Abstract

Propagation of light emitted by an instantaneous source located above a plane interface between two semi-infinite turbid media is considered using the diffusion approximation. Green functions are derived for an instantaneous line source and an instantaneous point source by the method of Bellman et al. [Philos. Mag. 40, 297 (1949)], which is based on integral transforms. Both two-dimensional and three-dimensional Green functions for diffuse light have been obtained in the form of single integrals that allow for fast calculation of the specific intensity in the whole space. The influence of the optical parameters of the two media (diffusion coefficients, absorptions, and refractive indices) on the shapes of the contour lines of the specific intensity is analyzed.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2004

F. Martelli, S. Del 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]

M. L. Shendeleva, "Green functions for diffuse photon-density waves generated by a line source in two nonabsorbing turbid media in contact," Appl. Opt. 43, 1638-1642 (2004).
[CrossRef] [PubMed]

M. L. Shendeleva, "Green functions for diffuse light in a medium comprising two turbid half-spaces," Appl. Opt. 43, 5334-5543 (2004).
[CrossRef] [PubMed]

2002

2001

2000

1999

A. Kienle and T. Glanzmann, "In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model," Phys. Med. Biol. 44, 2689-2702 (1999).
[CrossRef] [PubMed]

J. Ripoll and M. Nieto-Vesperinas, "Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces," J. Opt. Soc. Am. A 16, 1947-1957 (1999).
[CrossRef]

1998

1997

1996

1995

I. A. Vitkin, B. C. Wilson, and R. R. Anderson, "Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue," Appl. Opt. 34, 2973-2982 (1995).
[CrossRef] [PubMed]

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

1994

1990

1989

1949

R. Bellman, R, E. Marshak, and G. M. Wing, "Laplace transform solution of two-medium neutron ageing problem," Philos. Mag. 40, 297-308 (1949).

Abramowitz, M.

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Alexandrakis, G.

Anderson, R. R.

Avrillier, S.

Bays, R.

Bellman, R.

R. Bellman, R, E. Marshak, and G. M. Wing, "Laplace transform solution of two-medium neutron ageing problem," Philos. Mag. 40, 297-308 (1949).

Chance, B.

Contini, D.

Culver, J. P.

Del Bianco, S.

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

Dögnitz, N.

Farrell, T. J.

Feng, T.-C.

Glanzmann, T.

A. Kienle and T. Glanzmann, "In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model," Phys. Med. Biol. 44, 2689-2702 (1999).
[CrossRef] [PubMed]

Haskell, R. C.

Hielscher, A. H.

Jacques, S. L.

A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, and S. L. Jacques, "Time-resolved photon emission from layered turbid media," Appl. Opt. 35, 719-728 (1996).
[CrossRef] [PubMed]

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

Kienle, A.

A. Kienle and T. Glanzmann, "In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model," Phys. Med. Biol. 44, 2689-2702 (1999).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, "Noninvasive determination of the optical properties of two-layered turbid media," Appl. Opt. 37, 779-791 (1998).
[CrossRef]

Leontovich, M. A.

M. A. Leontovich, Introduction to Thermodynamics. Statistical Physics (Nauka, 1989) (in Russian).

Liu, H.

Marshak, R

R. Bellman, R, E. Marshak, and G. M. Wing, "Laplace transform solution of two-medium neutron ageing problem," Philos. Mag. 40, 297-308 (1949).

Martelli, F.

McAdams, M. S.

Nieto-Vesperinas, M.

Ntziachristos, V.

Pattanayak, D. N.

Patterson, M. S.

Pham, T. H.

Prat, J.

Ripoll, J.

Sassaroli, A.

Schmitt, J. M.

Shendeleva, M. L.

Spott, T.

Stegun, I.

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Svaasand, L. O.

Tinet, E.

Tittel, F. K.

Tromberg, B. J.

Tsay, T.-T.

Tualle, J.-M.

van den Bergh, H.

Vitkin, I. A.

Wagnières, G.

Walker, E. C.

Wall, R. T.

Wang, L.

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

Wilson, B. C.

Wing, G. M.

