## Laser light scattering in turbid media Part I: Experimental and simulated results for the spatial intensity distribution

Optics Express, Vol. 15, Issue 17, pp. 10649-10665 (2007)

http://dx.doi.org/10.1364/OE.15.010649

Acrobat PDF (1055 KB)

### Abstract

We investigate the scattering and multiple scattering of a typical laser beam (*λ*=800 nm) in the intermediate scattering regime. The turbid media used in this work are homogeneous solutions of monodisperse polystyrene spheres in distilled water. The two-dimensional distribution of light intensity is recorded experimentally, and calculated via Monte Carlo simulation for both forward and side scattering. The contribution of each scattering order to the total detected light intensity is quantified for a range of different scattering phase functions, optical depths, and detection acceptance angles. The Lorentz-Mie scattering phase function for individual particles is varied by using different sphere diameters (*D*=1 and 5 *µ*m). The optical depth of the turbid medium is varied (*OD*=2, 5, and 10) by employing different concentrations of polystyrene spheres. Detection angles of *θ _{a}
*=1.5° and 8.5° are considered. A novel approach which realistically models the experimental laser source is employed in this paper, and very good agreement between the experimental and simulated results is demonstrated. The data presented here can be of use to validate other modern Monte Carlo models, which generate high resolution light intensity distributions. Finally, an extrapolation of the Beer-Lambert law to multiple scattering is proposed based on the Monte Carlo calculation of the ballistic photon contribution to the total detected light intensity.

© 2007 Optical Society of America

## 1. Introduction

2. B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, “Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements,” Opt. Express **14**, 12473–12484 (2006). [CrossRef] [PubMed]

4. C. -K. Lee, C. -W. Sun, P. -L. Lee, H. -C. Lee, C. Yang, C. -P. Jiang, Y. -P. Tong, T. -C. Yeh, and J. -C. Hsieh, “Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling,” Opt. Express **13**, 8339–8348 (2005). [CrossRef] [PubMed]

9. D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express **10**, 159–170 (2002). [PubMed]

*t*is time,

*r⃗*is the vector position,

*s⃗*is the incident direction of propagation,

*f*(

*s⃗′*,

*s⃗*) is the scattering phase function derived from the appropriate scattering theory (e.g. Lorentz-Mie or Rayleigh-Gans theory),

*d*Ω′ is the solid angle spanning

*s⃗′*and

*c*is the speed of the light in the surrounding medium. The RTE can be summarized as follows: the change of radiance along a line of sight (Eq.(1) term (a)), corresponds to the loss of radiance due to the extinction of incident light (Eq.(1) term (b)) plus the amount of radiance that is scattered from all other directions

*s⃗′*, into the incident direction

*s⃗*(Eq.(1) term (c)) The total extinction represented by Eq.(1) term (b) equals the radiance lost due to scattering of the incident light in all other directions, minus the radiance that is absorbed at each light-droplet interaction. RTE is applicable for a wide range of turbid media; however the analytical solutions are only available in rather simple circumstances where assumptions and simplifications are introduced to reduce the equation to a more tractable form. Since there are no analytical solutions available to the transport equation in realistic cases, numerical techniques have been developed and utilized. The most versatile and widely used numerical solution is based on the statistical Monte Carlo (MC) technique [6].

7. L. Wang, S. L. Jacques, and L. Zheng, “MCML — Monte Carlo modelling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed . **47**, 131–146 (1995). [CrossRef] [PubMed]

9. D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express **10**, 159–170 (2002). [PubMed]

10. T. Girasole, C. Roze, B. Maheu, G. Grehan, and J. Menard, “Visibility distances in a foggy atmosphere: Comparisons between lighting installations by Monte Carlo simulation,” Int. Journal of Lighting Research and technology **30**, 29–36 (1998). [CrossRef]

12. I. R. Abubakirov and A. A. Gusev, “Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake,” Phys. Earth Planet. Inter . **64**, 52–67 (1990). [CrossRef]

12. I. R. Abubakirov and A. A. Gusev, “Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake,” Phys. Earth Planet. Inter . **64**, 52–67 (1990). [CrossRef]

13. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express **13**, 4420–4438 (2005). [CrossRef] [PubMed]

15. E. Berrocal, I. V. Meglinski, and M. C. Jermy, “New model for light propagation in highly inhomogeneous polydisperse turbid media with applications in spray diagnostics,” Opt. Express **13**, 9181–9195 (2005). [CrossRef] [PubMed]

16. E. Berrocal, D. Y. Churmakov, V. P. Romanov, M. C. Jermy, and I. V. Meglinski, “Crossed source/detector geometry for novel spray diagnostic: Monte Carlo and analytical results”, Appl. Opt . **44**, 2519–2529 (2005). [CrossRef] [PubMed]

*single scattering regime*the average number of scattering events ≤1 and the non-scattered ballistic photons are dominant. For off-axis detection, the single scattering approximation which assumes that photons have experienced only one scattering event prior arriving to the detector applies.

