## Effective scattering phase functions for the multiple scattering regime |

Optics Express, Vol. 19, Issue 5, pp. 4786-4794 (2011)

http://dx.doi.org/10.1364/OE.19.004786

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### Abstract

The propagation of light through turbid media is of fundamental interest in a number of areas of optical science including atmospheric and oceanographic science, astrophysics and medicine amongst many others. The angular distribution of photons after a single scattering event is determined by the scattering phase function of the material the light is passing through. However, in many instances photons experience multiple scattering events and there is currently no equivalent function to describe the resulting angular distribution of photons. Here we present simple analytic formulas that describe the angular distribution of photons after multiple scattering events, based only on knowledge of the single scattering albedo and the single scattering phase function.

© 2011 OSA

## 1. Introduction

6. M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. **235**, 1183–1192 (2007). [CrossRef]

7. N. Pfeiffer and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase functions,” Opt. Express **16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

*n*order scattering has

^{th}*g*. They also showed that the resulting

_{n}= g_{1}^{n}*n*order phase function takes the form of a HG phase function.

^{th}*θ*and

_{1}*θ*, a photon will propagate at an angle

_{2}*θ*to the original incident direction, whereand

_{n}*ψ*is the azimuthal angle of scattering for the second scattering event relative to the first. After two scattering events the photon will have a direction of propagation between

*θ*and

_{1}- θ_{2}*θ*(Fig. 1 ). Here we assume that the medium is infinite, isotropic and homogeneous.

_{1}+ θ_{2}7. N. Pfeiffer and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase functions,” Opt. Express **16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

7. N. Pfeiffer and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase functions,” Opt. Express **16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

^{th}order scattering, as determined from Monte Carlo simulations. Furthermore, we define an effective phase function for multiple scattering which describes the angular distribution of photons after all orders of scattering have been simulated (limited only by the number of photons used in the Monte Carlo simulations). We believe this name represents a logical extension of the classical, single scattering phase function.

## 2. Methods

8. P. Flatau, J. Piskozub, and J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Express **5**(5), 120–124 (1999). [CrossRef] [PubMed]

9. Z. Otremba and J. Piskozub, “Modelling the bidirectional reflectance distribution function (BRDF) of seawater polluted by an oil film,” Opt. Express **12**(8), 1671–1676 (2004). [CrossRef] [PubMed]

10. J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. **39**(12), 1872–1878 (2000). [CrossRef]

11. D. McKee, J. Piskozub, and I. Brown, “Scattering error corrections for in situ absorption and attenuation measurements,” Opt. Express **16**(24), 19480–19492 (2008). [CrossRef] [PubMed]

12. G. Marsaglia, *The Diehard Battery of Tests of Randomness* (1995), http://www.stat.fsu.edu/pub/diehard/.

^{127}(KISS by George Marsaglia) was used. In the setup used for this study, the modelled light source was a parallel light beam placed in a virtually unbounded space (no virtual photons reached the 3D model walls which were several thousands of optical depths from the light source). The beam is assumed to be vertical for simplicity. For each run, one billion (10

^{9}) virtual photons were used. The zenith angle of the virtual photon and the scattering order were recorded for each scattering event. This made possible the estimation of phase functions for each scattering order as well as the effective phase function (the latter using all scattering events).

13. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. **93**, 70–83 (1941). [CrossRef]

*g*is the mean cosine of the scattering phase function. Mobley [14] showed that

*g*= 0.924 gave a best-fit to the ‘particle phase function’ derived from Petzold’s [15] measurements of the scattering phase function of seawater. Fournier and Forand [16

16. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE **2258**, 194–201 (1994). [CrossRef]

17. J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE **3761**, 34–44 (1999). [CrossRef]

*μ*) and a real refractive index,

*n*,whereBy assuming a linear relationship between

_{r}*n*and

*μ*, and integrating Eq. (4), Mobley et al [18

18. I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. **182**, 012035 (2009). [CrossRef]

*B*=

_{p}*b*, as input. This formulation has been used widely in ocean optics and is adopted here. For the purpose of this paper the key difference between HG and FF scattering phase functions is that the former are defined on the asymmetry parameter, g, while the latter are not. This difference facilitates testing the general applicability of the Pfeiffer and Chapman [7

_{bp}/ b_{p}**16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

## 3. Results

_{1}< 0.99). In every case the angular distribution of

*n*order scattering was well described by a HG function with

^{th}*g*, closely reproducing the results of Pfeiffer and Chapman [7

_{n}= g_{1}^{n}**16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

16. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE **2258**, 194–201 (1994). [CrossRef]

17. J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE **3761**, 34–44 (1999). [CrossRef]

*g*is true for all orders of scattering up to and including 50th order scattering (the result is only limited by the number of photons used in the simulation). We note that this result is consistent with taking the mean of both sides of Eq. (2) over all values of

_{n}= g_{1}^{n}*ψ*.

