Abstract

A theoretical foundation is presented for the experimental determination of the scattering-phase matrix for nonspherical ice crystals. The scattering-phase matrix can be determined solely from four radiance measurements. Conditions for the phase matrix of a volume of ice crystals that are typical of those in cirrus clouds are presented.

© 1975 Optical Society of America

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References

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  1. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  2. R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
    [CrossRef] [PubMed]
  3. D. L. Coffeen and J. E. Hansen, in Proceedings of the Eighth International Symposium on Remote Sensing of Environment (Ann Arbor, Mich., 1972), p. 515.
  4. J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
    [CrossRef]
  5. K. N. Liou, J. Atmos. Sci. 31, 522 (1974).
    [CrossRef]
  6. K. N. Liou and H. Lahore, J. Appl. Meteor. 13, 257 (1974).
    [CrossRef]
  7. Atmospheric Radiation Working Group, Bull. Am. Meteor. Soc. 53, 950 (1972).
  8. P. Huffman and W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
    [CrossRef]
  9. B. S. Pritchard and W. G. Elliott, J. Opt. Soc. Am. 50, 191 (1960).
    [CrossRef]
  10. A. C. Holland and G. Gagne, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]
  11. R. C. Jones, J. Opt. Soc. Am. 37, 110 (1947).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 544.
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  14. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), p. 40.
  15. H. K. Weickmann, Ber. Deut. Wetterd. 6, 1 (1949).
  16. A. J. Heymsfield and R. G. Knollenberg, J. Atmos. Sci. 29, 1358 (1972).
    [CrossRef]
  17. D. O. Jayaweera and B. J. Mason, J. Fluid Mech. 22, 709 (1965).
    [CrossRef]
  18. A. Ono, J. Atmos. Sci. 26, 138 (1969).
    [CrossRef]
  19. F. Perrin, J. Chem. Phys. 10, 415 (1942).
    [CrossRef]

1974 (2)

K. N. Liou, J. Atmos. Sci. 31, 522 (1974).
[CrossRef]

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

1973 (1)

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

1972 (2)

A. J. Heymsfield and R. G. Knollenberg, J. Atmos. Sci. 29, 1358 (1972).
[CrossRef]

Atmospheric Radiation Working Group, Bull. Am. Meteor. Soc. 53, 950 (1972).

1971 (1)

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[CrossRef]

1970 (1)

1969 (2)

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

P. Huffman and W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

1965 (1)

D. O. Jayaweera and B. J. Mason, J. Fluid Mech. 22, 709 (1965).
[CrossRef]

1960 (1)

1949 (1)

H. K. Weickmann, Ber. Deut. Wetterd. 6, 1 (1949).

1947 (1)

1942 (1)

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 544.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), p. 40.

Coffeen, D. L.

D. L. Coffeen and J. E. Hansen, in Proceedings of the Eighth International Symposium on Remote Sensing of Environment (Ann Arbor, Mich., 1972), p. 515.

Conrath, B. J.

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Curran, R. J.

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Deirmendjian, D.

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

Elliott, W. G.

Gagne, G.

Hanel, R. A.

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Hansen, J. E.

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[CrossRef]

D. L. Coffeen and J. E. Hansen, in Proceedings of the Eighth International Symposium on Remote Sensing of Environment (Ann Arbor, Mich., 1972), p. 515.

Heymsfield, A. J.

A. J. Heymsfield and R. G. Knollenberg, J. Atmos. Sci. 29, 1358 (1972).
[CrossRef]

Holland, A. C.

Huffman, P.

P. Huffman and W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

Jayaweera, D. O.

D. O. Jayaweera and B. J. Mason, J. Fluid Mech. 22, 709 (1965).
[CrossRef]

Jones, R. C.

Knollenberg, R. G.

A. J. Heymsfield and R. G. Knollenberg, J. Atmos. Sci. 29, 1358 (1972).
[CrossRef]

Kunde, V. G.

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Lahore, H.

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

Liou, K. N.

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

K. N. Liou, J. Atmos. Sci. 31, 522 (1974).
[CrossRef]

Mason, B. J.

D. O. Jayaweera and B. J. Mason, J. Fluid Mech. 22, 709 (1965).
[CrossRef]

Ono, A.

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

Pearl, J. C.

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Perrin, F.

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

Pritchard, B. S.

Thursby, W. R.

P. Huffman and W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

van de Hulst, H. C.

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

Weickmann, H. K.

H. K. Weickmann, Ber. Deut. Wetterd. 6, 1 (1949).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 544.

Appl. Opt. (1)

Ber. Deut. Wetterd. (1)

H. K. Weickmann, Ber. Deut. Wetterd. 6, 1 (1949).

Bull. Am. Meteor. Soc. (1)

Atmospheric Radiation Working Group, Bull. Am. Meteor. Soc. 53, 950 (1972).

