The influence of the microscopic characteristics of a random medium on incoherent light transport
Optics Express, Vol. 15, Issue 6, pp. 2847-2872 (2007)
http://dx.doi.org/10.1364/OE.15.002847
Acrobat PDF (1116 KB)
Abstract
In this paper the influence of the microscopic characteristics of a random medium on non polarized, incoherent steady light transport (ISLT) is investigated. After close examination of current diffusion models, the source term in those models is modified, allowing a complete modelling of experimental and simulated radial dependance of backscattered and transmitted intensities for media thicknesses larger than the transport length. The new model only presents an additional source with respect to the elementary point source model. Thanks to more than 200 Monte-Carlo simulations, this parameter is correlated to the backscattering part of the Mie phase function. Incoherent Steady Light Transport measurements on two industrial emulsions at various volume fractions validate experimentally this correlation. This establishes a complete link between the microscopic characteristic of the random medium (size, optical indexes and volume fraction) and its macroscopic description in terms of diffusion and source parameters, openning new potential applications of the ISLT technique to, for example, the evaluation of the particles interaction potential in concentrated suspensions.
© 2007 Optical Society of America
1. Introduction
1. L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef]
2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]
3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
6. J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef]
9. S. Arridge, “Topical review: optical tomography in medical imaging,” Inv. Probl. 15,R41 (1999). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
14. X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Rand. Media 9,489 (1999). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
15. X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments.” J. Biomedical Opt. 8,608–617 (2003). [CrossRef]
16. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39,1580 (2000). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]
2. Experimental Setup and Methods
2.1. Experimental Setup
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
2.2. Materials
17. C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, “Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions.” Langmuir 17,5184–5188 (2001). [CrossRef]
18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]
18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]
3. Diffusion theory: The double source model
3.1. Radiative transfer basic hypothesis
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
3.1.1. Incoherent light transport
3.1.2. Non polarized light transport
6. J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef]
1. L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
3.2. Scalar radiative transfer
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
3.3. The diffuse source problem for a collimated beam
3.3.1. Intensity separation hypothesis: the exponential model
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
3.3.2. Source location hypothesis: the Haskell Model
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
3.3.3. The double source model
3.4. Boundary conditions
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
20. H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs,” Phys. Rev. Lett. 79,4369 (1997). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
3.5. Solution method
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
3.6. Comparison between the models
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
4. Monte Carlo simulations
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
22. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1,441–453 (1997). [CrossRef] [PubMed]
13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
23. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J 93,70 (1941). [CrossRef]
4.1. Parameters
15. X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments.” J. Biomedical Opt. 8,608–617 (2003). [CrossRef]
16. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39,1580 (2000). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
4.2. Monte Carlo Algorithm
4.3. Phase functions
23. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J 93,70 (1941). [CrossRef]
11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).
18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]
5. Results
5.1. A Universal Curve?
3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]
5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
5.2. Fit of experimental data: first validation of the double source model
5.2.1. Fit of the low r/l_{tr} region
5.2.2. Accuracy of the model for a finite size media
4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]
Backscattering.
Transmission.
5.3. Numerical simulations: an exploration of the microscopic parameter space
5.3.1. Determination of α: fit of the radial intensity distributions
5.3.2. Source relative strength: a measure of the backscattering phase function
Correlation with the anisotropy parameter g.
Correlation with the size parameter x.
Correlation with backscattering phase function P(π).