*θ*is given by

_{1}*n*order scattering can be calculated from the single scattering phase function by iterating through each order of scattering

^{th}*n,*usingwith

*θ*determined for each combination of

_{n}*θ*,

_{1}*θ*,

_{2}*ψ*using Eq. (2). The validity of this parameterisation is demonstrated in Fig. 5a where

*n*order HG phase functions are accurately reproduced up to order 50 using Eq. (6) and using the result of Pfeiffer and Chapman [7

^{th}**16**(18), 13637–13642 (2008). [CrossRef] [PubMed]

*n*order FF phase functions obtained from Monte Carlo simulations up to at least 50th order.

^{th}*n*order phase functions using Eq. (7) Figure 6 shows effective multiple scattering phase functions obtained from Monte Carlo simulations for a single FF scattering phase function and a range of scattering albedo values, match effective scattering phase functions calculated using Eq. (7) and

^{th}*n*order phase functions obtained from Monte Carlo simulations.

^{th}## 4. Conclusions

18. I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. **182**, 012035 (2009). [CrossRef]

*g*would be well placed. It is anticipated that these results will facilitate new insights into the effect of multiple scattering on a variety of applications such as generation of remote sensing signals, impact on optical measurements and interpretation of scattering signals from turbid media.

## Acknowledgments

## References and links

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

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

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

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

5. | A. A. Kokhanovsky, |

6. | M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. |

7. | N. Pfeiffer and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase functions,” Opt. Express |

8. | P. Flatau, J. Piskozub, and J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Express |

9. | Z. Otremba and J. Piskozub, “Modelling the bidirectional reflectance distribution function (BRDF) of seawater polluted by an oil film,” Opt. Express |

10. | J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. |

11. | D. McKee, J. Piskozub, and I. Brown, “Scattering error corrections for in situ absorption and attenuation measurements,” Opt. Express |

12. | G. Marsaglia, |

13. | L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. |

14. | C. D. Mobley, |

15. | T. J. Petzold, T. J. SIO Ref. 72–78, Scripps Institute of Oceanography (U. California, 1972). |

16. | G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE |

17. | J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE |

18. | I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: January 26, 2011

Manuscript Accepted: February 16, 2011

Published: February 25, 2011

**Virtual Issues**

Vol. 6, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Jacek Piskozub and David McKee, "Effective scattering phase functions for the multiple scattering regime," Opt. Express **19**, 4786-4794 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4786

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### References

- H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
- H. C. van de Hulst, Multiple Light Scattering, Vol. 1 (Academic Press, 1980).
- H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic Press, 1980).
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge Univ. Press, 2006).
- A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
- M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. 235, 1183–1192 (2007). [CrossRef]
- N. Pfeiffer and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase functions,” Opt. Express 16(18), 13637–13642 (2008). [CrossRef] [PubMed]
- P. Flatau, J. Piskozub, and J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Express 5(5), 120–124 (1999). [CrossRef] [PubMed]
- Z. Otremba and J. Piskozub, “Modelling the bidirectional reflectance distribution function (BRDF) of seawater polluted by an oil film,” Opt. Express 12(8), 1671–1676 (2004). [CrossRef] [PubMed]
- J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39(12), 1872–1878 (2000). [CrossRef]
- D. McKee, J. Piskozub, and I. Brown, “Scattering error corrections for in situ absorption and attenuation measurements,” Opt. Express 16(24), 19480–19492 (2008). [CrossRef] [PubMed]
- G. Marsaglia, The Diehard Battery of Tests of Randomness (1995), http://www.stat.fsu.edu/pub/diehard/ .
- L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]
- C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters. (Academic, 1994).
- T. J. Petzold, T. J. SIO Ref. 72–78, Scripps Institute of Oceanography (U. California, 1972).
- G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994). [CrossRef]
- J. L. Forand and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,” Proc. SPIE 3761, 34–44 (1999). [CrossRef]
- I. Turcu and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J. Phys.: Conf. Ser. 182, 012035 (2009). [CrossRef]

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