Abstract

Time-domain analytical solutions of the diffusion equation for photon migration through highly scattering two- and three-layered slabs have been obtained. The effect of the refractive-index mismatch with the external medium is taken into account, and approximate boundary conditions at the interface between the diffusive layers have been considered. A Monte Carlo code for photon migration through a layered slab has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the mean path length followed by photons inside each diffusive layer and the shape of the temporal profile of received photons, while discrepancies are observed for the continuous-wave reflectance or transmittance.

© 2002 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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2000 (2)

1999 (3)

1998 (5)

1997 (2)

1996 (1)

1995 (1)

1994 (2)

1992 (2)

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

1989 (3)

1988 (2)

1980 (1)

1979 (1)

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Alexandrakis, G.

Alianelli, L.

Aronson, R.

Arridge, S. R.

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Avrillier, S.

Bays, R.

Blumetti, C.

Bolin, F. P.

Bonner, R.

Carraresi, S.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, UK, 1959), Chap. XIV.

Chance, B.

Contini, D.

Cope, M.

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Dayan, I.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Delpy, D. T.

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Dognitz, N.

Essenpreis, M.

Fantini, S.

Farrel, T. J.

Farrell, T. J.

Feng, T. C.

Ference, R. J.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 7.

Firbank, M.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Franceschini, M. A.

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

K. Furutsu, “Diffusion equation derived from space-time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
[CrossRef]

Glanzmann, T.

A. Kienle, 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, T. Glanzmann, G. Wagnieres, H. van de Bergh, “Investigation of two-layered media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[CrossRef]

Graham, M. D.

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Gratton, E.

Haskell, R. C.

Havlin, S.

Hielscher, A. H.

Jacques, S. L.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, UK, 1959), Chap. XIV.

Keijer, M.

Kiefer, J.

Kienle, A.

Kudo, N.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Lin, L.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Liu, H.

Maier, J. S.

Martelli, F.

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25, 1508–1510 (2000).
[CrossRef]

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

F. Martelli, A. Sassaroli, Y. Yamada, “Analytical solution of the time-dependent photon diffusion equation for a layered medium,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 79–89 (1999).
[CrossRef]

McAdams, M. S.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 7.

Nieto-Vesperinas, M.

Niwayama, M.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Nossal, R.

Okada, E.

Patterson, M.

Patterson, M. S.

Paunescu, L. A.

Prat, J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Preuss, L. E.

Ripoll, J.

Roach, G. F.

G. F. Roach, Green’s Functions (Van Nostrand Reinhold, London, 1970), Chap. 9.

Sassaroli, A.

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25, 1508–1510 (2000).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, “Analytical solution of the time-dependent photon diffusion equation for a layered medium,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 79–89 (1999).
[CrossRef]

Say, T. T.

Schmitt, J. M.

Schweiger, M.

Shao, J.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Shiga, T.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Star, W. M.

Storchi, P. R. M.

Svaasand, L. O.

Taitelbaum, H.

Takatani, S.

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Taylor, R. C.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Tinet, E.

Tittel, F. K.

Tromberg, B. J.

Tualle, J.-M.

van de Bergh, H.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Wagnieres, G.

Walker, E. C.

Wall, R. T.

Weiss, G. H.

Wilson, B. C.

Yamada, Y.

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25, 1508–1510 (2000).
[CrossRef]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, “Analytical solution of the time-dependent photon diffusion equation for a layered medium,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 79–89 (1999).
[CrossRef]

Yamamoto, K.

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

Zaccanti, G.

Zhou, G. X.

Appl. Opt. (14)

M. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the non-invasive determination of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

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

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

R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layer media,” Appl. Opt. 27, 3382–3391 (1988).
[CrossRef] [PubMed]

H. Taitelbaum, S. Havlin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

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

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, H. van de Bergh, “Noninvasive determination of the optical properties of two-layered turbid medium,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

A. Kienle, T. Glanzmann, G. Wagnieres, H. van de Bergh, “Investigation of two-layered media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[CrossRef]

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

T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
[CrossRef]

M. A. Franceschini, S. Fantini, L. A. Paunescu, J. S. Maier, E. Gratton, “Influence of a superficial layer in the quantitative spectroscopic study of strongly scattering media,” Appl. Opt. 37, 7447–7458 (1998).
[CrossRef]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[CrossRef] [PubMed]

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Lett. (2)

Phys. Med. Biol. (2)

A. Kienle, 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]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Other (6)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 7.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, UK, 1959), Chap. XIV.

