## Successive order, multiple scattering of two-term Henyey-Greenstein phase functions

Optics Express, Vol. 16, Issue 18, pp. 13637-13642 (2008)

http://dx.doi.org/10.1364/OE.16.013637

Acrobat PDF (239 KB)

### Abstract

An analytic solution to the problem of determining photon direction after successive scatterings in an infinite, homogeneous, isotropic medium, where each scattering event is in accordance with a two-term Henyey-Greenstein phase function, is presented and compared against Monte Carlo simulation results. The photon direction is described by a probability density function of the dot product of the initial direction and the direction after multiple scattering events, and it is found that such a probability density function can be represented as a weighted series of one-term Henyey-Greenstein phase functions.

© 2008 Optical Society of America

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

2. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. **41**, 1035–1050 (2002). [CrossRef] [PubMed]

4. N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE **5695**, 136–147 (2005). [CrossRef]

*θ*> followed a definite pattern as the scattering order

*n*was increased. Specifically, it was found that with a one-term Henyey-Greenstein (OTHG) phase function, the mean scattering cosine decreased by a multiplier of the anisotropy factor

*g*at each successive scattering order. Such behavior is consistent with reported features of the OTHG phase function [5

5. I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008). [CrossRef]

6. W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter **9**, 9083–9096 (1997). [CrossRef]

*n*

^{th}scattering can also be expanded using the Legendre polynomials. This implies that the probability density function of the scattering cosine after multiple scatterings, when the single particle scattering is defined by a linear combination of Henyey-Greenstein phase functions, is also a linear combination of Henyey-Greenstein phase functions.

## 2. Single-particle probability density distributions

### 2.1 One-term Henyey-Greenstein phase function

_{hg}(cos

*θ*) describes the probability density that a photon with direction

**Ω**

_{0}will scatter to direction

**Ω**with scattering cosine cos

*θ*[8

8. T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. **51**, N313–N322 (2006). [CrossRef]

5. I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008). [CrossRef]

## 2.2 Two-term Henyey-Greenstein phase function

_{hg}

_{2}(cos

*θ*) is comprised of two weighted OTHG phase functions, each with different anisotropy factors

*g*

*,*

_{α}*g*

*, such that*

_{β}*α*+

*β*=1.

## 3. Successive scattering probability density distributions

### 3.1 One-term successive scattering probability density function

*n*

^{th}order phase function describing the scattering cosine after

*n*scattering events for a OTHG phase function has been shown to be [5

5. I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008). [CrossRef]

_{hg},

_{n}(cos

*θ*,

*g*) has such a form for

*n*=0, 1 and ∞. For

*n*=0, the initial photon direction,

*g*

^{0}=1. For

*n*=1, the first scattering event,

*g*

^{1}=

*g*. For, n=∞, fully scattered,

*g*

^{∞}=0. It is evident that the mean scattering cosine after

*n*successive scatterings is

*g*

*.*

^{n}## 3.2 Two-term successive scattering probability density function

*n*

^{th}order phase function describing the scattering cosine for a TTHG phase function can be expanded as

_{hg}

_{2}(cos

*θ*) is comprised of the summation of two OTHG phase functions with weights

*α*and

*β*. At the first scattering event, a photon has probability

*α*of scattering according to p

_{hg}(cos

*θ*,

*g*

*) and probability*

_{α}*β*of scattering according to p

_{hg}(cos

*θ*,

*g*

*).*

_{β}_{hg}(cos

*θ*,

*g*

*) still has probability*

_{α}*α*of scattering according to p

_{hg}(cos

*θ*,

*g*

*) again. The probability that the photon will scatter twice in succession according to p*

_{α}_{hg}(cos

*θ*,

*g*

*) is*

_{α}*α*

^{2}. In like manner, the probability that a photon will scatter twice in succession according to p

_{hg}(cos

*θ*,

*g*

*) is*

_{β}*β*

^{2}. The probability that a photon will scatter first according to p

_{hg}(cos

*θ*,

*g*

*) and then according to p*

_{α}_{hg}(cos

*θ*,

*g*

*) is*

_{β}*αβ*. The probability density function p

_{hg2},

_{2}(cos

*θ*) is the weighted sum of the four individual contributions.

