Abstract

The photon migration in two semi-infinite highly scattering media with different refractive indices is studied in the diffusion approximation for two sets of boundary conditions at the interface. In commonly used boundary conditions, the ratio of the intensity (fluence rate) to the squared refractive index is assumed continuous across an interface and the normal component of flux is required to be continuous. However, a more rigorous approach shows that the boundary condition for the intensity may be different. As was shown by Aronson [J. Opt. Soc. Am. A 12, 2532 (1995)] , the ratio of the intensity to the squared refractive index undergoes a jump across an interface that is proportional to the diffuse flux. A diffusion model with an instantaneous point source that can be solved analytically for both sets of boundary conditions is considered. The analytical solutions are derived and compared with the results of Monte Carlo simulations that take into account the reflections and refractions at the interface according to Fresnel’s formulas. It is shown that the analytical solutions with the Aronson boundary condition for intensity match the Monte Carlo results better than the solutions with a continuous ratio of the intensity to the squared refractive index.

© 2010 Optical Society of America

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References

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  1. M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co., 1914).
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  4. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
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    [CrossRef]
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    [CrossRef]
  10. J.-M. Tualle, E. Tinet, J. Prat, and S. Avrillier, “Light propagation near turbid-turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
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  11. J.-M. Tualle, J. Prat, E. Tinet, and S. Avrillier, “Real space Green’s function calculation for the solution of the diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
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  12. M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
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  16. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).
  17. L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  18. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

2008 (1)

J. Bouza-Dominguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78, 031926 (2008).
[CrossRef]

2007 (2)

J. S. Cassell and M. M. R. Williams, “Radiation transport and internal reflection in a two region, turbid sphere,” J. Quant. Spectrosc. Radiat. Transf. 104, 400–427 (2007).
[CrossRef]

M. L. Shendeleva, “Time-domain Green functions for diffuse light in two adjoining turbid half-spaces,” Appl. Opt. 46, 1641–1649 (2007).
[CrossRef] [PubMed]

2004 (2)

F. Martelli, S. D. Bianco, and G. Zaccanti, “Effect of the refractive index mismatch on light propagation through diffusive layered media,” Phys. Rev. E 70, 011907 (2004).
[CrossRef]

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

2002 (1)

2000 (2)

1999 (1)

1995 (2)

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (1)

1990 (1)

1988 (2)

M. Keijzer, W. M. Star, and P. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).

1986 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

1970 (1)

M.Abramovitz and I.A.Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

1914 (1)

M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co., 1914).

Aronson, R.

Avrillier, S.

Bianco, S. D.

F. Martelli, S. D. Bianco, and G. Zaccanti, “Effect of the refractive index mismatch on light propagation through diffusive layered media,” Phys. Rev. E 70, 011907 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Bouza-Dominguez, J.

J. Bouza-Dominguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78, 031926 (2008).
[CrossRef]

Cassell, J. S.

J. S. Cassell and M. M. R. Williams, “Radiation transport and internal reflection in a two region, turbid sphere,” J. Quant. Spectrosc. Radiat. Transf. 104, 400–427 (2007).
[CrossRef]

Faris, G. W.

Feng, T. -C.

Haskell, R. C.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Keijzer, M.

Martelli, F.

F. Martelli, S. D. Bianco, and G. Zaccanti, “Effect of the refractive index mismatch on light propagation through diffusive layered media,” Phys. Rev. E 70, 011907 (2004).
[CrossRef]

McAdams, M. S.

Nieto-Vesperinas, M.

Planck, M.

M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co., 1914).

Prahl, S. A.

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).

Prat, J.

Ripoll, J.

Schmitt, J. M.

Shendeleva, M. L.

M. L. Shendeleva, “Time-domain Green functions for diffuse light in two adjoining turbid half-spaces,” Appl. Opt. 46, 1641–1649 (2007).
[CrossRef] [PubMed]

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

Star, W. M.

