Abstract

The complete radiation field including polarization is calculated by the matrix operator method for scattering layers of various optical thicknesses. Results obtained for Rayleigh scattering are compared with those for scattering from a continental haze. Radiances calculated using Stokes vectors show differences as large as 23% compared to the approximate scalar theory of radiative transfer, while the same differences are only of the order of 0.1% for a continental haze phase function. The polarization of the reflected and transmitted radiation is given for a wide range of optical thicknesses of the scattering layer, for various solar zenith angles, and various surface albedos. Two entirely different types of neutral points occur for aerosol phase functions. Rayleigh-like neutral points (RNP) arise from the zero polarization in single scattering that occurs for all phase functions at scattering angles of 0° and 180°. For Rayleigh phase functions, the position of the RNP varies appreciably with the optical thickness of the scattering layer. At low solar elevations there may be four RNP. For a continental haze phase function the position of the RNP in the reflected radiation shows only a small variation with the optical thickness, and the RNP exists in the transmitted radiation only for extremely small optical thicknesses. Another type of neutral point (NRNP) exists for aerosol phase functions. It is associated with the zeros of the single scattered polarization, which occur between the end points of the curve; these are called non-Rayleigh neutral points (NRNP). There may be from zero to four of these neutral points associated with each zero of the single scattering curve. They occur over a range of azimuthal angles, unlike the RNP that are in the principal plane only. The position of these neutral points is given as a function of solar angle and optical thickness.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).
  2. Z. Sekera, Adv. Geophys. 3, 43 (1956).
    [CrossRef]
  3. Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Fluegge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
    [CrossRef]
  4. V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).
  5. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).
  6. B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
    [CrossRef]
  7. J. V. Dave, P. M. Furukawa, J. Opt. Soc. Am. 56, 394 (1966).
    [CrossRef]
  8. A. B. Kahle, J. Geophys. Res. 73, 7511 (1968).
    [CrossRef]
  9. A. B. Kahle, Astrophys. J. 151, 637 (1968).
    [CrossRef]
  10. H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
    [CrossRef]
  11. A. L. Fymat, K. D. Abhyankar, J. Geophys. Res. 76, 732 (1971).
    [CrossRef]
  12. C. N. Adams, G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 10, 341 (1970).
    [CrossRef]
  13. G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
    [CrossRef] [PubMed]
  14. G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer 13, 1081 (1973).
    [CrossRef]
  15. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
    [CrossRef] [PubMed]
  16. G. W. Kattawar, G. N. Plass, Appl. Opt. 10, 74 (1971).
    [CrossRef] [PubMed]
  17. G. W. Kattawar, G. N. Plass, C. N. Adams, Astrophys. J. 170, 371 (1971).
    [CrossRef]
  18. G. W. Kattawar, G. N. Plass, Appl. Opt. 11, 2851 (1972).
    [CrossRef] [PubMed]
  19. G. W. Kattawar, G. N. Plass, J. A. Guinn, J. Phys. Oceanogr. 3, 353 (1973).
    [CrossRef]
  20. G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 2489 (1969).
    [CrossRef] [PubMed]
  21. G. N. Plass, G. W. Kattawar, Appl. Opt. 9, 1122 (1970).
    [CrossRef] [PubMed]
  22. G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
    [CrossRef]
  23. G. N. Plass, G. W. Kattawar, Appl. Opt. 11, 2866 (1972).
    [CrossRef] [PubMed]
  24. D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, Appl. Opt. 11, 2684 (1972).
    [CrossRef] [PubMed]
  25. W. G. Blättner, H. G. Horak, D. G. Collins, M. B. Wells, Appl. Opt. 13, 534 (1974).
    [CrossRef] [PubMed]
  26. M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296, 321, 333 (1971).
  27. J. V. Dave, Appl. Opt. 9, 2673 (1970).
    [CrossRef] [PubMed]
  28. B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1971).
    [CrossRef]
  29. J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
    [CrossRef]
  30. E. deBary, Appl. Opt. 3, 1293 (1964).
    [CrossRef]
  31. D. Eschelbach, J. Quant. Spectrosc. Radiat. Transfer 11, 757 (1971).
    [CrossRef]
  32. G. W. Kattawar, G. N. Plass, F. E. Catchings, Appl. Opt. 12, 1071 (1973). The mode radius given on p. 1071 should be changed to 0.05.
    [CrossRef] [PubMed]
  33. G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 15, 61 (1975).
    [CrossRef]
  34. G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
    [CrossRef]
  35. G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 13, 145 (1973).
    [CrossRef]
  36. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  37. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]