R. Bellman, R, E. Marshak, and G. M. Wing, "Laplace transform solution of two-medium neutron ageing problem," Philos. Mag. 40, 297-308 (1949).

Yamada, Y.

Yodh, A. G.

Zaccanti, G.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, "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.

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. 1. Theory," Appl. Opt. 36, 4587-4599 (1997).
[CrossRef] [PubMed]

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, "Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium," Appl. Opt. 37, 7401-7409 (1998).
[CrossRef]

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, "Monte Carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain," Appl. Opt. 39, 2235-2244 (2000).
[CrossRef]

I. A. Vitkin, B. C. Wilson, and R. R. Anderson, "Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue," Appl. Opt. 34, 2973-2982 (1995).
[CrossRef] [PubMed]

A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, and S. L. Jacques, "Time-resolved photon emission from layered turbid media," Appl. Opt. 35, 719-728 (1996).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, "Noninvasive determination of the optical properties of two-layered turbid media," Appl. Opt. 37, 779-791 (1998).
[CrossRef]

T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, "Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance," Appl. Opt. 39, 4733-4745 (2000).
[CrossRef]

M. L. Shendeleva, "Green functions for diffuse photon-density waves generated by a line source in two nonabsorbing turbid media in contact," Appl. Opt. 43, 1638-1642 (2004).
[CrossRef] [PubMed]

M. L. Shendeleva, "Green functions for diffuse light in a medium comprising two turbid half-spaces," Appl. Opt. 43, 5334-5543 (2004).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed.

L. Wang, S. L. Jacques, and L. Zheng, "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

Opt. Commun.

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]

Philos. Mag.

R. Bellman, R, E. Marshak, and G. M. Wing, "Laplace transform solution of two-medium neutron ageing problem," Philos. Mag. 40, 297-308 (1949).

Phys. Med. Biol.

A. Kienle and T. Glanzmann, "In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model," Phys. Med. Biol. 44, 2689-2702 (1999).
[CrossRef] [PubMed]

Phys. Rev. E

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

Other

M. A. Leontovich, Introduction to Thermodynamics. Statistical Physics (Nauka, 1989) (in Russian).

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the model. Two turbid half-spaces with diffusion coefficients D j , absorption coefficients β j , and refractive indices n j (where j = 1, 2) are in contact along the interface at z = 0. A line or point source (indicated by A) is located at distance z 0 from the interface.

Fig. 2
Fig. 2

Influence of parameter γ in the case β 1 = β 2 = 0 and N = 1. D 1 v 1 = 1 cm 2 / s . The source is located at z 0 = 1   cm . Contour lines of the specific intensity are shown at t = 0.5 s. Dimensions on the axes are in centimeters.

Fig. 3
Fig. 3

(Color online) Influence of the refractive index mismatch for the case γ = 0.3 , β 1 = β 2 = 0 . D 1 v 1 = 1 cm 2 / s . The upper row shows contour lines of the reduced specific intensity ϕ / n 2 , and the lower row shows the contour lines of the specific intensity for N = 1.3 (left) and N = 0.65 (right) at t = 0.5 s. The source is located at z 0 = 1   cm . Dimensions on the axes are in centimeters.

Fig. 4
Fig. 4

Influence of the absorption for N = 1, γ = 0.3 . D 1 v 1 = 1 cm 2 / s . The contour lines of the specific intensity are shown for two nonabsorbing half-spaces (solid curves) and for the absorbing second medium with β 2 v 2 = 2 s 1 (dashed curves). The source is located at z 0 = 1   cm . Contour lines of the specific intensity are shown at t = 0.5 s. Dimensions on the axes are in centimeters.

