Abstract

The degree of polarization as well as the direction of the polarization are calculated by a Monte Carlo method for homogeneous layers. Two solar zenith angles and a range of optical thicknesses up to 10 are considered. The results are compared with calculations for single scattered photons. For a given pair of incident and scattered directions, there are only two possible values for the direction of the polarization for single scattering: it must be within or perpendicular to the scattering plane. The choice between these two values depends only on the sign of the element M in the first row and second column of the scattering matrix in the I, Q U, V representation. In most cases there is little change in the direction of the polarization when multiple scattering is taken into account, so that this quantity with respect to a meridian plane can usually be predicted from a very simple trigonometric relationship to good accuracy. The partial scattering angles at which M changes sign are unique for a particular aerosol model. Thus different size distributions and indices of refraction can be distinguished by measurements of the direction of polarization.

© 1972 Optical Society of America

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References

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  1. H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.
  2. J. E. Hansen, J. Atmos. Sci 28, 1400 (1971).
    [CrossRef]
  3. B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
    [CrossRef]
  4. J. V. Dave, Appl. Opt. 9, 2673 (1970).
    [CrossRef] [PubMed]
  5. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 415 (1968).
    [CrossRef] [PubMed]
  6. G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
    [CrossRef]
  7. G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 2489 (1969).
    [CrossRef] [PubMed]
  8. G. N. Plass, G. W. Kattawar, Appl. Opt. 9, 1122 (1970).
    [CrossRef] [PubMed]
  9. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
    [CrossRef] [PubMed]
  10. G. W. Kattawar, G. N. Plass, Appl. Opt. 10, 74 (1971).
    [CrossRef] [PubMed]
  11. G. W. Kattawar, G. N. Plass, C. N. Adams, Astroph. J. 170, 371 (1971).
    [CrossRef]
  12. G. A. Mikhaylov, M. A. Nazaraliyev, Izv. Ser. Fiz. Atmos. Oceanic 7, 377 (1971). [Bull., Atmos. Oceanic Phys. Ser. 7, 254 (1971)].
  13. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  14. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]
  15. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

1971 (6)

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

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

G. W. Kattawar, G. N. Plass, C. N. Adams, Astroph. J. 170, 371 (1971).
[CrossRef]

G. A. Mikhaylov, M. A. Nazaraliyev, Izv. Ser. Fiz. Atmos. Oceanic 7, 377 (1971). [Bull., Atmos. Oceanic Phys. Ser. 7, 254 (1971)].

G. W. Kattawar, G. N. Plass, Appl. Opt. 10, 74 (1971).
[CrossRef] [PubMed]

1970 (2)

1969 (1)

1968 (2)

1967 (1)

1964 (1)

Adams, C. N.

G. W. Kattawar, G. N. Plass, C. N. Adams, Astroph. J. 170, 371 (1971).
[CrossRef]

Browning, S. R.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Curran, R. J.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
[CrossRef]

Dave, J. V.

Deirmendjian, D.

Grossman, K.

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

Hansen, J. E.

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

Herman, B. M.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
[CrossRef]

Kattawar, G. W.

Mikhaylov, G. A.

G. A. Mikhaylov, M. A. Nazaraliyev, Izv. Ser. Fiz. Atmos. Oceanic 7, 377 (1971). [Bull., Atmos. Oceanic Phys. Ser. 7, 254 (1971)].

Nazaraliyev, M. A.

G. A. Mikhaylov, M. A. Nazaraliyev, Izv. Ser. Fiz. Atmos. Oceanic 7, 377 (1971). [Bull., Atmos. Oceanic Phys. Ser. 7, 254 (1971)].

Plass, G. N.

van de Hulst, H. C.

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

Appl. Opt. (8)

Astroph. J. (1)

G. W. Kattawar, G. N. Plass, C. N. Adams, Astroph. J. 170, 371 (1971).
[CrossRef]

Izv. Ser. Fiz. Atmos. Oceanic (1)

G. A. Mikhaylov, M. A. Nazaraliyev, Izv. Ser. Fiz. Atmos. Oceanic 7, 377 (1971). [Bull., Atmos. Oceanic Phys. Ser. 7, 254 (1971)].

J. Atmos. Sci (1)

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

J. Atmos. Sci. (2)

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

Other (2)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy, Eds. (Gordon and Breach, New York, 1968), pp. 35–55.

