Abstract

The K-distribution, widely used for investigation of fractal scattering in the atmosphere and from surfaces, is applied as a model for light propagation in biological tissue and cells. This leads to simple expressions for the scattering function, anisotropy function, phase function, reduced scattering coefficient, and scattering power. Compared with an alternative previously published model [Opt. Lett. 30, 3051 (2005) ], the range of allowable power laws is extended into the subfractal regime.

© 2006 Optical Society of America

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References

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  1. J. M. Schmitt and G. Kumar, Opt. Lett. 21, 1310 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. M. Xu and R. R. Alfano, Opt. Lett. 22, 3051 (2005).
    [CrossRef]
  4. M. V. Berry, J. Phys. A 12, 781 (1979).
    [CrossRef]
  5. B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
    [CrossRef]
  6. E. Jakeman, J. Opt. Soc. Am. 72, 1034 (1982).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2005 (1)

M. Xu and R. R. Alfano, Opt. Lett. 22, 3051 (2005).
[CrossRef]

2001 (1)

1996 (2)

1989 (1)

1982 (1)

1981 (1)

B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
[CrossRef]

1979 (1)

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

Alfano, R. R.

M. Xu and R. R. Alfano, Opt. Lett. 22, 3051 (2005).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

Booker, H. G.

B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
[CrossRef]

Jacques, S. L.

Jakeman, E.

Keller, J. B.

Kumar, G.

Marians, M.

B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
[CrossRef]

Moscoso, M.

Nishioka, N. S.

Papanicolaou, G.

Parsa, P.

Schmitt, J. M.

Sheppard, C. J. R.

C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
[CrossRef]

Uscinski, B. J.

B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
[CrossRef]

Xu, M.

M. Xu and R. R. Alfano, Opt. Lett. 22, 3051 (2005).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

Opt. Commun. (1)

C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London Ser. A (1)

B. J. Uscinski, H. G. Booker, and M. Marians, Proc. R. Soc. London Ser. A 374, 503 (1981).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Log/log plot of the power spectrum, normalized to unity for low frequencies, for (a) Xu and Alfano’s model for different values of parameter D f , (b) K-distribution model for different values of parameter D 3 .

Fig. 2
Fig. 2

(a) Anisotropy factor g predicted by the K-distribution model for different fractal dimensions. (b) Normalized phase function for the extreme fractal D 3 = 4 .

Fig. 3
Fig. 3

Log/log plot of the wavelength dependence of the reduced scattering coefficient μ s [calculated directly from Eq. (15) with the first factor in brackets suppressed] for different values of n. The slope of the curves for large k L gives the scattering power, b.

Tables (1)

Tables Icon

Table 1 Parameters for the Two Fractal Models, D f and D 3

Equations (15)

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R ( r ) ( r l max ) 4 D f Γ ( ( 4 D f ) , r l max ) ,
G ( K ) = 0 l max ϵ 2 l 3 π 2 ( 1 + K 2 l 2 ) 2 η 0 l 3 D f d l ,
G ( K ) = σ 2 0 1 ( 7 D f ) f 6 D f [ 1 + ( f K l max ) 2 ] 2 d f = σ 2 2 F 1 [ 2 , 7 D f 2 , 9 F f 2 , ( K l max ) 2 ] ,
σ 2 = ϵ 2 η 0 l max 7 D f ( 7 D f ) π 2 .
G ( K ) = 2 σ 2 ( K l max ) 2 { ln [ 1 ( K l max ) 2 ] ( K l max ) 2 1 1 + ( K l max ) 2 } .
G ( K ) = 3 σ 2 2 { arctan ( K l max ) ( K l max ) 3 1 ( K l max ) 2 [ 1 + ( K l max ) 2 ] } .
G ( K ) = σ 2 [ 1 + ( K l max ) 2 ] .
2 D 3 = 11 2 n , 2 n 5 .
D f = 2 D 3 4 .
R ( r ) = 2 Γ ( n 3 2 ) ( r 2 L ) n 3 2 K n 3 2 ( r L ) ,
G ( K ) = ( 2 π ) 1 2 σ 2 K 0 R ( r ) sin ( K r ) r d r = σ 2 ( 2 L ) n 3 2 Γ ( n ) Γ ( n ( 3 2 ) ) 1 [ 1 + ( K L ) 2 ] n .
S 2 ( μ ) G 2 [ k L ( 1 μ ) 1 2 ] 1 [ 1 + 2 ( k L ) 2 ( 1 μ ) ] n ,
g = ( 1 + μ 2 ) μ S ( θ ) 2 2 k 2 d Ω ( 1 + μ 2 ) S ( θ ) 2 2 k 2 d Ω = { [ 1 + 4 ( k L ) 2 ] n 1 [ 3 6 ( k L ) 2 ( n 4 ) + 8 ( k L ) 4 ( n 4 ) ( n 3 ) 8 ( k L ) 6 ( n 4 ) ( n 3 ) ( n 2 ) ] [ 3 + 6 ( k L ) 2 ( n + 2 ) + 8 ( k L ) 4 ( n 2 n + 6 ) + 8 ( k L ) 6 n ( n 2 5 n + 10 ) ] } ( 2 ( k L ) 2 ( n 4 ) { [ 1 + 2 ( k L ) 2 ( n + 1 ) + 4 ( k L ) 4 ( n 2 3 n + 4 ) ] [ 1 + 4 ( k L ) 2 ] n 1 [ 1 + 4 ( k L ) 4 ( n 3 ) ( n 2 ) 2 ( k L ) 2 ( n 3 ) ] } )
p ( θ ) = ( 1 + μ 2 ) S ( θ ) 2 2 k 2 1 + μ 2 2 [ 1 + 2 ( k L ) 2 ( 1 μ ) ] n .
μ s = ( 1 + μ 2 ) ( 1 μ ) μ S ( θ ) 2 2 k 2 d Ω = [ π 24 Γ ( n ) Γ ( n 3 2 ) ( 2 L ) n + 5 2 σ 2 ] × 3 64 ( n 4 ) ( n 3 ) ( n 2 ) ( n 1 ) ( k L ) 4 × ( { 3 4 ( k L ) 2 ( n 4 ) + 4 ( k L ) 4 ( n 4 ) ( n 3 ) } [ 1 + 4 ( k L ) 2 ] ( n 1 ) { 3 + 4 ( k L ) 2 [ ( 2 n + 1 ) + ( k L ) 2 ( 3 n 2 5 n + 8 ) + 4 ( k L ) 4 ( n 1 ) ( n 2 5 n + 8 ) ] } ) .

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