,
F η η m m = p l p l W S η l m ( p ) * S ¯ l m , l m ( p , p ) W E η l m ( p ) , μ = 0 2 π exp ( - i α μ ) w ( α ) d α ,
w ( α ) = w Θ ( α , β = 0 , γ = 0 ) .
I ¯ η η = N 16 π 2 k 2 r 2 | E 0 n | 2 m m m m F η η m m * F η η m m .
I ¯ η η = N 16 π 2 k 2 r 2 | E 0 n | 2 m m F η η m m * F η η m m
E int ( r α ) = p l m C α l m ( p ) J l m ( p ) ( n α k , r α ) ,
E sca = α p l m A α l m ( p ) H l m ( p ) ( n k , r α ) ,
E = p l m W α l m ( p ) J l m ( p ) ( n k , r α ) ,
W α l m ( p ) = p l m J α l m , l m ( p , p ) W l m ( p ) .
J α l m , 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 ( n k , R α 0 ) × C ( 1 , l , l ; - μ , m + μ ) ,
G l m , l m ( 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 = [ ( 2 λ + 1 ) ( 2 l + 1 ) 4 π ( 2 l + 1 ) ] 1 / 2 × C ( l , λ , l ; 0 , 0 ) C ( l , λ , l ; m , m - m ) .
M A = - W ,
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 ( n α k b α ) u l ( n k b α ) - ( 1 + n ¯ α δ p 2 ) u l ( n α k b α ) u l ( n k α ) ( 1 + n ¯ α δ p 1 ) u l ( n α k b α ) w l ( n k b α ) - ( 1 + n ¯ α δ p 2 ) u l ( n α k b α ) w l ( n k b α ) ,
n ¯ α = n n α - 1 ,             u l ( x ) = x j l ( x ) ,             w l ( x ) = x h l ( 1 ) ( x ) .
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 l + 1 - δ p p , m + μ ; l , m + μ ( n k , R α α ) × C ( 1 , l , l ; - μ , m + μ ) ,
A = - M - 1 W ,
A α l m ( p ) = - α p l m [ M - 1 ] α l m , α l m ( p , p ) W α l m ( p ) .
A l m ( p ) = α p l m J l m , α l m ( p , p ) A α l m ( p ) ;
S l m , l m ( p , p ) = α α q L M q L M J l m , α L M ( p , q ) × [ M - 1 ] α L M , α L M ( q , q ) J α L M , l m ( q , p ) .

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