Equations (73)

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1 v 1 ϕ 1 t D 1 ( 2 ϕ 1 x 2 + 2 ϕ 1 z 2 ) + β 1 ϕ 1 = q 0 v 1 δ ( t ) δ ( x ) δ ( z z 0 ) ,
1 v 2 ϕ 2 t D 2 ( 2 ϕ 2 x 2 + 2 ϕ 2 z 2 ) + β 2 ϕ 2 = 0 ,
ϕ 1 n 1 2 = ϕ 2 n 2 2 ,
D 1 ϕ 1 z = D 2 ϕ 2 z ,
v 1 v 2 = n 2 n 1 = N ,
D 1 D 2 = γ ,
N 2 ϕ 1 = ϕ 2 ,
γ 2 ϕ 1 z = ϕ 2 z .
u j = 0 ϕ j e p t d t .
2 u 1 x 2 + 2 u 1 z 2 p u 1 D 1 v 1 β 1 u 1 D 1 = 1 D 1 v 1 δ ( x ) δ ( z z 0 ) ,
2 u 2 x 2 + 2 u 2 z 2 p u 2 D 2 v 2 β 2 u 2 D 2 = 0 ,
N 2 u 1 = u 2 , γ 2 u 1 z = u 2 z .
ϕ S 2 D = 1 4 π D 1 v 1 t exp ( r 1 2 4 D 1 v 1 t β 1 v 1 t ) ,
r 1 = x 2 + ( z z 0 ) 2 .
u S = 1 2 π D 1 v 1 K 0 ( r 1 p + β 1 v 1 D 1 v 1 ) ,
K 0 ( a x 2 + z 2 ) = 0 exp ( z λ 2 + a 2 ) cos ( λ x ) d λ λ 2 + a 2
u 1 = 1 2 π D 1 v 1 K 0 ( r 1 p + β 1 v 1 D 1 v 1 ) + 0 f 1 ( λ ) exp ( z λ 2 + p + β 1 v 1 D 1 v 1 ) cos ( λ x ) d λ ,
u 2 = 0 f 2 ( λ ) exp ( z λ 2 + p + β 2 v 2 D 2 v 2 ) cos ( λ x ) d λ .
f 1 = e z 0 η 1 2 π D 1 v 1 η 1 ( 1 + 2 η 1 η 1 + μ η 2 ) ,
f 2 = N 2 e z 0 η 1 π D 1 v 1 ( η 1 + μ η 2 ) ,
η j = λ 2 + p + β j v j D j v j
μ = N 2 γ 2 .
u 1 = 1 2 π D 1 v 1 K 0 ( r 1 p + β 1 v 1 D 1 v 1 ) 1 2 π D 1 v 1 × K 0 ( r 2 p + β 1 v 1 D 1 v 1 ) + S 1 ,
S 1 = 1 π D 1 v 1 0 e ( z + z 0 ) η 1 cos ( λ x ) η 1 + μ η 2  d λ ,
r 2 = x 2 + ( z + z 0 ) 2 .
ϕ 1 = 1 4 π D 1 v 1 t exp ( r 1 2 4 D 1 v 1 t β 1 v 1 t ) 1 4 π D 1 v 1 t exp ( r 2 2 4 D 1 v 1 t β 1 v 1 t ) + W 1 ,
W 1 = L 1 { S 1 } ,
1 η 1 + μ η 2 = 0 e w ( η 1 + μ η 2 ) d w .
S 1 = 1 π D 1 v 1 0 E 1 cos ( λ x ) d λ ,
E 1 = 0 e η 1 ( w + z + z 0 ) e μ w η 2 d w .
L 1 { e a p } = a 4 π t 3 exp ( a 2 4 t ) ,
L 1 { f ( p ) g ( p ) } = 0 t F ( u ) G ( t u ) d u .
A 1 = L 1 ( E 1 ) = μ ν D 1 v 1 2 π t 0 1 exp [ λ 2 D 1 v 1 t ( 1 u + u ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 1 d u ,
  G 1 = 1 2 π [ D 1 v 1 t u ( 1 u ) ] 3 / 2 0 w ( w + z + z 0 ) × exp [ ( w + z + z 0 ) 2 4 D 1 v 1 t ( 1 u ) μ 2 ν 2 w 2 4 D 1 v 1 t u ] d w .
ν = γ N = D 1 v 1 D 2 v 2 .
0 ( w + a ) ( w + b ) exp [ k ( w + a ) 2 q ( w + b ) 2 ] d w = π 4 ( k + q ) 3 / 2 exp [ ( a b ) 2 k q k + q ] erfc ( a k + b q k + q ) × [ 1 2 k q ( a b ) 2 k + q ] + ( b k + a q ) 2 ( k + q ) 2 × exp [ ( a 2 k + b 2 q ) ] ,