*intermediate single-to-multiple scattering regime*operates when the average number of scattering events is between 2 and 9. In this regime, one dominant scattering order is clearly defined. No approximation can be made under such a regime.

*multiple scattering regime*is defined when the average number of scattering events is greater or equal to 10. In this regime, the relative amount of each scattering order tends to be equal and no dominant scattering order is apparent. The diffusion approximation can be applied in this regime.

*l*. The mean free path length, which is the average distance between two light-particle interactions, is inversely proportional to the extinction coefficient, such that:

_{fp}*µ*=

_{e}*µ*+

_{s}*µ*. Here,

_{a}*µ*is the scattering coefficient and

_{s}*µ*is the absorption coefficient. The optical depth (

_{a}*OD*) can be calculated by dividing the total length,

*l*, traversed by a light beam, by the mean free path length:

*l*. The classification of each scattering regime can be performed from the value of the optical depth as presented in Table 1.

## 2. Description of the experimental setup

*d*=2.55 mm when

_{a}*θ*=8.5

_{a}*°*and

*d*=2.61 mm when

_{b}*θ*=1.5°.

_{b}## 3. Description of the Monte Carlo simulation

13. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express **13**, 4420–4438 (2005). [CrossRef] [PubMed]

16. E. Berrocal, D. Y. Churmakov, V. P. Romanov, M. C. Jermy, and I. V. Meglinski, “Crossed source/detector geometry for novel spray diagnostic: Monte Carlo and analytical results”, Appl. Opt . **44**, 2519–2529 (2005). [CrossRef] [PubMed]

*l*, before each light-particle interaction is derived from the Beer-Lambert law and is calculated as a function of the extinction coefficient

*µ*using a random number ξ uniformly distributed between 0 and 1:

_{e}*l*=-ln(ξ)/

*µ*. The extinction coefficient is deduced such that:

_{e}*µ*=

_{e}*N*·

*σ*where

_{e}*N*is the number density and

*σ*is the extinction cross section of the scattering particles. At each particle interaction, photons can be either absorbed or scattered. If the particles are non-absorbing, the extinction coefficient is then equal to the scattering coefficient and the albedo Λ(Λ=

_{e}*µ*/(

_{s}*µ*+

_{s}*µ*) is equal to one. In the MC technique, independent scattering is assumed requiring a distance between individual particles greater than three times the radius of the particles [19]. The MC model treats light as a collection of distinct entities, and as a consequence, interference phenomena are neglected in the simulation. This requires a random distribution of the scattering particles and the absence of periodic structures within the turbid medium. After a scattering event, the photon’s new direction is selected based on a random number and the Cumulative Probability Density Function (CPDF) calculated from the appropriate scattering phase function

_{a}*f*. The scattering phase function is defined as a function of the properties of the scattering particles. Lorentz-Mie, Rayleigh-Gans [19–20] or Henyey-Greenstein [21

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

*θ*defined between 0 and

_{s}*π*is calculated from the inverse CPDF of

*f*by:

*θ*=CPDF

_{s}^{-1}(ξ), where ξ is once again a random number generated between 0 and 1. The resolution of the sampled scattering angle,

*θ*, equals 0.1°. The azimuthal scattering angle,

_{s}*φ*, is uniformly distributed between 0 and 2

_{s}*π*such that the scattered radiation is assumed to be independent to the orientation of the scattering particle with respect to the direction of the incident radiation (valid for spherical particles). When a new direction of propagation is defined, the position of the next scattering point is calculated again and the process is repeated until the photon is either absorbed or exits the medium at a boundary. The optimum number of photons employed in the simulation depends on the desired accuracy and on the detector characteristics. The final direction of propagation, the final position, the number of scatters, and the total path length are calculated for each light entity. If the conditions of detection are met (e.g. photon lies within the field of view of the detector with its trajectory within the acceptance angle), such data are written to disk. The process is repeated for a sufficiently large amount of photons such that the distribution of the light intensity impinging on the detector is accurately represented.

