Abstract

We discuss the anisotropic scattering of unpolarized light in optically dense random media and the flux analysis of an incoherent backscattered spotlight. We present a classic statistical approach based on the photon-diffusion approximation and Monte Carlo simulations to describe the anisotropic propagation of ballistic and long-path photons in a semi-infinite random medium with internal reflections. An imagery technique with high gray-level resolution is used to measure the surface flux density in the incoherent backscattered spotlight. We investigated light scattering from homogeneous suspensions of nonspherical alumina particles in water. We analyzed the particle volume fraction and the particle-size dependence of the surface flux density to determine the transport mean free path and the optical properties of scatterers from scaling laws that account for short-path photons and internal reflections.

© 1998 Optical Society of America

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References

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  1. E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).
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    [CrossRef]
  3. A. Cutolo, I. Rendina, U. Arena, A. Marzocchella, L. Massimilla, “Optoelectronic technique for the characterization of high concentration gas–solid suspension,” Appl. Opt. 29, 1317–1322 (1990).
    [CrossRef] [PubMed]
  4. H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
    [CrossRef]
  5. D. J. Lisher, M. Y. Louge, “Optical fiber measurements of particle concentration in dense suspensions: calibration and simulation,” Appl. Opt. 31, 5106–5113 (1992).
    [CrossRef]
  6. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.
    [CrossRef]
  8. P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
    [CrossRef]
  9. G. Maret, P. E. Wolf, “Static and dynamic multiple scattering of light,” Physica A 157, 293–300 (1989).
    [CrossRef]
  10. D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.
  11. G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
    [CrossRef]
  12. D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
    [CrossRef] [PubMed]
  13. P. N. Den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
  15. A. H. Gandjbakhche, P. Mills, P. Snabre, “Light scattering technique for the study of orientation and deformation of red blood cells in a concentrated suspension,” Appl. Opt. 33, 1070–1078 (1994).
    [CrossRef] [PubMed]
  16. A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
    [CrossRef]
  17. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  18. M. Ospeck, S. Fraden, “Influence of reflecting boundaries and finite interfacial thickness on the coherent backscattering cone,” Phys. Rev. E 49, 4578–4589 (1994).
    [CrossRef]
  19. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  20. I. Freund, “Surface reflections and multiple scattering in one, two, and three dimensions,” J. Opt. Soc. Am. A 11, 3274–3283 (1994).
    [CrossRef]
  21. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
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  23. P. Snabre, “Agrégation des globules rouges en présence de dextrane. Rhéologie des suspensions concentrées de particules déformables,” state doctorate dissertation (University Paris VII, Paris, 1988).
  24. A. Arhaliass, “Diffusion multiple de la lumière par les suspensions. Analyse par imagerie de la tache de rétrodiffusion,” doctorate dissertation (Institut National Polytechnique de Toulouse, Toulouse, France, 1994).
  25. P. G. de Gennes, Scaling Concepts in Polymer Physics, 2nd ed. (Cornell U. Press, Ithaca, N.Y., 1987).
  26. P. Snabre, “Multiple light scattering in random systems. Spatial distribution of the diffuse light intensity,” submitted to Eur. Phys. J.
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    [CrossRef]
  28. B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed sphere systems,” J. Thermophys. 1, 63–68 (1987).
    [CrossRef]
  29. U. Riebel, U. Krauter, “Extinction of radiations in sterically interacting systems of monodisperse particles,” Part. Part. Syst. Charact. 11, 212–221 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1994 (4)

A. H. Gandjbakhche, P. Mills, P. Snabre, “Light scattering technique for the study of orientation and deformation of red blood cells in a concentrated suspension,” Appl. Opt. 33, 1070–1078 (1994).
[CrossRef] [PubMed]

M. Ospeck, S. Fraden, “Influence of reflecting boundaries and finite interfacial thickness on the coherent backscattering cone,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

I. Freund, “Surface reflections and multiple scattering in one, two, and three dimensions,” J. Opt. Soc. Am. A 11, 3274–3283 (1994).
[CrossRef]

