Abstract

The ℐ-matrix method, as extended recently to randomly oriented scatterers [ J. Opt. Soc. Am. A 8, 871 ( 1991)], is used to calculate rigorously light scattering by size–shape distributions of randomly oriented axially symmetric particles. The computational scheme is described in detail along with a newly developed convergence procedure that enables one to substantially reduce computer time and storage requirements. It is demonstrated that the elements of the Stokes scattering matrix for a power law size distribution of randomly oriented moderately aspherical spheroids are much smoother than and differ substantially from those of equivalent monodisperse spheroids, and thus averaging over orientations does not eliminate the necessity of averaging over particle sizes. Numerical calculations are reported for volume-equivalent polydispersions of spheres and size–shape distributions of moderately aspherical spheroids with the index of refraction 1.5 + 0.02 i, which is typical of some maritime aerosols. The angular-scattering behavior of the ensembles of nonspherical particles is found to be greatly different from that of the equivalent polydisperse spheres. The size–shape distributions of spheroids exhibit stronger side scattering near 120° and weaker backscattering, the ratio F22/F11 of the elements of the scattering matrix substantially deviates from unity, and the element F33 is greatly different from F44. For size distributions of oblate and prolate spheroids of the same aspect ratio, the ratios F22/F11, F33/F11, and F34/F11 can differ substantially and, thus, are indicators of particle shape, whereas the angular patterns of the intensity (F11) and linear polarization (−F12/F11) are similar. For the size–shape distributions of moderately aspherical spheroids, the optical cross sections, the single-scattering albedo, and the asymmetry parameter of the phase function do not differ substantially from those of equivalent spheres. In general, the elements of the scattering matrix and optical cross sections are more shape dependent for larger particles.

© 1993 Optical Society of America

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References

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  1. S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,”Appl. Opt. 23, 1025–1031 (1984).
    [CrossRef] [PubMed]
  2. J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).
  3. W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2. Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
    [CrossRef] [PubMed]
  4. M. I. Mishchenko, “Infrared absorption by shape distributions of NH3 ice particles: an application to the Jovian atmosphere,” Earth Moon Planet 53, 149–156 (1991).
    [CrossRef]
  5. F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
    [CrossRef]
  6. C. F. Bohren, S. B. Singham, “Backscattering by nonspherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. D 96, 5269–5277 (1991).
    [CrossRef]
  7. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971);“Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979);B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
    [CrossRef]
  8. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975);D.-S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
    [CrossRef] [PubMed]
  9. V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1980).
  10. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  11. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  14. I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
    [CrossRef]
  15. H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974);“Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357–361 (1975);O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Atmos. Oceanic Phys. 12, 366–370 (1976).
    [CrossRef]
  16. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981);“On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
    [CrossRef]
  17. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983);C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).
  18. I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their ApplicationsPergamon, Oxford, 1963).
  19. J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).
  20. R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986);“The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
    [CrossRef]
  21. M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990);“Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
    [CrossRef]
  22. W. M. F. Wauben, J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 491–504 (1992).
    [CrossRef]
  23. P. W. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. dissertation (University of California, Los Angeles, Los Angeles, Calif.1973).
  24. W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, Md., 1986).
  25. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  26. P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
    [CrossRef]
  27. M. I. Mishchenko, “Interstellar light absorption by randomly oriented nonspherical dust grains,” Sov. Astron. Lett. 15, 299–302 (1989);“Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990);“Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990);N. G. Khlebtsov, “Orientational averaging of light scattering observables in the ℐ-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
    [CrossRef] [PubMed]
  28. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  29. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,”Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  30. P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).
  31. E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301 (1986).
    [CrossRef] [PubMed]
  32. R. J. Perry, A. J. Hant, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2700–2710 (1978).
    [CrossRef] [PubMed]
  33. W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
    [CrossRef]
  34. M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
    [CrossRef]
  35. A. C. Holland, G. Gagne, “The scattering of polarized light by polydisperse systems of irregular particles,”Appl. Opt. 9, 1113–1121 (1970).
    [CrossRef] [PubMed]
  36. R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).
  37. D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
    [CrossRef]
  38. V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
    [CrossRef]

1992 (2)

W. M. F. Wauben, J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 491–504 (1992).
[CrossRef]

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

1991 (3)

