Abstract

Light scattering properties of an assembly of randomly oriented, identical spheroidal particles are studied. A computation scheme has been developed to integrate the solution of Asano and Yamamoto for scattering from a homogeneous spheroid over all the particle orientations. The extinction and scattering cross sections, asymmetry factor, and scattering matrix elements are calculated for randomly oriented prolate and oblate spheroids and compared with both calculations for spheres and laboratory measurements, The scattering cross section, single scattering albedo, and asymmetry factor of spheroids tend to be larger than those for spheres of the same volume. The normalized scattering matrix has a symmetrical form with six independent elements. The angular scattering behavior of spheroids is found to be greatly different from that of spheres for side scattering to backscattering directions. In general, prolate and oblate spheroids of the same shape parameter have similar angular scattering patterns. The angular distribution of scattered intensity is characterized by strong forward scattering and weak backscattering. The linear polarization tends to be positive at intermediate scattering angles. The linear polarization and depolarization are discussed in application to scattering in the earth and planetary atmospheres.

© 1980 Optical Society of America

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References

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K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838, 852 (1979).
[CrossRef]

C. Rogers, P. G. Martin, Astrophys. J. 228, 450 (1979).
[CrossRef]

S. Asano, M. Sato, J. E. Hansen, NASA Conf. Publ. 2076, 265 (1979).

S. Asano, Appl. Opt. 18, 712 (1979).
[CrossRef] [PubMed]

P. Chýlek, R. G. Pinnick, Appl. Opt. 18, 1123 (1979).
[CrossRef] [PubMed]

M. Kotlarchyk, S.-H. Chen, S. Asano, Appl. Opt. 18, 2470 (1979).
[CrossRef] [PubMed]

1978 (7)

1977 (4)

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ. Ser. 5: 24, 89 (1977).

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

C. M. R. Platt, J. Appl. Meteorol. 17, 482 (1977).
[CrossRef]

K. Sassen, Appl. Opt. 16, 1332 (1977).
[CrossRef] [PubMed]

1976 (6)

R. G. Pinnick, D. E. Carroll, D. J. Hofmann, Appl. Opt. 15, 384 (1976).
[CrossRef] [PubMed]

P. Chýlek, J. Opt. Soc. Am. 66, 285 (1976).
[CrossRef]

R. H. Zerull, Contrib. Atmos. Phys. 49, 168 (1976).

P. Chylek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 106 (1976).

V. P. Dugin, S. O. Mirumyants, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 606 (1976).

1975 (3)

K. Sassen, Nature London 255, 316 (1975).
[CrossRef]

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
[PubMed]

1974 (5)

K.-N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

J. E. Hansen, J. W. Hovenier, J. Atmos. Sci. 31, 1137 (1974).
[CrossRef]

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

D. W. Cooper, J. W. Davis, R. L. Byers, J. Aerosol Sci. 5, 117 (1974).
[CrossRef]

1973 (1)

1972 (2)

1971 (4)

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, I. N. Plakhina, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 367 (1971).

R. M. Schotland, K. Sassen, R. Stone, J. Appl. Meteorol. 10, 1011 (1971).
[CrossRef]

K.-N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

1970 (2)

1969 (1)

D. L. Coffeen, Astron. J. 74, 446 (1969).
[CrossRef]

1965 (1)

G. I. Gorchakov, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 752 (1965).

1960 (2)

J. M. Greenberg, A. S. Meltzer, Astrophys. J. 132, 667 (1960).
[CrossRef]

B. S. Pritchard, W. G. Elliot, J. Opt. Soc. Am. 50, 191 (1960).
[CrossRef]

1948 (1)

V. Vouk, Nature London 162, 330 (1948).
[CrossRef]

Acquista, C.

Asano, S.

M. Kotlarchyk, S.-H. Chen, S. Asano, Appl. Opt. 18, 2470 (1979).
[CrossRef] [PubMed]

S. Asano, M. Sato, J. E. Hansen, NASA Conf. Publ. 2076, 265 (1979).

S. Asano, Appl. Opt. 18, 712 (1979).
[CrossRef] [PubMed]

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ. Ser. 5: 24, 89 (1977).

S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
[PubMed]

Barber, P. W.

Byers, R. L.

D. W. Cooper, J. W. Davis, R. L. Byers, J. Aerosol Sci. 5, 117 (1974).
[CrossRef]

Carroll, D. E.

Carswell, A. I.

Chen, S.-H.

Chylek, P.