Storchi, P. M.

Svaasand, L. O.

Tinet, E.

Tromberg, B. J.

Tsay, T. -T.

Tualle, J. -M.

Walker, E. C.

Wall, R. T.

Wang, L.

L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Williams, M. M. R.

J. S. Cassell and M. M. R. Williams, “Radiation transport and internal reflection in a two region, turbid sphere,” J. Quant. Spectrosc. Radiat. Transf. 104, 400–427 (2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Zaccanti, G.

F. Martelli, S. D. Bianco, and G. Zaccanti, “Effect of the refractive index mismatch on light propagation through diffusive layered media,” Phys. Rev. E 70, 011907 (2004).
[CrossRef]

Zeng, L.

L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zhou, G. X.

Appl. Opt. (2)

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zeng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (6)

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

J. S. Cassell and M. M. R. Williams, “Radiation transport and internal reflection in a two region, turbid sphere,” J. Quant. Spectrosc. Radiat. Transf. 104, 400–427 (2007).
[CrossRef]

Opt. Commun. (2)

J.-M. Tualle, E. Tinet, J. Prat, and S. Avrillier, “Light propagation near turbid-turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[CrossRef]

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

Phys. Rev. E (2)

F. Martelli, S. D. Bianco, and G. Zaccanti, “Effect of the refractive index mismatch on light propagation through diffusive layered media,” Phys. Rev. E 70, 011907 (2004).
[CrossRef]

J. Bouza-Dominguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78, 031926 (2008).
[CrossRef]

Other (4)

M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co., 1914).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

M.Abramovitz and I.A.Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the model: two semi-infinite media in contact along a plane interface at z = 0 . The diffusion coefficients and the refractive indices of the media are D 1 and n 1 for z > 0 and D 2 and n 2 for z < 0 . A random walk of a photon, emitted from the source denoted by S, is shown.

Fig. 2
Fig. 2

Relative difference of the intensities on both sides of the interface, ( ϕ 1 ϕ 2 ) / ( ϕ 1 + ϕ 2 ) z = 0 , versus time for solutions corresponding to the basic boundary conditions denoted by ( b ) and to the Aronson boundary conditions denoted by ( A ) . The properties of the media are μ s 1 = 20 cm 1 and n 1 = 1 , and μ s 2 = 10 cm 1 and n 2 = 2 , respectively. For the Aronson boundary conditions, θ = 0.3   cm .

Fig. 3
Fig. 3

The number of collisions of a photon in bins, collected during a time interval T = 1.667   ns at r = 0.5   cm for media with properties μ s 1 = 20 cm 1 and n 1 = 1 , and μ s 2 = 10 cm 1 and n 2 = 2 . The source is located at z 0 = 0.5   cm . Solid lines correspond to the analytical solutions with basic boundary conditions ( b ) , denoted by ( b ) , and with Aronson boundary conditions ( A ) , denoted by ( A ) . Here θ = 0.3   cm . The numerical data, averaged over 20,000 photons, are represented by dots.

Fig. 4
Fig. 4

Relative difference of the collision numbers in the bins nearest to the interface versus distance along the interface. The analytical solution corresponding to the basic boundary conditions is shown by a solid line denoted by ( b ) . The analytical solution corresponding to the Aronson boundary conditions is shown by a solid line denoted by ( A ) . Dots show the numerical data averaged over 20,000 photons. The parameters of the media are μ s 1 = 20 cm 1 and n 1 = 1 , and μ s 2 = 10 cm 1 and n 2 = 2 . The data are obtained for a time interval T = 1.667   ns . Here θ = 0.3   cm .

Fig. 5
Fig. 5

The number of collisions of a photon in bins collected during a time interval T = 3.333   ns at r = 0.5   cm for media with properties μ s 1 = 10 cm 1 and n 1 = 2 , and μ s 2 = 20 cm 1 and n 2 = 1 . The source is located at z 0 = 0.5   cm . Solid lines correspond to the analytical solutions with basic boundary conditions ( b ) and with Aronson boundary conditions ( A ) , which are almost identical. Here θ = 0.03   cm . The numerical data, averaged over 20,000 photons, are represented by dots.

