Abstract

A calculation is made of the luminance and polarization of light due to single and double reflections from the faces of particles in a surface composed of random, irregular particles using equations of electromagnetic waves and materials with a complex index of refraction. Some geometric properties of shadows are derived and used. Good agreement is obtained between these results and measurements of polarized light from Mars, Mercury, and the moon, including the phenomenon of negative polarization at small phase angles. Negative polarization is found to be caused by shade and shadows affecting the double-reflected rays. Graphical results are provided for materials of varied real and complex indices of refraction. The model can be used to calculate polarization and luminance of rough astronomical bodies and surfaces as a function of the viewing angle. Calculated ratios of single-reflected, double-reflected, and randomly diffused light can be related to the surface structure and optical properties of the material.

© 1975 Optical Society of America

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References

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  1. T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
    [CrossRef]
  2. A. Dollfus, E. Bowell, Astron. Astrophys. 10, 29 (1971).
  3. S. F. Pellicori, Appl. Opt. 10, 270 (1971).
    [CrossRef] [PubMed]
  4. A. Dollfus, Ann. Astrophys. 19, 83 (1956).
  5. E. Bowell, A. Dollfus, J. E. Geake, “Polarimetric Properties of the Lunar Surface and Its Interpretation,” in Proc. of Third Lunar Sci. Conf. (MIT Press, Cambridge, Mass., 1972).
  6. S. F. Pellicori, Technical Report 42, Optical Sci. Center, Univ. of Arizona (1969).
  7. E. Bowell, Polarimetric Studies—in Geology and Physics of the Moon (American Elsevier, New York, 1972), Chap. 9.
  8. B. Lyot, NASA Technical Translation TTF-187, Washington, D.C. (1964)and/or Annales de L'Observatoire de Paris, section Meudon8, No. 1 (1929).
  9. Y. Ohman, Ann. Stockholm Obs. 18, no. 8 (1955).
  10. T. McCoyd, J. Opt. Soc. Am. 57, 1345 (1967).
    [CrossRef]
  11. J. Hopfield, Science 151, 1380 (1966).
    [CrossRef] [PubMed]
  12. B. Hapke, J. Geophys. Res. 68, 4571 (1963).
    [CrossRef]
  13. J. B. Pollack, C. Sagan, Space Sci. Rev. 9, 243 (1969).
    [CrossRef]
  14. A. Dollfus, Nasa Technical Translation TTF-188, Washington, D.C. (1964);A. Dollfus, Thesis, Univ. of Paris (1955).
  15. S. Provin, Publ. Astron. Soc. Pac. 67, 115 (1955).
    [CrossRef]
  16. J. Veverka, Icarus 15, 454 (1971).
    [CrossRef]
  17. B. Zellner, Astron. J. 77, 183.5 (1972).
    [CrossRef]
  18. B. Zellner, Astron. J. 174, L107 (1972).
    [CrossRef]
  19. T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
    [CrossRef]
  20. T. Widorn, Ann. Univ. Stern W. Wien 27, 112 (1967).
  21. F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), p. 174.
  22. B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
    [CrossRef]
  23. S. F. Pellicori, Astron. J. 74, 1066 (1969).
    [CrossRef]

1974

B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
[CrossRef]

1972

B. Zellner, Astron. J. 77, 183.5 (1972).
[CrossRef]

B. Zellner, Astron. J. 174, L107 (1972).
[CrossRef]

1971

J. Veverka, Icarus 15, 454 (1971).
[CrossRef]

A. Dollfus, E. Bowell, Astron. Astrophys. 10, 29 (1971).

S. F. Pellicori, Appl. Opt. 10, 270 (1971).
[CrossRef] [PubMed]

1970

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

1969

S. F. Pellicori, Astron. J. 74, 1066 (1969).
[CrossRef]

J. B. Pollack, C. Sagan, Space Sci. Rev. 9, 243 (1969).
[CrossRef]

1967

T. McCoyd, J. Opt. Soc. Am. 57, 1345 (1967).
[CrossRef]

T. Widorn, Ann. Univ. Stern W. Wien 27, 112 (1967).

1966

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

1964

T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
[CrossRef]

1963

B. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

1956

A. Dollfus, Ann. Astrophys. 19, 83 (1956).

1955

Y. Ohman, Ann. Stockholm Obs. 18, no. 8 (1955).

S. Provin, Publ. Astron. Soc. Pac. 67, 115 (1955).
[CrossRef]

Bowell, E.

