Abstract

For a beam impinging on a scattering medium the diffusion approximation to the radiative transport equation is not valid for analyzing the radiance near the source, especially if the medium scatters strongly with a sharp forward peak. To analyze the radiance, we use the Fokker–Planck approximation to the radiative transport equation. Numerical results show a backscattered ring appearing around the beam center. It also appears in Monte Carlo simulations of the radiative transport equation. This ring is manifested from successive near-forward scattering events, so it requires a directional description. Therefore the diffusion approximation cannot predict this ring.

© 2004 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).
  2. T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
    [CrossRef]
  5. A. D. Kim and J. B. Keller, J. Opt. Soc. Am. A 20, 92 (2003).
    [CrossRef]
  6. J. E. Morel, Nucl. Sci. Eng. 89, 131 (1985).
  7. V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
    [CrossRef]
  8. A. D. Kim and M. Moscoso, Opt. Lett. 27, 1589 (2002).
    [CrossRef]

2003 (1)

2002 (1)

1998 (1)

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

1995 (1)

1992 (1)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

1990 (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
[CrossRef]

1985 (1)

J. E. Morel, Nucl. Sci. Eng. 89, 131 (1985).

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
[CrossRef]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).

Jacques, S. L.

Keller, J. B.

Kim, A. D.

Morel, J. E.

J. E. Morel, Nucl. Sci. Eng. 89, 131 (1985).

Moscoso, M.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
[CrossRef]

Tromberg, B. J.

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

Venugopalan, V.

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

Wang, L.

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
[CrossRef]

Wilson, B. C.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

You, J. S.

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, IEEE J. Quantum Electron. 26, 2166 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, Med. Phys. 19, 879 (1992).
[CrossRef] [PubMed]

Nucl. Sci. Eng. (1)

J. E. Morel, Nucl. Sci. Eng. 89, 131 (1985).

Opt. Lett. (1)

Phys. Rev. E (1)

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1996).

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Figures (4)

Fig. 1
Fig. 1

Contour plot of the backscattered integrated flux from a uniform half-space. The medium parameters are g=0.9 and σa/σs=0.01. The beam width is W0=0.5/σs+σa.

Fig. 2
Fig. 2

Backscattered integrated flux Fρ for g=0.85, 0.90, 0.95, and 0.98. All the other parameters are the same as for Fig. 1.

Fig. 3
Fig. 3

Diagram showing the effect of successive near-forward scattering events. The backscattering ring appears because these successive scattering events direct light away from the beam center.

Fig. 4
Fig. 4

The backscattered integrated flux for g=0.90 and 0.95 computed from the Fokker–Planck (FP) equation (dashed and solid curves) and Monte Carlo simulations of the radiative transport (RT) equation (dashed and solid curves with circles). All the other parameters are the same as for Fig. 1.

Equations (7)

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

ω·xΨ+σaΨ=LΨ,
LΨ=-σsΨ+Ωfω·ωΨω,rdω,
LΨ=12σs1-gΔΨ,
g=2π-1+1fμμdμ,
Ψ|z=0=δω-zˆexp-ρ2/W02,  ω·zˆ>0,
Fρ=ω·zˆ<0ω·-zˆΨω,r|z=0dω.
fω·ω=12πexp-1-ω·ω/1-exp-2/.

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