Abstract

The single particle phase function and the linear polarization for large stochastically deformed spheres have been calculated by Monte Carlo simulation using the geometrical optics approximation. The radius vector of a particle is assumed to obey a bivariate lognormal distribution with three free parameters: mean radius, its standard deviation and the coherence length of the autocorrelation function. All reflections/refractions which include sufficient energy have been included. Real and imaginary parts of the refractive index can be varied without any restrictions. Results and comparisons with some earlier less general theories are presented. Applications of this theory to the photometric properties of atmosphereless bodies and interplanetary dust are discussed.

© 1989 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. R. Schiffer, "The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to Zodiacal Light," Astron. Astrophys. 148, 347–358 (1985).
  4. F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).
  5. R. Schiffer and K. O. Thielheim, "A Scattering Model for the Zodiacal Light Particles," Astron. Astrophys. 116, 1–9 (1982).
  6. S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
    [CrossRef]
  7. D. W. Schuerman, Eds., Light Scattering by Irregularly Shaped Particles (Plenum, New York, 1980).
    [CrossRef]
  8. R. H. Giese and P. Lamy, Eds., Properties and Interactions of Interplanetary Dust (D. Reidel, Dordrecht, 1985).
    [CrossRef]
  9. K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  10. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).
  11. L. L. House and L. W. Avery, "The Monte-Carlo Technique Applied to Radiative Transfer," J. Quant. Spectrosc. Radiat. Transfer 9, 1957–1991 (1967).
  12. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 391–468.
  13. B. G. Smith, "Geometrical Shadowing of a Random Rough Surface," IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
    [CrossRef]
  14. R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).
  15. K. Weiss-Wrana, "Optical Properties of Interplanetary Dust: Comparison with Light Scattering by Larger Meteoric and Terrestrial Grains," Astron. Astrophys. 126, 240–250 (1983).
  16. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, MA, 1982), pp. 825–827.
  17. K. Muinonen, K. Lumme, J. Peltoniemi, and W. M. Irvine, "Light Scattering by Randomly Oriented Crystals," Appl. Opt. 28, 3051–3060 (1989).
    [CrossRef] [PubMed]
  18. K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).
  19. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

1989

1987

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

1985

R. Schiffer, "The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to Zodiacal Light," Astron. Astrophys. 148, 347–358 (1985).

1983

K. Weiss-Wrana, "Optical Properties of Interplanetary Dust: Comparison with Light Scattering by Larger Meteoric and Terrestrial Grains," Astron. Astrophys. 126, 240–250 (1983).

1982

R. Schiffer and K. O. Thielheim, "A Scattering Model for the Zodiacal Light Particles," Astron. Astrophys. 116, 1–9 (1982).

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

1978

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

1967

L. L. House and L. W. Avery, "The Monte-Carlo Technique Applied to Radiative Transfer," J. Quant. Spectrosc. Radiat. Transfer 9, 1957–1991 (1967).

B. G. Smith, "Geometrical Shadowing of a Random Rough Surface," IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

Avery, L. W.

L. L. House and L. W. Avery, "The Monte-Carlo Technique Applied to Radiative Transfer," J. Quant. Spectrosc. Radiat. Transfer 9, 1957–1991 (1967).

Bass, F. G.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Bowell, E.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

Darbinjan, R. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Elepov, B. S.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Fuks, I. M.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Fung, A. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, MA, 1982), pp. 825–827.

Giese, R. H.

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

House, L. L.

L. L. House and L. W. Avery, "The Monte-Carlo Technique Applied to Radiative Transfer," J. Quant. Spectrosc. Radiat. Transfer 9, 1957–1991 (1967).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Irvine, W. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 391–468.

Kargin, B. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Karttunen, H.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

Liou, K-N.

K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Lumme, K.

K. Muinonen, K. Lumme, J. Peltoniemi, and W. M. Irvine, "Light Scattering by Randomly Oriented Crystals," Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

Marchuk, G. I.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Mikhailov, G. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Moore, R. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, MA, 1982), pp. 825–827.

Muinonen, K.

K. Muinonen, K. Lumme, J. Peltoniemi, and W. M. Irvine, "Light Scattering by Randomly Oriented Crystals," Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

Mukai, S.

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

Mukai, T.

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

Nazaraliev, M. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

Ono, T.

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

Peltoniemi, J.

K. Muinonen, K. Lumme, J. Peltoniemi, and W. M. Irvine, "Light Scattering by Randomly Oriented Crystals," Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

Schiffer, R.

R. Schiffer, "The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to Zodiacal Light," Astron. Astrophys. 148, 347–358 (1985).

