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

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  1. 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]
  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," Appl. 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: timeresolved 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 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).
  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. Lett. 93, 073903 (2004).
    [CrossRef]

2003 (2)

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]

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]

2001 (1)

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]

2000 (1)

1999 (3)

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]

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]

S. Arridge, "Topical review: optical tomography in medical imaging," Inv. Probl. 15, R41 (1999).
[CrossRef]

1998 (1)

1997 (4)

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).

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]

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]

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]

1995 (1)

1994 (2)

1985 (1)

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]

1976 (1)

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]

1941 (1)

L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941).
[CrossRef]

Alfana, R.

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).

Aronson, R.

Arridge, S.

S. Arridge, "Topical review: optical tomography in medical imaging," Inv. Probl. 15, R41 (1999).
[CrossRef]

Asakura, T.

M. Dogariu and T. Asakura, "Reflectance properties of finite-size turbid media, "Waves Rand. Media 4, 429-439 (1994).
[CrossRef]

Baravian, C.

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]

Bartel, S.

Bibette, J.

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]

Bigio, I. J.

Cariou, J.

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]

Caton, F.

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]

de Vries, P.

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]

Dillet, J.

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]

Dogariu, A.

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).

Dogariu, M.

M. Dogariu and T. Asakura, "Reflectance properties of finite-size turbid media, "Waves Rand. Media 4, 429-439 (1994).
[CrossRef]

Dolne, T.

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).

Durian, D.

Eick, A.

Eick, A. A.

Freyer, J.

Freyer, J. P.

Gorria, P.

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]

Goubault, C.

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]

Greenstein, J.

L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941).
[CrossRef]

Guern, Y.

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]

Haskell, R.

Henyey, L.

L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941).
[CrossRef]

Hielscher, A.

Hielscher, A. H.

Intes, X.

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]

Ishimaru, A.

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]

Jeune, B. L.

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]

Johnson, C. C.

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]

Johnson, T.

Kienle, A.

Kopf, H. J.

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]

Lagendijk, A.

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]

Leal-Calderon, F.

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]

Liu, F.

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).

Lotrian, J.

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]

Maret, G.

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]

Mendez-Alcaraz, J.

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]

Mourant, J. R.

Mujat, C.

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).

Olea, D.

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]

Patterson, M.

Pays, K.

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]

Pellen, F.

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]

Polishchuk, A.

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).

Popescu, G.

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).

Reynolds, L.

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]

Rojas-Ochoa, L. F.

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]

Rudnick, J.

Saenz, J. J.

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]

Scheffold, F.

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]

Schmitt, V.

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]

Schurtenberger, P.

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]

Shen, D.

Sprik, R.

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]

Sun, C.-W.

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]

Svaasand, L.

Wang, L.

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]

Wang, X.

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]

Wolf, P. E.

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]

Yang, C.-C.

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]

App. Opt. (1)

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]

Appl. Opt. (2)

Astrophys. J (1)

L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941).
[CrossRef]

Europhys. Lett. (1)

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]

Inv. Probl. (1)

S. Arridge, "Topical review: optical tomography in medical imaging," Inv. Probl. 15, R41 (1999).
[CrossRef]

J. Biomedical Opt. (1)

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]

J. Opt. Soc. Am. A (5)

Langmuir (1)

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]

Opt. Express (1)

Phys. Rev. Lett. (2)

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]

Waves Rand. Media (2)

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]

M. Dogariu and T. Asakura, "Reflectance properties of finite-size turbid media, "Waves Rand. Media 4, 429-439 (1994).
[CrossRef]

Other (7)

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. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

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).

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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Figures (17)

Fig. 1.
Fig. 1.

Sketch of the experimental system.

Fig. 2.
Fig. 2.

Droplets radii distributions for the three emulsions.

Fig. 3.
Fig. 3.

Comparison of the different models. Contiuous line: Haskell model (eq. 7); Squares and triangles: double source model (eq. 8); Dashed and dotted lines: exponential source model (eq. 6)

Fig. 4.
Fig. 4.

Example of the differences between the 3 phase functions. The logarithm of the phase function is plotted as a function of the scattering angle. Black curve: monodisperse Mie calculation; red curve: polydisperse Mie calculation with σ= 0.2; blue curve: Henyey-Greenstein approximation.

Fig. 5.
Fig. 5.

Radial intensity distribution for several dilutions of emulsion EC-06a in dimensional units. The first 4 radial points correspond to the laser beam.

