Abstract

Measurements have been made to determine all sixteen elements of the Mueller scattering matrix for two types of nonspherical particles. Rounded particles of ammonium sulfate and nearly cubic particles of sodium chloride in the 0.1–1.0-μm size range have been prepared by nebulizing salt water solutions and drying the droplets. Scanning electron micrographs are used to determine size distributions used in Mie calculations of all matrix elements. The expected symmetry of the scattering matrices across the diagonal was confirmed, and the expected eight of the sixteen elements were found to be zero within measurement accuracy. The rounded particles were found accurately to obey Mie theory, while the cubic particles were poorly described by Mie theory for some matrix elements and some angles. Total intensity and linear polarization measurements are presented also for a series of increasing sizes of rounded and cubic particles. A discussion of the effect of nonsphericity on the various matrix elements is given, and applications of these results are given to analysis of particle properties in the laboratory, the clouds of Venus, reflection nebulae, the zodiacal light, and atmospheric particulates.

© 1978 Optical Society of America

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References

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  1. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
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    [Crossref]
  5. G. Mie, Ann. Phys. 25, 377 (1908).
    [Crossref]
  6. P. Debye, Ann. Phys. 30, 59 (1909).
  7. A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
    [Crossref]
  8. A. J. Hunt, D. R. Huffman, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 435 (1974).
  9. A. C. Holland, G. Gagné, Appl. Opt. 9, 1113 (1970).
    [Crossref] [PubMed]
  10. M. B. Denton, D. B. Swartz, Rev. Sci. Instrum. 45, 81 (1974).
    [Crossref]
  11. D. D. Lobdell, J. Acoust. Soc. Am. 43, 229 (1968).
    [Crossref]
  12. R. J. Perry, M.A. Thesis, U. Arizona (1977).
  13. J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Report 320-3236 (1968).
  14. D. E. Gray, Ed. American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. G-43, G-114.
  15. A. J. Hunt, Ph.D. Thesis, U. Arizona (1974).
  16. R. G. Pinnick, D. E. Carroll, D. J. Hofmann, Appl. Opt. 15, 384 (1976).
    [Crossref] [PubMed]
  17. V. Vouk, Nature 162, 330 (1948).
    [Crossref] [PubMed]
  18. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  19. Science Spectrum, P.O. Box 3003, Santa Barbara, Calif. 93105, for example. Bibliographies are available from this company.
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    [Crossref] [PubMed]
  21. J. E. Hansen, J. W. Hovenier, J. Atmos. Sci. 31, 1137 (1974).
    [Crossref]
  22. D. R. Huffman, Adv. Phys. 26, 141 (1977).
    [Crossref]
  23. B. Zellner, Ph.D. Thesis, U. Arizona (1973).
  24. B. Zellner, Interstellar Dust and Related Topics, Reidel (Boston, 1970), p. 109.
  25. C. Leinert, Space Sci. Rev. 18, 281 (1975).
    [Crossref]
  26. R. H. Giese, Planet. Space Sci. 21, 513 (1973).
    [Crossref]
  27. R. H. Zerull, R. H. Giese, K. Weiss, Appl. Opt. 16, 777 (1977).
    [PubMed]
  28. R. H. Zerull, R. H. Giese, Planets, Stars, and Nebulae Studied With Photopolarimetry (U. Arizona Press, Tucson, 1974), p. 901.
  29. G. Ward, K. M. Cushing, R. D. McPeters, A. E. S. Green, Appl. Opt. 12, 2585 (1973).
    [Crossref] [PubMed]
  30. G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
    [Crossref]
  31. R. Eiden, Appl. Opt. 10, 749 (1971).
    [Crossref] [PubMed]
  32. R. Eiden, Appl. Opt. 5, 569 (1966).
    [Crossref] [PubMed]

1977 (2)

1976 (1)

1975 (1)

C. Leinert, Space Sci. Rev. 18, 281 (1975).
[Crossref]

1974 (4)

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

A. J. Hunt, D. R. Huffman, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 435 (1974).

M. B. Denton, D. B. Swartz, Rev. Sci. Instrum. 45, 81 (1974).
[Crossref]

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

1973 (3)

A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[Crossref]

R. H. Giese, Planet. Space Sci. 21, 513 (1973).
[Crossref]

G. Ward, K. M. Cushing, R. D. McPeters, A. E. S. Green, Appl. Opt. 12, 2585 (1973).
[Crossref] [PubMed]

1971 (2)

J. E. Hansen, A. Arking, Science 171, 669 (1971).
[Crossref] [PubMed]

R. Eiden, Appl. Opt. 10, 749 (1971).
[Crossref] [PubMed]

1970 (1)

1968 (1)

D. D. Lobdell, J. Acoust. Soc. Am. 43, 229 (1968).
[Crossref]

1966 (1)

1948 (2)

H. Mueller, J. Opt. Soc. Am. 38, 661 (1948).

V. Vouk, Nature 162, 330 (1948).
[Crossref] [PubMed]

1942 (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

1909 (1)

P. Debye, Ann. Phys. 30, 59 (1909).

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[Crossref]

Arking, A.