A. Dollfus, E. Bowell, Astron. Astrophys. 10, 29 (1971).

E. Bowell, Polarimetric Studies—in Geology and Physics of the Moon (American Elsevier, New York, 1972), Chap. 9.

E. Bowell, A. Dollfus, J. E. Geake, “Polarimetric Properties of the Lunar Surface and Its Interpretation,” in Proc. of Third Lunar Sci. Conf. (MIT Press, Cambridge, Mass., 1972).

Coffeen, T.

T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
[CrossRef]

Dollfus, A.

A. Dollfus, E. Bowell, Astron. Astrophys. 10, 29 (1971).

A. Dollfus, Ann. Astrophys. 19, 83 (1956).

E. Bowell, A. Dollfus, J. E. Geake, “Polarimetric Properties of the Lunar Surface and Its Interpretation,” in Proc. of Third Lunar Sci. Conf. (MIT Press, Cambridge, Mass., 1972).

A. Dollfus, Nasa Technical Translation TTF-188, Washington, D.C. (1964);A. Dollfus, Thesis, Univ. of Paris (1955).

Geake, J. E.

E. Bowell, A. Dollfus, J. E. Geake, “Polarimetric Properties of the Lunar Surface and Its Interpretation,” in Proc. of Third Lunar Sci. Conf. (MIT Press, Cambridge, Mass., 1972).

Gehrels, T.

B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
[CrossRef]

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
[CrossRef]

Gradie, J.

B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
[CrossRef]

Hapke, B.

B. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

Hopfield, J.

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

Lyot, B.

B. Lyot, NASA Technical Translation TTF-187, Washington, D.C. (1964)and/or Annales de L'Observatoire de Paris, section Meudon8, No. 1 (1929).

McCoyd, T.

Ohman, Y.

Y. Ohman, Ann. Stockholm Obs. 18, no. 8 (1955).

Owings, D.

T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
[CrossRef]

Pellicori, S. F.

S. F. Pellicori, Appl. Opt. 10, 270 (1971).
[CrossRef] [PubMed]

S. F. Pellicori, Astron. J. 74, 1066 (1969).
[CrossRef]

S. F. Pellicori, Technical Report 42, Optical Sci. Center, Univ. of Arizona (1969).

Pollack, J. B.

J. B. Pollack, C. Sagan, Space Sci. Rev. 9, 243 (1969).
[CrossRef]

Provin, S.

S. Provin, Publ. Astron. Soc. Pac. 67, 115 (1955).
[CrossRef]

Roemer, E.

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

Sagan, C.

J. B. Pollack, C. Sagan, Space Sci. Rev. 9, 243 (1969).
[CrossRef]

Sears, F. W.

F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), p. 174.

Taylor, E.

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

Veverka, J.

J. Veverka, Icarus 15, 454 (1971).
[CrossRef]

Widorn, T.

T. Widorn, Ann. Univ. Stern W. Wien 27, 112 (1967).

Zellner, B.

B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
[CrossRef]

B. Zellner, Astron. J. 77, 183.5 (1972).
[CrossRef]

B. Zellner, Astron. J. 174, L107 (1972).
[CrossRef]

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

Ann. Astrophys.

A. Dollfus, Ann. Astrophys. 19, 83 (1956).

Ann. Stockholm Obs.

Y. Ohman, Ann. Stockholm Obs. 18, no. 8 (1955).

Ann. Univ. Stern W. Wien

T. Widorn, Ann. Univ. Stern W. Wien 27, 112 (1967).

Appl. Opt.

Astron J.

T. Gehrels, T. Coffeen, D. Owings, Astron J. 69, 826 (1964).
[CrossRef]

Astron. Astrophys.

A. Dollfus, E. Bowell, Astron. Astrophys. 10, 29 (1971).

Astron. J.

B. Zellner, Astron. J. 77, 183.5 (1972).
[CrossRef]

B. Zellner, Astron. J. 174, L107 (1972).
[CrossRef]

T. Gehrels, E. Roemer, E. Taylor, B. Zellner, Astron. J. 75, 186 (1970).
[CrossRef]

B. Zellner, T. Gehrels, J. Gradie, Astron. J. 79, 1100 (1974).
[CrossRef]

S. F. Pellicori, Astron. J. 74, 1066 (1969).
[CrossRef]

Icarus

J. Veverka, Icarus 15, 454 (1971).
[CrossRef]

J. Geophys. Res.

B. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

J. Opt. Soc. Am.

Publ. Astron. Soc. Pac.