U. Riebel, U. Krauter, “Extinction of radiations in sterically interacting systems of monodisperse particles,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

1993 (1)

1992 (3)

H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
[CrossRef]

D. J. Lisher, M. Y. Louge, “Optical fiber measurements of particle concentration in dense suspensions: calibration and simulation,” Appl. Opt. 31, 5106–5113 (1992).
[CrossRef]

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

1991 (2)

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1990 (2)

1989 (3)

G. Maret, P. E. Wolf, “Static and dynamic multiple scattering of light,” Physica A 157, 293–300 (1989).
[CrossRef]

E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).

A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

1988 (1)

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

1987 (2)

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

1983 (1)

L. Rizzuti, P. L. Yue, “The measurement of light transmission through an irradiated fluidized bed,” Chem. Eng. Sci. 38, 1241–1248 (1983).
[CrossRef]

1982 (1)

1979 (1)

1962 (1)

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

Alfano, R. R.

Arena, U.

Arhaliass, A.

A. Arhaliass, “Diffusion multiple de la lumière par les suspensions. Analyse par imagerie de la tache de rétrodiffusion,” doctorate dissertation (Institut National Polytechnique de Toulouse, Toulouse, France, 1994).

Bohren, C.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Chaikin, P. M.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Cutolo, A.

de Gennes, P. G.

P. G. de Gennes, Scaling Concepts in Polymer Physics, 2nd ed. (Cornell U. Press, Ithaca, N.Y., 1987).

de Vries, P.

A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Den Outer, P. N.

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

Durian, D. J.

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

Fraden, S.

M. Ospeck, S. Fraden, “Influence of reflecting boundaries and finite interfacial thickness on the coherent backscattering cone,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Freund, I.

I. Freund, “Surface reflections and multiple scattering in one, two, and three dimensions,” J. Opt. Soc. Am. A 11, 3274–3283 (1994).
[CrossRef]

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Gandjbakhche, A. H.

Hamazaki, H.

H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
[CrossRef]

Hartge, E. U.

E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Huffman, D. R.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Ishimaru, A.

Krauter, U.

U. Riebel, U. Krauter, “Extinction of radiations in sterically interacting systems of monodisperse particles,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Kuga, Y.

Lagendijk, A.

P. N. Den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Lisher, D. J.

Louge, M. Y.

Malitson, I. H.

Maret, G.

G. Maret, P. E. Wolf, “Static and dynamic multiple scattering of light,” Physica A 157, 293–300 (1989).
[CrossRef]

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Marzocchella, A.

Massimilla, L.

Maynard, R.

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

Mills, P.

Miyanami, K.

H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
[CrossRef]

Nieuwenhuizen, Th. M.

Ospeck, M.

M. Ospeck, S. Fraden, “Influence of reflecting boundaries and finite interfacial thickness on the coherent backscattering cone,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Rendina, I.

Rensner, D.

E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).

Riebel, U.

U. Riebel, U. Krauter, “Extinction of radiations in sterically interacting systems of monodisperse particles,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Rizzuti, L.

L. Rizzuti, P. L. Yue, “The measurement of light transmission through an irradiated fluidized bed,” Chem. Eng. Sci. 38, 1241–1248 (1983).
[CrossRef]

Snabre, P.

A. H. Gandjbakhche, P. Mills, P. Snabre, “Light scattering technique for the study of orientation and deformation of red blood cells in a concentrated suspension,” Appl. Opt. 33, 1070–1078 (1994).
[CrossRef] [PubMed]

P. Snabre, “Multiple light scattering in random systems. Spatial distribution of the diffuse light intensity,” submitted to Eur. Phys. J.

P. Snabre, “Agrégation des globules rouges en présence de dextrane. Rhéologie des suspensions concentrées de particules déformables,” state doctorate dissertation (University Paris VII, Paris, 1988).

Tang, G. C.

Tien, C. L.

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

Tojo, K.

H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
[CrossRef]

Twersky, V.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

Vreeker, R.