J. E. Hansen, A. Arking, Science 171, 669 (1971).
[Crossref] [PubMed]

Blifford, I. H.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

Carroll, D. E.

Cushing, K. M.

Dave, J. V.

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Report 320-3236 (1968).

Debye, P.

P. Debye, Ann. Phys. 30, 59 (1909).

Denton, M. B.

M. B. Denton, D. B. Swartz, Rev. Sci. Instrum. 45, 81 (1974).
[Crossref]

Eiden, R.

Gagné, G.

Giese, R. H.

R. H. Zerull, R. H. Giese, K. Weiss, Appl. Opt. 16, 777 (1977).
[PubMed]

R. H. Giese, Planet. Space Sci. 21, 513 (1973).
[Crossref]

R. H. Zerull, R. H. Giese, Planets, Stars, and Nebulae Studied With Photopolarimetry (U. Arizona Press, Tucson, 1974), p. 901.

Gillette, D. A.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

Grams, G. W.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

Green, A. E. S.

Hansen, J. E.

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

J. E. Hansen, A. Arking, Science 171, 669 (1971).
[Crossref] [PubMed]

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.

D. R. Huffman, Adv. Phys. 26, 141 (1977).
[Crossref]

A. J. Hunt, D. R. Huffman, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 435 (1974).

A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[Crossref]

Hunt, A. J.

A. J. Hunt, D. R. Huffman, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 435 (1974).

A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[Crossref]

A. J. Hunt, Ph.D. Thesis, U. Arizona (1974).

Kerker, M.

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

Leinert, C.

C. Leinert, Space Sci. Rev. 18, 281 (1975).
[Crossref]

Lobdell, D. D.

D. D. Lobdell, J. Acoust. Soc. Am. 43, 229 (1968).
[Crossref]

McPeters, R. D.

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[Crossref]

Mueller, H.

H. Mueller, J. Opt. Soc. Am. 38, 661 (1948).

Perrin, F.

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

Perry, R. J.

R. J. Perry, M.A. Thesis, U. Arizona (1977).

Pinnick, R. G.

Russell, P. B.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).

Swartz, D. B.

M. B. Denton, D. B. Swartz, Rev. Sci. Instrum. 45, 81 (1974).
[Crossref]

van de Hulst, H. C.

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

Vouk, V.

V. Vouk, Nature 162, 330 (1948).
[Crossref] [PubMed]

Ward, G.

Weiss, K.

Zellner, B.

B. Zellner, Ph.D. Thesis, U. Arizona (1973).

B. Zellner, Interstellar Dust and Related Topics, Reidel (Boston, 1970), p. 109.

Zerull, R. H.

R. H. Zerull, R. H. Giese, K. Weiss, Appl. Opt. 16, 777 (1977).
[PubMed]

R. H. Zerull, R. H. Giese, Planets, Stars, and Nebulae Studied With Photopolarimetry (U. Arizona Press, Tucson, 1974), p. 901.

Adv. Phys. (1)

D. R. Huffman, Adv. Phys. 26, 141 (1977).
[Crossref]

Ann. Phys. (2)

G. Mie, Ann. Phys. 25, 377 (1908).
[Crossref]

P. Debye, Ann. Phys. 30, 59 (1909).

Appl. Opt. (6)

J. Acoust. Soc. Am. (1)

D. D. Lobdell, J. Acoust. Soc. Am. 43, 229 (1968).
[Crossref]

J. Appl. Meteorol. (1)

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, J. Appl. Meteorol. 13, 459 (1974).
[Crossref]

J. Atmos. Sci. (1)

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

J. Chem. Phys. (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

J. Opt. Soc. Am. (1)

H. Mueller, J. Opt. Soc. Am. 38, 661 (1948).

Jpn. J. Appl. Phys. (1)

A. J. Hunt, D. R. Huffman, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 435 (1974).

Nature (1)

V. Vouk, Nature 162, 330 (1948).
[Crossref] [PubMed]

Planet. Space Sci. (1)

R. H. Giese, Planet. Space Sci. 21, 513 (1973).
[Crossref]

Rev. Sci. Instrum. (2)

M. B. Denton, D. B. Swartz, Rev. Sci. Instrum. 45, 81 (1974).
[Crossref]

A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[Crossref]

Science (1)

J. E. Hansen, A. Arking, Science 171, 669 (1971).
[Crossref] [PubMed]

Space Sci. Rev. (1)

C. Leinert, Space Sci. Rev. 18, 281 (1975).
[Crossref]

Other (11)

R. H. Zerull, R. H. Giese, Planets, Stars, and Nebulae Studied With Photopolarimetry (U. Arizona Press, Tucson, 1974), p. 901.

B. Zellner, Ph.D. Thesis, U. Arizona (1973).

B. Zellner, Interstellar Dust and Related Topics, Reidel (Boston, 1970), p. 109.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).

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

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

Science Spectrum, P.O. Box 3003, Santa Barbara, Calif. 93105, for example. Bibliographies are available from this company.