R. Schiffer and K. O. Thielheim, "A Scattering Model for the Zodiacal Light Particles," Astron. Astrophys. 116, 1–9 (1982).

Smith, B. G.

B. G. Smith, "Geometrical Shadowing of a Random Rough Surface," IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

Thielheim, K. O.

R. Schiffer and K. O. Thielheim, "A Scattering Model for the Zodiacal Light Particles," Astron. Astrophys. 116, 1–9 (1982).

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, MA, 1982), pp. 825–827.

van de Hulst, H. C.

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

Weiss, K.

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

Weiss-Wrana, K.

K. Weiss-Wrana, "Optical Properties of Interplanetary Dust: Comparison with Light Scattering by Larger Meteoric and Terrestrial Grains," Astron. Astrophys. 126, 240–250 (1983).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Zerull, R. H.

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

Appl. Opt.

Astron. Astrophys.

R. H. Giese, K. Weiss, R. H. Zerull, and T. Ono, "Large Fluffy Particles: A Possible Explanation of the Optical Properties of Interplanetary Dust," Astron. Astrophys. 65, 265–272 (1978).

K. Weiss-Wrana, "Optical Properties of Interplanetary Dust: Comparison with Light Scattering by Larger Meteoric and Terrestrial Grains," Astron. Astrophys. 126, 240–250 (1983).

R. Schiffer, "The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to Zodiacal Light," Astron. Astrophys. 148, 347–358 (1985).

R. Schiffer and K. O. Thielheim, "A Scattering Model for the Zodiacal Light Particles," Astron. Astrophys. 116, 1–9 (1982).

Bull. Am. Astron. Soc.

K. Lumme, K. Muinonen, J. Peltoniemi, H. Karttunen, and E. Bowell, "A Possible Explanation for the Anomalously Sharp Opposition Effects," Bull. Am. Astron. Soc. 19, 850 (1987).

IEEE Trans. Antennas Propag.

B. G. Smith, "Geometrical Shadowing of a Random Rough Surface," IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

L. L. House and L. W. Avery, "The Monte-Carlo Technique Applied to Radiative Transfer," J. Quant. Spectrosc. Radiat. Transfer 9, 1957–1991 (1967).

Moon and Planets

S. Mukai, T. Mukai, K. Weiss, and R. H. Zerull, "Scattering of Radiation by a Large Particle with a Random Rough Surface," Moon and Planets 26, 197–208 (1982).
[CrossRef]

Other

D. W. Schuerman, Eds., Light Scattering by Irregularly Shaped Particles (Plenum, New York, 1980).
[CrossRef]

R. H. Giese and P. Lamy, Eds., Properties and Interactions of Interplanetary Dust (D. Reidel, Dordrecht, 1985).
[CrossRef]

K-N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, New York, 1980).

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 391–468.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, MA, 1982), pp. 825–827.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

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Figures (7)

Fig. 1
Fig. 1

Computer-generated silhouettes of typical rough particles: (a)σ = 0.1, ρ = 0.5; (b)σ = 0.5, ρ = 1.5, where σ is the normalized standard deviation of the radius and ρ is the standard deviation of the slope.

Fig. 2
Fig. 2

Small deformations from the sphere. This figure illustrates clearly how fast even a small roughness changes the scattering behavior. The first particle is almost spherical; the only effects of roughness are a small opposition spike and the moving of rainbows closer to each other. The seocnd particle is slightly deformed and the third has visible irregularities. Note the great change in both the phase function and polarization.

Fig. 3
Fig. 3

Typical rough particles with m = 1.55. As in Fig. 2 we have the nearly spherical particle first as a comparison base. The two other particles are intermediately and extremely rough. The difference between spherical and irregular particles is very clear. An additional increase of roughness does not give anything new within this theory.

Fig. 4
Fig. 4

Roughness of the particle is now fixed and the scattering has been calculated with different refractive indices.

Fig. 5
Fig. 5

Dependence of the phase function and the polarization on the imaginary part of the refractive index. The size parameter x = 500.

Fig. 6
Fig. 6

Comparison of the present model with the experiment by Giese et al.14 Unlike in the previous figures, the diffraction is included in our results using a simple circular aperture theory. The dashed line in the upper figure (very near the solid line) is the function fit to our results [Eq. (28)].

Fig. 7
Fig. 7

Comparison of the present theory with experiment by Weiss-Wrana.15 The agreement is satisfactory.