Fig. 6
Fig. 6

Relative difference of the collision numbers in the bins nearest to the interface versus distance along the interface. The analytical solution corresponding to the basic boundary conditions is shown with a solid line denoted by ( b ) . The analytical solution corresponding to the Aronson boundary conditions is shown with a solid line denoted by ( A ) . Dots depict the numerical data averaged over 20,000 photons. The parameters of the media are μ s 1 = 10 cm 1 and n 1 = 2 , and μ s 2 = 20 cm 1 and n 2 = 1 . The data are obtained for a time interval T = 3.333   ns . Here θ = 0.03   cm .

Fig. 7
Fig. 7

Collision numbers in the bins versus time. Solid lines denoted by ( A ) and ( b ) represent analytical solutions calculated as w ( A ) Δ V Δ t and w ( b ) Δ V Δ t , respectively, for bins centered at z = 0.15   cm and r = 0.45   cm . The difference between these solutions is denoted by dif, and the relative difference, ( w ( A ) w ( b ) ) / w ( b ) , is denoted by reldif. The dots depict the Monte Carlo data averaged over 50,000 photons. The dimensions of the bins are Δ r = 0.1   cm , Δ z = 0.1   cm , and Δ t = 0.033   ns . Here θ = 0.3   cm .

Equations (51)