*g*

*=*

_{α}*g*

*=*

_{β}*g*and comparing to the OTHG phase function, it is easily shown that

*g*

*=*

_{αβ}*g*

*=*

_{βα}*g*

_{α}*g*

*.*

_{β}_{hg}

_{2},

*(cos*

_{n}*θ*) for

*n*=0 to 3, based on the above analysis.

*n*successive scatterings, the probability that a photon has scattered according to a particular sequence of applications of p

_{hg}(cos

*θ*,

*g*

*) and p*

_{α}_{hg}(cos

*θ*,

*g*

*) is*

_{β}*α*

^{a}*β*

*, where*

^{b}*a*+

*b*=

*n*, and that binomial coefficients specify the number of possible paths to get to order

*n*that contain only

*i*occurrences of

*α*branches. The final form of p

_{hg}

_{2},

*(cos*

_{n}*θ*) is thus:

_{hg}

_{2},

*(cos*

_{n}*θ*) is a weighted summation of

*n*+1 terms of OTHG phase functions. In a similar manner, the mean scattering cosine after

*n*successive scatterings is

## 4. Monte Carlo simulation

### 4.1 Method

_{hg2},

*(cos*

_{n}*θ*) and confirm correct behavior with p

_{hg},

*(cos*

_{n}*θ*), the Photon Transport Simulator software was modified to record photon direction information after each scattering event. Each photon was scattered up to 100 times according to either the OTHG or TTHG phase function. After each scattering event, a photon density map (a table of scattering cosine versus scattering order) was updated. Once all photons had been launched, the photon density map was post-processed to determine the scattering cosine probability density distribution as a function of scattering order. All simulations were run with 10

^{8}photons on a 3 GHz PC operating under Windows XP.

## 4.2 Photon scattering

*φ*is uniformly distributed between 0 and 2π, while the scattering angle

*θ*follows the p

_{hg}(cos

*θ*) probability distribution.

*θ*, p

_{hg}(cos

*θ*) was sampled with a uniform random variable

*ξ*∈[0..1].

_{hg}(cos

*θ*,

*g*

*) is*

_{α}*α*and according to p

_{hg}(cos

*θ*,

_{g}

*) is*

_{β}*β*=1-

*α*. Therefore, first a uniform random variable

*χ*∈[0..1] was used to determine which phase function to use [11], and then the selected phase function was sampled with a uniform random variable

*ξ*.

## 4.3 Recording of results

*n*and 804 bins of scattering cosine cos

*θ*, was created to record photon density with respect to scattering order and scattering cosine. The cos

*θ*bins were more finely grained as cos

*θ*→1 in order to capture photons sufficiently accurately when

*g*→1.

**Ω**with the photons initial direction

**Ω**

_{0}and the number of scattering events

*n*for that photon.

## 5. Results

### 5.1 One-term successive scattering

*g*showed good correlation between p

_{hg},

*(cos*

_{n}*θ*) and the recorded values. Representative results are shown in Figs. 1 and 2.

*g*=0.9 at scattering orders of

*n*=1..4. This anisotropy factor was selected as it is often used to represent many common biological media. It can be seen that the probability density distribution p

_{hg},

*(cos*

_{n}*θ*) from Eq. (5) closely matches the density distribution obtained from the simulation.

*n*=10, 20, .., 50, and has an expanded vertical scale. As expected, as

*n*increases, the scattering cosine more closely becomes uniform and p

_{hg},

*(cos*

_{n}*θ*) approaches a value of ½.

*r*between the Monte Carlo simulation data and p

_{hg},

*(cos*

_{n}*θ*) is greater than 0.9999 for

*n*≤10 and greater than 0.92 for

*n*≤50. Limited testing shows that simulating larger numbers of photons improves

*r*by reducing statistical fluctuations.

## 5.2 Two-term successive scattering

*α*,

*β*, and anisotropy factors

*g*

*,*

_{α}*g*

*showed good correlation between p*

_{β}_{hg}

_{2},

*(cos*

_{n}*θ*) and the recorded values. Representative results are shown in Figs. 3 and 4.

*α*=0.9,

*g*

*=0.9,*

_{α}*β*=0.1,

*g*

*=0 at scattering orders*

_{β}*n*=1..4 and 10. This phase function represents scattering through human dermis [9].