G 1 = 1 [ u + μ 2 ν 2 ( 1 u ) ] 3 / 2 exp ( μ 2 ν 2 ( z + z 0 ) 2 4 D 1 v 1 t ( u + μ 2 ν 2 ( 1 u ) ) ) × Erfc [ ( z + z 0 ) u 2 D 1 v 1 t ( 1 u ) u + μ 2 ν 2 ( 1 u ) ] × [ 1 μ 2 ν 2 ( z + z 0 ) 2 2 D 1 v 1 t ( u + μ 2 ν 2 ( 1 u ) ) ] + μ 2 ν 2 ( z + z 0 ) u ( 1 u ) 2 π D 1 v 1 t ( u + μ 2 ν 2 ( 1 u ) ) 2 exp [ ( z + z 0 ) 2 4 D 1 v 1 t ( 1 u ) ] .
W 1 = L 1 ( S 1 ) = 1 π D 1 v 1 0 A 1 cos ( λ x ) d λ ,
0 e λ 2 a cos ( b λ ) d λ = π 2 a exp ( b 2 4 a ) .
W 1 = μ ν 4 π D 1 v 1 t 0 1 exp [ x 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 1 d u 1 u + u / ν 2 .
ϕ 1 2 D = 1 4 π D 1 v 1 t exp [ r 1 2 4 D 1 v 1 t β 1 v 1 t ] 1 4 π D 1 v 1 t exp [ r 2 2 4 D 1 v 1 t β 1 v 1 t ] + μ ν 4 π D 1 v 1 t 0 1 exp [ x 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 1 d u 1 u + u / ν 2 ,
u 2 = N 2 π D 1 v 1 0 E 2 cos ( λ x ) d λ ,
E 2 = 0 exp [ η 1 ( w + z 0 ) η 2 μ ( w z μ ) ] d w .
A 2 = L 1 ( E 2 ) = μ ν D 1 v 1 2 π t 0 1 exp [ λ 2 D 1 t ( 1 u + u ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 2 d u ,
G 2 = 1 2 π [ D 1 v 1 t u ( 1 u ) ] 3 / 2 0 ( w + z 0 ) ( w z μ ) × exp [ ( w + z 0 ) 2 4 D 1 v 1 t ( 1 u ) μ 2 ν 2 ( w z / μ ) 2 4 D 1 v 1 t u ] d w .
G 2 = 1 ( u + μ 2 ν 2 ( 1 u ) ) 3 / 2 [ 1 ν 2 ( μ z 0 + z ) 2 2 D 1 v 1 t ( u + μ 2 ν 2 ( 1 u ) ) ] × exp [ ν 2 ( μ z 0 + z ) 2 4 D 1 v 1 t ( u + μ 2 ν 2 ( 1 u ) ) ] × erfc [ z 0 u μ ν 2 z ( 1 u ) 2 D 1 v 1 t u ( 1 u ) u + μ 2 ν 2 ( 1 u ) ] + ( z u + z 0 μ 3 ν 2 ( 1 u ) ) μ D 1 v 1 t u ( 1 u ) [ u + μ 2 ν 2 ( 1 u ) ] 2 × exp [ z 2 ν 2 ( 1 u ) + z 0 2 u 4 D 1 v 1 t u ( 1 u ) ] .
ϕ 2 = N 2 π D 1 v 1 0 A 2 cos ( λ x ) d λ .
ϕ 2 2 D = μ ν N 2 4 π D 1 v 1 t 0 1 exp ( x 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ) × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 2 d u 1 u + u / ν 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
=   1 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 ,
ϕ S 3 D = 1 8 ( π D 1 v 1 t ) 3 / 2 exp ( 1 2 4 D 1 v 1 t β 1 v 1 t ) ,
1 = ρ 2 + ( z z 0 ) 2 ,
U S = 1 4 π D 1 v 1 1 exp ( 1 p + β 1 v 1 D 1 v 1 ) .