P. Chylek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Chýlek, P.

Coffeen, D. L.

D. L. Coffeen, Astron. J. 74, 446 (1969).
[CrossRef]

Cooper, D. W.

D. W. Cooper, J. W. Davis, R. L. Byers, J. Aerosol Sci. 5, 117 (1974).
[CrossRef]

Cox, S. K.

Cuzzi, J. N.

J. B. Pollack, J. N. Cuzzi, in Proceedings, Third Conference on Atmospheric Radiation, 28–30 June (American Meteorological Society, New York, 1978).

Davis, J. W.

D. W. Cooper, J. W. Davis, R. L. Byers, J. Aerosol Sci. 5, 117 (1974).
[CrossRef]

Dedrick, K. G.

K. G. Dedrick, A. R. Hessing, G. J. Johnson, IEEE Trans. Antennas Propag. AP-26, 420 (1978).
[CrossRef]

Dugin, V. P.

V. P. Dugin, S. O. Mirumyants, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 606 (1976).

Elliot, W. G.

Gagne, G.

Golovanev, V. A.

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

Gorchakov, G. I.

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

G. I. Gorchakov, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 752 (1965).

Grams, G. W.

P. Chýlek, G. W. Grams, Icarus 36, 198 (1978).
[CrossRef]

P. Chylek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Greenberg, J. M.

J. M. Greenberg, A. S. Meltzer, Astrophys. J. 132, 667 (1960).
[CrossRef]

Hansen, J. E.

S. Asano, M. Sato, J. E. Hansen, NASA Conf. Publ. 2076, 265 (1979).

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

J. E. Hansen, J. W. Hovenier, J. Atmos. Sci. 31, 1137 (1974).
[CrossRef]

K.-N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Heintzenberg, J.

J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).

Hessing, A. R.

K. G. Dedrick, A. R. Hessing, G. J. Johnson, IEEE Trans. Antennas Propag. AP-26, 420 (1978).
[CrossRef]

Hodkinson, J. R.

J. R. Hodkinson, in Electromagnetic Scattering ICES, M. Kerker, Ed. (Macmillan, New York, 1963).

Hofmann, D. J.

Holland, A. C.

Hovenier, J. W.

J. E. Hansen, J. W. Hovenier, J. Atmos. Sci. 31, 1137 (1974).
[CrossRef]

Huffman, D. R.

Huffman, P. J.

P. J. Huffman, J. Atmos. Sci. 27, 1207 (1970).
[CrossRef]

Hunt, A. J.

Irvine, W. M.

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

Johnson, G. J.

K. G. Dedrick, A. R. Hessing, G. J. Johnson, IEEE Trans. Antennas Propag. AP-26, 420 (1978).
[CrossRef]

Kadyshevich, Ye. A.

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 106 (1976).

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, I. N. Plakhina, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 367 (1971).

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), pp. 19 and 20.

Kotlarchyk, M.

Lahore, H.

K.-N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

Liou, K.-N.

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838, 852 (1979).
[CrossRef]

K.-N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[CrossRef]

K.-N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Lyubovtseva, Yu. S.

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 106 (1976).

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, I. N. Plakhina, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 367 (1971).

Martin, P. G.

C. Rogers, P. G. Martin, Astrophys. J. 228, 450 (1979).
[CrossRef]

Meltzer, A. S.

J. M. Greenberg, A. S. Meltzer, Astrophys. J. 132, 667 (1960).
[CrossRef]

Mirumyants, S. O.

V. P. Dugin, S. O. Mirumyants, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 606 (1976).

Nakajima, T.

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ. Ser. 5: 24, 89 (1977).

Pal, S. R.

Perry, R. J.

Pinnick, R. G.

Plakhina, I. N.

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, I. N. Plakhina, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 367 (1971).

Platt, C. M. R.

C. M. R. Platt, J. Appl. Meteorol. 17, 482 (1977).
[CrossRef]

Pollack, J. B.

J. B. Pollack, J. N. Cuzzi, in Proceedings, Third Conference on Atmospheric Radiation, 28–30 June (American Meteorological Society, New York, 1978).

Pritchard, B. S.

Quiney, R. G.

Rogers, C.

C. Rogers, P. G. Martin, Astrophys. J. 228, 450 (1979).
[CrossRef]

Rozenberg, G. V.

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 106 (1976).

G. I. Gorchakov, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 752 (1965).

Sassen, K.