*α*=0.96,

*g*

*=0.9,*

_{α}*β*=0.04,

*g*

*=-0.26 at scattering orders*

_{β}*n*=1..4 and 10. This phase function has significant backscattering and matches the first three moments of the seawater-Petzold particle phase function [12

12. R. A. Leathers and N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. **36**, 8685–8698 (1997). [CrossRef]

_{hg2},

*(cos*

_{n}*θ*) from Eq. (12) closely matches the density distribution obtained from the simulations. The correlation coefficient r is greater than 0.995 for all results shown.

## 6. Discussion and conclusions

_{hg},

*(cos*

_{n}*θ*) and p

_{hg2},

*(cos*

_{n}*θ*) and demonstrate that the probability density function for the n

^{th}scattering cosine can be calculated directly, without resorting to numerical simulation.

^{th}order TTHG phase function can be exactly represented as a weighted sum of OTHG phase functions.

_{hg2},

*(cos*

_{n}*θ*) is equal to the scattering order

*n*plus one - as the complexity of the solution is linear with scattering order, it is possible to calculate p

_{hg2},

*(cos*

_{n}*θ*) rapidly, even as

*n*becomes large. This is an improvement over the 2

^{n}terms suggested by Eq. (6).

## References and links

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

2. | C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. |

3. | L. Wang and H. Wu, |

4. | N. Pfeiffer and G.H. Chapman, “Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium,” Proc. SPIE |

5. | I. Turcu and R. Bratfalean, “Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function,” J. Opt. A: Pure Appl. Opt.10, (2008). [CrossRef] |

6. | W. E. Vargas and G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys.: Condens. Matter |

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

8. | T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, “The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics,” Phys. Med. Biol. |

9. | S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular Dependence of HeNe laser Light Scattering by Human Dermis,” Laser Life Sci. |

10. | V. I. Haltrin
, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in |

11. | S. A. Prahl, |

12. | R. A. Leathers and N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. |

**OCIS Codes**

(290.4210) Scattering : Multiple scattering

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 12, 2008

Revised Manuscript: August 11, 2008

Manuscript Accepted: August 13, 2008

Published: August 20, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Nick Pfeiffer and Glenn H. Chapman, "Successive order, multiple scattering of two-term Henyey-Greenstein phase functions," Opt. Express **16**, 13637-13642 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13637

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### References

- L.G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941). [CrossRef]
- C. D. Mobley, L. K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002). [CrossRef] [PubMed]
- L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (John Wiley and Sons, 2007), Ch. 3.
- N. Pfeiffer and G. H. Chapman, "Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic Photon Fractions with Depth in an Isotropic Medium," Proc. SPIE 5695, 136-147 (2005). [CrossRef]
- I. Turcu and R. Bratfalean, "Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function," J. Opt. A: Pure Appl. Opt. 10, (2008). [CrossRef]
- W. E. Vargas and G. A. Niklasson, "Forward-scattering ratios and average pathlength parameter in radiative transfer models," J. Phys. Condens. Matter 9, 9083-9096 (1997). [CrossRef]
- H. C. van de Hulst, Multiple Light Scattering, Vol 1 (Academic, 1980).
- T. Binzoni1, T. S. Leung, A. H. Gandjbakhche, D. Rüfenacht, and D. T. Delpy, "The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics," Phys. Med. Biol. 51, N313-N322 (2006). [CrossRef]
- S. L. Jacques, C. A. Alter, and S. A. Prahl, "Angular Dependence of HeNe laser Light Scattering by Human Dermis," Laser Life Sci. 1, 309-333 (1987).
- V. I. Haltrin, "Two-term Henyey-Greenstein light scattering phase function for seawater," in IGARSS �??99: Proceeding of the International Geoscience and Remote Sensing Symposium, T. I. Stein, ed. (IEEE, 1999), pp. 1423-1425.
- S. A. Prahl, Light Transport in Tissue, App. A1, PhD Thesis, (University of Texas at Austin, 1988).
- R. A. Leathers and N. J. McCormick, "Ocean inherent optical property estimation from irradiances," Appl. Opt. 36, 8685-8698 (1997). [CrossRef]

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