## Monte Carlo study of pathlength distribution of polarized light in turbid media

Optics Express, Vol. 15, Issue 3, pp. 1348-1360 (2007)

http://dx.doi.org/10.1364/OE.15.001348

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### Abstract

Photon pathlength distributions as a function of the number of scattering events in cylindrical turbid samples are studied using a polarization-sensitive Monte Carlo model with linearly polarized light input. Sample scattering causes extensive depolarization, yielding a photon field comprised of polarized and depolarized sub-populations. It is found that the pathlength of polarization-preserving photons is distributed within a defined spatial range with strong angular dependence. This pathlength, averaged over the range, is 2-3X smaller than the one averaged over the widely-spread range of all (polarized + depolarized) collected photons. It is also demonstrated that changes in optical properties of the media affect the pathlength distributions.

© 2007 Optical Society of America

## 1. Introduction

1. G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. **36**,349–359 (1989). [CrossRef]

5. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” App. Opt. **28**,2331–2336 (1989). [CrossRef]

8. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. **10**,824-830 (1983). [CrossRef] [PubMed]

12. L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A **10**,1746–1752 (1993). [CrossRef]

_{L}[13–21

13. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**,213–220 (2004). [CrossRef] [PubMed]

13. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**,213–220 (2004). [CrossRef] [PubMed]

14. D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express **13**,148–163 (2005). [CrossRef] [PubMed]

## 2. Theory: Monte Carlo simulations in cylindrical geometry

13. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**,213–220 (2004). [CrossRef] [PubMed]

14. D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express **13**,148–163 (2005). [CrossRef] [PubMed]

15. B. Kaplan, G. Ledanois, and B. Drévillon, “Muller Matrix of dense polystyrene latex sphere suspensions: measurements and Monte Carlo simulation,” Appl. Opt. **40**,2769–2777 (2001). [CrossRef]

16. F. Jaillon and H. Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. **42**,3290–3296 (2003). [CrossRef] [PubMed]

14. D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express **13**,148–163 (2005). [CrossRef] [PubMed]

^{2}) and acceptance angle (48°) similar to our experimental system. The detected photons can be further binned based on the number of scattering events they undergo within the sample, which is the key investigation method employed in this study.

21. X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt. **11**,041105 (2006). [CrossRef] [PubMed]

^{2}detection area on the surface of the cylinder. The detection angle θ, which is the angle between forward direction and the normal of the surface detection element, varies from 0° to 180°. The vertical position of the surface detection element z ranges from -20 mm to +20 mm, with the signs indicating the relative position with respect to the horizontal incident plane. The samples simulate water suspensions of 4.1 μm diameter polystyrene microspheres. For most simulations (except Fig. 8), the scattering coefficient of the medium μ

_{s}is set to 100 cm

^{-1}. The scattering coefficient range is chosen to approximate typical turbidity of biological tissue. The refractive index of the scattering particles is 1.59. Calculated from Mie theory [22], the scattering anisotropy g (a measure of the amount of forward direction retained after a single scattering event, described by mean value of cos γ, where γ is scattering angle) is 0.88 and the scattering efficiency Q

_{sca}is 2.71. The absorption coefficient is set to 0.00326 cm

^{-1}for water. To reduce the statistical uncertainty of the values obtained from the Monte Carlo simulations, a large number (10

^{9}) of horizontally polarized photons are launched in the simulations.

_{N}, β

_{LN}and L

_{N}) and cumulative ones (I

_{total}, β

_{Ltotal}, L

_{total}and N

_{total}). I

_{N}is the summed number of photons in bin N, including both polarized and depolarized ones. β

_{LN}is the surviving linear polarization fraction of those photons, and L

_{N}is their average pathlength. I

_{total}, β

_{Ltotal}and L

_{total}are the corresponding parameters for the bin Total. N

_{total}is the average number of scattering events of the total collected photons,

## 3. Results and discussion

### 3.1 Photon intensity distributions

_{N}distribution. The photons (comprised of both polarized and depolarized sub-populations) are not evenly distributed in N, and the shape of the distribution is detection-direction dependent. In backward direction, θ=180°, the photon intensity peaks at N=1 (note the logarithmic scale on vertical axis), and decreases rapidly with N value. It indicates that single-scattering dominates the detected signal in the exact back-scattered geometry. For smaller angles, the intensity distributions are not monotonically decreasing, but instead show broad peaks. Both the minimum number of scattering events needed for photons to escape from a certain geometry and the most likely number of scattering events contributing to the detected signal can be estimated from the presented data. Taking θ=151° curve for example, photons have to be scattered at least 8 times before they can exit, while the photons which undergo ~25 scattering events have more chances to re-emit in this geometry than photons with other scattering histories.