1 4 π D 1 v 1 1 exp ( 1 p + β 1 v 1 D 1 v 1 ) = 1 4 π D 1 v 1 0 e | z z 0 | η 1 × J 0 ( λ ρ ) λ d λ η 1 ,
U 1 = 1 4 π D 1 v 1 1 exp ( 1 p + β 1 v 1 D 1 v 1 ) 1 4 π D 1 v 1 2 exp ( 2 p + β 1 v 1 D 1 v 1 ) + 1 2 π D 1 v 1 0 e ( z + z 0 ) η 1 J 0 ( λ ρ ) η 1 + μ η 2 λ d λ
U 2 = N 2 2 π D 1 v 1 0 e z η 2 z 0 η 1 J 0 ( λ ρ ) η 1 + μ η 2 λ d λ
2 = ρ 2 + ( z + z 0 ) 2 .
0 λ J 0 ( λ ρ ) e a λ 2 d λ = 1 2 a exp ( ρ 2 4 a ) .
ϕ 1 3 D = 1 ( 4 π D 1 v 1 t ) 3 / 2 exp [ 1 2 4 D 1 v 1 t β 1 v 1 t ] 1 ( 4 π D 1 v 1 t ) 3 / 2 exp [ 2 2 4 D 1 v 1 t β 1 v 1 t ] + μ ν ( 4 π D 1 v 1 t ) 3 / 2 0 1 exp [ ρ 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 1 d u ( 1 u + u / ν 2 ) ,
ϕ 2 3 D = μ ν N 2 ( 4 π D 1 v 1 t ) 3 / 2 0 1 exp [ ρ 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × exp [ β 1 v 1 t ( 1 u ) β 2 v 2 t u ] G 2 d u ( 1 u + u / ν 2 ) ,
+ ϕ j 3 D d y = ϕ j 2 D
  ϕ 1 2 D = 1 4 π D 1 v 1 t exp [ r 1 2 4 D 1 v 1 t ] 1 4 π D 1 v 1 t exp [ r 2 2 4 D 1 v 1 t ] + μ ν 4 π D 1 v 1 t 0 1 exp ( x 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ) × G 1 d u 1 u + u / ν 2
ϕ 2 2 D = μ ν N 2 4 π D 1 v 1 t 0 1 exp ( x 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ) × G 2 d u 1 u + u / ν 2
0 1 G 1 d u = 2 μ ν ( 1 + μ ν ) exp ( ( z + z 0 ) 2 4 D 1 v 1 t ) ,
0 1 G 2 d u = 2 μ ν ( 1 + μ ν ) exp ( ( ν z z 0 ) 2 4 D 1 v 1 t ) ,
ϕ 1 2 D = 1 4 π D 1 v 1 t exp ( r 1 2 4 D 1 v 1 t ) + 1 N 3 1 + N 3 1 4 π D 1 v 1 t exp ( r 2 2 4 D 1 v 1 t ) ,
ϕ 2 2 D = N 2 1 + N 3 1 2 π D 1 v 1 t exp ( r 1 2 4 D 1 v 1 t )
  ϕ 1 3 D = 1 ( 4 π D 1 v 1 t ) 3 / 2 exp [ 1 2 4 D 1 v 1 t ] 1 ( 4 π D 1 v 1 t ) 3 / 2 exp [ 2 2 4 D 1 v 1 t ] + μ ν ( 4 π D 1 v 1 t ) 3 / 2 0 1 exp [ ρ 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × G 1 d u ( 1 u + u / ν 2 )
ϕ 2 3 D = μ ν N 2 ( 4 π D 1 v 1 t ) 3 / 2 0 1 exp [ ρ 2 4 D 1 v 1 t ( 1 u + u / ν 2 ) ] × G 2 d u ( 1 u + u / ν 2 )
  ϕ 1 3 D = 1 ( 4 π D 1 v 1 t ) 3 / 2 exp ( 1 2 4 D 1 v 1 t ) + 1 N 3 1 + N 3 1 ( 4 π D 1 v 1 t ) 3 / 2 exp ( 2 2 4 D 1 v 1 t ) ,
ϕ 2 3 D = 2 N 2 1 + N 3 1 ( 4 π D 1 v 1 t ) 3 / 2 exp ( 1 2 4 D 1 v 1 t ) .
α cr = arcsin ( γ N )

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