Abstract

A three-parameter model based on the Whittle–Matérn correlation family is used to describe continuous random refractive-index fluctuations. The differential scattering cross section is derived from the index correlation function using nonscalar scattering formulas within the Born approximation. Parameters such as scattering coefficient, anisotropy factor, and spectral dependence are derived from the differential scattering cross section for this general class of functions.

© 2009 Optical Society of America

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References

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2007

2006

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

2005

P. Guttorp and T. Gneiting, National Research Center for Statistics and the Environment-Technical Report Series (2005).

M. Xu and R. R. Alfano, Opt. Lett. 30, 3051 (2005).
[CrossRef] [PubMed]

2000

1997

J. R. Mourant, T. Fuselier, J. Boyer, T. M. Johnson, and I. J. Bigio, Appl. Opt. 36, 949 (1997).
[CrossRef] [PubMed]

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

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

1996

Aalders, M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Alfano, R. R.

Andersson-Engels, S.

Bartek, M.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Beek, J. F.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Bevilacqua, F.

Bigio, I. J.

Blokland, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Boyer, J.

Dam, J. S.

Fuselier, T.

Gneiting, T.

P. Guttorp and T. Gneiting, National Research Center for Statistics and the Environment-Technical Report Series (2005).

Guttorp, P.

P. Guttorp and T. Gneiting, National Research Center for Statistics and the Environment-Technical Report Series (2005).

Ishimaru, A.

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

Johnson, T. M.

Kumar, J. M.

Mourant, J. R.

Paulsen, K.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Pham, T. H.

Pickering, J. W.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Pogue, B.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Posthumus, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Schmitt, J. M.

Sheppard, C. J. R.

Spott, T.

Sterenborg, H. J. C. M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Tromberg, B. J.

van Gemert, M. J. C.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Wang, M.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Wells, W.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Xu, M.

Appl. Opt.

J. Biomed. Opt.

M. Bartek, M. Wang, W. Wells, K. Paulsen, and B. Pogue, J. Biomed. Opt. 11, 064007 (2006).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. van Gemert, Phys. Med. Biol. 42, 2255 (1997).
[CrossRef] [PubMed]

Other

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

P. Guttorp and T. Gneiting, National Research Center for Statistics and the Environment-Technical Report Series (2005).

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Figures (5)

Fig. 1
Fig. 1

Index correlation functions for some values of m.

Fig. 2
Fig. 2

Above, example of a differential scattering cross section plotted in spherical coordinates. The incident wave propagates from left to right, and the polarization is such that electric field is in the vertical plane. The dimple is located at the origin. Below, comparison of the rotationally averaged σ up (inner) corresponding to unpolarized incidence and the scalar wave approximation σ sw (outer) for k l c = 0.1 (isotropic scattering) shown left, and k l c = 1 (forward scattering) shown right.

Fig. 3
Fig. 3

Wavelength-normalized scattering coefficient μ s k as a function of wavelength normalized index correlation length k l c . Inset, the normalized mean-free path k l s = k μ s dependence. It should be noted that although no limit is shown for μ s , the value is inherently limited by the requirement that l s > l c . This weak scattering limit requires that d n 2 ( k l c ) 2 1 .

Fig. 4
Fig. 4

Reduced scattering coefficient as a function of index correlation length (each normalized by wavelength).

Fig. 5
Fig. 5

Plots of μ s versus k l c with and without the dipole factor.

Equations (7)

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B n ( r ) = d n 2 2 5 2 m Γ ( m ( 3 / 2 ) ) ( r l c ) m 3 2 K m 3 2 ( r l c ) ,
Φ ( k s ) = F { B n ( r ) } = d n 2 l c 3 Γ ( m ) ( 1 + k s 2 l c 2 ) m π 3 2 Γ ( m ( 3 2 ) ) .
σ ( θ , ϕ ) = 2 π k 4 ( 1 sin 2 ( θ ) cos 2 ( ϕ ) ) Φ ( 2 k sin ( θ 2 ) ) = 2 d n 2 k 4 l c 3 Γ ( m ) π Γ ( m 3 2 ) ( 1 sin 2 ( θ ) cos 2 ( ϕ ) ) ( 1 + [ 2 k l c sin ( θ 2 ) ] 2 ) m .
μ s k = d n 2 π Γ ( m 3 ) 2 k 3 l c 3 Γ ( m ( 3 / 2 ) ) [ ( 1 + ( 2 k 2 l c 2 ( m 2 ) 1 ) × 2 k 2 l c 2 ( m 3 ) ) ( 1 + 2 k 2 l c 2 ( 1 + m ) + 4 k 4 l c 4 ( 4 + ( m 3 ) m ) ) ( 1 + 4 k 2 l c 2 ) 1 m ] .
μ s k = { 16 π Γ ( m ) 3 Γ ( m 3 2 ) d n 2 ( k l c ) 3 if k l c 1 2 π Γ ( m 1 ) Γ ( m 3 2 ) d n 2 k l c if k l c 1 & m > 1 } .
g = [ ( 1 + 4 k 2 l c 2 ) m ( 3 + 2 k 2 l c 2 ( m 4 ) ( 3 4 k 2 l c 2 × ( k 2 l c 2 ( m 2 ) 1 ) ( m 3 ) ) ) ( 1 + 4 k 2 l c 2 ) × ( 3 + 6 k 2 l c 2 ( 2 + m ) + 8 k 6 l c 6 m ( 10 + ( m 5 ) m ) + 8 k 4 l c 4 ( 6 + ( m 1 ) m ) ) ] [ 2 k 2 l c 2 ( m 4 ) ( ( 1 + 4 k 2 l c 2 ) m ( 1 2 k 2 l c 2 × ( 2 k 2 l c 2 ( m 2 ) 1 ) ( m 3 ) ) + ( 1 + 4 k 2 l c 2 ) ( 1 + 2 k 2 l c 2 ( 1 + m ) + 4 k 4 l c 4 ( 4 + ( m 3 ) m ) ) ) ] .
g = { ( 4 / 5 ) m ( k l c ) 2 if k l c 1 1 1 2 ( m 2 ) ( k l c ) 2 if k l c 1 & m > 2 1 b 2 3 2 m ( k l c ) 2 2 m if k l c 1 & 1 < m < 2 where b = ( m 1 ) ( 8 + m ( m 5 ) ) ( 4 m ) ( 6 + m ( 3 m ) ) } .

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