_{total}as a function of detection angle is plotted in Fig. 2(b). It shows general increase of photon intensity with detection angle (i.e., detected intensity is higher in the backwards hemisphere), but lacks detailed description of photon scattering interactions, represented by I

_{N}.

^{2}+U

^{2}+V

^{2})

^{1/2}/I (I, Q, U and V are elements of the Stokes vector of detected photons) [22], has similar value to β

_{L}, for example, for θ=180° DOP

_{N=1}=0.22492493, β

_{L1}=0.22491533 and for θ=135° DOP

_{N=22}=0.55322760, β

_{L22}=0.55322686. It implies that the low β

_{LN}value (< 100%) is the result of true depolarization, and not due to transformation of the polarization form (linearly polarized to elliptically/circularly polarized light). However, the low β

_{LN}value at 180°, N=1 (~ 23%) is surprising in that single scattering by identical particles with spherical symmetry should not decrease the degree of polarization of 100% polarized incident light, although the nature of polarization may be changed [2

2. M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. **35**,2234–2239 (1996). [CrossRef]

_{L1}value for θ=180° is dependent on sample (anisotropy factor g and scatterer radius r) and detection system (acceptance angle ψ). Different g values were obtained by varying r while keeping the scattering coefficient constant in the simulations. Note that g is not unique with scatterer size r. Therefore, samples with the same g may be simulated differently, as our Monte Carlo simulations use the full calculated Mie scattering phase function. For example, for the same g=0.88 as in our current simulations, but with smaller scatterer size [r=0.33 μm, versus 2.05 μm in Fig. 2(a)], β

_{L1}value reaches ~65% for ψ=48° [in contrast to ~23% in Fig. 3(a)]. For θ=180° and ψ approaching zero, β

_{L1}could be as high as ~94% for g=0.74 (r=0.2 μm). So the degree of polarization after a single scattering interaction appears to vary widely in magnitude, and to depend on medium properties and detection geometry. Further studies with circularly polarized light input and birefringent media properties are under way to check if these simulation results are seen under varying conditions, and to help interpret this phenomenon.

_{L}).

_{L}I

_{N}) in the total collected photon populations I

_{total}. Presented in Fig. 4(a) are plots of the ratio R

_{N}of indexed surviving polarized photon intensity to the total collected photon intensity as a function of number of scattering events. R

_{N}is calculated from

^{52}

_{N=17}R

_{N}(135°,z=0), is equal to the cumulative surviving linear polarization fraction β

_{Ltotal}[7.5% for this case, see numerical labels on Fig. 3(a)], which means all the surviving polarized photons detected at that angle are contained in bins N=17 to N=52. For θ=180°, polarized photons are distributed between N=1 and N=20. The peak intensity (at N=1) is about four times higher than the one at θ=135° (N=32), implying the potential advantage of backwards detection geometries for enhanced polarization preservation.

_{Ltotal}at different detection angles. The differential β

_{LN}and cumulative β

_{Ltotal}show opposite angular trends: β

_{LN}was shown [Fig. 3(b)] to decrease with detection angle θ, while β

_{Ltotal}increases with θ. The increase of β

_{Ltotal}with detection angle in Fig. 4(b) is the result of the larger portion of depolarized photons of high N values at smaller angles. For example, at θ=135° the total average number of scattering events N

_{total}=102, compared with N

_{total}=17 at θ=180°.

### 3.2 Photon pathlength distributions

_{s}(0.01 cm). However, the average pathlengths for smaller detection angles deviate from the main trend, especially at smaller N. For instance, the 121° curve starts converging at N~45, whereas the 158° merges at N~22. Sample geometry effects can explain this, as shown in Fig. 5(b). The flat parts of the curves in Fig. 5(b) correspond to the linear regions in Fig.5(a). The reference line is the direct distance between entrance and exit of the sample (OP in the top view of the sample cylinder) at all detection angles [see inset in Fig. 5(b)]. This reference line sets lower pathlength limit, in that only photons that are able to travel at least that pathlength can escape in a given direction. For example, N~20 scattering events are sufficient for photons to exit at θ=180°, 173° and 165° [flat region of N=20 curve in Fig. 5(b)], but are insufficient for the photons to escape at θ=135°. Only those that travel a longer pathlength after being scattered N~20 times can exit at that angle. However, these photons only account for a small fraction of total photon population at θ=135° [see 135° curve in Fig. 2(a)]. This geometry effect on pathlength is also seen when simulations are performed off the incident plane (z≠0, results not shown).

