Abstract

We recently examined properties of the surface emission profiles of particles (photons) injected into a turbid medium consisting of two layers [ R. Nossal, J. Kiefer, G. H. Weiss, R. F. Bonner, H. Taitelbaum, and S. Havlin, “ Photon Migration in Layered Media,” Appl. Opt. 27, 3382– 3391 ( 1988)]. The two layers differ in the coefficient that appears when internal absorption is modeled in terms of Beer’s law. The model relates to the injection of laser radiation into tissue for diagnostic or therapeutic purposes. Results of our earlier work were derived from extensive computer simulations. In the present paper we discuss a simple analytical approximation to the surface intensity profile valid when the absorptivity of the upper layer is greater than that of the lower.

© 1989 Optical Society of America

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References

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  1. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “A Model for Photon Diffusion in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423 (1987).
    [CrossRef] [PubMed]
  2. R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon Migration in Layered Media,” Appl. Opt. 27, 3382–3391 (1988).
    [CrossRef] [PubMed]
  3. S. Havlin, D. Ben-Avraham, “Transport in Disordered Media,” Adv. Phys. 36, 695 (1987).
    [CrossRef]
  4. H. Taitelbaum, “Characteristics of Photon Migration in Turbid Media,” M.Sc. Thesis, Bar-Ilan U. (1988).

1988

1987

Ben-Avraham, D.

S. Havlin, D. Ben-Avraham, “Transport in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

Bonner, R.

Bonner, R. F.

Havlin, S.

Kiefer, J.

Nossal, R.

Taitelbaum, H.

Weiss, G. H.

Adv. Phys.

S. Havlin, D. Ben-Avraham, “Transport in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Other

H. Taitelbaum, “Characteristics of Photon Migration in Turbid Media,” M.Sc. Thesis, Bar-Ilan U. (1988).

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Figures (4)

Fig. 1
Fig. 1

Two-dimensional projection of a semi-infinite medium consisting of two layers. The thickness of the upper layer is taken equal to D, and the absorption coefficient in the upper and lower layers, respectively, are taken equal to μ1 and μ2.

Fig. 2
Fig. 2

Graphs of the exact and approximate values of the function logΓ(ρ) for μ1 > μ2 for different values of the layer thickness D. The approximations calculated using Eq. (12) are indicated by dashed lines, while those found by the exact enumeration method3 are given as solid lines. The parameters used for these calculation are (a) μ1 = 0.2,μ2 = 0.01 and (b) μ1 = 0.4,μ2 = 0.1.

Fig. 3
Fig. 3

Comparison of values of 〈n|ρ〉 as a function of ρ calculated by the exact enumeration method3 (solid lines) and the approximation of this paper (dashed lines) for μ1 = 0.4 and μ2 = 0.1 for several values of the thickness.

Fig. 4
Fig. 4

Curves of logΓ(ρ) as a function of ρ obtained by the exact enumeration method3 for μ1 = 0.01 and μ2 = 0.2. The results closely resemble those given in Ref. 1 for a homogeneous medium.

Equations (21)

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n 1 + n 2 = n .
Γ ( n , ρ | μ ) = F ( n , ρ ) exp ( μ n ) ,
F ( n , ρ ) 3 2 ( 2 π n ) 3 / 2 [ 1 exp ( 6 / n ) ] exp ( 3 ρ 2 2 n ) .
Γ ( n 1 , n 2 , ρ | μ 1 , μ 2 ) = F ( n , ρ ) exp [ μ 1 n 1 μ 2 n 2 ] = F ( n , ρ ) exp [ μ 2 n n 1 ( μ 1 μ 2 ) ] ,
Q n 1 / 2 ( D ) 3 π n 1 exp ( 3 n 1 D 2 μ 1 n 1 2 ) ,
n 1 = m = 5 . 2 D / μ 1 .
Γ ( ρ ) n = 1 m Γ ( n , 0 , ρ | μ 1 , 0 ) + n = m + 1 Γ ( m , n m , ρ | μ 1 , μ 2 ) .
Γ ( ρ ) 0 m Γ ( n , 0 , ρ | μ 1 , 0 ) d n + m Γ ( m , n m , ρ | μ 1 , μ 2 ) d n .
Γ ( n , 0 , ρ | μ 1 , 0 ) = Γ ( n , ρ | μ 1 ) ,
exp ( 3 ρ 2 2 n μ 1 n )
n m 1 = ρ 3 2 μ 1 ,
n m 2 = 3 ρ 2 + 6 2 μ 1 .
Γ ( ρ ) 1 4 π ρ 2 { 6 μ 1 exp ( ρ 6 μ 1 ) + 6 μ 2 exp [ ρ 6 μ 2 m ( μ 1 μ 2 ) ] } .
6 μ 1 exp [ ρ 6 μ 1 ] > 6 μ 2 exp [ ρ 6 μ 2 m ( μ 1 μ 2 ) ] .
ρ < 5 . 2 6 D ( 1 + μ 2 μ 1 ) ,
n | ρ [ 0 m n Γ ( n , 0 , ρ | μ 1 , 0 ) d n + m n Γ ( m , n m , ρ | μ 1 , μ 2 ) d n ] / Γ ( ρ ) .
n | ρ 3 ρ 6 μ 1 [ 1 + μ 1 μ 2 A ( ρ ) 1 + A ( ρ ) ] ,
A ( ρ ) = μ 2 μ 1 exp [ ρ ( 6 μ 1 6 μ 2 ) m ( μ 1 μ 2 ) ] .
A ( ρ ) 0 when ρ ( 5 . 2 / 6 ) D ( 1 + μ 2 / μ 1 ) ,
A ( ρ ) when ρ ( 5 . 2 / 6 ) D ( 1 + μ 2 / μ 1 ) .
μ = α D 2 μ 1 + μ 2 α D 2 + 1 ,

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