Abstract

Consider light scattering by a small volume element filled with randomly positioned particles, the far-field modified uncorrelated single-scattering approximation (MUSSA) leads to the incoherent summation of the phase matrices of particles in the volume. The validity of the MUSSA is revisited in this paper to include the variation of the particles’ positions. Analytical results show that the MUSSA does not require the distance between any pair of particles in the volume to be larger than what is required in the single-scattering approximation (SSA). Instead, it requires the dimension of the volume to be large compared to the incident wavelength. The new results also make the requirements of MUSSA easier to be met. We also analyze energy conservation for the MUSSA.

© 2007 Optical Society of America

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References

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  2. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. D. M. O’Brien, "Accelerated quasi Monte Carlo integration of the radiative transfer equation," J. Quant. Spectrosc. Radiat. Transf.  48, 41-59 (1992).
    [CrossRef]
  13. E. P. Zege, I. L. Katsev, and I. N. Polonsky, "Multicomponent approach to light propagation in clouds and mists," Appl. Opt.,  32, 2803-2812 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
  15. K. N. Liou, An Introduction to Atmospheric Radiation, Second Edition, (Academic Press, New York, 2002).
  16. M. I. Mishchenko, "Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics," Appl. Opt. 41, 7114-7134 (2002).
    [CrossRef] [PubMed]
  17. M. I. Mishchenko, L. D. Travis and A. A. Lacis, Multiple Scattering of Light by Particles, (Cambridge University Press, Cambridge, UK, 2006).
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    [CrossRef]
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  21. O. Munoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, "Experimental determination of scattering matrices of randomly oriented flay ash and clay particles at 442 and 633 nm," J. Geophys. Res. 106, 22833- 22844 (2001).
    [CrossRef]
  22. J. W. Hovenier, H. Volten, O. Munoz, W. J. van der Zande, and L. B. F. M. Waters, "Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives," J. Quant. Spectrosc. Radiat. Transf. 79/80, 741-755 (2003).
    [CrossRef]

2004

2003

J. W. Hovenier, H. Volten, O. Munoz, W. J. van der Zande, and L. B. F. M. Waters, "Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives," J. Quant. Spectrosc. Radiat. Transf. 79/80, 741-755 (2003).
[CrossRef]

2002

2001

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, "Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm," J. Geophys. Res. 106, 17375-17402 (2001).
[CrossRef]

O. Munoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, "Experimental determination of scattering matrices of randomly oriented flay ash and clay particles at 442 and 633 nm," J. Geophys. Res. 106, 22833- 22844 (2001).
[CrossRef]

1998

K. F. Evans, "The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci.  55, 429-446 (1998).
[CrossRef]

1993

1992

D. M. O’Brien, "Accelerated quasi Monte Carlo integration of the radiative transfer equation," J. Quant. Spectrosc. Radiat. Transf.  48, 41-59 (1992).
[CrossRef]

F. Weng, "A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere-I. theory," J. Quant. Spectrosc. Radiat. Transf. 47, 19-33 (1992).
[CrossRef]

F. Weng, "A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere-II. Application," J. Quant. Spectrosc. Radiat. Transf. 47, 35-42 (1992).
[CrossRef]

1988

1986

R. D. M. Garcia and C. E. Siewert, "A generalized spherical harmonics solution for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transf. 36, 401-423 (1986).
[CrossRef]

1970

C. N. Adams and G. W. Kattawar, "Solutions of the equation of radiative transfer by an invariant imbedding approach," J. Quant. Spectrosc. Radiat. Transf. 10, 341-366 (1970).
[CrossRef]

1968

Appl. Opt.

J. Atmos. Sci.

K. F. Evans, "The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci.  55, 429-446 (1998).
[CrossRef]

J. Geophys. Res.

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, "Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm," J. Geophys. Res. 106, 17375-17402 (2001).
[CrossRef]

O. Munoz, H. Volten, J. F. de Haan, W. Vassen, J. W. Hovenier, "Experimental determination of scattering matrices of randomly oriented flay ash and clay particles at 442 and 633 nm," J. Geophys. Res. 106, 22833- 22844 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

F. Weng, "A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere-I. theory," J. Quant. Spectrosc. Radiat. Transf. 47, 19-33 (1992).
[CrossRef]

F. Weng, "A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere-II. Application," J. Quant. Spectrosc. Radiat. Transf. 47, 35-42 (1992).
[CrossRef]