1975 (1)

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

1974 (1)

1973 (6)

G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, F. E. Catchings, Appl. Opt. 12, 1071 (1973). The mode radius given on p. 1071 should be changed to 0.05.
[CrossRef] [PubMed]

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 13, 145 (1973).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. A. Guinn, J. Phys. Oceanogr. 3, 353 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

1972 (3)

1971 (8)

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

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

A. L. Fymat, K. D. Abhyankar, J. Geophys. Res. 76, 732 (1971).
[CrossRef]

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

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296, 321, 333 (1971).

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

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

D. Eschelbach, J. Quant. Spectrosc. Radiat. Transfer 11, 757 (1971).
[CrossRef]

1970 (5)

C. N. Adams, G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 10, 341 (1970).
[CrossRef]

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[CrossRef]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[CrossRef]

G. N. Plass, G. W. Kattawar, Appl. Opt. 9, 1122 (1970).
[CrossRef] [PubMed]

J. V. Dave, Appl. Opt. 9, 2673 (1970).
[CrossRef] [PubMed]

1969 (1)

1968 (3)

G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
[CrossRef] [PubMed]

A. B. Kahle, J. Geophys. Res. 73, 7511 (1968).
[CrossRef]

A. B. Kahle, Astrophys. J. 151, 637 (1968).
[CrossRef]

1967 (1)

1966 (1)

1964 (1)

1956 (1)

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[CrossRef]

Abhyankar, K. D.

A. L. Fymat, K. D. Abhyankar, J. Geophys. Res. 76, 732 (1971).
[CrossRef]

Adams, C. N.

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

C. N. Adams, G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 10, 341 (1970).
[CrossRef]

Binstock, J.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

Blattner, W. G.

Blättner, W. G.

Browning, S. R.

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

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[CrossRef]

Catchings, F. E.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

Collins, D. G.

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Curran, R. J.

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

Dave, J. V.

J. V. Dave, Appl. Opt. 9, 2673 (1970).
[CrossRef] [PubMed]

J. V. Dave, P. M. Furukawa, J. Opt. Soc. Am. 56, 394 (1966).
[CrossRef]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

deBary, E.

Deirmendjian, D.

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

Eschelbach, D.

D. Eschelbach, J. Quant. Spectrosc. Radiat. Transfer 11, 757 (1971).
[CrossRef]

Furukawa, P. M.

Fymat, A. L.

A. L. Fymat, K. D. Abhyankar, J. Geophys. Res. 76, 732 (1971).
[CrossRef]

Guinn, J. A.

G. W. Kattawar, G. N. Plass, J. A. Guinn, J. Phys. Oceanogr. 3, 353 (1973).
[CrossRef]

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]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[CrossRef]

Hitzfelder, S. J.

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

Horak, H. G.

Howell, H. B.

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[CrossRef]

Jacobowitz, H.

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[CrossRef]

Kahle, A. B.

A. B. Kahle, Astrophys. J. 151, 637 (1968).
[CrossRef]

A. B. Kahle, J. Geophys. Res. 73, 7511 (1968).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 13, 145 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
[CrossRef] [PubMed]

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

G. W. Kattawar, G. N. Plass, F. E. Catchings, Appl. Opt. 12, 1071 (1973). The mode radius given on p. 1071 should be changed to 0.05.
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, J. A. Guinn, J. Phys. Oceanogr. 3, 353 (1973).
[CrossRef]

G. W. Kattawar, G. N. Plass, Appl. Opt. 11, 2851 (1972).
[CrossRef] [PubMed]

G. N. Plass, G. W. Kattawar, Appl. Opt. 11, 2866 (1972).
[CrossRef] [PubMed]

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

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

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

G. N. Plass, G. W. Kattawar, Appl. Opt. 9, 1122 (1970).
[CrossRef] [PubMed]

C. N. Adams, G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 10, 341 (1970).
[CrossRef]

G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 2489 (1969).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
[CrossRef] [PubMed]

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

Plass, G. N.

Sekera, Z.

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[CrossRef]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Fluegge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
[CrossRef]

Tanaka, M.

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296, 321, 333 (1971).

Wells, M. B.