A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Werther, J.

E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).

Wolf, P. E.

G. Maret, P. E. Wolf, “Static and dynamic multiple scattering of light,” Physica A 157, 293–300 (1989).
[CrossRef]

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

Yoo, K. M.

Yue, P. L.

L. Rizzuti, P. L. Yue, “The measurement of light transmission through an irradiated fluidized bed,” Chem. Eng. Sci. 38, 1241–1248 (1983).
[CrossRef]

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Appl. Opt. (4)

Chem. Eng. Sci. (1)

L. Rizzuti, P. L. Yue, “The measurement of light transmission through an irradiated fluidized bed,” Chem. Eng. Sci. 38, 1241–1248 (1983).
[CrossRef]

J. Chem. Eng. Technol. (1)

E. U. Hartge, D. Rensner, J. Werther, “Measuring techniques for gas/solid fluidized bed reactors. Solids concentration and velocity patterns in circulating fluidized beds,” J. Chem. Eng. Technol. 61, 1781–1789 (1989).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Phys. (France) (1)

P. E. Wolf, G. Maret, E. Akkermans, R. Maynard, “Optical coherent backscattering by random media: an experimental study,” J. Phys. (France) 49, 63–75 (1988).
[CrossRef]

J. Thermophys. (1)

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

Part. Part. Syst. Charact. (1)

U. Riebel, U. Krauter, “Extinction of radiations in sterically interacting systems of monodisperse particles,” Part. Part. Syst. Charact. 11, 212–221 (1994).
[CrossRef]

Phys. Lett. A (1)

A. Lagendijk, R. Vreeker, P. de Vries, “Influence of internal reflections on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Phys. Rev. A (2)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Ospeck, S. Fraden, “Influence of reflecting boundaries and finite interfacial thickness on the coherent backscattering cone,” Phys. Rev. E 49, 4578–4589 (1994).
[CrossRef]

Physica A (1)

G. Maret, P. E. Wolf, “Static and dynamic multiple scattering of light,” Physica A 157, 293–300 (1989).
[CrossRef]

Powder Technol. (1)

H. Hamazaki, K. Tojo, K. Miyanami, “Measurement of local solids concentration in a suspension by an optical method,” Powder Technol. 70, 93–96 (1992).
[CrossRef]

Science (1)

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple light scattering probes of foam structure and dynamics,” Science 252, 686–688 (1991).
[CrossRef] [PubMed]

Z. Phys. B (1)

G. Maret, P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Other (9)

D. J. Pine, D. A. Weitz, G. Maret, P. E. Wolf, E. Herbolzheimer, P. M. Chaikin, Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed. (World Scientific, Singapore, 1989), pp. 312–372.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

P. Snabre, “Agrégation des globules rouges en présence de dextrane. Rhéologie des suspensions concentrées de particules déformables,” state doctorate dissertation (University Paris VII, Paris, 1988).

A. Arhaliass, “Diffusion multiple de la lumière par les suspensions. Analyse par imagerie de la tache de rétrodiffusion,” doctorate dissertation (Institut National Polytechnique de Toulouse, Toulouse, France, 1994).

P. G. de Gennes, Scaling Concepts in Polymer Physics, 2nd ed. (Cornell U. Press, Ithaca, N.Y., 1987).

P. Snabre, “Multiple light scattering in random systems. Spatial distribution of the diffuse light intensity,” submitted to Eur. Phys. J.

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Figures (13)

Fig. 1
Fig. 1

Probability P(ρ, N) of a photon exit at distance ρ from the emission point versus the reduced number N/ N* of scattering events in a nonabsorbing semi-infinite medium (K = 0). Symbols, Monte Carlo simulations; solid curves, calculated from Eq. (10). g = 0.9, p b = 0: ○, ρ = 2l*; □, ρ = 5l*; ●, ρ = 10l*; ■, ρ = 20l*.

