Abstract

We discuss several outstanding theoretical problems in optical diffusion in random media. Specifically, we discuss which of several diffusion theories most closely approximates exact solutions of the equation of transfer. We consider a plane wave impinging upon a plane-parallel slab of a random medium as a model problem to compare the diffusion theories with a numerical solution of the equation of transfer for continuous-wave, pulsed, and photon density waves. In addition, we discuss the validity of the diffusion approximation for a variety of parameter settings to ascertain the diffusion approximation’s applicability to imaging biological media.

© 1998 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1997).
  2. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  3. A. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef] [PubMed]
  4. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random media and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  5. K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
    [CrossRef]
  6. A. Ya. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
    [CrossRef]
  7. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [CrossRef]
  8. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for noninvasive measurements of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  9. K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef]
  10. M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
    [CrossRef]
  11. Q. Ma, A. Ishimaru, “Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding medium,” Radio Sci. 25, 419–426 (1990).
    [CrossRef]
  12. V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
    [CrossRef]

1997 (3)

1994 (2)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random media and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
[CrossRef]

1990 (2)

Q. Ma, A. Ishimaru, “Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding medium,” Radio Sci. 25, 419–426 (1990).
[CrossRef]

K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef]

1989 (3)

1978 (1)

Alfano, R. R.

A. Ya. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
[CrossRef]

K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef]

Chance, B.

Durian, D. J.

Foo, K. M.

K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef]

Furutsu, K.

K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
[CrossRef]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random media and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Gutman, S.

Ishimaru, A.

Q. Ma, A. Ishimaru, “Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding medium,” Radio Sci. 25, 419–426 (1990).
[CrossRef]

A. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989).
[CrossRef] [PubMed]

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1997).

Jacques, S. L.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Kisselev, V. B.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
[CrossRef]

Lax, M.

Liu, F.

K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef]

Ma, Q.

Q. Ma, A. Ishimaru, “Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding medium,” Radio Sci. 25, 419–426 (1990).
[CrossRef]

Patterson, M. S.

Perona, G.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
[CrossRef]

Polishchuk, A. Ya.

Prahl, S. A.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Roberti, L.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
[CrossRef]

Rudnick, J.

Welch, A. J.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Wilson, B. C.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random media and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transf. (1)

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 51, 603–614 (1994).
[CrossRef]

Lasers Surg. Med. (1)

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random media and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

K. M. Foo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef]

Radio Sci. (1)

Q. Ma, A. Ishimaru, “Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding medium,” Radio Sci. 25, 419–426 (1990).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1997).

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Figures (9)

Fig. 1
Fig. 1

Diagram of the plane-parallel problem.

Fig. 2
Fig. 2

Continuous wave: transmitted and backscattered fluxes for W0 = 0.99 and d = 3.0 cm. Here and in subsequent figures, Rad. Tr. means radiative transfer.

Fig. 3
Fig. 3

Continuous wave: transmitted and backscattered fluxes for W0 = 0.85 and d = 3.0 cm.

Fig. 4
Fig. 4

10-ps pulse: amplitude of transmitted and backscattered fluxes for W0 = 0.99, d = 3.0 cm, and t 0 = 100 ps.

Fig. 5
Fig. 5

10-ps pulse: amplitude of transmitted and backscattered fluxes for W0 = 0.85, d = 3.0 cm, and t 0 = 100 ps.

Fig. 6
Fig. 6

Density wave: amplitude and phase of transmitted fluxes for W0 = 0.99 and d = 3.0 cm.

Fig. 7
Fig. 7

Density wave: amplitude and phase of backscattered fluxes for W0 = 0.99 and d = 3.0 cm.

Fig. 8
Fig. 8

Density wave: amplitude and phase of transmitted fluxes for W0 = 0.85 and d = 3.0 cm.

Fig. 9
Fig. 9

Density wave: amplitude and phase of backscattered fluxes for W0 = 0.85 and d = 3.0 cm.

Tables (1)

Tables Icon

Table 1 Parameter Values for the Various Diffusion Theories

Equations (24)

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τ = ρ σ t z ,
μ   d I τ ,   μ d τ + 1 - i   ω n τ 0 I τ ,   μ = W 0 2 - 1 1   p 0 μ ,   μ × I τ ,   μ d μ + ε ri τ ,   μ ,
ω n = ω d t 0 ,     t 0 = d / c
p 0 μ ,   μ = 1 2 π 0 2 π 1 2 π 0 2 π   p μ ,   ϕ ,   μ ,   ϕ d ϕ d ϕ ,
p μ ,   ϕ ,   μ ,   ϕ = 1 - g 2 1 + g 2 - 2 g   cos   ξ 3 / 2
cos   ξ = μ μ + 1 - μ 2 1 - μ 2 cos ϕ - ϕ
g = 4 π   p μ ,   ϕ ,   μ ,   ϕ cos   ξ d μ d ϕ 4 π   p μ ,   ϕ ,   μ ,   ϕ d μ d ϕ
ε ri τ ,   μ = 1 4 π   p 0 μ ,   1 exp - η τ ,
η = 1 - i   ω n / τ 0 .
I τ = 0 ,   μ = 0 ,     0 < μ 1 , I τ = τ 0 ,   μ = 0 ,       - 1 μ < 0 .
I r ,   s ˆ = U d r + 3 4 π F d r · s ˆ + ,
U d r = 1 / 2   4 π   I r ,   s ˆ d Ω
F d r = 4 π   I r ,   s ˆ s ˆ d Ω
| F d τ |     U d ,
d 2 U d τ d τ 2 + κ 2 U d τ = Q 0 τ ,
κ 2 = 3 α   ω n 2 τ 0 2 + i   ω n τ 0 W 0 1 - g + β 1 - W 0 - 1 - W 0 W 0 1 - g + γ 1 - W 0
Q 0 τ = 3 4 π W 0 g W 0 - η 1 + g exp - η τ .
U d = 3 4 π g W 0 - η F d + 3 4 π   g W 0 exp - η τ z ˆ .
U d - h   d U d d τ + 1 2 π   Q 1 = 0 ,     τ = 0 , U d + h   d U d d τ - 1 2 π   Q 1 = 0 ,     τ = τ 0 ,
h = 2 / 3 g W 0 - η - 1 ,
Q 1 τ = g W 0 g W 0 - η exp - η τ .
F T = 2 π   0 1   I τ 0 ,   μ μ d μ ,
F B = 2 π   - 1 0   I 0 ,   μ μ d μ ,
g 0 ,     W 0 1 ,     τ 0     1 .

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