_{N}/N), which we shall term “the unit pathlength”. The ratio is not generally equal to the calculated mean free path 1/μ

_{s}=0.01 cm (see the reference line), but asymptotically approaches it at large N values. This behavior can be interpreted as follows. Mean free path of photons is a statistical estimation of the distance which photons travel between successive scattering events, equal to L

_{N}/(N+1). It can be approximated to L

_{N}/N when N is large enough so that the “1” in the denominator can be neglected, which can help explain some deviation at low (N≤10) values. With the increase of N, the effect is diminishing and L

_{N}/N gradually approaches 1/μ

_{s}. For the smaller detection angles (θ<173°), another more important mechanism is involved: the geometry induced longer pathlength as discussed above [see Fig. 5(b)]. It makes the difference between L

_{N}/N and 1/μ

_{s}larger than that at the high θ values. The smaller θ value curves approach 180° and 173° results when the geometry effect decreases at high N values.

_{N}is polarization-independent. For a given number of scattering events N, polarized and unpolarized photons, escaping from same detection direction traverse similar pathlengths (provided that N is small enough such that the polarized subpopulation still exits). Therefore, the pathlength distribution as a function of number of scattering events, discussed above in Fig. 5(a), can be used to represent pathlength distribution of polarized photons. Combining Fig. 4(a) with Fig. 5(a), we get distribution of polarized photons as a function of pathlength, shown in Fig. 7(a). It reveals that the polarized photons have a defined pathlength range for a given detection geometry. For example, at θ=135°, the pathlength of the polarized subpopulation ranges from ~ 0.33 cm to ~ 0.57 cm. Here, the photon population threshold for the upper and lower pathlength limits is set to 10% of the peak value. The range can be made smaller by raising the threshold. We define the average of this pathlength range as the characteristic polarization pathlength, L

_{CP}. That is,

_{total},

_{CP}(using 10% of peak value threshold) as a function of θ, and compares it with the total average pathlength L

_{total}. Both curves show similar trends, with pathlengths decreasing with detection angle θ. However, L

_{total}is 2-3X longer than L

_{CP}. If pathlength is used to estimate the penetration depth of injected photons, L

_{CP}will be more appropriate than L

_{total}to describe the interaction extent of polarization-preserving subpopulation.

### 3.3 Effects of optical properties on pathlength

_{s}results in longer pathlength because of increasing the mean free path. The anisotropy factor g doesn’t have as much influence on the pathlength as μ

_{s}, especially at backward directions (θ=180° and 173°). When the lumped optical property reduced scattering coefficient μ

_{s}’, which incorporates scattering coefficient μ

_{s}and anisotropy g by μ

_{s}’=μ

_{s}(1-g), increases from 6 cm

^{-1}to 25 cm

^{-1}, the pathlength varies in a non-monotonic way. In general, for any optical property set, the pathlength (and hence the sampling volume) decreases as the detection direction approaches the exact backward geometry (θ=180°).

## 4. Conclusion

_{s}~100 cm

^{-1}). The pathlength of the escaping photons spreads out due to multiple scattering. However, the pathlength of the surviving polarized photons is distributed in a more defined range, the average over which is 2-3X smaller than the average pathlength of all collected photons, which is dominated by the longer pathlengths of depolarized photon populations. Therefore, the average over this range, defined as characteristic polarization pathlength, is more descriptive as the pathlength of the polarization-preserving photon subpopulation. The strong angular dependence of the characteristic polarization pathlength suggests that penetration depth of injected photons, as estimated by pathlength distribution, can be controlled by adjusting the detection angle. The simulation results demonstrate that the change in scattering coefficient μ

_{s}and anisotropy factor g of turbid media impacts the photon pathlength distribution.

## Acknowledgments

## References and links

1. | G. H. Weiss, R. Nossal, and R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. |

2. | M. Dogariu and T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. |

3. | Y. Tsuchiya, “Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law,” Phys. Med. Biol. |

4. | Y. Liu, Y. L. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express |

5. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” App. Opt. |

6. | S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. |

7. | G. Popescu and A. Dogariu, “Dynamic light scattering in subdiffusive regimes,” Appl. Opt. |

8. | B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. |

9. | S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues-I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. |

10. | X. Wang, G. Yao, and L. V. Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. |

11. | X. Wang, L. V. Wang, C. Sun, and C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. |

12. | L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A |

13. | D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. |

14. | D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express |

15. | B. Kaplan, G. Ledanois, and B. Drévillon, “Muller Matrix of dense polystyrene latex sphere suspensions: measurements and Monte Carlo simulation,” Appl. Opt. |