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ϕ 1 n 1 2 = ϕ 2 n 2 2 ,
D 1 ϕ 1 z = D 2 ϕ 2 z ,
ϕ 1 n 1 2 = ϕ 2 n 2 2 + θ n 2 2 ϕ 2 z ,
θ = D 2 C ( n 2 / n 1 ) ,
θ = D 2 n 2 2 n 1 2 C ( n 1 n 2 ) .
1 v 1 ϕ 1 t D 1 ( 2 ϕ 1 x 2 + 2 ϕ 1 y 2 + 2 ϕ 1 z 2 ) + β 1 ϕ 1 = q 0 v 1 δ ( t ) δ ( x ) δ ( y ) δ ( z z 0 ) ,
1 v 2 ϕ 2 t D 2 ( 2 ϕ 2 x 2 + 2 ϕ 2 y 2 + 2 ϕ 2 z 2 ) + β 2 ϕ 2 = 0 ,
u j = 0 ϕ j e p t d t .
ϕ s = 1 8 ( π D 1 v 1 t ) 3 / 2 e { [ r 2 + ( z z 0 ) 2 ] / 4 D 1 v 1 t } β 1 v 1 t ,
u s = 1 4 π D 1 v 1 r 2 + ( z z 0 ) 2 e r 2 + ( z z 0 ) 2 ( p + β 1 v 1 ) / D 1 v 1 ,
e a r 2 + z 2 r 2 + z 2 = 0 e z λ 2 + a 2 J 0 ( λ r ) λ d λ λ 2 + a 2 ,
u s = 1 4 π D 1 v 1 0 e | z z 0 | η 1 J 0 ( λ r ) λ η 1 d λ ,
η j = λ 2 + p + β j v j D j v j .
u 1 = 1 4 π D 1 v 1 R 1 e R 1 ( p + β 1 v 1 ) / D 1 v 1 + 0 f 1 e z η 1 J 0 ( λ r ) λ d λ ,
u 2 = 0 f 2 e z η 2 J 0 ( λ r ) λ d λ ,
u 1 = u 2 N 2 + θ N 2 u 2 z ,
γ 2 u 1 = u 2 ,
N = n 2 n 1 ,     γ = D 1 D 2 .
f 1 = e z 0 η 1 4 π D 1 v 1 η 1 ( 1 + 2 η 1 ( 1 + θ η 2 ) μ η 2 + η 1 ( 1 + θ η 2 ) ) ,
f 2 = N 2 e z 0 η 1 2 π D 1 v 1 [ μ η 2 + η 1 ( 1 + θ η 2 ) ] ,
μ = N 2 γ 2 .
D 1 v 1 = D 2 v 2 ,     β 1 v 1 = β 2 v 2
γ = 1 N ,     β 2 β 1 = N ,
u 1 = 1 4 π D 1 v 1 R 1 e R 1 ( p + β 1 v 1 ) / D 1 v 1 1 4 π D 1 v 1 R 2 e R 2 ( p + β 1 v 1 ) / D 1 v 1 + 1 2 π D 1 v 1 0 ( 1 + θ η 1 ) e ( z + z 0 ) η 1 J 0 ( λ r ) λ ( N 3 + 1 + θ η 1 ) η 1 d λ ,
u 2 = N 2 2 π D 1 v 1 0 e ( z z 0 ) η 1 J 0 ( λ r ) λ ( N 3 + 1 + θ η 1 ) η 1 d λ ,
L 1 { e a p p ( b + p ) } = e b 2 t + a b   Erfc ( a 2 t + b t ) ,
0 λ J 0 ( λ r ) e a λ 2 d λ = 1 2 a exp ( r 2 4 a ) .
ϕ 1 ( A ) = q 0 ( 4 π D 1 v 1 t ) 3 / 2 { e R 1 2 / 4 D 1 v 1 t + e R 2 2 / 4 D 1 v 1 t 2 N 3 π D 1 v 1 t θ exp [ ( ( 1 + N 3 ) D 1 v 1 t θ + ( z + z 0 ) 2 D 1 v 1 t ) 2 ] e R 2 2 / 4 D 1 v 1 t   Erfc ( ( z + z 0 ) 2 D 1 v 1 t + ( 1 + N 3 ) D 1 v 1 t θ ) } ,
ϕ 2 ( A ) = q 0 N 2 4 π D 1 v 1 t θ exp [ ( ( 1 + N 3 ) D 1 v 1 t θ + ( z 0 z ) 2 D 1 v 1 t ) 2 ] e R 1 2 / 4 D 1 v 1 t   Erfc ( ( z 0 z ) 2 D 1 v 1 t + ( 1 + N 3 ) D 1 v 1 t θ ) ,
ϕ 1 ( b ) = q 0 ( 4 π D 1 v 1 t ) 3 / 2 { e R 1 2 / 4 D 1 v 1 t + ( 1 N 3 ) ( 1 + N 3 ) e R 2 2 / 4 D 1 v 1 t } ,