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838, 852 (1979).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

K. Sassen, Appl. Opt. 16, 1332 (1977).
[CrossRef] [PubMed]

K. Sassen, Nature London 255, 316 (1975).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

R. M. Schotland, K. Sassen, R. Stone, J. Appl. Meteorol. 10, 1011 (1971).
[CrossRef]

Sato, M.

S. Asano, M. Sato, J. E. Hansen, NASA Conf. Publ. 2076, 265 (1979).

Schotland, R. M.

R. M. Schotland, K. Sassen, R. Stone, J. Appl. Meteorol. 10, 1011 (1971).
[CrossRef]

Sidorov, V. N.

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

Stone, R.

R. M. Schotland, K. Sassen, R. Stone, J. Appl. Meteorol. 10, 1011 (1971).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 42–57, 110, 111, 172–183, and 200–233.

Vouk, V.

V. Vouk, Nature London 162, 330 (1948).
[CrossRef]

Wang, D.-S.

Welch, R. M.

Yamamoto, G.

Yemilenko, A. A.

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

Zerull, R. H.

R. H. Zerull, Contrib. Atmos. Phys. 49, 168 (1976).

Appl. Opt. (13)

Astron. J. (1)

D. L. Coffeen, Astron. J. 74, 446 (1969).
[CrossRef]

Astrophys. J. (2)

J. M. Greenberg, A. S. Meltzer, Astrophys. J. 132, 667 (1960).
[CrossRef]

C. Rogers, P. G. Martin, Astrophys. J. 228, 450 (1979).
[CrossRef]

Contrib. Atmos. Phys. (2)

R. H. Zerull, Contrib. Atmos. Phys. 49, 168 (1976).

J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).

Icarus (2)

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

P. Chýlek, G. W. Grams, Icarus 36, 198 (1978).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. G. Dedrick, A. R. Hessing, G. J. Johnson, IEEE Trans. Antennas Propag. AP-26, 420 (1978).
[CrossRef]

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (5)

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, I. N. Plakhina, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 367 (1971).

Ye. A. Kadyshevich, Yu. S. Lyubovtseva, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 106 (1976).

V. P. Dugin, S. O. Mirumyants, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, 606 (1976).

G. I. Gorchakov, G. V. Rozenberg, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 1, 752 (1965).

V. A. Golovanev, G. I. Gorchakov, A. A. Yemilenko, V. N. Sidorov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 857 (1971).

J. Aerosol Sci. (1)

D. W. Cooper, J. W. Davis, R. L. Byers, J. Aerosol Sci. 5, 117 (1974).
[CrossRef]

J. Appl. Meteorol. (5)

R. M. Schotland, K. Sassen, R. Stone, J. Appl. Meteorol. 10, 1011 (1971).
[CrossRef]

K.-N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 13, 923 (1974).
[CrossRef]

C. M. R. Platt, J. Appl. Meteorol. 17, 482 (1977).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

J. Atmos. Sci. (5)

J. E. Hansen, J. W. Hovenier, J. Atmos. Sci. 31, 1137 (1974).
[CrossRef]

K.-N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

K. Sassen, K.-N. Liou, J. Atmos. Sci. 36, 838, 852 (1979).
[CrossRef]

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[CrossRef]

P. J. Huffman, J. Atmos. Sci. 27, 1207 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

NASA Conf. Publ. (1)

S. Asano, M. Sato, J. E. Hansen, NASA Conf. Publ. 2076, 265 (1979).

Nature London (2)

K. Sassen, Nature London 255, 316 (1975).
[CrossRef]

V. Vouk, Nature London 162, 330 (1948).
[CrossRef]

Sci. Rep. Tohoku Univ. Ser. 5 (1)

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ. Ser. 5: 24, 89 (1977).

Science (1)

P. Chylek, G. W. Grams, R. G. Pinnick, Science 193, 480 (1976).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Other (4)

J. B. Pollack, J. N. Cuzzi, in Proceedings, Third Conference on Atmospheric Radiation, 28–30 June (American Meteorological Society, New York, 1978).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 42–57, 110, 111, 172–183, and 200–233.

J. R. Hodkinson, in Electromagnetic Scattering ICES, M. Kerker, Ed. (Macmillan, New York, 1963).

M. Kerker, The Scattering of Light (Academic, New York, 1969), pp. 19 and 20.

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Figures (14)

Fig. 1
Fig. 1

Geometry of the scattering description for an arbitrarily oriented spheroid in the XYZ coordinate system, where the incident wave vector (i)k is in the polar axis OZ. Orientation of the spheroid is specified by the incidence angle ζ and azimuth angle χ. The direction of scattered wave vector (s)k is defined by the scattering angle Θ and azimuth angle Φ in the XYZ system and by (θ,ϕ) in the body framed system or the xyz coordinate system.