16. | F. Jaillon and H. Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. |

17. | J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. |

18. | R. J. McNichols and G. L. Coté, “Optical glucose sensing in biological fluids: an overview,” J. Biomed. Opt. |

19. | M. Mehrübeoǧlu, N. Kehtarnavaz, S. Rastegar, and L. V. Wang, “Effect of molecular concentrations in tissue-simulating phantoms on images obtained using diffuse reflectance polarimetry,” Opt. Express |

20. | G. L. Coté, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. |

21. | X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt. |

22. | C. F. Bohren and D. R. Huffman, |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(170.5280) Medical optics and biotechnology : Photon migration

(170.7050) Medical optics and biotechnology : Turbid media

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: October 27, 2006

Revised Manuscript: January 16, 2007

Manuscript Accepted: January 17, 2007

Published: February 5, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Xinxin Guo, Michael F. G. Wood, and Alex Vitkin, "Monte Carlo study of pathlength distribution of polarized light in turbid media," Opt. Express **15**, 1348-1360 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-1348

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### References

- G. H. Weiss, R. Nossal, and R. F. Bonner, "Statistics of penetration depth of photons re-emitted from irradiated tissue," J. Mod. Opt. 36, 349-359 (1989). [CrossRef]
- M. Dogariu and T. Asakura, "Photon pathlength distribution from polarized backscattering in random media," Opt. Eng. 35, 2234-2239 (1996). [CrossRef]
- Y. Tsuchiya, "Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law," Phys. Med. Biol. 46, 2067-2084 (2001). [CrossRef] [PubMed]
- Y. Liu, Y. L. Kim, X. Li, and V. Backman, "Investigation of depth selectivity of polarization gating for tissue characterization," Opt. Express 13, 601-611 (2005). [CrossRef] [PubMed]
- M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989). [CrossRef]
- S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992). [CrossRef] [PubMed]
- G. Popescu and A. Dogariu, "Dynamic light scattering in subdiffusive regimes," Appl. Opt. 40, 4215-4221 (2001). [CrossRef]
- B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983). [CrossRef] [PubMed]
- S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, "Monte Carlo modeling of light propagation in highly scattering tissues-I: model predictions and comparison with diffusion theory," IEEE Trans. Biomed. Eng. 36, 1162-1168 (1989). [CrossRef] [PubMed]
- X. Wang, G. Yao, and L. V. Wang, "Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose," Appl. Opt. 41, 792-801 (2002). [CrossRef] [PubMed]
- X. Wang, L. V. Wang, C. Sun, and C. Yang, "Polarized light propagation through scattering media:time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003). [CrossRef] [PubMed]
- L. Wang and S. L. Jacques, "Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media," J. Opt. Soc. Am. A 10, 1746-1752 (1993). [CrossRef]
- D. Côté and I. A. Vitkin, "Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms," J. Biomed. Opt. 9, 213-220 (2004). [CrossRef] [PubMed]
- D. Côté and I. A. Vitkin, "Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations," Opt. Express 13, 148-163 (2005). [CrossRef] [PubMed]
- B. Kaplan, G. Ledanois, and B. Drévillon, "Muller Matrix of dense polystyrene latex sphere suspensions:measurements and Monte Carlo simulation," Appl. Opt. 40, 2769-2777 (2001). [CrossRef]
- F. Jaillon and H. Saint-Jalmes, "Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media," Appl. Opt. 42, 3290-3296 (2003). [CrossRef] [PubMed]
- J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, "Use of polarized light to discriminate short-path photons in a multiply scattering medium," Appl. Opt. 31, 6535-6546 (1992). [CrossRef] [PubMed]
- R. J. McNichols and G. L. Coté, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5-16 (2000). [CrossRef] [PubMed]
- M. Mehrübeoğlu, N. Kehtarnavaz, S. Rastegar, and L. V. Wang, "Effect of molecular concentrations in tissue-simulating phantoms on images obtained using diffuse reflectance polarimetry," Opt. Express 3, 286-297 (1998). [CrossRef] [PubMed]
- G. L. Coté, M. D. Fox, and R. B. Northrop, "Noninvasive optical polarimetric glucose sensing using a true phase measurement technique," IEEE Trans. Biomed. Eng. 39, 752-756 (1992). [CrossRef] [PubMed]
- X. Guo, M. F. G. Wood, and I. A. Vitkin, "Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter," J. Biomed. Opt. 11, 041105 (2006). [CrossRef] [PubMed]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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