Abstract

We obtain and analyze simple analytical formulas for asymmetry parameters and absorption cross sections of large, nonspherical particles. The formulas are based on the asymptotic properties of these characteristics at strong and weak absorption of radiation inside particles. The absorption cross section depends on parameter ϕ, which determines the value of the light-absorption cross section for weakly absorbing particles. It is larger for nonspherical scatterers. The asymmetry parameter depends on two parameters. The first is the asymmetry parameter g0 of a nonspherical, transparent particle with the same shape as an absorbing one. The second parameter, β, determines the strength of the influence of light absorption on the value of the asymmetry parameter. Parameter β is larger for nonspherical particles. One can find these three parameters (ϕ, g0, and β) using a ray-tracing code (RTC) for nonabsorbing and weakly absorbing particles. The RTC can then be used to check the accuracy of the equations at any absorption for hexagonal cylinders and spheroids. It is found that the error of computing the absorption cross section and 1 − g (g is the asymmetry parameter) is less than 20% at the refractive index of particles n = 1.333. Values for asymmetry parameters of large, nonabsorbing, spheroidal particles with different aspect ratios are tabulated for the first time to our knowledge. They do not depend on the size of particles and can serve as an independent check of the accuracy of T-matrix codes for large parameters.

© 1997 Optical Society of America

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References

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  1. K. S. Shifrin, “Scattering of light in a turbid medium,” (NASA, Washington, D.C., 1967).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  5. A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
    [CrossRef]
  6. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  7. Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
    [CrossRef]
  8. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
    [CrossRef]
  9. J. L. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
    [CrossRef]
  12. A. Macke, M. I. Mischenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
    [CrossRef] [PubMed]
  13. H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.
  14. J. Lenoble, ed., Radiative transfer in scattering and absorbing atmospheres: standard computational procedures, (A. Deepak, Hampton, Va., 1985), p. 65.
  15. K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
    [CrossRef]
  16. M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse, rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
    [CrossRef]
  17. A. Macke, M. I. Mischenko, “Applicability of regular particle shapes in light scattering calculations for atmospheric ice particles,” Appl. Opt. 35, 4291–4296 (1996).
    [CrossRef] [PubMed]
  18. F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
    [CrossRef]
  19. D. L. Mitchel, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on cloud morphology,” J. Atmos. Sci. 51, 817–832 (1994).
    [CrossRef]
  20. V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
    [CrossRef]
  21. D. J. Brown, P. J. Felton, “Direct measurement of concentration and size for particles of different shapes using laser light diffraction,” Chem. Eng. Res. Des. 63, 125–132 (1985).

1996 (3)

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

A. Macke, M. I. Mischenko, “Applicability of regular particle shapes in light scattering calculations for atmospheric ice particles,” Appl. Opt. 35, 4291–4296 (1996).
[CrossRef] [PubMed]

1995 (3)

1994 (4)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse, rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

D. L. Mitchel, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on cloud morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

1992 (1)

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

1989 (2)

J. L. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

1985 (1)

D. J. Brown, P. J. Felton, “Direct measurement of concentration and size for particles of different shapes using laser light diffraction,” Chem. Eng. Res. Des. 63, 125–132 (1985).

1969 (1)

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

1948 (1)

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

Arnott, W. P.

D. L. Mitchel, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on cloud morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

Bohren, C. F.

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

Brown, D. J.

D. J. Brown, P. J. Felton, “Direct measurement of concentration and size for particles of different shapes using laser light diffraction,” Chem. Eng. Res. Des. 63, 125–132 (1985).

Bryant, F. D.

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

Carlson, B. E.

Felton, P. J.

D. J. Brown, P. J. Felton, “Direct measurement of concentration and size for particles of different shapes using laser light diffraction,” Chem. Eng. Res. Des. 63, 125–132 (1985).

Huffman, D. R.

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

Irvine, W. M.

Kokhanovsky, A. A.

Latimer, P.

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

Liou, K. N.

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Lumme, K.

Macke, A.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Minnis, P.

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Mischenko, M. I.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse, rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Mitchel, D. L.

D. L. Mitchel, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on cloud morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

Mueller, J.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Muinonen, K.

Peltoniemi, J. L.

Raschke, E.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, “Scattering of light in a turbid medium,” (NASA, Washington, D.C., 1967).

Takano, Y.

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse, rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

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

Vouk, V.

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

Zege, E. P.

Appl. Opt. (3)

Atmos. Res. (1)

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Chem. Eng. Res. Des. (1)

D. J. Brown, P. J. Felton, “Direct measurement of concentration and size for particles of different shapes using laser light diffraction,” Chem. Eng. Res. Des. 63, 125–132 (1985).

J. Atmos. Sci. (5)

D. L. Mitchel, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on cloud morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

J. Colloid Interface Sci. (1)

F. D. Bryant, P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. 30, 291–304 (1969).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse, rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

Nature (London) (1)

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

Opt. Commun. (1)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Opt. Lett. (1)

Other (5)

K. S. Shifrin, “Scattering of light in a turbid medium,” (NASA, Washington, D.C., 1967).

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).

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

J. Lenoble, ed., Radiative transfer in scattering and absorbing atmospheres: standard computational procedures, (A. Deepak, Hampton, Va., 1985), p. 65.

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Figures (4)

Fig. 1
Fig. 1

Dependence of asymmetry parameter g0 of nonabsorbing spheroids on the value of ξ at different values of the refractive index of particles n.

Fig. 2
Fig. 2

Dependence of function β on the value of ξ at different values of the refractive index of particles n.

Fig. 3
Fig. 3

Dependence of function ϕ on the value of ξ at different values of the refractive index of particles n.

Fig. 4
Fig. 4

Dependence of the relative error of approximate formulas for computing 1 − ω, 1 − g, s, and Cpr at n = 1.333, (a) ξ = 0.5 and (b) ξ = 2.0, on the value of the imaginary part of the refractive index χ.

Tables (6)

Tables Icon

Table 1 Dependence of Values R and g on the Refractive Index of Particles n

Tables Icon

Table 2 Surface Areas and Volumes of Spheroids and Hexagonal Cylindersa

Tables Icon

Table 3 Value of g0 for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

Tables Icon

Table 4 Values of β(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

Tables Icon

Table 5 Values of ϕ(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

Tables Icon

Table 6 Values of ψ, β, and g0 for Hexagonal Cylinders at n = 1.333 and Different Values of the Ratio ν = L/la

Equations (11)

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C ext = 2 S ,
C abs = ϕ ( n , ξ ) α V ,
ϕ ( n , ξ ) = n 2 ( 1 - b 3 ) ,
C abs = [ 1 - R ( n , ξ ) ] S ,
C abs = [ 1 - R ( n , ξ ) ] { 1 - exp [ - ψ ( n , ξ ) c ] } S ,
ψ ( n , ξ ) = 2 ϕ ( n , ξ ) 3 [ 1 - R ( n , ξ ) ] .
C abs = 1 4 [ 1 - R ( n ) ] { 1 - exp [ - ψ ( n , ξ ) c ] } Σ ,
ψ ( n , ξ ) = 2 C abs 3 α V [ 1 - R ( n ) ] .
C pr = ( 1 - g ) C ext + g C abs .
g = g ( n ) - [ g ( n ) - g 0 ( n ) ] exp [ - β ( n ) c ] ,
a = L 2 + 2 L / s ,

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