*S*, is modeled from the experimental matrix of the incident laser beam as illustrated in Fig. 3. This matrix is obtained from the EM-CCD camera by imaging the surface of a cell containing only distilled water (without polystyrene spheres). Using this technique, the exact experimental source is considered and any irregularity in the laser beam profile is accounted for within the modeling, allowing a more realistic MC simulation. Computed photons are recorded at the exit position, provided detection conditions are met. This implies that the angle between the vector normal to the detection face (front face or side face) and the vector direction of the photons must be within the acceptance angle,

*θ*(see Fig. 3). The absorption of 800 nm light within the polystyrene sphere solution is negligible, so a real (non-absorbing) index of refraction is assumed for the scatterers, such that

_{a}*n*=1.578-0.0

*i*. The same assumption applies for the surrounding medium composed of distilled water, where

*n*=1.33-0.0

*i*. The resulting Lorentz-Mie scattering phase function of a single polystyrene sphere is illustrated in Fig. 3, for the two diameters

*D*=1 and 5

*µ*m. For each simulation, 3 billion photons are sent through the scattering medium and the resultant computational time is ~7.5 hours at optical depth

*OD*=2 and ~25 hours at

*OD*=10, when using a modern Intel(R) Core(TM) 2 CPU 6600 @ 2.40GHz processor. The relative speed of computation is then on the order of 9

*µ*s/photon (at

*OD*=2) and 30

*µ*s/photon (at

*OD*=10).

## 4. Results and comparison for the forward scattering detection

### 4.1. Polystyrene spheres of 1 µm diameter:

*µ*m diameter. The 2D intensity distribution is shown on the front face for the large detection acceptance angle

*θ*=8.5° in Fig. 5(a). By increasing the optical depth from

_{a}*OD*=2 to

*OD*=10, the light intensity transmitted through the scattering sample is reduced and the laser beam profile diffuses.

*OD*=2, the amount of light crossing the sample reaches a maximum value of 19% of the initial intensity. This result is found both experimentally and in the simulation. The Beer-Lambert law predicts a lower transmission of 13.5% in the same conditions. At

*OD*=5, the simulated results differ somewhat from the experimental results (the maximum light transmission equals 2.2% experimentally versus 2.8% for the simulation) and the differences with the Beer-Lambert calculation (

*I*/

_{f}*I*=0.67%) become significant. It is also seen that the laser beam starts to diffuse and its FWHM is 1.28 times wider than its original value. At

_{i}*OD*=10, the laser beam is now highly diffused with a FWHM equal to 1.85

*d*. The maximum transmission corresponds to 0.18% experimentally, 0.52% with the simulation and the Beer-Lambert law predicts only 0.0045%.

_{a}*θ*=1.5° (see Fig. 5 (b)), it is observed that the detected light intensity decreases while the incident laser beam profile tends to diffuse less. The FWHM is now 1.61 larger from its initial width and the light transmission equals ~0.025%. It is also apparent that at smaller detection acceptance angles, the number of photons detected is significantly reduced, resulting to a deterioration of the spatial resolution in the MC image. This is an artifact of the number of photons used in the computation. These results show that the amount of light intensity detected (experimentally and via simulation) differs considerably from that calculated by the Beer-Lambert formula. These divergences increase when increasing

_{a}*OD*and for large

*θ*. Discrepancies between the experimental and simulated results occur principally at

_{a}*OD*=10.

*P*(0) of the ballistic photons can be observed by comparing Fig. 6(a) and 6(b).

*θ*=8.5°

_{a}*P*(0) equals 65%, 18% and 0, 6% for the respective optical depths

*OD*=2,

*OD*=5 and

*OD*=10. By reducing the detection acceptance angle to

*θ*=1.5°, the amount of multiply scattered light detected is reduced and

_{a}*P*(0) increases significantly, reaching 98%, 87% and 16% (for

*OD*=2,

*OD*=5 and

*OD*=10 respectively). At the same time, the contribution of the high scattering orders (n≥4) increases smoothly with increasing

*OD*while the contribution of the low scattering orders (n≤3) is reduced abruptly.