C. F. Bohren, S. B. Singham, “Backscattering by nonspherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. D 96, 5269–5277 (1991).
[CrossRef]

M. I. Mishchenko, “Infrared absorption by shape distributions of NH3 ice particles: an application to the Jovian atmosphere,” Earth Moon Planet 53, 149–156 (1991).
[CrossRef]

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

1990 (1)

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990);“Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

1989 (1)

M. I. Mishchenko, “Interstellar light absorption by randomly oriented nonspherical dust grains,” Sov. Astron. Lett. 15, 299–302 (1989);“Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990);“Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990);N. G. Khlebtsov, “Orientational averaging of light scattering observables in the ℐ-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

1986 (3)

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986);“The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

1984 (1)

1983 (1)

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983);C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

1981 (3)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981);“On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
[CrossRef]

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301 (1986).
[CrossRef] [PubMed]

1980 (1)

1978 (1)

1977 (1)

P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

1976 (1)

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

1975 (1)

1974 (2)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974);“Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357–361 (1975);O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Atmos. Oceanic Phys. 12, 366–370 (1976).
[CrossRef]

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971);“Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979);B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

1970 (1)

1959 (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

1948 (1)

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Asano, S.

Barber, P.

Barber, P. W.

S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,”Appl. Opt. 23, 1025–1031 (1984).
[CrossRef] [PubMed]

P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

P. W. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. dissertation (University of California, Los Angeles, Los Angeles, Calif.1973).

Bohren, C. F.

C. F. Bohren, S. B. Singham, “Backscattering by nonspherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. D 96, 5269–5277 (1991).
[CrossRef]

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

Bosma, P. B.

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

de Haan, J. F.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

Domke, H.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974);“Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357–361 (1975);O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Atmos. Oceanic Phys. 12, 366–370 (1976).
[CrossRef]

Fry, E. S.

Gagne, G.

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986);“The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

Gelfand, I. M.

I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their ApplicationsPergamon, Oxford, 1963).

Ghoul, W. A.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Glantz, M.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Hammond, S.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Hant, A. J.

Hill, A. C.

Hill, C.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Hill, S. C.

S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,”Appl. Opt. 23, 1025–1031 (1984).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Holland, A. C.

Hovenier, J. W.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

W. M. F. Wauben, J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 491–504 (1992).
[CrossRef]

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983);C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

Huffman, D. R.

Jaggard, D. L.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Kattawar, G. W.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Kuik, F.

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

Kušcer, I.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

McClain, W. M.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Minlos, R. A.

I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their ApplicationsPergamon, Oxford, 1963).

Mishchenko, M. I.

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

M. I. Mishchenko, “Infrared absorption by shape distributions of NH3 ice particles: an application to the Jovian atmosphere,” Earth Moon Planet 53, 149–156 (1991).
[CrossRef]

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990);“Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

M. I. Mishchenko, “Interstellar light absorption by randomly oriented nonspherical dust grains,” Sov. Astron. Lett. 15, 299–302 (1989);“Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990);“Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990);N. G. Khlebtsov, “Orientational averaging of light scattering observables in the ℐ-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

Mugnai, A.

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2. Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, Md., 1986).

Perry, R. J.

Ribaric, M.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Rosswog, F.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Salzman, G. C.

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Sato, M.

Shapiro, Z. Ya.

I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their ApplicationsPergamon, Oxford, 1963).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Shorthill, R. W.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986);“The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981);“On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
[CrossRef]

Singham, M. K.

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Singham, S. B.

C. F. Bohren, S. B. Singham, “Backscattering by nonspherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. D 96, 5269–5277 (1991).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

Stammes, P.

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

Stuart, D.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Taggart, B.

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

van de Hulst, H. C.

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

van der Mee, C. V. M.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983);C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

Vouk, V.

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971);“Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979);B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Wauben, W. M. F.

W. M. F. Wauben, J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 491–504 (1992).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2. Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, Md., 1986).

Yeh, C.

Zerull, R. H.