Fig. 2
Fig. 2

Averaged extinction cross sections normalized by the cross-sectional area π r V 2 of the volume equivalent spheres as a function of the equivalent size parameter 2πrV/λ for randomly oriented prolate and oblate spheroids with the refractive index m ˜ = 1.33 and the shape parameter a/b = 2, 3, and 5. The extinction cross sections for spheres are also shown. For spheres, the size distribution Eq. (28) is used with req = rV and veff = 0.01.

Fig. 3
Fig. 3

Averaged extinction cross sections normalized by the cross-sectional area π r G 2 of the area equivalent spheres or the extinction efficiency factors of randomly oriented oblate spheroids with m ˜ = 1.33 and a/b = 2, 3, and 5 as a function of the size parameter 2πa/λ. Values calculated with the anomalous diffraction approximation are compared.

Fig. 4
Fig. 4

Average cross sections for extinction, scattering, and absorption, normalized by the cross-sectional area π r V 2 of the volume equivalent spheres as a function of the equivalent size parameter 2πrV/λ for randomly oriented absorbing prolate and oblate spheroids with m ˜ = 1.33 + 0.05i and a/b = 5. The cross sections for spheres, calculated with req = rV and veff = 0.01 in the size distribution Eq. (28), are also shown.

Fig. 5
Fig. 5

Averaged asymmetry factor cos Θ ¯, as a function of size parameter 2πrV/λ for the volume equivalent spheres, for randomly oriented prolate and oblate spheroids with m ˜ = 1.33 and a/b = 2, 3, and 5. cos Θ ¯ for spheres is also shown and calculated with req = rV and veff = 0.01 in the size distribution Eq. (28).

Fig. 6
Fig. 6

Angular distribution of elements of the normalized scattering matrix for randomly oriented prolate spheroids with m ˜ = 1.33, α = 15, and a/b = 5: (a) represents the normalized phase function P11; (b) represents P22/P11 and the degree of linear polarization −P12/P11; (c) represents P43/P11; and (d) represents P33/P11 and P44/P11. The scattering matrix elements for the area equivalent spheres, shown by dotted lines, are calculated with req = rG and veff = 0.05 in the size distribution Eq. (28).

Fig. 7
Fig. 7

Angular distribution of elements of the normalized scattering matrix for randomly oriented prolate spheroids with m ˜ = 1.33, α = 15, and a/b = 2: (a) represents the normalized phase function P11; (b) represents P22/P11 and the degree of linear polarization −P12/P11; (c) represents P43/P11; and (d) represents P33/P11 and P44/P11. The scattering matrix elements for the area equivalent spheres, shown by dotted lines, are calculated with req = rG and veff = 0.05 in the size distribution Eq. (28).

Fig. 8
Fig. 8

Angular distribution of elements of the normalized scattering matrix for randomly oriented oblate spheroids with m ˜ = 1.33, α = 15, and a/b = 5: (a) represents the normalized phase function P11; (b) represents P22/P11 and the degree of linear polarization −P12/P11; (c) represents P43/P11; and (d) represents P33/P11 and P44/P11. The scattering matrix elements for the area equivalent spheres, shown by dotted lines, are calculated with req = rG and veff = 0.05 in the size distribution Eq. (28).

Fig. 9
Fig. 9

Angular distribution of elements of the normalized scattering matrix for randomly oriented oblate spheroids with m ˜ = 1.33, α = 15, and a/b = 2: (a) represents the normalized phase function P11; (b) represents P22/P11 and the degree of linear polarization −P12/P11; (c) represents P43/P11; and (d) represents P33/P11 and P44/P11. The scattering matrix elements for the area equivalent spheres, shown by dotted lines, are calculated with req = rG and veff = 0.05 in the size distribution Eq. (28).

Fig. 10
Fig. 10

Angular distribution of the normalized phase function P11 and the degree of linear polarization −P12/P11 (insert) for randomly oriented oblate spheroids with m ˜ = 1.33 and a/b = 5 for three sizes: α = 5, 10, and 20.

Fig. 11
Fig. 11

Contour map of the percent of polarization, −100 P12/P11, for single scattering of unpolarized incident light as functions of the scattering angle Θ and particle size for randomly oriented prolate spheroids with m ˜ = 1.44 and a/b = 2. Positive polarization regions are shaded.