G. F. Roach, Green’s Functions (Van Nostrand Reinhold, London, 1970), Chap. 9.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

F. Martelli, A. Sassaroli, Y. Yamada, “Analytical solution of the time-dependent photon diffusion equation for a layered medium,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 79–89 (1999).
[CrossRef]

M. Niwayama, L. Lin, J. Shao, T. Shiga, N. Kudo, K. Yamamoto, “Quantitative measurement of muscle oxygenation by NIRS: Analysis of the influence of a subcutaneous fat layer and skin,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 291–299 (1999).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometrical scheme and symbols used (a) for a two-layered infinite slab and (b) for a three-layered infinite slab.

Fig. 2
Fig. 2

Time-resolved reflectance R from a two-layered slab and time-resolved mean path lengths l0 and l1 followed by received photons in the first and in the second layer, respectively. s0=2 mm, s1=100 mm, μa1=μa0=0, μs0=1 mm-1, nd=1.4, n0=1. (a) μs1=0.5 mm-1, (b) μs1=2 mm-1.

Fig. 3
Fig. 3

As in Fig. 2, with s0=4 mm, s1=100 mm, μa1=μa0=0, μs0=1 mm-1. (a) μs1=0.5 mm-1, (b) μs1=2 mm-1.

Fig. 4
Fig. 4

As in Fig. 2, with s0=8 mm, s1=100 mm, μa1=μa0=0, μs0=1 mm-1. (a) μs1=0.5 mm-1, (b) μs1=2 mm-1.

Fig. 5
Fig. 5

Ratio RDE/RMC of the cw reflectance for a two-layered slab versus the source–receiver distance ρ. The symbols RDE and RMC represent the cw reflectance by the integral of Eq. (13) and by MC simulations, respectively. The figure refers to s0=4 mm, s1=100 mm, μs0=1 mm-1. The data are plotted for two values of the absorption coefficient of the layers, μa1=μa0=0 for the bottom figure and μa1=μa0=0.01 mm-1 for the top figure, and for two values of μs1, 2 mm-1 (triangles) and 0.5 mm-1 (squares).

Fig. 6
Fig. 6

As in Fig. 2, with s0=4 mm, s1=100 mm, μs1=μs0=1 mm-1, μa0=0, μa1=0.01 mm-1.

Fig. 7
Fig. 7

Time-resolved transmittance through a two-layered slab with s0=6.67 mm, s1=13.33 mm, ρ=0, μs1=μs0=1 mm-1, μa0=0, μa1=0.01 mm-1. The symbols are the MC results. The thick curve represents the analytical solution of the DE [Eq. (14)]. The curves Hom A and Hom B represent the solutions of the DE for a homogeneous slab 20 mm thick, μs=1 mm-1, and μa=0 and μa=0.01 mm-1, respectively.

Fig. 8
Fig. 8

Time-resolved reflectance from a three-layered slab with s0=6.67 mm, s1=6.67 mm, s2=6.67 mm, ρ=18 mm, μs2=μs1=μs0=1 mm-1, μa0=0, μa1=0.01 mm-1, μa2=0. The symbols are the MC results. The thick curve represents the analytical solution of the DE. The curve denoted Hom represents the reflectance from a homogeneous slab 20 mm thick, μs=1 mm-1 and μa=0 obtained with the DE.