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

Appl. Opt. (7)

Astron. Astrophys. (2)

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987);P. Stammes, J. F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983);C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

Astrophys. J. (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981);“On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
[CrossRef]

Astrophys. Space Sci. (1)

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974);“Fourier expansion of the phase matrix for Mie scattering,” Z. Meteorol. 25, 357–361 (1975);O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Atmos. Oceanic Phys. 12, 366–370 (1976).
[CrossRef]

Atmos. Environ. (1)

D. L. Jaggard, C. Hill, R. W. Shorthill, D. Stuart, M. Glantz, F. Rosswog, B. Taggart, S. Hammond, “Light scattering from particles of regular and irregular shape,” Atmos. Environ. 15, 2511–2519 (1981).
[CrossRef]

Beitr. Phys. Atmos. (1)

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

Earth Moon Planet (1)

M. I. Mishchenko, “Infrared absorption by shape distributions of NH3 ice particles: an application to the Jovian atmosphere,” Earth Moon Planet 53, 149–156 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

J. Chem. Phys. (2)

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

M. K. Singham, S. B. Singham, G. C. Salzman, “The scattering matrix for randomly oriented particles,” J. Chem. Phys. 85, 3807–3815 (1986).
[CrossRef]

J. Geophys. Res. D (1)

C. F. Bohren, S. B. Singham, “Backscattering by nonspherical particles: a review of methods and suggested new approaches,” J. Geophys. Res. D 96, 5269–5277 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (4)

F. Kuik, J. F. de Haan, J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 477–489 (1992).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986);“The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990);“Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 46, 171–181 (1991).
[CrossRef]

W. M. F. Wauben, J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer 47, 491–504 (1992).
[CrossRef]

Nature (London) (1)

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Opt. Acta (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971);“Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979);B. Peterson, S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Sov. Astron. Lett. (1)

M. I. Mishchenko, “Interstellar light absorption by randomly oriented nonspherical dust grains,” Sov. Astron. Lett. 15, 299–302 (1989);“Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990);“Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990);N. G. Khlebtsov, “Orientational averaging of light scattering observables in the ℐ-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (10)

I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and Their ApplicationsPergamon, Oxford, 1963).

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1980).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

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

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

P. W. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. dissertation (University of California, Los Angeles, Los Angeles, Calif.1973).

W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, Md., 1986).

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

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Figures (9)

Fig. 1
Fig. 1

Elements of the scattering matrix for a moderately wide power law size distribution of randomly oriented oblate spheroids with d = 1.5 and effective equal-volume-sphere radius reff = 0.9102 μm (solid curves) and randomly oriented monodisperse oblate spheroids with d = 1.5 and equal-volume-sphere radius r = 0.9102 μm (dashed curves).

Fig. 2
Fig. 2

Elements of the scattering matrix for the power law size distribution of prolate spheroids with rmin = 0.5 μm, rmax = 1.5 μm, and d = 0.6 (thin solid curves), 0.7 (dashed curves), and 0.83 (dotted-dashed curves), and for the same size distribution of spherical particles (thick solid curves).

Fig. 3
Fig. 3

Elements of the scattering matrix as in Fig. 2 but for oblate spheroids widh d = 1.7 (thin solid curves), 1.4 (dashed curves), and 1.2 (dotted-dashed curves).

Fig. 4
Fig. 4

Elements of the scattering matrix as in Fig. 2 but for shape averages over the prolate (dashed curves) and oblate (dotted-dashed curves) shapes, as well as the total averages over the six spheroidal shapes (thin solid curves).

Fig. 5
Fig. 5

Elements of the scattering matrix as in Fig. 2 but for smaller particles with rmin = 0.2 μm and rmax = 0.6 μm.

Fig. 6
Fig. 6

Elements of the scattering matrix as in Fig. 3 but for the smaller particles.

Fig. 7
Fig. 7

Elements of the scattering matrix as in Fig. 4 but for the smaller particles.

Fig. 8
Fig. 8

Linear polarization – F12/F11 for the power law size distribution of randomly oriented prolate spheroids with rmin = 0.2 μm, rmax = 0.6 μm, and d = 1/3 (dashed curve) and 1/2 (dotted-dashed curve).

Fig. 9
Fig. 9

Linear polarization as in Fig. 8 but for oblate spheroids with d = 3 (dashed curve) and 2 (dotted–dashed curve).