Fig. 12
Fig. 12

Normalized phase functions at forward scattering, P11(0°), and backscattering, P11(180°), as a function of the size parameter 2/πa/λ for randomly oriented prolate spheroids with m ˜ = 1.33 and a/b = 2. The depolarization components at backscattering, ½[P11(180°) − P22(180°)], are also shown. P11(0°) and P11(180°) are compared with those for the area equivalent spheres; the latter are calculated with req = rG and veff = 0.01 in the size distribution Eq. (28).

Fig. 13
Fig. 13

Backscattering depolarization ratio, Eq. (30), as a function of size parameter 2πrV/λ for the volume equivalent spheres for randomly oriented prolate and oblate spheroids with m ˜ = 1.33 and a/b = 2 and 5.

Fig. 14
Fig. 14

Angular distribution of the linear depolarization ratios for randomly oriented oblate spheroids with the same size parameter, α = 15, but with different shape parameters, a/b = 2 (left-hand side) and a/b = 5 (right-hand side). The linear depolarization ratios, δH and δV, for the incidence of light polarized parallel and perpendicular to the scattering plane are shown by solid and dotted lines, respectively.

Tables (1)

Tables Icon

Table I Ratios of the Radii rV and rG of the Volume and Area Equivalent Spheres, Respectively, to the Semimajor Axis a of Spheroids

Equations (31)