Tables (1)

Tables Icon

Table I Summary of Computed Results

Equations (29)

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I s ( Ω ) = σ ext ω 0 r 2 d 2 Ω P ( θ ) 4 π I i ( Ω ) ,
4 π d 2 Ω P 11 ( θ ) 4 π = 1 .
β ( r , Ω , m ) I ( r , Ω , m ) + Ω · I ( r , Ω , m ) = m d Ω W ( Ω , m r , Ω , m ) β s ( r , Ω , m ) I ( r , Ω , m ) + Ψ ( r , Ω , m ) ,
I = K ̂ I + I 0 ,
K ̂ = m d Ω d r exp { [ τ ( r , Ω , m ) τ ( r , Ω , m ) ] } β ( r , Ω , m ) × W ( Ω , m | r , Ω , m ) δ ( r r + | r r | Ω ) ,
I 0 = d r exp { [ τ ( r , Ω , m ) τ ( r , Ω , m ) ] } × δ ( r r + | r r | Ω ) Ψ ( r , Ω , m ) ,
I ( r , Ω , m ) = n K ̂ n I 0 = I 0 + I 1 + I 2 + I 3 + I 4 + = I 0 + I s .
f ( h ) d h = d h 2 π β h exp [ [ ln ( h / a ) + ½ β 2 ] 2 / 2 β 2 ] ,
C ( ) = 1 1 2 2 ξ 2 ,
g 1 ( tan γ 1 ) d tan γ 1 = d tan γ 1 2 π ρ exp ( tan 2 γ 1 / 2 ρ 2 ) ,
g ( tan γ , ζ ) d tan γ d ζ = tan γ d tan γ d ζ 2 π ρ 2 exp ( tan 2 γ / 2 ρ 2 ) ,
β s ( r , Ω , m 1 ) d l = cot ψ d tan γ 1 dhf ( r ) g 1 ( tan γ 1 ) 0 r dhf ( h ) ,
β s ( r , Ω , m 1 ) d l = f ( r ) F ( r ) [ ρ 2 g 1 ( cot ψ ) + cot ψ G 1 ( cot ψ ) ] sin ψ d l ,
β s ( r , Ω , m 2 ) d l = f ( r ) 1 F ( r ) [ ρ 2 g 1 ( cot ψ ) cot ψ G 1 ( cot ψ ) ] sin ψ d l ,
g eff ( tan γ ψ ) d tan γ d ζ = d tan γ d ζ Q ( cot ψ ) cos ι cos γ g ( tan γ ) ,
Q ( t ) = t d tan γ d ζ cos ι cos γ g ( tan γ ) .
W ( Ω , m r , Ω , m ) = K ( Φ 2 ) { R T } ( α i , α t , m , m ) K ( Φ 1 ) × g eff ( tan η | ψ ) d tan γ d ζ d Ω ,
K = [ 1 0 0 0 0 cos 2 Φ sin 2 Φ 0 0 sin 2 Φ cos 2 Φ 0 0 0 0 1 ] .
cos Φ 1 = n · Ω 0 + Ω · Ω 0 cos α i 1 Ω · Ω 0 sin α i , sin Φ 1 = sign ( Ω 0 · n × Ω ) 1 cos 2 Φ 1 ,
cos Φ 2 = n · Ω 0 + Ω · Ω 0 cos α i 1 Ω · Ω 0 sin α i , sin Φ 1 = sign ( Ω 0 · n × Ω ) 1 cos 2 Φ 2 ,
cos α i = 1 Ω · Ω 2 ,
n = Ω Ω 2 cos α i .
cos α i = | m Ω · Ω m | m 2 + m 2 2 m m Ω · Ω ,
cos α t = | m Ω · Ω m | m 2 + m 2 2 m m Ω · Ω , n = m Ω m Ω m cos α i m cos α t .
R = 1 2 [ | r | 2 + | r | 2 | r | 2 | r | 2 0 0 | r | 2 | r | 2 | r | 2 + | r | 2 0 0 0 0 2 Re [ r * r ] 2 Im [ r * r ] 0 0 2 Im [ r * r ] 2 Re [ r * r ] ] ,
R = m cos α t 2 m cos α i [ | t | 2 + | t | 2 | t | 2 | t | 2 0 0 | t | 2 | t | 2 | t | 2 + | t | 2 0 0 0 0 2 Re [ t * t ] 2 Im [ t * t ] 0 0 2 Im [ t * t ] 2 Re [ t * t ] ] .
cos γ = n · r / r , cos ζ = cos ψ cos γ + cos α 1 sin ψ sin γ .
P 11 = k = k min k max a k exp ( k cos θ ) ,
ω ¯ 0

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