S. Provin, Publ. Astron. Soc. Pac. 67, 115 (1955).
[CrossRef]

Science

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

Space Sci. Rev.

J. B. Pollack, C. Sagan, Space Sci. Rev. 9, 243 (1969).
[CrossRef]

Other

A. Dollfus, Nasa Technical Translation TTF-188, Washington, D.C. (1964);A. Dollfus, Thesis, Univ. of Paris (1955).

F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), p. 174.

E. Bowell, A. Dollfus, J. E. Geake, “Polarimetric Properties of the Lunar Surface and Its Interpretation,” in Proc. of Third Lunar Sci. Conf. (MIT Press, Cambridge, Mass., 1972).

S. F. Pellicori, Technical Report 42, Optical Sci. Center, Univ. of Arizona (1969).

E. Bowell, Polarimetric Studies—in Geology and Physics of the Moon (American Elsevier, New York, 1972), Chap. 9.

B. Lyot, NASA Technical Translation TTF-187, Washington, D.C. (1964)and/or Annales de L'Observatoire de Paris, section Meudon8, No. 1 (1929).

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Figures (15)

Fig. 1
Fig. 1

Simulated curves of lunar polarization obtained by using surface structure parameters. Measurements from Dollfus and Bowell.2

Fig. 2
Fig. 2

Light enters a particulate surface material, is reflected once, and escapes.

Fig. 3
Fig. 3

A light ray enters the particulate surface material, is reflected twice at P1 and P2, and escapes.

Fig. 4
Fig. 4

The geometry used to compute the Fresnel factors for doubly reflected rays. Inset: the path crosses when angle θ is in the forward hemisphere.

Fig. 5
Fig. 5

The three planes of incidence are determined by three ray vectors: D0, D1,and D2.

Fig. 6
Fig. 6

Computed polarization from a particle cloud due to single and double reflection without shadowing or structure effects.

Fig. 7
Fig. 7

The retrodirected light from a rough surface has different angular widths for different types of rays.

Fig. 8
Fig. 8

(A) Double reflections with the intermediate path parallel to the plane of incidence always produces positively polarized light. (B) Reflections with the intermediate path perpendicular to the plane of incidence always produces negatively polarized light.

Fig. 9
Fig. 9

Typical contributions to polarization by single and double reflected light paths.

Fig. 10
Fig. 10

A pitted backscattering surface showing doubly reflected rays: (a) left and right paths are permitted; (b) backward paths are permitted; (c) forward paths are blocked.

Fig. 11
Fig. 11

Computed polarization of light from a particle cloud with shadowing effects included.

Fig. 12
Fig. 12

The effect on polarization of assuming different ratios of single and double reflected light.

Fig. 13
Fig. 13

The single-reflected, double-reflected, and unpolarized light components that make up the lunar polarization curves of Fig. 1 and the photofunction of Fig. 14.

Fig. 14
Fig. 14

Photometric functions corresponding to the lunar polarization curves of Fig. 1.

Fig. 15
Fig. 15

The four types of coplanar paths for double-reflected light shown with the same phase angle g. Types C and D lead to tunnel paths.

Tables (1)

Tables Icon

Table I Planes of Figure 4

Equations (29)

