## The far-field modified uncorrelated single-scattering approximation in light scattering by a small volume element

Optics Express, Vol. 15, Issue 13, pp. 8479-8485 (2007)

http://dx.doi.org/10.1364/OE.15.008479

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### 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

## 1. Introduction

4. G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. , **7**, 415–419, (1968). [CrossRef] [PubMed]

5. 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, (1968). [CrossRef]

8. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate method radiative trasfer in multiple scattering and emitting layered media,” Appl. Opt. , **27**, 2502–2509, (1988). [CrossRef]

9. 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]

10. 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]

11. 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]

12. D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spec-trosc. 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]

14. K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transf.,” J. Atmos. Sci. , **55**, 429–446, (1998). [CrossRef]

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]

18. M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, “Single scattering by a small volume element,” J. Opt. Soc. Am. A **21**, 71–87 (2004) [CrossRef]

18. M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, “Single scattering by a small volume element,” J. Opt. Soc. Am. A **21**, 71–87 (2004) [CrossRef]

## 2. Formulation

18. M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, “Single scattering by a small volume element,” J. Opt. Soc. Am. A **21**, 71–87 (2004) [CrossRef]

*V*filled with

*N*particles. The origin of the local coordinates is defined near the center of gravity of the small volume element. Note that

*r*̂ and

*s*̂ are the unit vectors in the scattering and incident directions, respectively. Under the far-field SSA, the total amplitude matrix of the group

**S**(

*r*̂,

*s*̂) can be expressed in terms of the amplitude matrices

**S**

_{i}(

*r*̂,

*s*̂) of the constituent particles (see Eq. 7.2.9) in [17]):

*k*

_{1}is the wave number in the surrounding medium and

**R**

_{i}is the coordinate vector of particle i. The conditions of applicability of Eq. (1) are summarized as:

*r*is the distance between the observation point and the origin of the local coordinates;

*L*is the largest linear dimension of the volume element and

*a*is the smallest circuscribing sphere of particle

_{i}*i*. Please see Refs. [17, 18

**21**, 71–87 (2004) [CrossRef]

**S**

_{i′}is independent of position;

*V*is the total volume of the small volume element; and

*dV*

_{i′}=

*dx*

_{i′}

*dy*

_{i′}

*dz*

_{i′}is the differential volume element for the coordinates of particle

*i*′.

*i*′ moves freely in a spherical space whose center is at particle

*i*. Therefore

*dV*

_{i′}=

*dx*

_{i′}

*dy*

_{i′}

*dz*

_{i′}=

*d*(

*x*

_{i′}-

*x*)

_{i}*d*(

*y*

_{i′}-

*y*)

_{i}*d*(

*z*

_{i′}-

*z*) =

_{i}*D*

^{2}sin(α)

*dDd*α

*d*ϕ, where

*D*, α, ϕ are the spherical coordinates of the vector

**R**

_{i′}-

**R**. Now the average of the phase factor in Eq. (7) can be evaluated as:

_{i}*l*is the minimal distance between particle

*i*and particle

*i*′ to ensure the far-field single-scattering approximation;

*L*is the largest dimension of the small volume. The angular integral gives:

*s*̂ -

*r*̂| = 2sin(θ/2) has been used. Equation (9) is the same as the result of Mishchenko et al. [17, 18

**21**, 71–87 (2004) [CrossRef]

*θ*is the scattering angle and

*V*= 4π(

*L*

^{3}-

*l*

^{3})/3 is used.

## 3. Discussions

**21**, 71–87 (2004) [CrossRef]

*D*as

*d*in Eq. (9). The first observation of Eq. (9) and (10) is that

*f*(

*θ*) is real. If

*f*(

*θ*) is sufficiently small, both the real and imaginary parts of the interference term in Eq. (7) can be ignored. The condition is satisfied when

*k*

_{1}

*L*≫ 1 and

*L*≫

*l*. With the condition of

*k*

_{1}

*L*≫ 1 and

*L*≫

*l*, the first term in the numerator of Eq. (10) is much larger than the remaining terms. The

*k*

_{1}

*l*term in the denominator is also negligible comparing to the

*k*

_{1}

*L*term. Hence

*f*(

*θ*) ~ (

*k*

_{1}

*L*)

^{-2}for large

*k*

_{1}

*L*in Eq. (10). It is also noteworthy to mention that Eq. (10) is an infinitesimal of higher order than the previous result of Eq. (9

9. 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*(

*θ*) ~ (

*k*

^{1}

*d*)

^{-1}. In addition, Eq. (10) does not depend on the distance

*d*between particles. As a consequence the conditions of MUSSA will not be expressed in terms of the mean distance. It is not necessary to assume that the distance between an arbitrary pair of particles is roughly equal to the mean distance of all pairs of particles. Also, Eq. (10) has the same feature that

*f*(

*θ*) → 1 as

*θ*→ 0 as Eq.(9), if we expand cos(2

*k*

_{1}

*y*sin(

*θ*/2)) and sin(2

*k*

_{1}

*y*sin(

*θ*/2)) in terms of the small quantity 2

*k*

_{1}

*y*sin(

*θ*/2), where

*y*can be either

*L*or

*l*.