R. D. M. Garcia and C. E. Siewert, "A generalized spherical harmonics solution for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transf. 36, 401-423 (1986).
[CrossRef]

D. M. O’Brien, "Accelerated quasi Monte Carlo integration of the radiative transfer equation," J. Quant. Spectrosc. Radiat. Transf.  48, 41-59 (1992).
[CrossRef]

C. N. Adams and G. W. Kattawar, "Solutions of the equation of radiative transfer by an invariant imbedding approach," J. Quant. Spectrosc. Radiat. Transf. 10, 341-366 (1970).
[CrossRef]

J. W. Hovenier, H. Volten, O. Munoz, W. J. van der Zande, and L. B. F. M. Waters, "Laboratory studies of scattering matrices for randomly oriented particles: potentials, problems, and perspectives," J. Quant. Spectrosc. Radiat. Transf. 79/80, 741-755 (2003).
[CrossRef]

Other

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications, 1 and 2. (Academic Press, 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, Berlin, 1980).

J. W. Hovenier, "Measuring scattering matrices of small particles at optical wavelengths," in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 355-365.

M. I. Mishchenko, L. D. Travis and A. A. Lacis, Multiple Scattering of Light by Particles, (Cambridge University Press, Cambridge, UK, 2006).

K. N. Liou, An Introduction to Atmospheric Radiation, Second Edition, (Academic Press, New York, 2002).

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

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

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon Press, Oxford, 1965).

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Figures (1)

Fig. 1.
Fig. 1.

f(θ) as a function of θ for k 1 L = 15 and 60.

Equations (23)

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S ( r ̂ , s ̂ ) = i = 1 N exp ( i k 1 R i ( s ̂ r ̂ ) ) S i ( r ̂ , s ̂ ) ,
k 1 r 1 ,
r L 2 ,
r k 1 L 2 8 ,
r L k 1 a i π , for i = 1 , , N ,
i = 1 N [ S i ( r ̂ , s ̂ ) ] k l [ S i ( r ̂ , s ̂ ) ] pq * + i′ ( i ) = 1 N [ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * exp [ i k 1 ( R i R i′ ) ( s ̂ r ̂ ) ] .
i = 1 N [ S i ( r ̂ , s ̂ ) ] k l [ S i ( r ̂ , s ̂ ) ] pq * + i′ ( i ) = 1 N [ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * 1 V ʃ dV i′ exp [ i k 1 ( R i R i′ ) ( s ̂ r ̂ ) ] ,
1 V ʃ dV i′ ( exp [ i k 1 ( R i R i′ ) ( s ̂ r ̂ ) ]
= 1 V l L D 2 dD 0 π sin ( α ) d α 0 2 π d ϕ exp [ i k 1 D s ̂ r ̂ cos ( α ) ]
1 4 π 0 π sin ( α ) d α 0 2 π d ϕ exp [ i k 1 D s ̂ r ̂ cos ( α ) ]
= sin ( k 1 D s ̂ r ̂ ) k 1 D s ̂ r ̂
= sin ( 2 k 1 D sin ( θ 2 ) ) 2 k 1 D sin ( θ 2 ) .
f ( θ ) = 1 V ʃ dV i′ ( exp [ i k 1 ( R i R i′ ) ( s ̂ r ̂ ) ]
= 1 V l L DdD 4 π sin ( k 1 D s ̂ r ̂ ) k 1 s ̂ r ̂
= 3 ( 2 k 1 L sin ( θ 2 ) cos ( 2 k 1 L sin ( θ 2 ) ) + sin ( 2 k 1 L sin ( θ 2 )
+ 2 k 1 l sin ( θ 2 ) cos ( 2 k 1 l sin ( θ 2 ) ) sin ( 2 k 1 l sin ( θ 2 ) ) / ( [ 2 sin ( θ 2 ) ] 3 k 1 3 ( L 3 l 3 ) ) ,
θ = θ 0 = 2 arcsin [ 4.49341 ( 2 k 1 L ) ] .
L a i , i = 1 , , N .
i ( i ) = 1 N [ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * f ( θ ) ~ ( N 1 ) [ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * f ( θ ) .
N ( k 1 L ) 2 1 ,
C = N d Ω [ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * f ( θ ) 0 .
[ S i ( r ̂ , s ̂ ) ] k l [ S i′ ( r ̂ , s ̂ ) ] pq * = n = 0 n max w n P n ( cos ( θ ) ) ,
C = N 2 π n = 0 n max w n 0 π P n ( cos ( θ ) ) f ( θ ) sin ( θ ) d θ .

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