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H = H 0 exp ( X / K ) W / m 2 ,
Δ B = H 0 ( 1 / 4 π ) [ exp ( L 1 / K ) ] ( Δ L 1 / K ) F F W / ( m 2 sr ) .
Δ B = H 0 ( 1 / 4 π ) [ exp ( L 1 / K ) ] ( F F ) [ exp ( L 2 / K ) ] ( Δ L 1 / K ) .
B 1 = [ H 0 / cos ( a 2 ) ] ( 1 / 4 π ) F F 0 { exp [ D / K cos ( a 1 ) ] } { exp [ D / K cos ( a 2 ) ] } ( d D / K ) = H 0 ( 1 / 4 π ) F F [ cos ( a 1 ) ] cos ( a 1 ) + cos ( a 2 ) W / ( m 2 sr ) .
a 1 = g λ a 2 = λ ,
Δ B = H 0 exp ( D 0 / K ) PROB 1 ( Δ D 0 / K ) W / ( m 2 sr ) ,
Δ H = H 0 exp ( D 0 / K ) exp ( D 1 / K ) ( 1 / 4 π ) ( Δ D 0 / K ) ( Δ ϕ sin ϕ Δ θ ) ,
Δ B 2 = H 0 exp [ ( D 0 D 1 D 2 ) / K ] ( 1 / 4 π ) 2 Δ ϕ sin ϕ Δ θ ( Δ D 0 ) Δ D 2 / K 2 .
B 2 = H 0 exp [ ( D 0 D 1 D 2 ) / K ] sin ϕ ( F M ) d D 0 d D 1 d ϕ d θ / 4 π K 2 .
B 2 = H 0 D 0 = 0 D 1 = 0 L IM θ = 0 2 π ϕ = 0 π exp ( D 0 ) exp ( D 1 ) exp ( D 2 ) ( F M ) sin ϕ Δ θ Δ ϕ Δ D 0 Δ D 1 / 4 π 2 ,
( F M 1 F M 2 ) = | M 20 | ( C 2 ) | M 12 | ( C 1 ) ( I 1 I 2 ) = | F M | ( I 1 I 2 ) .
C 1 = [ F F 1 ( A I 1 ) F F 2 ( A I 1 ) ] .
M 12 = | cos 2 ( θ 12 ) sin 2 ( θ 12 ) sin 2 ( θ 12 ) cos 2 ( θ 12 ) | .
C 2 = [ F F 1 ( A I 2 ) F F 2 ( A I 2 ) ] .
M 20 = | cos 2 ( θ 20 ) sin 2 ( θ 20 ) sin 2 ( θ 20 ) cos 2 ( θ 20 ) | .
A I 1 = 1 2 [ π arcos ( D 0 D 1 ) ] , A I 2 = ϕ / 2 , θ 12 = arcos ( D 0 × D 1 [ D 0 × D 1 ] ) ( D 1 × D 2 [ D 1 × D 2 ] ) , θ 20 = θ .
D 0 = [ sin ( g ) , 0 , cos ( g ) ] , D 1 = ( sin ϕ cos θ , sin ϕ sin θ , cos ϕ ) , D 2 = ( 0 , 0 , 1 ) .
A I 1 = 1 2 { π arccos [ sin ( g ) sin ( ϕ ) cos ( θ ) + cos ( g ) cos ( ϕ ) ] } , A I 2 = ϕ / 2 , θ 20 = θ , cos ( θ 12 ) = cos ( g ) sin ( ϕ ) sin ( g ) cos ( ϕ ) cos ( θ ) { 1 [ sin ( g ) sin ( ϕ ) cos ( θ ) + cos ( g ) cos ( ϕ ) ] 2 } 1 / 2 ,
F F 1 = ( E 0 p / E 0 p ) 2 = | m 2 A | 2 | m 2 + A | 2 = [ Re ( m 2 ) Re ( A ) ] 2 + [ Im ( m 2 ) Im ( A ) ] 2 [ Re ( m 2 ) + Re ( A ) ] + [ Im ( m 2 ) Im ( A ) ] 2
F F 2 = ( E 0 n / E 0 n ) 2 = | 1 A | 2 / | 1 + A | 2 = [ 1 Re ( A ) ] 2 + [ Im ( A ) ] 2 [ 1 + Re ( A ) ] 2 + [ Im ( A ) ] 2 ,
A = ( m 2 sin 2 i 1 sin 2 i ) 1 / 2 .
Singles polarization = B 11 B 12 B ( Total ) , Doubles polarization = F M 1 F M 2 B ( Total ) .
SHAG = { 1 + exp ( g / GOS ) 2 } ,
if cos θ > 0.707 , SHAD = 1 , if cos θ < 0.707 , SHAD = 1 + cos θ { 1 exp [ ( 2 g / GOS ) 2 ] } .
K 1 B 1 ( g ) + K 2 B 2 ( g ) + K 3 UNPOL ( g , A ) = Observed B ,
require : d ( UNPOL ) d g < d d g ( B 1 + B 2 ) ,
UNPOL = ( A b ) cos ( g λ ) / [ cos ( g λ ) + cos λ ] ,
EXCLUDE , IF ϕ < g ( cos θ ) and , IF cos θ = + , EXCLUDE , IF ϕ > ( π + g cos θ ) and , IF cos θ = .
Polarization = B ( g ) B ( g ) B ( g ) + B ( g ) + 1 unpol ( g ) .

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