Fig. 2
Fig. 2

Dimensionless surface flux F S (ρ)l* 2 in backscattered spot light versus the reduced distance ρ/l* for a semi-infinite medium. Monte Carlo simulations with g = 0.5, g f = 0.8, g b = -0.5, and p b = 0.23: ○, K = 10-6; □, K = 10-4; ●, K = 10-3; ■, K = 10-2; △, K = 0.05. Solid curves are calculated from Eq. (11).

Fig. 3
Fig. 3

Dimensionless surface flux F S (ρ)l* 2 in backscattered spot light versus reduced distance ρ/l* for a semi-infinite medium. Monte Carlo simulations with g = 0.9, g f = 0.95, g b = -0.5, and p b = 0.104: ○, K = 10-6; □, K = 10-4; ●, K = 10-3; ■, K = 10-2; △, K = 0.05. Solid curves are calculated from relations (14) and (15).

Fig. 4
Fig. 4

Influence of glass wall thickness D on the backscattered spot light for alumina particles in water (d = 112 μm, ϕ = 10%, shutter speed ω = 66 Hz, camera sight angle θ0 = 10°, glass wall thickness D = 5 mm or D = 12 mm).

Fig. 5
Fig. 5

Influence of wall thickness D on dimensionless surface flux F S (ρ)l* 2 in backscattered spot light for a nonabsorbing semi-infinite medium limited by glass walls in air. Monte Carlo simulations with g = 0, g f = 0.5, g b = -0.5, p b = 0.5, n = 1.33, and n w = 1.5: ●, D = 0; ○, D = 10l*; □, D = 50l*. The solid curve is calculated from relations (14) and (15).

Fig. 6
Fig. 6

Normalized angular distribution Q(θ) of the diffuse scattered photons at the inner interface x = 0 in a semi-infinite and nonabsorbing medium. Monte Carlo simulations with g = 0.8, g b = -0.5, p b = 0.5, n = 1, n w = 1, and D = 0: ○, p b = 0; □, p b = 0.0714; ●, p b = 0.133. The solid curve represents the normalized angular distribution Q(θ) = (cos θ)3/2.

Fig. 7
Fig. 7

Reflection coefficient R(n) of photons at the outer interface x = -D versus refractive index n of fluid for a semi-infinite medium limited by glass walls in air. Monte Carlo simulations for isotropic scatterers ○; g = 0, g f = 0.5, g b = -0.5, p b = 0.5, n v = 1.5) or anisotropic scatterers ●; g = 0.8, g f = 0.9, g b = -0.5, p b = 0.0714, n w = 1.5). The solid curves are calculated from Eq. (16) and the angle-dependent Fresnel formula with either Q(θ) = cos θ for isotropic scattering or Q(θ) = (cos θ)3/2 for anisotropic scattering.

Fig. 8
Fig. 8

Schematic representation of the experimental setup for imagery visualization of the incoherent backscattered spot light.

Fig. 9
Fig. 9

Succession of four frames of the incoherent backscattered spotlight for alumina particles in water (d = 112 μm, ϕ = 20%, glass wall thickness D = 12 mm, camera sight angle θ0 = 10°) recorded at video frequency under variable shutter speed.

Fig. 10
Fig. 10

Radial gray-level distribution f(ρ, θ0) in backscattered spot light (solid curve) obtained from analysis of a succession of five frames acquired at variable shutter speeds for alumina particles in air (d = 90 μm, ϕ = 60%, glass wall thickness D = 5 mm, camera sight angle θ0 = 10°). Shutter speeds (in hertz) were □, ω = 61145; ○, ω = 7200; ¤, ω = 850; △, ω = 250; and dotted line, ω = 66.

Fig. 11
Fig. 11

Influence of camera sight angle θ0 on the normalized surface flux density F S *(ρ) determined from imagery experiments for a reference Kodak white suspension: ○, θ0 = 5°; □, θ0 = 30°; ●, θ0 = 70°.

