Abstract

The percent polarization of light reflected from a rough surface has been calculated using a two-dimensional, two-layer, dielectric model and the geometrical-optics approximation. The choice of this model was prompted by a desire to investigate the dependence of the percent polarization upon internal scattering and re-emission of light which has penetrated the top surface. The parameters of the model are the indices of refraction of the two layers, and the distribution of slopes of the top surface of the upper layer. Polarization curves that are qualitatively similar to lunar and laboratory-sample signatures (i.e., the percent polarization is negative at small phase angles and positive at larger phase angles) can be obtained with this model. The curves are very sensitive to changes of the distribution of slopes or the index of refraction of the upper layer, but not, for the cases considered, to changes of the index of refraction of the lower layer. For the range of parameters considered in this paper, a variation of the index of refraction of the upper layer produces a change of the albedo; the sign of the change is opposite to that of the change of the percent polarization at high phase angles (~90°). However, a variation of the structure (distribution of slopes) of the top surface produces changes of the albedo and percent polarization of the same sign.

© 1967 Optical Society of America

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References

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  1. B. Lyot, Annales de l’Observatoire de Paris Section de Meudon, VIII No. 1 (1929), NASA Technical Translation: NASA TT F-187.
  2. A. Dollfus, “Study of the Planets by Means of the Polarization of Their Light,” thesis, University of Paris, May1955, NASA Technical Translation: NASA TT F-188.
  3. D. L. Coffeen, Astron. J. 70, 403 (1965).
    [Crossref]
  4. W. Egan, L. Smith, and G. McCoyd, Grumman Research Department Report RE-250 (May1966).
  5. W. Egan and H. Hallock, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (Institute of Science and Technology, Univ. of Mich., Ann Arbor, Mich.1966), p. 671.
  6. K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
    [Crossref]
  7. J. J. Hopfield, Science 151, 1380 (1966).
    [Crossref] [PubMed]
  8. V. P. Rvachev and V. K. Polyanskii, Opt. Spectry (USSR) 18, 594 (1965).
  9. K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).
  10. D. Clarke, Monthly Notices Royal Astron. Soc. 130, 83 (1965).

1966 (3)

K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
[Crossref]

J. J. Hopfield, Science 151, 1380 (1966).
[Crossref] [PubMed]

K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).

1965 (3)

D. Clarke, Monthly Notices Royal Astron. Soc. 130, 83 (1965).

V. P. Rvachev and V. K. Polyanskii, Opt. Spectry (USSR) 18, 594 (1965).

D. L. Coffeen, Astron. J. 70, 403 (1965).
[Crossref]

Birkebak, R. C.

Clarke, D.

D. Clarke, Monthly Notices Royal Astron. Soc. 130, 83 (1965).

Coffeen, D. L.

D. L. Coffeen, Astron. J. 70, 403 (1965).
[Crossref]

Dollfus, A.

A. Dollfus, “Study of the Planets by Means of the Polarization of Their Light,” thesis, University of Paris, May1955, NASA Technical Translation: NASA TT F-188.

Durrani, S. H.

K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).

Egan, W.

W. Egan, L. Smith, and G. McCoyd, Grumman Research Department Report RE-250 (May1966).

W. Egan and H. Hallock, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (Institute of Science and Technology, Univ. of Mich., Ann Arbor, Mich.1966), p. 671.

Hallock, H.

W. Egan and H. Hallock, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (Institute of Science and Technology, Univ. of Mich., Ann Arbor, Mich.1966), p. 671.

Hopfield, J. J.

J. J. Hopfield, Science 151, 1380 (1966).
[Crossref] [PubMed]

Koepsel, W. W.

K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).

Krishen, K.

K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).

Lyot, B.

B. Lyot, Annales de l’Observatoire de Paris Section de Meudon, VIII No. 1 (1929), NASA Technical Translation: NASA TT F-187.

McCoyd, G.

W. Egan, L. Smith, and G. McCoyd, Grumman Research Department Report RE-250 (May1966).

Polyanskii, V. K.

V. P. Rvachev and V. K. Polyanskii, Opt. Spectry (USSR) 18, 594 (1965).

Rvachev, V. P.

V. P. Rvachev and V. K. Polyanskii, Opt. Spectry (USSR) 18, 594 (1965).

Smith, L.

W. Egan, L. Smith, and G. McCoyd, Grumman Research Department Report RE-250 (May1966).

Sparrow, E. M.

Torrance, K. E.

Astron. J. (1)

D. L. Coffeen, Astron. J. 70, 403 (1965).
[Crossref]

IEEE Intern. Convention Record (1)

K. Krishen, W. W. Koepsel, and S. H. Durrani, IEEE Intern. Convention Record 14, Part 4, p. 21 (1966).

J. Opt. Soc. Am. (1)

Monthly Notices Royal Astron. Soc. (1)

D. Clarke, Monthly Notices Royal Astron. Soc. 130, 83 (1965).

Opt. Spectry (USSR) (1)

V. P. Rvachev and V. K. Polyanskii, Opt. Spectry (USSR) 18, 594 (1965).

Science (1)

J. J. Hopfield, Science 151, 1380 (1966).
[Crossref] [PubMed]

Other (4)

W. Egan, L. Smith, and G. McCoyd, Grumman Research Department Report RE-250 (May1966).

W. Egan and H. Hallock, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (Institute of Science and Technology, Univ. of Mich., Ann Arbor, Mich.1966), p. 671.

