Abstract

The three-dimensional radiative transfer equation is solved in the spatial frequency domain for modeling the light propagation due to a spatially modulated light source obliquely incident on a semi-infinite uniform medium. The dependence of the derived solution on the spatial frequencies as well as on position and direction is found analytically. The main computational procedure arises from the determination of several constants obtained by a system of linear equations. The obtained equations are verified and illustrated by comparisons with Monte Carlo simulations and the diffusion approximation, respectively.

© 2012 Optical Society of America

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  1. D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, and B. J. Tromberg, Opt. Lett. 30, 1354 (2005).
    [CrossRef]
  2. A. Bassi, C. D’Andrea, G. Valentini, R. Cubeddu, and S. Arridge, Opt. Lett. 33, 2836 (2008).
    [CrossRef]
  3. V. Lukic, V. A. Markel, and J. C. Schotland, Opt. Lett. 34, 983 (2009).
    [CrossRef]
  4. C. D’Andrea, N. Ducros, A. Bassi, S. Arridge, and G. Valentini, Biomed. Opt. Express 1, 471 (2010).
    [CrossRef]
  5. A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Opt. Soc. Am. A 25, 2833 (2008).
    [CrossRef]
  6. D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
    [CrossRef]
  7. A. R. Gardner and V. Venugopalan, Opt. Lett. 36, 2269 (2011).
    [CrossRef]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).
  9. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).
  10. J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
    [CrossRef]
  11. K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, New York, 1967).
  12. A. Liemert and A. Kienle, J. Opt. Soc. Am. A 29, 1475 (2012).
    [CrossRef]
  13. A. Kienle, “Lichtausbreitung in biologischem Gewebe,” dissertation (University of Ulm, 1995).

2012 (1)

2011 (1)

2010 (1)

2009 (3)

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

V. Lukic, V. A. Markel, and J. C. Schotland, Opt. Lett. 34, 983 (2009).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

2008 (2)

2005 (1)

Arridge, S.

Ayers, F. R.

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

Bassi, A.

Bevilacqua, F. P.

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, and B. J. Tromberg, Opt. Lett. 30, 1354 (2005).
[CrossRef]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, New York, 1967).

Cubeddu, R.

Cuccia, D. J.

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Opt. Soc. Am. A 25, 2833 (2008).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, and B. J. Tromberg, Opt. Lett. 30, 1354 (2005).
[CrossRef]

D’Andrea, C.

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).

Ducros, N.

Durkin, A. J.

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Opt. Soc. Am. A 25, 2833 (2008).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, and B. J. Tromberg, Opt. Lett. 30, 1354 (2005).
[CrossRef]

Gardner, A. R.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).

Kienle, A.

A. Liemert and A. Kienle, J. Opt. Soc. Am. A 29, 1475 (2012).
[CrossRef]

A. Kienle, “Lichtausbreitung in biologischem Gewebe,” dissertation (University of Ulm, 1995).

Liemert, A.

Lukic, V.

Markel, V. A.

Martelli, F.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).

Schotland, J. C.

Tromberg, B. J.

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Opt. Soc. Am. A 25, 2833 (2008).
[CrossRef]

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, and B. J. Tromberg, Opt. Lett. 30, 1354 (2005).
[CrossRef]

Valentini, G.

Venugopalan, V.

Weber, J. R.

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

Zaccanti, G.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, New York, 1967).

Biomed. Opt. Express (1)

J. Appl. Phys. (1)

J. R. Weber, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, J. Appl. Phys. 105, 102028 (2009).
[CrossRef]

J. Biomed. Opt. (1)

D. J. Cuccia, F. P. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, J. Biomed. Opt. 14, 024012 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (4)

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue (SPIE Press, Bellingham, 2010).

A. Kienle, “Lichtausbreitung in biologischem Gewebe,” dissertation (University of Ulm, 1995).

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, New York, 1967).

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Figures (3)

Fig. 1.
Fig. 1.

Reflectance versus spatial frequency due to a modulated light source that is incident perpendicularly on a semi-infinite medium having an absorption coefficient of μ a = 0.01 mm 1 .

Fig. 2.
Fig. 2.

Reflectance versus spatial frequency due to a modulated light source that is incident perpendicularly on a semi-infinite medium having a scattering coefficient of μ s = 10 mm 1 .

Fig. 3.
Fig. 3.

Reflectance amplitude and spatial phase shift versus spatial frequency caused by a modulated light source obliquely incident on the semi-infinite medium.

Equations (14)

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s ^ · I ( r , s ^ ) + μ t I ( r , s ^ ) = μ s f ( s ^ · s ^ ) I ( r , s ^ ) d 2 s ,
I ( ρ , z = 0 , s ^ ) = exp ( i k · ρ ) δ ( s ^ s ^ 0 ) , s ^ · z ^ > 0 ,
I ( r , s ^ ) = exp ( i k · ρ ) I ( z , s ^ ) ,
cos θ I ( z , s ^ ) z = [ i k sin θ cos ( ϕ ϕ k ) + μ t ] I ( z , s ^ ) + μ s f ( s ^ · s ^ ) I ( z , s ^ ) d 2 s
I ( z = 0 , s ^ ) = δ ( s ^ s ^ 0 ) , s ^ · z ^ > 0 .
I ( z , s ^ ) = e ξ z I ( s ^ ) ,
[ s ^ · Q + μ t ] I ( s ^ ) = μ s f ( s ^ · s ^ ) I ( s ^ ) d 2 s
I ( z , s ^ ) = λ i > 0 C i ( k ) exp [ Q i ( k ) z ] × l = M N m = l l ψ l m ( k λ i ) Y l m ( s ^ ) exp ( i m ϕ k ) ,
λ i > 0 C i ( k ) l = l ¯ N ψ l m ( k λ i ) R l l m = Re { Y l m ( θ 0 , ϕ 0 ϕ k ) } ,
R ( k ) = cos θ 0 4 π 3 i = 1 N C i ( k ) ψ 10 ( k λ i ) ,
Φ ( k , z ) = 4 π i = 1 N + 1 2 C i ( k ) ψ 00 ( k λ i ) exp [ Q i ( k ) z ] .
R ( k x ) = μ s 1 + 2 Q D μ t + Q cos θ 0 + i k sin θ 0 ( μ t + Q cos θ 0 ) 2 + ( k sin θ 0 ) 2 ,
D d 2 Φ ( z ) d z 2 ( μ a + D k 2 ) Φ ( z ) = S ( z )
f ( s ^ · s ^ ) = 1 4 π 1 g 2 [ 1 + g 2 2 g ( s ^ · s ^ ) ] 3 / 2

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