Fig. 12
Fig. 12

a, Normalized surface flux density F S *(ρ) in backscattered spot light for alumina particles in water and a particle volume fraction ranging from 2% to 60% (d = 112 μm, camera sight angle θ0 = 10°, glass wall thickness D = 12 mm). b, Experimental data plotted in dimensionless form F S *(ρ/l*)l* 2 and theoretical reduced flux density F S (ρ/l*)l* 2 derived from relations (14) and (15) with g = 0.75 and K = 3 × 10-5.

Fig. 13
Fig. 13

a, Normalized surface flux density F S *(ρ) in backscattered spot light for alumina particles in water and a mean particle size ranging from 60 to 112 μm (ϕ = 20%, camera sight angle θ0 = 10°, glass wall thickness D = 12 mm). b, Experimental data plotted in dimensionless form F S *(ρ/l*)l* 2 and theoretical reduced flux density F S (ρ/l*)l* 2 derived from relations (14) and (15) with g = 0.75 and K = 3 × 10-5.

Equations (21)

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l = 1 S e = 2 σ = d 3 ϕ ,
g = cos   θ = 4 π   ψ θ ,   φ cos   θ d Ω 4 π   ψ θ ,   φ d Ω ,
g f = cos   θ 0 θ π / 2 = 0 π / 2   H θ cos   θ d θ 0 π / 2   H θ d θ , g b = cos   θ π / 2 θ π = π / 2 π   H θ cos   θ d θ π / 2 π   H θ d θ ,
g f β f = 0 β f   H f θ cos   θ d θ 0 β f   H f θ d θ = sin 2   β f 2 1 - cos   β f , g b β b = π - β b π   H b θ cos   θ d θ π - β b π   H b θ d θ = sin 2   β b 2 1 - cos   β b .
W i = W i - 1 + Y ¯ ¯ i U i ,     Y ¯ ¯ i = Y ¯ ¯ i - 1 X ¯ ¯ i ,
l * i   cos   θ i l l 1 - g .
P ρ ,   N d ρ = N - 3 / 2 exp - 3 ρ 2 ε ρ N 2 d ρ ,
P S ρ ,   N 2 π ρ d ρ d N = N / N * - 3 / 2 exp - 3 ρ 2 ε ρ N 2 2 π ρ d ρ d N N * ρ N 2 ,
ρ N = l * N / N * 1 / 2 ,     ε = 4 ,
P S ρ ,   N = 1 l * 2 N N * - 5 / 2 1 - g exp - 3 4 ρ l * 2 N * N .
P S ρ ,   N ,   K = 1 l * 2 N / N * - 5 / 2 1 - g × exp - 3 4 ρ l * 2 N * N exp - KN ,
F S ρ ,   g ,   K = N = 1   P S ρ ,   N ,   K d N N = 1 d N   0   2 π ρ P S ρ ,   N ,   K = 0 d ρ .
F S ρ ,   g ,   K exp - α KN c N 0 N / N * - 5 / 2 d N N * N / N * - 5 / 2 d N   0 ρ 0   2 π ρ d ρ ,
N c ρ / l * N * / K .
F S ρ ,   g ,   K l * π ρ 3 exp - α ρ / l * KN * ,     ρ > 4 l * .
F S ρ = γ ρ / l * 0.6 1 / ρ 2 × exp - α ρ l * KN * ,     ρ < 4 l * ,
R = 0 π / 2   r e θ w ,   n w 1 - r i θ ,   n ,   n w Q θ sin   θ d θ 0 π / 2   Q θ sin   θ d θ ,
F S ρ = Q r 0 θ 0 Q 0 θ 0 f ρ ,   θ 0 0   2 π ρ f r ρ ,   θ 0 d ρ ,
Q 0 θ 0 = Q θ tan   θ tan   θ 0 ,     sin   θ 0 = n   sin   θ .
F S * ρ = 1 1 - R Q r 0 θ 0 Q 0 θ 0 f ρ ,   θ 0 0   2 π ρ f r ρ ,   θ 0 d ρ .
F S ρ ρ / l * 0.6 1 / ρ 2 ϕ 1 - g 0.6 d 0.6 ρ 1.4 ,     ρ < 4 l * ,

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