Fig. 6.
Fig. 6.

Scaled radial intensity distribution for a few dilutions of emulsion EC06-a in non dimensional units. The continuous lines are fits using the double source model (“DSF”). The exponential model (g=0 and g=0.9) and the Haskell model are also displayed.

Fig. 7.
Fig. 7.

Comparison of the double source and exponential models(g=0) to experiments for several depths for emulsion STT063 at a volume fraction of 63% (ltr = 0.25mm, α= 0.09 and β = 0).

Fig. 8.
Fig. 8.

Comparison of the double source model to experiments in the transmission geometry for several depths.

Fig. 9.
Fig. 9.

Intensity curves as a function of the size parameter x. Points corresponds to numerical simulations while lines correspond to the double-source model.

Fig. 10.
Fig. 10.

α plotted as a function of the anisotropy parameter g. left) monodisperse phase function. right)polydisperse phase function (σ= 0.2). HG: Henyey Greenstein phase function.

Fig. 11.
Fig. 11.

α plotted as a function of the size parameter x. left)monodisperse phase function. right)polydisperse phase function (σ = 0.2).HG corresponds to the Henyey Green-stein phase function.

Fig. 12.
Fig. 12.

α plotted as a function of the backscattering probability P(π). left)monodisperse phase function. right)polydisperse phase function (σ = 0.2).

Fig. 13.
Fig. 13.

Structure factors S(θ) for different size parameters at a volume fraction of ϕ= 0.3.

Fig. 14.
Fig. 14.

Dilute (Mie) and P-Y phase functions for 4 different sizes. Top left x=0.1; Bottom left: x=1; Top right, x=5; Bottom right, x=50.

Fig. 15.
Fig. 15.

Evolution with the volume fraction of: (Top Left) the phase function, (Top Right) p(π), (Bottom Left) the anisotropy parameter g, (Bottom Right) Relative scattering cross section. The size parameter is x=1 for all figures.

Fig. 16.
Fig. 16.

Evolution of the transport length ltr with the volume fraction ϕ for (left) the EC06-01a emulsion and (right) the STT046 emulsion.

Fig. 17.
Fig. 17.

Evolution of the transport length ltr with the volume fraction ϕ.

Tables (1)

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Table 1. Size distribution parmeters for the emulsions.

Equations (15)

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1 2 π σ d exp ( ( ln ( d ) ln ( d ¯ ) ) 2 2 σ 2 )
s ̂ I ( r , s ̂ ) = 1 l t I r s ̂ + 1 l s 4 π p ( s ̂ , s ̂ ) I ( r , s ̂ ) d Ω
I d ( r , s ̂ ) = 1 4 π ϕ ( r ) + 3 4 π F ( r ) . s ̂ + h . o . t
ϕ ( r ) = 4 π I d ( r , s ̂ ) d Ω and F ( r ) = 4 π I d ( r , s ̂ ) s ̂ d Ω
Δ ϕ ( r ) 3 1 l a l tr ϕ ( r ) = S ( r )
1 l tr = 1 l a + 1 g l s
S r z = S 0 ( g ) Exp ( z l t ) δ ( r = 0 )
R ( ρ ) * l tr 2 = 0.0398 ( 1 + ( r l tr ) 2 ) 3 2 + 0.0928 ( 5.444 + ( r l tr ) 2 ) 3 2 +
+ ( 0.0597 ( 1 + ( r l tr ) 2 ) 1 2 0.0597 ( 5.444 + ( r l tr ) 2 ) 1 2 )
S r z = [ αδ ( z = 0 ) + ( 1 α ) δ ( z = l tr ) ] δ ( r = 0 )
{ ϕ ( r ) ϕ ( r ) h 0 z = 0 in z = 0 ϕ ( r ) + ϕ ( r ) h d z = 0 in z = d
Δ ϕ ˜ ρ ξ β 2 ϕ ˜ ρ ξ = S ( ρ )
{ ϕ ˜ ρ ξ ϕ ˜ ρ ξ h 0 ξ = 0 in ξ = 0 ϕ ˜ ρ ξ + ϕ ˜ ρ ξ h τ ξ = 0 in ξ = τ
p HG ( θ ) = 1 2 1 g 2 ( 1 2 g cos ( θ ) + g 2 ) 3 2
p conc ( θ ) = p Mie ( θ ) S ( θ ) 4 π p Mie ( θ ) S ( θ ) d Ω

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