B. Lyot, Annales de l’Observatoire de Paris Section de Meudon, VIII No. 1 (1929), NASA Technical Translation: NASA TT F-187.

A. Dollfus, “Study of the Planets by Means of the Polarization of Their Light,” thesis, University of Paris, May1955, NASA Technical Translation: NASA TT F-188.

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Figures (8)

Fig. 1
Fig. 1

Two-layer, two-dimensional, rough-surface dielectric model.

Fig. 2
Fig. 2

Position (ξ) of local normal of a facet which reflects light into the direction β: ξ = 1 2 ( ψ + β ) ψ = 1 2 ( β ψ ).

Fig. 3
Fig. 3

Path of a ray through the medium n2 from the entrance facet (ξ = δ) to the exit facet (ξ = η).

Fig. 4
Fig. 4

Percent polarization vs phase angle for a uniform distribution with facets at every 1 2 °; viewing angle 10°, detector acceptance half angle 1°; dots represent n2 = 1.2, n3 = 1.4, albedo = 0.010; crosses represent n2 = 2.56, n3 = 1.4, albedo = 0.19.

Fig. 5
Fig. 5

Percent polarization vs phase angle for a uniform distribution with facets at every 1 2 °; viewing angle 10°, detector acceptance half angle 1°; dots represent n2 = 1.5, n3 = 1.4, albedo = 0.031; crosses represent n2 = 1.5, n3 = 2.56, albedo = 0.082.

Fig. 6
Fig. 6

Percent polarization vs phase angle for a uniform distribution; viewing angle 10°, detector acceptance half angle 1°, n2 = 1.96, n3 = 1.4; dots represent facets at every 1°, albedo = 0.066; crosses represent facets at every 1 2 °, albedo = 0.093.

Fig. 7
Fig. 7

Percent polarization vs phase angle for three distributions with facets at every 1 2 °; viewing angle 10°, detector acceptance half angle 1°, n2 = 2.56, n3 = 1.7; dots represent distribution I, albedo = 0.096; crosses represent distribution II, albedo = 0.14; triangles represent uniform distribution, albedo = 0.16.

Fig. 8
Fig. 8

Incident (j) and refracted (k) ray directions at the exit facet (η). The symbols, j, k, η, also represent the angles which these lines make with the vertical (n0).

Equations (34)

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α = φ + ψ .
ν = 1 2 ( φ ψ ω ) μ = 1 2 ( φ ψ + ω ) ,
ν ξ μ
θ i = ξ + ψ .
G ξ , r ( 1 ) = | R r ( θ i ) | 2 F ξ cos θ i ,
| R ( θ i ) | 2 = | sin ( θ i θ t ) sin ( θ i + θ t ) | 2 θ i 0 | n 2 1 n 2 + 1 | 2 | R ( θ i ) | 2 = | tan ( θ i θ t ) tan ( θ i + θ t ) | 2 θ i 0 | n 2 1 n 2 + 1 | 2 ,
n 2 sin θ t = sin θ i .
G r ( 1 ) = ξ | R r ( θ i ) | 2 F ξ cos θ i ,
ψ < 0 ( ψ + 90 ) ξ 90 ψ = 0 90 ξ 90 ψ > 0 90 ξ 90 ψ .
sin ( ψ + δ ) = n 2 sin ( j + δ ) .
sin ( η k ) = n 2 sin ( η j ) .
G ψ , δ , η , r ( 2 ) = | T δ , r | 2 | R r | 2 | T η , r | 2 S η P j , η cos θ η ,
G ψ , δ , r ( 2 ) = η G ψ , δ , η , r ( 2 ) ,
G ψ , r ( 2 ) = δ G ψ , δ , r ( 2 ) .
P j , η = F η cos ( η j ) / β F β cos ( β j ) ,
( 90 j ) β 90 j > 0 90 β 90 + j j < 0.
W ψ , δ , η = W ψ , δ [ cos ( j η ) ] 1 ,
W ψ , δ = F s cos ( j + δ ) .
S η = the smaller of { F η or W ψ , δ , η
G r = G r ( 1 ) + G r ( 2 ) ,
P = ( G G ) / ( G + G ) × 100 % .
d I / d β = C I 0 cos β ,
d I / d β = cos β .
I 2 ω = 2 0 ω cos β d β = 2 sin ω .
A = ( G + G ) / 2 H cos φ sin ω ,
H = ξ = 90 ° 90 ° F ξ cos ξ
A = ( G + G ) / 200 cos φ sin ω .
90 ξ 90 .
tan μ = sin ( φ ω ) n 2 sin j / cos ( φ ω ) n 2 cos j
tan ν = sin ( φ + ω ) n 2 sin j / cos ( φ + ω ) n 2 cos j ,
η k = 90 ° ,
sin ( η j ) = n 1 / n 2 .
cos ( j k ) grazing = sin ( η j ) = n 1 / n 2 .
n d sin ( j k ) d sin ( η j ) = { [ 1 sin 2 ( η j ) ] 1 2 [ n 2 sin 2 ( η j ) ] 1 2 } + sin 2 ( η j ) { [ n 2 sin 2 ( η j ) ] 1 2 [ 1 sin 2 ( η j ) ] 1 2 } ,