### 4.2. Polystyrene spheres of 5 µm diameter:

*µ*m diameter. The intensity distribution profile is shown on the front face for the large detection acceptance angle

*θa*=8.5° in Fig. 7(a). At

*OD*=2 the light transmission reaches a maximum value of ~36%. This is observed both experimentally and via simulation.

*OD*=5, this maximum equals 7% experimentally and 9% via simulation. Finally, at

*OD*=10,

*I*/

_{f}*I*equals 0.9% experimentally and 1.4% via simulation.

_{i}*µ*m diameter. The FWHM of the laser beam intensity profile in this case is 1.02

*d*, 1.10

_{a}*d*and 1.25

_{a}*d*for the respective

_{a}*OD*=2,

*OD*=5 and

*OD*=10. For the small detection acceptance angle

*θ*=1.5° and for

_{a}*OD*=10, Fig. 7(b) shows both the FWHM 1.13 is times greater than the initial width, and the number of photons detected is reduced by a factor of ~10. However, contrary to the previous case with the 1

*µ*m particles, the resultant statistics are sufficient for good spatial resolution in the MC image. The scattering phase function of the 5

*µ*m polystyrene spheres is characterized by a significant forward scattering lobe, as shown in Fig. 3(b). This scattering feature is responsible for the increase of the light intensity scattered and multiply scattered in the forward direction. As a result, divergences with the Beer-Lambert calculations increase; whereas the broadening effect of the incident laser beam is reduced.

*θ*=8.5°, the contribution of higher scattering orders is important even for the low optical depth

_{a}*OD*=2. By decreasing

*θ*from 8.5° to 1.5°, the detection of ballistic photons is once again largely improved: From 31.6% to 80.1% at

_{a}*OD*=2, from 5.6% to 43.2% at

*OD*=5 and from 0.23% to 4.9% at

*OD*=10.

*OD*. Conversely, at the lower detection acceptance angle, the dominant scattering order is n=0 for low optical depth. At higher optical depths each scattering order tends to have an equal contribution.

*θ*=8.5°. Conversely, when

_{a}*θ*=1.5°, the detected intensity is smaller for the simulated data. These differences are due to several factors. First, in the experiment an amount of light is reflected by the walls of the glass sample within the turbid medium. These internal reflection phenomena are not calculated in the present simulation. Additionally, computed photons are recorded only if the angle between the vector normal to the detection face and the direction of the exiting photons is within the acceptance angle. These selection criteria approximate the light collection efficiency of a real system, but do not account for the spatial filtering effects of the actual detector and imaging optics. A ray-tracing model which simulates photon propagation from the exit plane, through the collection optics, to the detector is required for better agreement between the experimental and simulated results. Finally, errors in the measurement of the experimental optical depth (estimated to a maximum of 5%) contribute to the observed quantitative differences. Despite these factors, the experimental and simulated transmission intensities are of the same order of magnitude under all conditions. By plotting normalized intensity, as shown on the right side of Fig. 5 and Fig. 7, it is apparent that the simulated profile remains true to the experimental results in all cases.

_{a}### 4.3. Extrapolation of the Beer-Lambert transmission to large detection acceptance angle and high optical depth:

*I*, equals the sum of the non-scattered light intensity,

_{f}*I*, from the ballistic photons, plus the light intensity,

_{b}*I*, from scattered and multiply scattered photons:

_{ms}*k*corresponds to the contribution of scattered and multiply scattered light over the contribution of non-scattered light. As described previously, P(0) is the contribution of ballistic photons and the contribution of scattered and multiply scattered photons equals P(

*tot*)-P(0). For a normalized distribution where P(

*tot*)=1, the multiply scattered photon contribution becomes, 1-P(0), and the coefficient

*k*equals:

*I*along a line-of-sight as a function of the optical depth. In highly scattering environments, some photons which are scattered away from the optical axis undergo a succession of scattering events and are eventually redirected along the original path of the incident light. Experimentally, the amount of multiply-scattered light detected increases with the detection acceptance angle. The contribution of

_{i}*I*is not considered in the Beer-Lambert law, which applies only to the number of ballistic photons crossing the scattering sample, such that:

_{ms}*P*(0) which is the inverse probability density of the ballistic photon contribution.

*OD*=2,

*OD*=5 and

*OD*=10.

