Abstract

We demonstrate that the angular distribution of light diffusely backscattered from an opaque slab depends not only on sample boundary reflectivity but also, in contrast to previous results for diffusely transmitted light, on the anisotropy of the scattering events. This influence of scattering anisotropy is modeled within diffusion theory by a discontinuity in the photon concentration at the source point that is proportional to the average cosine of the scattering angle. The resulting predictions are compared with random walk simulations and with measurements of transmitted and backscattered intensity versus angle for glass frits and aqueous suspensions of polystyrene spheres held in air or immersed in a water bath. Predicted distributions capture the features of experimental and simulation data to within 1% for the best case of high reflectivity and weak anisotropy and to within 10% for the worst case of low reflectivity and strong anisotropy.

© 1997 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  3. M. U. Vera, D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  5. D. J. Durian“Influence of boundary reflection and refraction on diffusive photontransport,” Phys. Rev. E 50, 857–866 (1994).
    [CrossRef]
  6. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
    [CrossRef]
  7. D. J. Durian, “Two-stream theory of diffusing light spectroscopies,” Physica A 229, 218–235 (1996).
    [CrossRef]
  8. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strongabsorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [CrossRef]
  9. 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]
  10. A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in stronglyscattering media,” Phys. Lett. A 136, 81–88 (1989).
    [CrossRef]
  11. I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherentbackscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990).
    [CrossRef]
  12. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  13. R. C. Haskell, L. O. Svaasand, Tsay Tsong-Tseh, Feng Ti-Chen, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  14. I. Freund, “Surface reflections and multiple scattering in one, two, and threedimensions,” J. Opt. Soc. Am. A 11, 3274–3283 (1994).
    [CrossRef]
  15. T. M. Nieuwenhuizen, J. M. Luck, “Skin layer of diffusive media,” Phys. Rev. E 48, 569–588 (1993).
    [CrossRef]
  16. G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusiontheory,” Ann. Nuc. En. 22, 787–817 (1995).
    [CrossRef]
  17. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  18. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).
  19. P. A. Lemieux, M. U. Vera, D. J. Durian, “Telegrapher theory of diffusing-light spectroscopies including ballistic transport and scattering anisotropy,” preprint (1997) available from the authors at the address on the title page.
  20. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  21. D. Eliyahu, M. Rosenbluh, I. Freund, “Angular intensity and polarization dependence of diffuse transmissionthrough random media,” J. Opt. Soc. Am. A 10, 477–491 (1993).
    [CrossRef]
  22. E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
    [CrossRef]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  24. D. E. Grey, American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–109.
  25. I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
    [CrossRef]

1997 (1)

1996 (3)

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

M. U. Vera, D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian, “Two-stream theory of diffusing light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

1995 (2)

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusiontheory,” Ann. Nuc. En. 22, 787–817 (1995).
[CrossRef]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

1994 (3)

1993 (2)

1992 (1)

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

1991 (1)

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 (1)

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherentbackscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990).
[CrossRef]

1989 (1)

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

1987 (1)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1985 (1)

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

1941 (1)

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

Amic, E.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

Aronson, R.

Berkovits, R.

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherentbackscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

De Vries, P.

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

Durian, D. J.

D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strongabsorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
[CrossRef]

D. J. Durian, “Two-stream theory of diffusing light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

M. U. Vera, D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian“Influence of boundary reflection and refraction on diffusive photontransport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

P. A. Lemieux, M. U. Vera, D. J. Durian, “Telegrapher theory of diffusing-light spectroscopies including ballistic transport and scattering anisotropy,” preprint (1997) available from the authors at the address on the title page.

Eliyahu, D.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Freund, I.

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

D. Eliyahu, M. Rosenbluh, I. Freund, “Angular intensity and polarization dependence of diffuse transmissionthrough random media,” J. Opt. Soc. Am. A 10, 477–491 (1993).
[CrossRef]

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

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherentbackscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990).
[CrossRef]

Ganapol, B. D.

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusiontheory,” Ann. Nuc. En. 22, 787–817 (1995).
[CrossRef]

Greenstein, J. L.

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

Grey, D. E.

D. E. Grey, American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–109.

Grigull, U.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Haskell, R. C.

Henyey, L. G.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Lagendijk, A.

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

Lemieux, P. A.

P. A. Lemieux, M. U. Vera, D. J. Durian, “Telegrapher theory of diffusing-light spectroscopies including ballistic transport and scattering anisotropy,” preprint (1997) available from the authors at the address on the title page.

Luck, J. M.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

T. M. Nieuwenhuizen, J. M. Luck, “Skin layer of diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

McAdams, M. S.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Nieuwenhuizen, T. M.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

T. M. Nieuwenhuizen, J. M. Luck, “Skin layer of diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[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]

Pomraning, G. C.

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusiontheory,” Ann. Nuc. En. 22, 787–817 (1995).
[CrossRef]

Rosenbluh, M.

Rudnick, J.

Straub, J.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Svaasand, L. O.

Thormahlen, I.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Ti-Chen, Feng

Tromberg, B. J.

Tsong-Tseh, Tsay

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Vera, M. U.

M. U. Vera, D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

P. A. Lemieux, M. U. Vera, D. J. Durian, “Telegrapher theory of diffusing-light spectroscopies including ballistic transport and scattering anisotropy,” preprint (1997) available from the authors at the address on the title page.

Vreeker, R.