Tables (3)

Tables Icon

Tables 1 Parameters nmax1 and nmax2 for Chebyshev Particles and Spheroids With mr = 1.5 + 0.02 i, x = 10, and Δ =10−3

Tables Icon

Table 2 Optical Cross Sections for Scattering Csca, Extinction Cext, and Absorption Cabs, Albedo for Single Scattering ϖ, and Asymmetry Parameter of the Phase Function g for Polydisperse Spheres and Size and Size–Shape Distributions of Volume Equivalent Spheroids With rmin = 0.5 μm and rmax = 1.5 μm

Tables Icon

Table 3 Optical Cross Sections for Scattering, Csca, Extinction, Cext, and Absorption, Cabs, Albedo for Single Scattering ϖ, and Asymmetry Parameter of the Phase Function g for rmln = 0.2 μm and rmax = 0.6 μm

Equations (27)

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I s = C sca 4 π R 2 F ( ϑ ) I i ,
I s = [ I s Q s U s V s ] , I i = [ I i Q i U i V i ] ,
F ( ϑ ) = [ F 11 ( ϑ ) F 12 ( ϑ ) 0 0 F 12 ( ϑ ) F 22 ( ϑ ) 0 0 0 0 F 33 ( ϑ ) F 34 ( ϑ ) 0 0 F 34 ( ϑ ) F 44 ( ϑ ) ] ,
F 11 ( ϑ ) = s = 0 s max a 1 s P 00 s ( cos ϑ ) ,
F 22 ( ϑ ) + F 33 ( ϑ ) = s = 2 s max ( a 2 s + a 3 s ) P 22 s ( cos ϑ ) ,
F 22 ( ϑ ) F 33 ( ϑ ) = s = 2 s max ( a 2 s a 3 s ) P 2 , 2 s ( cos ϑ ) ,
F 44 ( ϑ ) = s = 0 s max a 4 s P 00 s ( cos ϑ ) ,
F 12 ( ϑ ) = s = 2 s max b 1 s P 02 s ( cos ϑ ) ,
F 34 ( ϑ ) = s = 2 s max b 2 s P 02 s ( cos ϑ ) ,
C sca = j = 1 J p j r min r max d r n ( r ) C sca j ( r ) ,
C ext = j = 1 J p j r min r max d r n ( r ) C ext j ( r ) ,
a i s = 1 C sca j = 1 J p j r min r max d r n ( r ) C sca j ( r ) [ a i s ( r ) ] j , i = 1 , 4 ,
b i s = 1 C sca j = 1 J p j r min r max d r n ( r ) C sca j ( r ) [ b i s ( r ) ] j , i = 1 , 2 .
C abs = C ext C sca ,
ϖ = C sca / C ext ,
g = 1 2 1 + 1 d ( cos ϑ ) F 11 ( ϑ ) cos ϑ = a 1 1 / 3 .
T mnm n i j = δ m m T mnm n i j , | m | n , n n max , i , j = 1 , 2 ,
max [ | C 1 ( n max 1 ) C 1 ( n max 1 1 ) C 1 ( n max 1 ) | , | C 2 ( n max 1 ) C 2 ( n max 1 1 ) C 2 ( n max 1 ) | ] 0.1 Δ ,
C 1 ( n max ) = 2 π k 2 Re n = 1 n max ( 2 n + 1 ) ( T 0 n 0 n 11 + T 0 n 0 n 22 ) ,
C 2 ( n max ) = 2 π k 2 n = 1 n max ( 2 n + 1 ) [ | T 0 n 0 n 11 | 2 + | T 0 n 0 n 22 | 2 ] ;
max [ | C 1 ( n max 2 ) C 1 ( n max 1 ) C 1 ( n max 1 ) | , | C 2 ( n max 2 ) C 2 ( n max 1 ) C 2 ( n max 1 ) | ] 0.1 Δ .
C ext = 2 π k 2 Re n = 1 n max m = n n [ T mnmn 11 + T mnmn 22 ] ,
C sca = 2 π k 2 n = 1 n max n = 1 n max m = max ( n , n ) max ( n , n ) i , j = 1 , 2 | T mnm n i j | 2 .
n ( r ) = 2 r min 2 r max 2 r max 2 r min 2 r 3 , r [ r min , r max ] ,
r eff = r max r min ln ( r max / r min ) 0.9102 μ m ,
ν eff = r max + r min 2 ( r max r min ) ln ( r max / r min ) 1 0.0986 .
| F 33 F 44 | F 11 F 22

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