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Figures (18)

Fig. 1
Fig. 1

Polarization of reflected radiation as a function of the cosine of the nadir angle, μ. The cosine of the solar zenith angle μ0 = 0.81915. The results have been averaged over the azimuth angle ϕ measured from the incident plane over a range 0° < ϕ < 30° on the left side and 150° < ϕ < 180° on the right side. The solar horizon is on the left side of the figure, the nadir at the center, and the antisolar horizon on the right side. The surface albedo is taken as zero in all cases. Results are given for single scattering and for multiple scattering with optical thickness τ = 0.001, 0.1, 1, 10 for the haze C, nimbostratus, and ice crystal models.

Fig. 2
Fig. 2

Polarization of reflected radiation for 60° < ϕ < 120° and μ0 = 0.81915.

Fig. 3
Fig. 3

Direction of polarization χ of reflected radiation as a function of μ for μ0 = 0.81915 for 0° < ϕ < 30° and 150° < ϕ < 180°. Results are given for single scattering and for multiple scattering with τ = 0.01 and 0.1.

Fig. 4
Fig. 4

Direction of polarization of reflected radiation for 30° < ϕ < 60° and 120° < ϕ < 150° and μ0 = 0.81915.

Fig. 5
Fig. 5

Direction of polarization of reflected radiation for 60° < ϕ < 120° and μ0 = 0.81915.

Fig. 6
Fig. 6

Polarization of reflected radiation for μ0 = 0.1 and 0° < ϕ < 30° and 150° < ϕ < 180°.

Fig. 7
Fig. 7

Polarization of reflected radiation for 60° < ϕ < 120° and μ0 = 0.1.

Fig. 8
Fig. 8

Direction of polarization of reflected radiation for 0° < ϕ < 30° and 150° < ϕ < 180° and μ0 = 0.1.

Fig. 9
Fig. 9

Direction of polarization of reflected radiation for 60° < ϕ < 120° and μ0 = 0.1.

Fig. 10
Fig. 10

Polarization of transmitted radiation as a function of the zenith angle, μ for 0° < ϕ < 30° and 150° < ϕ < 180° and μ0 = 0.81915. The solar horizon is on the left side of the figure, the zenith at the center, and the anti-horizon on the right side. The direction of the unscattered solar photons is between the solar horizon and the zenith.

Fig. 11
Fig. 11

Polarization of transmitted radiation for 60° < ϕ < 120° and μ0 = 0.81915.

Fig. 12
Fig. 12

Direction of polarization of transmitted radiation for 0° < ϕ < 30° and 150° < ϕ < 180° and μ0 = 0.81915.

Fig. 13
Fig. 13

Direction of polarization of transmitted radiation for 30° < ϕ < 60° and 120° < ϕ < 150° and μ0 = 0.81915.

Fig. 14
Fig. 14

Direction of polarization of transmitted radiation for 60° < ϕ < 120° and μ0 = 0.81915.

Fig. 15
Fig. 15

Polarization of transmitted radiation for 0° < ϕ < 30° and 150° < ϕ < 180° and μ0 = 0.1.

Fig. 16
Fig. 16

Polarization of transmitted radiation for 60° < ϕ < 120° and μ0 = 0.1.

Fig. 17
Fig. 17

Direction of polarization of transmitted radiation for 0° < ϕ < 30° and 150° < ϕ < 180° and μ0 = 0.1.

Fig. 18
Fig. 18

Direction of polarization of transmitted radiation for 60° < ϕ < 120° and μ0 = 0.1.

Tables (1)

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Table I Direction of Plane of Polarization

Equations (9)

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P = - Q / I = ( I r - I l ) / ( I r + I l ) ,
P = ( Q 2 + U 2 + V 2 ) 1 2 / I .
tan 2 χ = U / Q .
n ( r ) = A r 5 exp ( - 0.1 r ) ,
R = | M + M - 0 0 M - M + 0 0 0 0 S 21 - D 21 0 0 D 21 S 21 |
tan 2 χ = U / Q = tan 2 i 2 ,
χ = i 2 ± 1 2 n π .
cot i 2 = ( sin θ cot θ 0 - cos θ cos ϕ ) / sin ϕ .
χ = ± tan - 1 [ sin ϕ / ( sin θ cot θ 0 - cos θ cos ϕ ) ] ± 1 2 n π ,

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