Adv. Geophys. (1)

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[CrossRef]

Appl. Opt. (13)

Astrophys. J. (2)

A. B. Kahle, Astrophys. J. 151, 637 (1968).
[CrossRef]

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

J. Atmos. Sci. (6)

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[CrossRef]

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

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

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[CrossRef]

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

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

J. Geophys. Res. (2)

A. L. Fymat, K. D. Abhyankar, J. Geophys. Res. 76, 732 (1971).
[CrossRef]

A. B. Kahle, J. Geophys. Res. 73, 7511 (1968).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296, 321, 333 (1971).

J. Opt. Soc. Am. (1)

J. Phys. Oceanogr. (1)

G. W. Kattawar, G. N. Plass, J. A. Guinn, J. Phys. Oceanogr. 3, 353 (1973).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (5)

D. Eschelbach, J. Quant. Spectrosc. Radiat. Transfer 11, 757 (1971).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

C. N. Adams, G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 10, 341 (1970).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, J. Quant. Spectrosc. Radiat. Transfer 13, 145 (1973).
[CrossRef]

Other (5)

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

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Fluegge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
[CrossRef]

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (27)

Fig. 1
Fig. 1

Upward radiance reflected from a plane parallel layer with Rayleigh phase matrix as a function of the cosine of the nadir angle μ. The cosine of the solar zenith angle μ0 = 1; the surface albedo A = 0. Curves are given for layers with various values of the optical thickness τ. The incoming solar flux is normalized to unity in a direction perpendicular to the solar beam.

Fig. 2
Fig. 2

Downward radiance transmitted through a plane parallel layer with Rayleigh scattering as a function of the cosine of the zenith angle μ. μ0 = 1; A = 0.

Fig. 3
Fig. 3

ΔR/R (where R is exact radiance from Stokes vector; ΔR is the exact radiance minus the approximate scalar radiance) for the reflected light from a Rayleigh scattering layer for μ0 = 1, A = 0.

Fig. 4
Fig. 4

ΔR/R (see caption to Fig. 3) for the transmitted light through a Rayleigh scattering layer for μ0 = 1, A = 0.

Fig. 5
Fig. 5

Single scattering function obtained from phase matrix for haze L (continental haze) as a function of the scattering angle.

Fig. 6
Fig. 6

Upward radiance reflected from a plane parallel haze L layer as a function of the cosine of the nadir angle μ for μ0 = 0.85332, A = 0. The left-hand portion of the graph is for an azimuthal angle ϕ = 0° and the right-hand portion for ϕ = 180°. The solar horizon is at the left of the figure, the nadir at the center followed by the antisolar point, and the antisolar horizon at the right. Curves are given for layers with various values of the optical thickness τ.

Fig. 7
Fig. 7

Downward radiance transmitted through a haze L layer as a function of the cosine of the zenith angle μ for μ0 = 0.85332, A = 0. The azimuthal angle ϕ = 0° for the left-hand portion of the graph and 180° for the right. The solar horizon is at the left of the figure, followed by the solar point, the zenith, and the antisolar horizon on the right.

Fig. 8
Fig. 8

Upward radiance for haze L, μ0 = 0.18817, A = 0, and ϕ = 0° and 180° (see caption to Fig. 6).

Fig. 9
Fig. 9

Downward radiance for haze L, μ0 = 0.18817, A = 0, and ϕ = 0° and 180° (see caption to Fig. 7).

Fig. 10
Fig. 10

Polarization of the upward radiation for Rayleigh scattering for μ0 = 0.18817, A = 0, and ϕ = 0° and 180° for layers of various optical thicknesses (see caption to Fig. 6).

Fig. 11
Fig. 11

Position of Babinet, Brewster, and Arago neutral points for upward radiation reflected from a Rayleigh scattering layer when μ0 = 0.85332, 0.11333, and 0.03785 and A = 0 and 1. The abscissa is the optical thickness of the reflecting layer. The ordinate is the cosine of the nadir angle chosen with a positive sign for the Brewster and Babinet points (ϕ = 180°) and chosen with a negative sign for the Arago points (ϕ = 0°). The Brewster point appears between the antisolar direction and the antisolar horizon, and the Arago point appears between the solar horizon and the nadir. When a curve giving the position of a neutral point passes through μ = 0 as the optical thickness increases, the actual Brewster neutral point jumps from the antisolar horizon to an Arago neutral point on the solar horizon.

Fig. 12
Fig. 12

Position of Babinet, Brewster, and Arago neutral points for upward radiation reflected from a Rayleigh scattering layer when μ0 = 0.53786 and 0.18817 and A = 0, 0.2, and 1 (see caption to Fig. 11).