Tables (1)

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Table 1 Parameter μsfit Retrieved from the Fit on MC Data a

Equations (42)

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1νt-D2+μaΦd(r, t)=Q(r, t),
Q(r, t)=δ(x)δ(y)δ(z-z0)δ(t),
G(r, r, t-T)=l=1ν ϕzl(z)ϕzl(z)Nl×exp[-Kzl2Dν(t-T)]×exp-(ρ-ρ)24Dν(t-T)4πDν(t-T)×exp[-μaν(t-T)],
Φi(r, r, t-T)=Φzi(z, z, t-T)Φρi(ρ, ρ, t-T)×exp[-μaiνi(t-T)],
Φzi(z, z, t-T)=l=1νi1/3ϕzli(z)ϕzli(z)Nl×exp[-Kzli2Diνi(t-T)],
ϕzli(z)=blisin (Kzliz+γzli),
Φρi(ρ, ρ, t-T)=νi2/3exp-(ρ-ρ)24Diνi(t-T)4πDiνi(t-T).
ΦρiΦρi+1,
Φi(r, t, s0, s1)
=l=1νblisin(Kzliz+γzli)sin(Kzl0z0+γzl0)Nl
×exp(-Kzli2Diνt)
×exp-ρ24Diνt4πDiνtexp(-μaiνt),i=0,1,
Φ0(r),0z<s0;Φ1(r),s0z<s0+s1.
1D0Kzl0tan (Kzl0s0+γzl0)
=1D1Kzl1tan (Kzl1s0+γzl1),
Kzl12=Kzl02D0+(μa0-μa1)D1,
γzl0=2AD0Kzl0,
γzl1=-Kzl1[(s0+s1)+2AD1].
bl0=1,bl1=sin(Kzl0s0+γzl0)sin(Kzl1s0+γzl1).
Nl=-2AD0s0ϕzl02(z)dz+s0s0+s1+2ADϕzl12(z)dz,
Nl=s02+γzl02Kzl0-14Kzl0sin[2(Kzl0s0+γzl0)]-bl122Kzl1 (Kzl1s0+γzl1)+bl124Kzl1sin[2(Kzl1s0+γzl1)].
R(ρ, t, s0, s1)=l=1νD0Kzl0cos(γzl0)sin(Kzl0z0+γzl0)Nl×exp(-Kzl02D0νt) exp-ρ24D0νt4πD0νtexp(-μa0νt),
T(ρ, t, s0, s1)=l=1νD1Kzl1bl1×cos(Kzl1(s0+s1)+γzl1)sin(Kzl0z0+γzl0)Nl×exp(-Kzl12D1νt) exp-ρ24D1νt4πD1νtexp(-μa1νt).
li(t)=- ln P(μai, t)μai,
ln[R(ρ2, t)/R(ρ1, t)]=-(ρ22-ρ12)/4D0νt.
ln[R(ρ2, t)/R(ρ1, t)]=A-(ρ22-ρ12)/4D0νt.
W=exp-iμaili.
Φi(r, t, s0, s1, s2)
=l=1νblisin(Kzliz+γzli)sin(Kzl0z0+γzl0)Nl×exp(-Kzli2Diνt) ×exp[-ρ2/(4Diνt)]4πDiνtexp(-μaiνt),i=0, 1, 2,
Φ0(r), 0z<s0;Φ1(r), s0z<s0+s1;Φ2(r), s0+s1z<s0+s1+s2.
1D0Kzl0tan(Kzl0s0+γzl0)
=1D1Kzl1tan(Kzl1s0+γzl1),
1D1Kzl1tan[Kzl1(s0+s1)+γzl1]
=1D2Kzl2tan[Kzl2(s0+s1)+γzl2],
Kzl12=Kzl02D0+(μa0-μa1)D1,
Kzl22=Kzl12D1+(μa1-μa2)D2,
γzl0=2AD0Kzl0,
γzl2=-Kzl2[(s0+s1+s2)+2AD2].
bl0=1,bl1=sin(Kzl0s0+γzl0)sin(Kzl1s0+γzl1),
bl2=bl1sin(Kzl1(s0+s1)+γzl1)sin(Kzl2(s0+s1)+γzl2).
Nl=-2AD0s0ϕzl02(z)dz+s0s0+s1ϕzl12(z)dz+s0+s1s0+s1+s2+2AD2ϕzl22(z)dz,
Nl=s02+γzl02Kzl0+bl12s12-bl222Kzl2 [Kzl2(s0+s1)+γzl2]-14Kzl0sin [2(Kzl0s0+γzl0)]-bl124Kzl1(sin [2[Kzl1(s0+s1)+γzl1]]- sin [2(Kzl1s0+γzl1)])+bl224Kzl2×sin {2[Kzl2(s0+s1)+γzl2]}.

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