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in stronglyscattering 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]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

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]

Am. J. Phys. (1)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Ann. Nuc. En. (1)

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusiontheory,” Ann. Nuc. En. 22, 787–817 (1995).
[CrossRef]

Astrophys. J. (1)

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

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

J. Phys. A (1)

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature,and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Phys. Lett. A (1)

A. Lagendijk, R. Vreeker, P. De Vries, “Influence of internal reflection on diffusive transport in stronglyscattering 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. B (1)

I. Freund, R. Berkovits, “Surface reflections and optical transport through random media: coherentbackscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41, 496–503 (1990).
[CrossRef]

Phys. Rev. E (3)

D. J. Durian“Influence of boundary reflection and refraction on diffusive photontransport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

M. U. Vera, D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

T. M. Nieuwenhuizen, J. M. Luck, “Skin layer of diffusive media,” Phys. Rev. E 48, 569–588 (1993).
[CrossRef]

Physica A (1)

D. J. Durian, “Two-stream theory of diffusing light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

Other (7)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

P. A. Lemieux, M. U. Vera, D. J. Durian, “Telegrapher theory of diffusing-light spectroscopies including ballistic transport and scattering anisotropy,” preprint (1997) available from the authors at the address on the title page.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

D. E. Grey, American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–109.

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Figures (6)

Fig. 1
Fig. 1

Normalized diffusing photon concentration profile versus distance in from the edge of a semi-infinite opaque slab in units of the transport mean free path l*. Solid curves are the histogram results of random walk computer simulations, and dotted curves are the profiles predicted from Eq. (2.3). For each case of constant boundary reflectivity R=0, 1/2 results are shown for three different cases of scattering anisotropy g=0, 0.5, 0.9. The isotropic scattering result g=0 always rises most slowly while the extreme anisotropic scattering result g=0.9 always rises most quickly with depth into the slab.

Fig. 2
Fig. 2

Comparison of simulations and predictions for the angular distribution P(μ) versus μ of light diffusely backscattered and transmitted at an angle cos-1 μ from the normal. Backscattering is depicted in the region μ<0 while transmission is depicted in the region μ>0 for cases of boundary reflectivity R=0, 1/2 and anisotropy g=0, 0.9. Circles, crosses, and triangles represent random walk simulations with steps taken according to the Mie, Henyey–Greenstein and fixed angle scattering form factors, respectively. The solid curve is the diffusion theory prediction for P(μ) according to Eqs. (2.6) for the corresponding boundary reflectivity and scattering anisotropy in each case.

Fig. 3
Fig. 3

Comparison of simulations and predictions for the angular distribution P(μ)/μ versus μ of light diffusely backscattered and transmitted at an angle cos-1 μ from the normal. Backscattering is depicted in the region μ<0 while transmission is depicted in the region μ>0 for cases of boundary reflectivity R=0, 1/2 and anisotropy g=0, 0.9. Circles, crosses, and triangles represent random walk simulations with steps taken according to the Mie, Henyey–Greenstein and fixed angle scattering form factors, respectively. The solid curve is the diffusion theory prediction P(μ)/μ, Eqs. (2.6), for the corresponding boundary reflectivity and scattering anisotropy in each case.

Fig. 4
Fig. 4

Angular distribution of light diffusely transmitted through and backscattered from a glass frit at angle cos-1 μ from the normal. Backscattering is depicted in the region μ<0 while transmission is depicted in the region μ>0. Symbols represent data collected with S- and P-polarized detection demonstrating no observed polarization dependence. The solid curve is a fit according to the diffusion theory prediction Eqs. (2.6) with ze=1.85 and g=0.9, assuming no refraction and angle- and polarization-independent boundary reflectivity. For comparison, predictions for the same boundary reflectivity but different scattering anisotropies are shown as long dashes (g =0) and short dashes (g=0.99).

Fig. 5
Fig. 5

Angular distribution of light diffusely transmitted through and backscattered from aqueous suspensions of polystyrene spheres placed in glass cells and held in air or immersed in a water bath as labeled using unpolarized detection. Backscattering is depicted in the region μ<0 while transmission is depicted in the region μ>0. For each case of boundary reflectivity, labeled by the exterior medium, and scattering anisotropy, labeled by sphere size, solid curves depict data for 2-mm slabs illuminated at λ=488.0 and 632.8 nm, symbols represent random walk simulation results with successive steps taken according to the Mie scattering form factor, and dashed curves show the diffusion theory prediction of Eqs. (2.6a) and (2.6b).

Fig. 6
Fig. 6

Angular distribution of S- and P-polarized light diffusely transmitted through and backscattered from aqueous suspensions of polystyrene spheres placed in glass cells held in air or immersed in a water bath as labeled. Data are shown by solid curves, while the diffusion theory predictions of Eqs. (2.6a) and (2.6b) for the corresponding polarization-dependent boundary reflectivity, labeled by the exterior medium, and scattering anisotropy, labeled by sphere size, are shown by dashed curves for S-polarized light and dotted–dashed curves for P-polarized light.

Tables (1)

Tables Icon

Table 1 Light-Scattering Parameters Predicted by Mie Theory for the Measured Aqueous Suspensions of Polystyrene Spheres As a Function of Sphere Diameter D, Volume Fraction ϕ, and Incident Wavelength λ

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

P(μe)μeze+μi1-g+μi[1-R(μi)]backscattering(ze+μi)[1-R(μi)]transmission,
ϕzp(z)=ϕ0D0 z+zez<zpzp+ze+gz>zp,
ze=23 1+R21-R1,Rn=01(n+1)μnR(μ)dμ,
D0ϕ0 ϕ(z)=D0ϕ0 0ϕzp(z)exp[-zpl*/ls]dzpl*/ls=1+ze-exp[-z/(1-g)],
PD(μe)dμedμi0r2drϕ(r) μir2 exp[-r][1-RD(μi)],
ϕ(r)
ze+rμitransmission1+ze-exp[-rμi/(1-g)]backscattering,
 
PD(μe)μe(ze+μi)[1-RD(μi)]transmissionze+μi1-g+μi[1-RD(μi)]backscattering,

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