*f*(

*θ*)s defined by Eq. (10) and (9) as functions of

*θ*. In Fig. 1, ”

*k*

_{1}

*d*= 15” and ”

*k*

_{1}

*d*= 60” are for Eq.(9); and ”

*k*

_{1}

*L*= 15” and ”

*k*

_{1}

*L*= 60” are for Eq.(10). We set

*l*= 0 in Eq. (10) to have an equivalent comparison with Eq. (9). It is obvious Eq. (10) has similar oscillation features as Eq.(9). The amplitude profile of Eq. (10) is smaller than Eq.(9), which is a direct consequence of

*f*(

*θ*) ~ (

*k*

_{1}

*L*)

^{-2}for large

*k*

_{1}

*L*. Another fact is that the first zero of Eq. (10) is slightly larger than Eq. (9) for both the cases shown. If

*l*= 0, the first zero of Eq.(10) is the solution of

*x*= tan(

*x*), where

*x*= 2

*k*

_{1}

*L*sin(

*θ*/2). The first solution is: 4.49341 = 2

*k*

_{1}

*L*sin(

*θ*/2), which is:

*θ*=

*θ*

_{0}= 2 arcsin[π/(2

*k*

_{1}

*L*)] and 4.49341 > π, the first zero of Eq.(10) is always slightly larger than Eq. (9). To merge the forward interference peak into the diffraction peak of large single particles,

*θ*

_{0}≪ 4/(

*k*

_{1}

*a*) is required, where

*a*is the largest dimension of a single particle, which leads to:

*N*- 1 ~

*N*for large

*N*is used.

*s*̂ =

*r*̂, Eq. (1) shows that the amplitude scattering matrix of the small volume element is an incoherent summation of those of the constituent particles. From the optical theorem for extinction, the total extinction cross section or matrix of the small volume element also has this feature. However, the total scattering cross section of the small volume element is equal to the integration of Eq. (6) over solid angles, which contains an interference term. To make the MUSSA satisfies energy conservation, Mishchenko et al. (pp. 151, Ref. [17]) concludes that the integration of the interference term has to go to zero; namely,

**S**

_{i}(

*r*̂,

*s*̂]

_{kl}[

**S**

_{i′}(

*r*̂,

*s*̂]

^{*}

_{pq}in terms of Legendre functions:

*P*is the Legendre functions of order

_{n}*n*;

*nmax*is the maximum order of the expansion; and

*w*is the expansion coefficient. Substituting Eq.(16) into Eq. (15), we have:

_{n}*c*= ʃ

_{n}^{π}

_{0}

*P*(cos(

_{n}*θ*))

*f*(

*θ*)sin(

*θ*)

*dθ*up to

*n*= 20. For each order

*n*and

*c*decrease with increasing

_{n}*k*

_{1}

*L*. It shows that the energy contained in the interference term is quite small if

*k*

_{1}

*L*is large enough. We conclude that energy conservation is also ensured by Eq. (14).

## 4. Summary

## Acknowledgment

## References and links

1. | V. Kourganoff, |

2. | S. Chandrasekhar, |

3. | R. W. Preisendorfer, |

4. | G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. , |

5. | C. N. Adams and G. W. Kattawar, “Solutions of the equation of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transf. , |

6. | H. C. van de Hulst, |

7. | G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, |

8. | K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate method radiative trasfer in multiple scattering and emitting layered media,” Appl. Opt. , |

9. | 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. , |

10. | 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. , |

11. | 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. , |

12. | D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spec-trosc. Radiat. Transf. , |

13. | E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. , |

14. | K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transf.,” J. Atmos. Sci. , |

15. | K. N. Liou, |

16. | M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. |

17. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

18. | M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, “Single scattering by a small volume element,” J. Opt. Soc. Am. A |

19. | J. W. Hovenier, ”Measuring scattering matrices of small particles at optical wavelengths,” in |

20. | H. Volten, O. Mun͂oz, 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. |

21. | O. Mun͂oz, H. Volten, J. F. de Haan, W. Vassen, and J. W. Hovenier, ”Experimental determination of scattering matrices of randomly oriented flay ash and clay particles at 442 and 633 nm,” J. Geophys. Res. |

22. | J. W. Hovenier, H. Volten, O. Mun͂oz, 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. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(030.5620) Coherence and statistical optics : Radiative transfer

(290.4210) Scattering : Multiple scattering

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: June 5, 2007

Revised Manuscript: June 18, 2007

Manuscript Accepted: June 19, 2007

Published: June 22, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Peng-Wang Zhai, George W. Kattawar, and Ping Yang, "The far-field modified uncorrelated single-scattering approximation in light scattering by a small volume element," Opt. Express **15**, 8479-8485 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8479

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### References

- 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).
- G. N. Plass and G. W. Kattawar, "Monte Carlo calculations of light scattering from clouds," Appl. Opt. 7, 415- 419 (1968). [CrossRef] [PubMed]
- 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]
- 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).
- K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate method radiative trasfer in multiple scattering and emitting layered media," Appl. Opt., 27, 2502-2509 (1988). [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]
- 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]
- 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]
- K. F. Evans, "The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998). [CrossRef]
- K. N. Liou, An Introduction to Atmospheric Radiation, Second Edition, (Academic Press, New York, 2002).
- 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]
- M. I. Mishchenko, L. D. Travis and A. A. Lacis, Multiple Scattering of Light by Particles, (Cambridge University Press, Cambridge, UK, 2006).
- M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, "Single scattering by a small volume element," J. Opt. Soc. Am. A 21, 71-87 (2004) [CrossRef]
- 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.
- 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. 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]

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