Fig. 13
Fig. 13

Polarization of the downward radiation transmitted through Rayleigh scattering layers of various optical thicknesses for μ0 = 0.18817, A = 0, and ϕ = 0° and 180° (see caption to Fig. 7).

Fig. 14
Fig. 14

Position of Babinet and Brewster neutral points for downward radiation transmitted through a Rayleigh scattering layer when μ0 = 0.85332 and A = 0, 0.2, and 1. The ordinate is the cosine of the zenith angle, and the abscissa is the optical thickness of the transmitting layer.

Fig. 15
Fig. 15

Position of Babinet, Brewster, and Arago neutral points for downward radiation transmitted through a Rayleigh scattering layer when μ0 = 0.53786 and A = 0, 0.2, and 1. The abscissa is the optical thickness of the reflecting layer. The ordinate is the cosine of the zenith angle chosen with a positive sign for the Brewster and Babinet points (ϕ = 0°) and chosen with a negative sign for the Arago points (ϕ = 180°). The Brewster point appears between the solar direction and the solar horizon, while the Arago point appears between the antisolar horizon and the zenith. In this case the Arago points exist only when A = 1.

Fig. 16
Fig. 16

Position of Babinet, Brewster, and Arago neutral points for downward radiation transmitted through a Rayleigh scattering layer when μ0 = 0.18817 and A = 0 and 1 (see caption to Fig. 15).

Fig. 17
Fig. 17

Position of Babinet, Brewster, and Arago neutral points for downward radiation transmitted through a Rayleigh scattering layer when μ0 = 0.03785 and A = 0 and 1 (see caption to Fig. 15).

Fig. 18
Fig. 18

Polarization of single scattered photons from haze L. The two insets show the polarization for scattering angles near 0°. The abscissa is the cosine of the scattering angle.

Fig. 19
Fig. 19

Angles at which polarization of single scattered photons from haze L is zero. Each curve is for a particular value of the cosine of the solar zenith angle μ0. The ordinate is the cosine of the zenith angle μ, and the abscissa is the azimuthal angle ϕ. The upper part of the figure is for photons reflected into an upward direction, while the lower part is for photons transmitted through the scattering layer in a downward direction.

Fig. 20
Fig. 20

Polarization of the upward radiation for haze L, μ0 = 0.85332, A = 0, and ϕ = 0° and 180° (upper curves) and ϕ = 30° and 150° (lower curves) for layers of various optical thicknesses (see caption to Fig. 6).

Fig. 21
Fig. 21

Position of Rayleigh-like neutral points for photons reflected from a haze L layer. Curves are given for five different solar zenith angles. The ordinate is the cosine of the nadir angle, and the abscissa is the optical thickness of the reflecting layer.

Fig. 22
Fig. 22

Polarization of the upward and downward radiation for haze L for μ0 = 0.85332, A = 0, and ϕ = 90° for layers of various optical thicknesses (see captions to Figs. 6 and 7).

Fig. 23
Fig. 23

Polarization of the upward radiation for haze L, μ0 = 0.188166, A = 0, and ϕ = 0° and 180° for layers of various optical thicknesses (see caption to Fig. 6).

Fig. 24
Fig. 24

Polarization of the downward radiation for haze L, μ0 = 0.8533, A = 0, and ϕ = 0° and 180° for layers of various optical thicknesses (see caption to Fig. 7).

Fig. 25
Fig. 25

Polarization of the downward radiation for haze L for μ0 = 0.85332, A = 0, and ϕ = 30° and 150° for layers of various optical thicknesses (see caption to Fig. 7).

Fig. 26
Fig. 26

Polarization of the downward radiation for haze L for μ0 = 0.188166, A = 0, and ϕ = 0° and 180° for layers of various optical thicknesses (see caption to Fig. 7).

Fig. 27
Fig. 27

Position of non-Rayleigh-like neutral points for photons transmitted through a haze L layer. Curves are given for six different solar zenith angles. The ordinate is t]he cosine of the zenith angle, and the abscissa is the optical thickness of the scattering layer.

Tables (3)

Tables Icon

Table I Angles for Maximum Radiance and Polarization

Tables Icon

Table II Comparison of Exact and Approximate Scalar Theory Radiance Values

Tables Icon

Table III Non-Rayleigh Neutral Points

Metrics