R. J. Perry, M.A. Thesis, U. Arizona (1977).

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Report 320-3236 (1968).

D. E. Gray, Ed. American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. G-43, G-114.

A. J. Hunt, Ph.D. Thesis, U. Arizona (1974).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the apparatus for producing aerosol particles and measuring their scattering properties. Droplets of salt water produced in the ultrasonic nebulizer are carried through the drying furnace by a flow of nitrogen gas. Particles exit into the polarization-modulated laser beam where light at various angles θ is collected and detected by the photomultiplier tube. Further details of the particle production and measurement system are found in the text and the references.

Fig. 2
Fig. 2

Scanning electron microscope photographs showing the characteristic nonspherical shapes of the (NH4)2SO4 and NaCl aerosol particles. Below are presented the histograms of the particle size distributions with the mean radius and variance of each distribution as determined from the photographs. The particle distributions were produced by nebulizing and drying solutions of 80.2 g/liter NaCl and 78.2 g/liter (NH4)2SO4. The solid curves drawn over the histograms show the Gaussian distributions used in the theoretical predictions in Fig. 3.

Fig. 3
Fig. 3

Measurements relating to all sixteen scattering matrix elements for NaCl and (NH4)2SO4. In each element the scattering angle varies from 0° to 180°. The solid lines represent the measurements, and the dashed lines represent the Mie calculations using the Gaussian size distributions of Fig. 2. The vertical scales for the first column elements are logarithmic. The remaining matrix elements are on a linear scale from −1 to +1. In those elements where it was necessary to subtract S12/S11 to obtain the desired matrix element as discussed in the text, the measurement is indicated by M and the desired matrix element by S.

Fig. 4
Fig. 4

The total intensity S11 on a Log10 scale as a function of scattering angle for different Mie size parameters X. Solid lines represent the measurements. The dashed lines show the Mie theory calculations based on the volume equivalent size parameter X obtained from SEM photographs. The dot–dash curves indicate theoretical calculations with size parameters XB found to be a better approximation to the normalized matrix elements in the forward-scattering angles. The scattering angle from 0° to 180° is presented on the bottom scale. The exact placement of the measurements relative to the calculations is arbitrary, as is the absolute value of the logarithmic scale. The Log10 scale is presented fully for the bottom measurement and is offset vertically by one decade for the remaining measurements.

Fig. 5
Fig. 5

The normalized matrix elements S12/S11 as a function of scattering angle for different Mie size parameters. In each element the scattering angle varies from 0° to 180°, and the vertical scale varies from −0.5 to +0.5. The solid line represents the measurements. The dashed lines show the Mie theory calculations based on the volume equivalent size parameter X obtained from SEM photographs. The dot–dash curves indicate the theoretical calculations with size parameter XB found to be a better approximation to the measurements in the forward-scattering angles. These measurements clearly show the characteristic differences in the scattering properties of the rounded and cubic aerosol particles and the regions of the largest deviation from the theoretical calculations for spheres.

Equations (11)

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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 l E r * E r E l * .
[ I s Q s U s V s ] 1 k 2 R 2 [ S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ] [ I o Q o U o V o ] .
[ S 11 S 12 S 13 S 14 S 12 S 22 S 23 S 24 S 13 S 23 S 33 S 34 S 14 S 24 S 34 S 44 ] .
[ S 11 S 12 0 0 S 12 S 22 0 0 0 0 S 33 S 34 0 0 S 34 S 44 ] .
[ S 11 S 12 0 0 S 12 S 11 0 0 0 0 S 33 S 34 0 0 S 34 S 33 ] .
log 10 S 11 = log 10 [ I ( θ ) sin θ sin 10 ° I ( 10 ° ) ] .
[ S 11 S 12 / S 11 S 13 / S 11 S 14 / S 11 S 21 / S 11 S 22 / S 11 S 23 / S 11 S 24 / S 11 S 31 / S 11 S 32 / S 11 S 33 / S 11 S 34 / S 11 S 14 / S 11 S 42 / S 11 S 43 / S 11 S 44 / S 11 ] .
[ S 11 S 12 / S 11 S 13 / S 11 S 14 / S 11 ( A ) (B) S 21 + S 11 S 22 + S 12 S 11 + S 21 S 23 + S 13 S 11 + S 21 S 24 + S 14 S 11 + S 21 S 31 + S 11 S 32 + S 12 S 11 + S 31 S 33 + S 13 S 11 + S 31 S 34 + S 14 S 11 + S 31 ( C ) (D) S 41 + S 11 S 42 + S 12 S 11 + S 41 S 43 + S 13 S 11 + S 41 S 44 + S 14 S 11 + S 41 ] .
y = 1 σ 2 π exp [ 1 2 ( r r ¯ σ ) 2 ] ,
NaCl 1.54 + i 0 at 6328 Å , 1.59 + i 0 at 3250 Å , ( NH 4 ) 2 SO 4 1.52 + i 0 at 6328 Å , 1.54 + i 0 at 3250 Å .
S 22 / S 11 = S 22 ( 1 + S 12 S 11 ) S 12 S 11

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