*P*(0) is plotted in Fig.9 as a function of

*OD*for the 1

*µ*m and 5

*µ*m sphere diameters and for the detections acceptance angles

*θ*=8.5° and

_{a}*θ*=1.5°. The simulation data shows that 1/

_{a}*P*(0) increases exponentially with

*α*·

*OD*. This increase becomes more apparent at large detection acceptance angles and for scattering particles which exhibit a dominant forward scattering lobe (larger particles). From these results, the Beer-Lambert relation can be modified and written such that the light intensity from multiply scattered photons is considered:

^{β}## 5. Results and comparison for the side scattering detection

*µ*m diameter (Fig. 10). The second set of comparison concerns the polystyrene spheres of 5

*µ*m diameter (Fig. 11). In both cases, the 2D light intensity distribution is shown on the side face for the large detection acceptance angle

*θ*=8.5° at optical depths

_{a}*OD*=2,

*OD*=5 and

*OD*=10 (Fig. 10(a) and Fig. 11(a)). The small detection acceptance angle

*θ*=1.5° is considered only for

_{a}*OD*=10 (Fig. 10(b) and Fig. 11(b)). The intensity profile along the vertical axis at Y=5 mm is also displayed in these figures.

*OD*=2 to

*OD*=10, the distance of photon penetration along the incident direction is reduced. Also, the circular cross-section of the incident laser beam becomes wider due to an increased number of scattering events within the cell. At equal

*OD*it is seen that the broadening of the laser beam operates more efficiently for the 1

*µ*m particles; whereas, for the 5

*µ*m particles, photons tend to penetrate further within the cell. Furthermore, the light intensity detected is larger for the 1

*µ*m particles than for the 5

*µ*m particles. These effect are due to the larger forward scattering characteristics of the 5

*µ*m spheres and to the larger amount of light scattered at ~90° for the 1

*µ*m particles (see the respective scattering phase functions illustrated in Fig.4).

*OD*, Fig. 12(a), (b), (c) and (d) show similar contributions. For the cases considered here, it is notable that the value of

*OD*consistently remains close or equal to the dominant scattering order. It is deduced from these results that the optical depth is the only parameter that influences the contribution of the scattering orders. The value of the detection acceptance is insignificant on both the distribution of light intensity by scattering order and the relative spatial light intensity distribution. A reduction of

*θ*leads only to a reduction of light intensity.

_{a}## 6. Conclusion

## Acknowledgments

## References and links

1. | R. M. Measures, Laser Remote Sensing: Fundamentals and applications (Krieger, Florida, 1992). |

2. | B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, “Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements,” Opt. Express |

3. | V. V. Tuchin (ed.), |

4. | C. -K. Lee, C. -W. Sun, P. -L. Lee, H. -C. Lee, C. Yang, C. -P. Jiang, Y. -P. Tong, T. -C. Yeh, and J. -C. Hsieh, “Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling,” Opt. Express |

5. | E. Berrocal, Multiple scattering of light in optical diagnostics of dense sprays and other complex turbid media (PhD Thesis, Cranfield University, 2006). |

6. | I. Sobol, The Monte Carlo method, (The University of Chicago Press, 1974). |

7. | L. Wang, S. L. Jacques, and L. Zheng, “MCML — Monte Carlo modelling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed . |

8. | D. Y. Churmakov, Multipurpose computational model for modern optical diagnostics and its biomedical applications (PhD Thesis, Cranfield University, 2005). |

9. | D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express |

10. | T. Girasole, C. Roze, B. Maheu, G. Grehan, and J. Menard, “Visibility distances in a foggy atmosphere: Comparisons between lighting installations by Monte Carlo simulation,” Int. Journal of Lighting Research and technology |

11. | J. Piskozub, D. Stramski, E. Terrill, and W. K. Melville, “Influence of Forward and Multiple Light Scatter on the Measurement of Beam Attenuation in Highly Scattering Marine Environments,” Appl. Opt . |

12. | I. R. Abubakirov and A. A. Gusev, “Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake,” Phys. Earth Planet. Inter . |

13. | J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express |

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

15. | E. Berrocal, I. V. Meglinski, and M. C. Jermy, “New model for light propagation in highly inhomogeneous polydisperse turbid media with applications in spray diagnostics,” Opt. Express |

16. | E. Berrocal, D. Y. Churmakov, V. P. Romanov, M. C. Jermy, and I. V. Meglinski, “Crossed source/detector geometry for novel spray diagnostic: Monte Carlo and analytical results”, Appl. Opt . |