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[ E l E r ] R O P = exp { i [ k R - k ( i ) · z ] } i k R [ A 2 A 3 A 4 A 1 ] [ E l 0 E r 0 ] Q O R ,
A 1 = T 11 ( θ , ϕ ) A 2 = T 22 ( θ , ϕ ) A 3 = - T 12 ( θ , ϕ ) A 4 = T 21 ( θ , ϕ ) } .
[ I Q U V ] R O P = 1 k 2 R 2 F ( θ , ϕ ) [ I 0 Q 0 U 0 V 0 ] Q O R ,
I = E l E l * + E r E r * Q = E l E l * - E r E r * U = E l E r * + E r E l * V = i ( E r E l * - E l E r * ) } ,
F = [ ½ ( M 2 + M 3 + M 4 + M 1 ) ½ ( M 2 - M 3 + M 4 - M 1 ) S 23 + S 41 D 23 + D 41 ½ ( M 2 + M 3 - M 4 - M 1 ) ½ ( M 2 - M 3 - M 4 + M 1 ) S 23 - S 41 D 23 - D 41 S 24 + S 31 S 24 - S 31 S 21 + S 34 D 21 - D 34 D 42 + D 13 D 42 - D 13 D 12 + D 43 S 21 - S 34 ] .
[ I Q U V ] Q O P = 1 k 2 R 2 Z [ I 0 Q 0 U 0 V 0 ] X Z ,
Z ( Θ , Φ ; ζ , χ ) = L ( π - γ ) · F ( θ , ϕ ) · L ( - χ ) ,
L ( π - α ) = L ( - α ) = [ 1 0 0 0 0 cos 2 α - sin 2 α 0 0 sin 2 α cos 2 α 0 0 0 0 1 ] .
cos θ = cos Θ · cos ζ + sin Θ · sin ζ · cos ( χ - Φ ) ,
cos ϕ = cos Θ · sin ζ - sin Θ · cos ζ · cos ( χ - Φ ) ± sin θ ,
cos γ = cos ζ · sin Θ - sin ζ · cos Θ · cos ( χ - Φ ) ± sin θ ,
F ( Θ ) ¯ = 1 4 π 0 2 π 0 π Z ( Θ , 0 ; ζ , χ ) sin ζ d ζ d χ .
F ¯ = [ f ˜ 11 f ˜ 12 0 0 f ˜ 12 f ˜ 22 0 0 0 0 f ˜ 33 f ˜ 43 0 0 f ˜ 43 f ˜ 44 ] .
C sca ¯ = 4 π I / I 0 R 2 d Ω ,
I / I 0 = 1 k 2 R 2 [ f ˜ 11 + f ˜ 12 Q 0 I 0 ] = 1 4 π k 2 R 2 [ f 11 sin ζ d ζ d χ + Q 0 I 0 ( cos 2 χ · f 12 + sin 2 χ · f 13 ) sin ζ d ζ d χ ] ,
C sca ¯ = ½ 0 π / 2 [ C 1 , sca ( ζ ) + C 2 , sca ( ζ ) ] sin ζ d ζ .
C ext ¯ = ½ 0 π / 2 [ C 1 , ext ( ζ ) + C 2 , ext ( ζ ) ] sin ζ d ζ .
cos Θ ¯ · C sca ¯ = 4 π I / I 0 cos Θ · R 2 d Ω ,
cos Θ = cos ζ cos θ + sin ζ sin θ cos ϕ .
cos Θ ¯ = 1 2 C sca ¯ 0 π / 2 { cos ζ [ ( cos Θ 1 C 1 , sca ) A + ( cos Θ 2 C 2 , sca ) A ] + sin ζ [ ( cos Θ 1 C 1 , sca ) B + ( cos Θ 2 C 2 , sca ) B ] } sin ζ d ζ ,
( cos Θ C sca ) A = π k 2 m = 0 ( 1 + δ 0 m ) ( n = m n = m ( α m n α m n * + β m n β m n * ) × { r = 0 , 1 2 ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! d r m n [ ( m + r ) ( m + r + 2 ) ( 2 m + r + 1 ) ( 2 m + 2 r + 3 ) d r + 1 m n + r ( m + r + 1 ) ( m + r - 1 ) ( 2 m + 2 r - 1 ) d r - 1 m n ˙ ] } ) + π k 2 m = 1 n = m m ( α m n β m n * + α m n * β m n ) [ r = 0 , 1 2 ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! ( d r m n ) 2 ] ,
( cos Θ C sca ) B = π 2 k 2 m = 0 ( 1 + δ 0 m ) n = m n = m + 1 { ( α m n α m + 1 , n * + β m n β m + 1 , n * ) × { r = 0 , 1 2 ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! d r m n [ ( m + r ) ( m + r + 2 ) ( 2 m + r + 1 ) ( 2 m + r + 2 ) ( 2 m + 2 r + 3 ) × d r m + 1 , n - r ( r - 1 ) ( m + r - 1 ) ( m + r + 1 ) ( 2 m + 2 r - 1 ) d r - 2 m + 1 , n ] } - ( α m n β m + 1 , n * + α m + 1 , n * β m n ) [ r = 0 , 1 2 r ( 2 m + r + 1 ) ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! d r m n d r - 1 m + 1 , n ] } + π 2 k 2 m = 1 ( 1 + δ 1 m ) n = m n = m - 1 { ( α m n α m - 1 , n * + β m n β m - 1 , n * ) × { r = 0 , 1 2 ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! d r m n · [ ( m + r - 1 ) ( m + r + 1 ) ( 2 m + 2 r - 1 ) d r m - 1 , n - ( m + r ) ( m + r + 2 ) ( 2 m + 2 r + 3 ) d r + 2 m - 1 , n ] } - ( α m n β m - 1 , n * + α m - 1 , n * β m n ) [ r = 0 , 1 2 ( 2 m + r ) ! ( 2 m + 2 r + 1 ) r ! d r m n d r + 1 m - 1 , n ] } .
4 π P 11 ( cos Θ ) d Ω / 4 π = 1.
k 2 C sca ¯ 4 π P i j = f ˜ i j             ( i , j = 1 , 4 ) .
[ I Q U V ] = C sca ¯ 4 π R 2 [ P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 - P 43 0 0 P 43 P 44 ] [ I 0 Q 0 U 0 V 0 ] .
[ I l I r U V ] = C sca ¯ 4 π R 2 [ Q 11 Q 12 0 0 Q 12 Q 22 0 0 0 0 Q 33 - Q 43 0 0 Q 43 Q 44 ] [ I l 0 I r 0 U 0 V 0 ] .
P 11 = ½ ( Q 11 + 2 Q 12 + Q 22 ) P 22 = ½ ( Q 11 - 2 Q 12 + Q 22 ) P 12 = ½ ( Q 11 - Q 22 ) , P 33 = Q 33 P 43 = Q 43 , P 44 = Q 44 } .
n ( r ) = const · r 1 - 3 v eff / v eff · exp ( - r r eq v eff ) ,
Δ = ( 1 - P 22 / P 11 ) = 2 Q 12 ½ ( Q 11 + 2 Q 12 + Q 22 ) .
δ = [ P 11 ( 180° ) - P 22 ( 180° ) ] / [ P 11 ( 180° ) + P 22 ( 180° ) ] ,
[ exp ( i ɛ ) 0 0 exp ( i ɛ ) ] ,

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