J. Appl. Meteor. (1)

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

J. Atmos. Sci. (5)

A. J. Heymsfield and R. G. Knollenberg, J. Atmos. Sci. 29, 1358 (1972).
[CrossRef]

A. Ono, J. Atmos. Sci. 26, 138 (1969).
[CrossRef]

P. Huffman and W. R. Thursby, J. Atmos. Sci. 26, 1073 (1969).
[CrossRef]

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[CrossRef]

K. N. Liou, J. Atmos. Sci. 31, 522 (1974).
[CrossRef]

J. Chem. Phys. (1)

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

J. Fluid Mech. (1)

D. O. Jayaweera and B. J. Mason, J. Fluid Mech. 22, 709 (1965).
[CrossRef]

J. Opt. Soc. Am. (2)

Science (1)

R. J. Curran, B. J. Conrath, R. A. Hanel, V. G. Kunde, and J. C. Pearl, Science 182, 381 (1973).
[CrossRef] [PubMed]

Other (5)

D. L. Coffeen and J. E. Hansen, in Proceedings of the Eighth International Symposium on Remote Sensing of Environment (Ann Arbor, Mich., 1972), p. 515.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 544.

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

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), p. 40.

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

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Equations (24)

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E l = a l e i ω t , E r = a r e i δ e i ω t
E ( t ; ψ , ) = E l cos ψ + E r e i sin ψ = a l cos ψ e i ω t + a r sin ψ e i ( δ + ) e i ω t .
I ( ψ , ) = E ( t ; ψ , ) E * ( t ; ψ , ) ,
I ( ψ , ) = a l 2 cos 2 ψ + a r 2 sin 2 ψ + 2 a l a r cos ( δ + ) sin ψ cos ψ = a l 2 cos 2 ψ + a r 2 sin 2 ψ + 1 2 sin 2 ψ ( 2 a l a r × cos δ cos 2 a l a r sin δ sin ) .
I = a l 2 + a r 2 , Q = a l 2 a r 2 , U = 2 a l a r cos δ , V = 2 a l a r sin δ .
I ( ψ , ) = 1 2 [ I + Q cos 2 ψ + ( U cos V sin ) sin 2 ψ ] .
I = I ( 0 ° , 0 ) + I ( 90 ° , 0 ) , Q = I ( 0 ° , 0 ) I ( 90 ° , 0 ) , U = I ( 45 ° , 0 ) I ( 135 ° , 0 ) , V = [ I ( 45 ° , π / 2 ) I ( 135 ° , π / 2 ) ] .
[ E l E r ] = [ A 2 A 3 A 4 A 1 ] [ E r 0 E l 0 ] ,
A j = e i k r i k r S j , j = 1 , 2 , 3 , 4 .
I ( Ө ) = C M ( Ө ) F 0 ,
2 π 0 π M i j ( Ө ) sin Ө d Ө = 1 , i , j = 1 , 2 , 3 , 4 .
[ E l E r ] = [ cos ϕ sin ϕ sin ϕ cos ϕ ] [ E l E r ] .
I = L ( ϕ ) I ,
L ( ϕ ) = [ 1 0 0 0 0 cos 2 ϕ sin 2 ϕ 0 0 sin 2 ϕ cos 2 ϕ 0 1 0 0 1 ] .
I ( Ө , ϕ 1 , ϕ 2 ) = C L ( ϕ 2 ) M ( Ө ) L ( ϕ 1 ) F 0 ,
P ( θ , θ ; φ , φ ) = L ( ϕ 2 ) M ( Ө ) L ( ϕ 1 ) .
M ( Ө ) I ( Ө ) 2 π 0 π I ( Ө ) sin Ө d Ө I ( Ө ) .
M 11 = I 1 , M 12 = Q 1 , M 13 = U 1 , M 14 = V 1 .
M 21 = I 2 I 1 , M 22 = Q 2 Q 1 , M 23 = U 2 U 1 , M 24 = V 2 V 1 .
M 31 = I 3 I 1 , M 32 = Q 3 Q 1 , M 33 = U 3 U 1 , M 34 = V 3 V 1 .
M 41 = I 4 I 1 , M 42 = Q 4 Q 1 , M 43 = U 4 U 1 , M 44 = V 4 V 1 .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] .
[ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 13 M 23 M 33 M 34 M 14 M 24 M 34 M 44 ] .
[ M 11 M 12 0 0 M 12 M 22 0 0 0 0 M 33 M 34 0 0 M 34 M 44 ] .