5.3.3. Conclusions: determination of P(π) and influence of the phase function
6. Validation: influence of particle pair correlations
6.1. Background: Influence of particle pair correlations on the scattering properties of a random medium
6.1.1. Influence on the phase function
24. L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]
24. L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]
6.1.2. Influence on the transport parameters
25. L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, “Photonic Properties of Strongly Correlated Colloidal Liquids,” Phys. Rev. Letters 93,073 903 (2004). [CrossRef]
6.2. Evolution of the experimental transport length
6.3. Validation of the α- p(π) correlation
12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]
7. Conclusion
Appendix A: Calulation of the double source model
References and links
1. | L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef] |
2. | R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef] |
3. | M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef] |
4. | D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef] |
5. | R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef] |
6. | J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef] |
7. | A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997). |
8. | A. Polishchuk, T. Dolne, F. Liu, and R. Alfana, “Averaged and most probable photon paths in random media,” J. Opt. Soc. Am. A 22,430 (1997). |
9. | S. Arridge, “Topical review: optical tomography in medical imaging,” Inv. Probl. 15,R41 (1999). [CrossRef] |
10. | J. Paasschens, On the transmission of light through random media, Ph.D. thesis, Leiden University, Netherlands (1997). |
11. | S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988). |
12. | C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef] |
13. | A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef] |
14. | X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Rand. Media 9,489 (1999). [CrossRef] |
15. | X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments.” J. Biomedical Opt. 8,608–617 (2003). [CrossRef] |
16. | S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39,1580 (2000). [CrossRef] |
17. | C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, “Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions.” Langmuir 17,5184–5188 (2001). [CrossRef] |
18. | F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef] |
19. | P. E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. 55,2696–2699 (1985). [CrossRef] [PubMed] |
20. | H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs,” Phys. Rev. Lett. 79,4369 (1997). [CrossRef] |
21. | G. Popescu, C. Mujat, and A. Dogariu, “vidence of scattering anisotropy effects on boundary conditions of the diffusion equation,” Phys. Rev. E 61,04 8264 (2005). |
22. | A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1,441–453 (1997). [CrossRef] [PubMed] |
23. | L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J 93,70 (1941). [CrossRef] |
24. | L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef] |
25. | L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, “Photonic Properties of Strongly Correlated Colloidal Liquids,” Phys. Rev. Letters 93,073 903 (2004). [CrossRef] |
OCIS Codes
(110.7050) Imaging systems : Turbid media
(290.1350) Scattering : Backscattering
(290.1990) Scattering : Diffusion
(290.4210) Scattering : Multiple scattering
(290.5850) Scattering : Scattering, particles
(290.7050) Scattering : Turbid media
ToC Category:
Coherence and Statistical Optics
History
Original Manuscript: August 22, 2006
Revised Manuscript: December 1, 2006
Manuscript Accepted: December 26, 2006
Published: March 19, 2007
Virtual Issues
Vol. 2, Iss. 4 Virtual Journal for Biomedical Optics
Citation
F. Caton, C. Baravian, and J. Mougel, "The influence of the microscopic characteristics of a random medium on incoherent light transport," Opt. Express 15, 2847-2872 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-2847
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References
- L. Reynolds, C. C. Johnson, and A. Ishimaru, "Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters," Appl. Opt. 15, 2059 (1976). [CrossRef]
- R. Aronson, "Boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2532 (1995). [CrossRef]
- M. Dogariu and T. Asakura, "Reflectance properties of finite-size turbid media, "Waves Rand. Media 4, 429-439 (1994). [CrossRef]
- D. Durian and J. Rudnick, "Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source," J. Opt. Soc. Am. A 16, 837 (1999). [CrossRef]
- R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727 (1994). [CrossRef]
- J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, "Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics," Appl. Opt. 37, 3586 (1998). [CrossRef]
- A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).
- A. Polishchuk, T. Dolne, F. Liu, and R. Alfana, "Averaged and most probable photon paths in random media," J. Opt. Soc. Am. A 22, 430 (1997).
- S. Arridge, "Topical review: optical tomography in medical imaging," Inv. Probl. 15, R41 (1999). [CrossRef]
- J. Paasschens, On the transmission of light through random media, Ph.D. thesis, Leiden University, Netherlands (1997).
- S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html (1988).
- C. Baravian, F. Caton, and J. Dillet, "Steady light transport under flow: Characterization of evolving dense random media," Phys. Rev. E 71, 066 603 (2005). [CrossRef]
- A. Kienle and M. Patterson, "Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium," J. Opt. Soc. Am. A 14, 246 (1997). [CrossRef]
- X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, "Localization of the virtual point source used in the diffusion approximation to model a collimated beam source," Waves Rand. Media 9, 489 (1999). [CrossRef]
- X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: timeresolved Monte Carlo simulations and experiments," J. Biomedical Opt. 8, 608-617 (2003). [CrossRef]
- S. Bartel and A. H. Hielscher, "Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media," Appl. Opt. 39, 1580 (2000). [CrossRef]
- C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, "Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions," Langmuir 17, 5184-5188 (2001). [CrossRef]
- F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, "Monodisperse fragmentation in emulsions: Mechanisms and kinetics," Europhys. Lett. 61, 708-714 (2003). [CrossRef]
- P. E. Wolf and G. Maret, "Weak Localization and Coherent Backscattering of Photons in Disordered Media," Phys. Rev. Lett. 55, 2696-2699 (1985). [CrossRef] [PubMed]
- H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs," Phys. Rev. Lett. 79, 4369 (1997). [CrossRef]
- G. Popescu and C. Mujat and A. Dogariu, "vidence of scattering anisotropy effects on boundary conditions of the diffusion equation," Phys. Rev. E 61, 04 8264 (2005).
- A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, "Diffuse backscattering Mueller matrices of highly scattering media," Opt. Express 1, 441-453 (1997). [CrossRef] [PubMed]
- L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941). [CrossRef]
- L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]
- L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, "Photonic Properties of Strongly Correlated Colloidal Liquids," Phys. Rev. Lett. 93, 073903 (2004). [CrossRef]
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