Abstract

An analytic phase function that reduces to the Rayleigh phase function for the scattering of unpolarized light is presented and compared with the traditional Henyey–Greenstein phase function. Comparisons between the proposed phase function and the phase function for three of Deirmendjian’s polydispersions are shown and applications to radiative transfer are demonstrated.

© 1992 Optical Society of America

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Errata

William M. Cornette and Joseph G. Shanks, "Physically reasonable analytic expression for the single-scattering phase function: errata," Appl. Opt. 34, 641-641 (1995)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-34-4-641

References

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  1. R. M. Goody, Y. L. Yung, Atmospheric Radiation: Theoretical Basis, 2nd ed. (Oxford U. Press, New York, 1989).
  2. K. -N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  3. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  4. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  5. D. Deirmendjian, “Scattering and polarization properties of polydispersed suspensions with partial absorption,” in Proceedings of the Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963).
  6. D. Diermendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  7. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  8. F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).
  9. M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
    [CrossRef]
  10. C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
    [CrossRef]

1986

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

1981

C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
[CrossRef]

1941

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Abreu, L. W.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Acquista, C.

C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
[CrossRef]

Anderson, G. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Bohren, C. F.

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

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Clough, J. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Deirmendjian, D.

D. Deirmendjian, “Scattering and polarization properties of polydispersed suspensions with partial absorption,” in Proceedings of the Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963).

Diermendjian, D.

D. Diermendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Gallery, W. O.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Goody, R. M.

R. M. Goody, Y. L. Yung, Atmospheric Radiation: Theoretical Basis, 2nd ed. (Oxford U. Press, New York, 1989).

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

House, F.

C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
[CrossRef]

Huffman, D. R.

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

Jafolla, J.

C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
[CrossRef]

King Harshvardhan, M. D.

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Liou, K. -N.

K. -N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Yung, Y. L.

R. M. Goody, Y. L. Yung, Atmospheric Radiation: Theoretical Basis, 2nd ed. (Oxford U. Press, New York, 1989).

Astrophys. J.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Atmos. Sci.

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

C. Acquista, F. House, J. Jafolla, “N-stream approximation to radiative transfer,” J. Atmos. Sci. 38, 1446–1451 (1981).
[CrossRef]

Other

R. M. Goody, Y. L. Yung, Atmospheric Radiation: Theoretical Basis, 2nd ed. (Oxford U. Press, New York, 1989).

K. -N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

D. Deirmendjian, “Scattering and polarization properties of polydispersed suspensions with partial absorption,” in Proceedings of the Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963).

D. Diermendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

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

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, J. A. Clough, “User’s guide to lowtran 7,” Rep. AFGL-TR-88-0177 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1988).

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Figures (8)

Fig. 1
Fig. 1

Comparison of double Henyey–Greenstein phase function (solid curve) to Rayleigh phase function (dashed curve).

Fig. 2
Fig. 2

(a)–(d) Comparison of Henyey–Greenstein (solid curves) and proposed (dashed curves) phase functions.

Fig. 3
Fig. 3

Deirmendjian polydispersions for three particle-size distributions. Total concentration is 1 cm−3 for each size distribution.

Fig. 4
Fig. 4

(a)–(e) Deirmendjian Haze C phase functions. Mie (solid curve) versus Henyey–Greenstein (long-dashed curves) versus proposed (short-dashed curves).

Fig. 5
Fig. 5

(a)–(e) Deirmendjian Haze M phase functions. Mie (solid curves) versus Henyey–Greenstein (long-dashed curves) versus proposed (short-dashed curves).

Fig. 6
Fig. 6

(a)–(e) Deirmendjian Cloud C.1 phase functions. Mie (solid curves) versus Henyey–Greenstein (long-dashed curves) versus proposed (short-dashed curves).

Fig. 7
Fig. 7

(a)–(f) Angular backscatter fraction, β(μ0). Haze C, pluses; Haze M, asterisks; Cloud C.1, open circles; Henyey–Greenstein, solid curves; proposed, dashed curves.

Fig. 8
Fig. 8

Average backscatter fraction, β ¯. Haze C, pluses; Haze M, asterisks; Cloud C. 1, open circles; Henyey–Greenstein, solid curve; proposed, dashed curve.

Tables (2)

Tables Icon

Table I Particle-Size Distributionsa

Tables Icon

Table II Particle Optical Propertiesa

Equations (18)

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P HG ( μ , g ) = 1 - g 2 ( 1 + g 2 - 2 g μ ) 3 / 2 ,
- 1 1 0 2 π d ϕ d μ P HG ( μ , g ) = 4 π ,
μ 1 4 π - 1 1 0 2 π d ϕ d μ μ P HG ( μ , g ) = g .
P HG ( μ , g ) = l = 0 ( 2 l + 1 ) g l P l ( μ ) .
P R ( μ ) = 3 4 ( 1 + μ 2 ) .
P DHG ( μ ; f , g 1 , g 2 ) = ( 1 - f ) P HG ( μ , g 1 ) + f P HG ( μ , g 2 ) ,             g 1 > 0 , g 2 < 0 ,
μ = ( 1 - f ) g 1 + f g 2 ,             f [ 0 , 1 ] .
P ( μ , g ) = 3 2 1 - g 2 2 + g 2 1 + μ 2 ( 1 + g 2 - 2 g μ ) 3 / 2 ,
μ = g 3 ( 4 + g 2 ) 5 ( 2 + g 2 ) ,
g = 5 9 μ - ( 4 3 - 25 81 μ 2 ) x - 1 / 3 + x 1 / 3 ,
x = 5 9 μ + 125 729 μ 3 + ( 64 27 - 325 243 μ 2 + 1250 2187 μ 4 ) 1 / 2 .
P ( μ , g ) = 3 2 1 2 + g 2 l = 0 ( l ( l - 1 ) 2 l - 1 g l - 2 + [ 5 l 2 - 1 2 l - 1 + ( l + 1 ) 2 2 l + 3 ] g l + ( l + 1 ) ( l + 2 ) 2 l + 3 g l + 2 ) P l ( μ ) .
N ( r ) = a r α exp ( - b r γ ) ,
β ( μ 0 ) = 1 4 π 0 1 0 2 π ϕ ( cos ψ ) d θ d μ ,
β ¯ = 0 1 β ( μ 0 ) d μ 0 .
P ( μ ) = l = 0 ω l P l ( μ ) ,
β ( μ 0 ) = 1 2 - 1 4 π ω 0 l = 0 ( - 1 ) l Γ ( l + 1 2 ) Γ ( l + 2 ) ω 2 l + 1 P 2 l + 1 ( μ o ) ,
β ¯ = 1 2 - 1 8 π ω 0 l = 0 [ Γ ( l + 1 2 ) Γ ( l + 2 ) ] 2 ω 2 l + 1 .

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