17. | E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser light scattering in turbid media Part II: Spatial and temporal analysis of individual scattering orders via Monte Carlo simulation,” Opt. Express, manuscript in preparation (to be submitted). |

18. | X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang, and X.-H. Xin, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol . |

19. | H. C. van de Hulst, Light scattering by small particles (Dover, N.Y., 1981). |

20. | C. Bohren and D. Huffman, Absorption and scattering of light by small particles (Wiley, N.Y., 1983). |

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

**OCIS Codes**

(290.4020) Scattering : Mie theory

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: April 4, 2007

Revised Manuscript: July 19, 2007

Manuscript Accepted: July 20, 2007

Published: August 8, 2007

**Virtual Issues**

Vol. 2, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Edouard Berrocal, David L. Sedarsky, Megan E. Paciaroni, Igor V. Meglinski, and Mark A. Linne, "Laser light scattering in turbid media Part I: Experimental and simulated results for the spatial intensity distribution," Opt. Express **15**, 10649-10665 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-17-10649

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### References

- R. M. Measures, Laser Remote Sensing: Fundamentals and applications (Krieger, Florida, 1992).
- B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, "Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements," Opt. Express 14, 12473-12484 (2006). [CrossRef] [PubMed]
- V. V. Tuchin (ed.), Handbook of Optical Biomedical Diagnostics, (SPIE Press, Bellingham, WA, 2002).
- C. -K. Lee, C. -W. Sun, P. -L. Lee, H. -C. Lee, C. Yang, C. -P. Jiang, Y. -P. Tong, T. -C. Yeh, and J. -C. Hsieh, "Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling," Opt. Express 13, 8339-8348 (2005). [CrossRef] [PubMed]
- E. Berrocal, Multiple scattering of light in optical diagnostics of dense sprays and other complex turbid media (PhD Thesis, Cranfield University, 2006).
- I. Sobol, The Monte Carlo method, (The University of Chicago Press, 1974).
- L. Wang, S. L. Jacques, L. Zheng, "MCML - Monte Carlo modelling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995). [CrossRef] [PubMed]
- D. Y. Churmakov, Multipurpose computational model for modern optical diagnostics and its biomedical applications (PhD Thesis, Cranfield University, 2005).
- D. Boas, J. Culver, J. Stott, and A. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10, 159-170 (2002). [PubMed]
- T. Girasole, C. Roze, B. Maheu, G. Grehan and J. Menard, "Visibility distances in a foggy atmosphere: Comparisons between lighting installations by Monte Carlo simulation," Int. Journal of Lighting Research and technology 30, 29-36 (1998). [CrossRef]
- J. Piskozub, D. Stramski, E. Terrill, and W. K. Melville, "Influence of Forward and Multiple Light Scatter on the Measurement of Beam Attenuation in Highly Scattering Marine Environments," Appl. Opt. 43, 4723-4731 (2004). [CrossRef] [PubMed]
- I. R. Abubakirov, A. A. Gusev, "Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake," Phys. Earth Planet. Inter. 64, 52-67 (1990). [CrossRef]
- J. Ramella-Roman, S. Prahl, and S. Jacques, "Three Monte Carlo programs of polarized light transport into scattering media: part I," Opt. Express 13, 4420-4438 (2005). [CrossRef] [PubMed]
- D. Côté and I. 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]
- E. Berrocal, I. V. Meglinski and M. C. Jermy, "New model for light propagation in highly inhomogeneous polydisperse turbid media with applications in spray diagnostics," Opt. Express 13, 9181-9195 (2005). [CrossRef] [PubMed]
- E. Berrocal, D. Y. Churmakov, V. P. Romanov, M. C. Jermy and I. V. Meglinski, "Crossed source/detector geometry for novel spray diagnostic: Monte Carlo and analytical results", Appl. Opt. 44, 2519-2529 (2005). [CrossRef] [PubMed]
- E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski and M. A. Linne, "Laser light scattering in turbid media Part II: Spatial and temporal analysis of individual scattering orders via Monte Carlo simulation," Opt. Express, manuscript in preparation (to be submitted).
- X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003). [CrossRef]
- H. C. van de Hulst, Light scattering by small particles (Dover, N.Y., 1981).
- C. Bohren, D. Huffman, Absorption and scattering of light by small particles (Wiley, N.Y., 1983).
- L. G. Henyey, J. L. Greenstein, "Diffuse radiation in the galaxy", Astrophys. J. 93, 70-83 (1941). [CrossRef]

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