ϕ 2 ( b ) = q 0 ( 4 π D 1 v 1 t ) 3 / 2 2 N 2 ( 1 + N 3 ) e R 1 2 / 4 D 1 v 1 t ,
Erfc [ a + b x ] e ( a + b x ) 2 ( 1 b x π a b 2 x 2 π + ) ,
( ϕ 1 ϕ 2 ϕ 1 + ϕ 2 ) z = 0 ( b ) = 1 N 2 1 + N 2 .
( ϕ 1 ϕ 2 ϕ 1 + ϕ 2 ) z = 0 ( A ) 1 ,     as   t 0 ,
( ϕ 1 ϕ 2 ϕ 1 + ϕ 2 ) z = 0 ( A ) 1 N 2 1 + N 2 ,     as   t .
p ¯ j μ s j = ln ( ξ ) ,
R ̃ 12 = 1 2 [ ( n 2   cos   φ 1 n 1   cos   φ 2 n 2   cos   φ 1 + n 1   cos   φ 2 ) 2 + ( n 1   cos   φ 1 n 2   cos   φ 2 n 1   cos   φ 1 + n 2   cos   φ 2 ) 2 ] ,
sin   φ 1 sin   φ 2 = n 2 n 1 .
p ̃ 1 μ s 1 + p ̃ 2 μ s 2 = ln ( ξ ) ,
w 1 ( b ) = μ s 1 v 1 ( 3 μ s 1 ) 3 / 2 ( 4 π v 1 t ) 3 / 2 { e 3 μ s 1 R 1 2 / 4 v 1 t + ( 1 N 3 ) ( 1 + N 3 ) e 3 μ 1 R 2 2 / 4 v 1 t } ,
w 2 ( b ) = μ s 1 v 1 ( 3 μ s 1 ) 3 / 2 ( 4 π v 1 t ) 3 / 2 2 N ( 1 + N 3 ) e 3 μ s 1 R 1 2 / 4 v 1 t ,
μ s 1 n 1 = μ s 2 n 2 .
w 1 ( b ) μ s 1 n 1 2 = w 2 ( b ) μ s 2 n 2 2
w 1 ( A ) = μ s 1 v 1 ( 3 μ s 1 ) 3 / 2 ( 4 π v 1 t ) 3 / 2 { e 3 μ s 1 R 1 2 / 4 v 1 t + e 3 μ s 1 R 2 2 / 4 v 1 t 2 N 3 π v 1 t θ 3 μ s 1 exp [ ( ( N 3 + 1 ) v 1 t θ 3 μ s 1 + ( z + z 0 ) 3 μ s 1 2 v 1 t ) 2 ] e 3 μ s 1 R 2 2 / 4 v 1 t   Erfc ( ( z + z 0 ) 3 μ s 1 2 v 1 t + ( 1 + N 3 ) v 1 t θ 3 μ s 1 ) } ,
w 2 ( A ) = 3 μ s 1 2 N 4 π t θ exp [ ( ( 1 + N 3 ) v 1 t θ 3 μ s 1 + ( z z 0 ) 3 μ s 1 2 v 1 t ) 2 ] e 3 μ s 1 R 1 2 / 4 v 1 t   Erfc ( ( z 0 z ) 3 μ s 1 2 v 1 t + ( 1 + N 3 ) v 1 t θ 3 μ s 1 ) ,
W j = 0 T w j d t .
W 1 ( b ) = 3 μ s 1 2 4 π { 1 R 1 Erfc ( R 1 3 μ s 1 2 v 1 T ) + 1 R 2 ( 1 N 3 ) ( 1 + N 3 ) Erfc ( R 2 3 μ s 1 2 v 1 T ) } ,
W 2 ( b ) = 3 μ s 1 2 N 2 π R 1 ( 1 + N 3 ) Erfc ( R 1 3 μ s 1 2 v 1 T ) ,
W 1 ( A ) = 3 μ s 1 2 4 π { 1 R 1 Erfc ( R 1 3 μ s 1 4 v 1 T ) + 1 R 2 Erfc ( R 2 3 μ s 1 4 v 1 T ) 2 N 3 θ 0 T exp [ ( ( 1 + N 3 ) u v 1 θ 3 μ s 1 + ( z + z 0 ) 3 μ s 1 2 u v 1 ) 2 3 μ s 1 R 2 2 4 v 1 u 2 ] Erfc ( ( 1 + N 3 ) u v 1 θ 3 μ s 1 + ( z + z 0 ) 3 μ s 1 2 u v 1 ) d u u } ,
W 2 ( A ) = 3 μ s 1 2 N 2 π θ 0 T { exp [ ( ( 1 + N 3 ) u v 1 θ 3 μ s 1 + ( z 0 z ) 3 μ s 1 2 u v 1 ) 2 3 μ s 1 R 1 2 4 v 1 u 2 ] Erfc ( ( 1 + N 3 ) u v 1 θ 3 μ s 1 + ( z 0 z ) 3 μ s 1 2 u v 1 ) d u u } ,
K ( z ) = W 1 ( z ) W 2 ( z ) W 1 ( z ) + W 2 ( z ) ,

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