Abstract

We have derived the space–time Green’s function for the diffusion equation in layered turbid media, starting from the case of a planar interface between two random scattering media. This new approach for working directly in real space permits highly efficient numerical processing, which is a decisive criterion for the feasibility of the inverse problem in biomedical optics. The results obtained by this method in the case of a two-layered medium are compared with Monte Carlo simulations.

© 2000 Optical Society of America

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  1. B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
    [CrossRef]
  2. T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, H. v. d. Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
  6. H. Taitelbaum, S. Halvin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
    [CrossRef] [PubMed]
  7. 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]
  8. M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
    [CrossRef]
  9. E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [CrossRef] [PubMed]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  11. B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957).
  12. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  13. J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.
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    [CrossRef] [PubMed]
  15. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  16. E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
    [CrossRef]
  17. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  18. S. E. Skipetrov, R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A 217, 181–185 (1996).
    [CrossRef]
  19. E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
    [CrossRef]

1999

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

1998

1997

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

1996

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

S. E. Skipetrov, R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A 217, 181–185 (1996).
[CrossRef]

E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[CrossRef]

1995

1994

1990

1989

1986

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

1979

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]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Amic, E.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

Aronson, R.

Avrillier, S.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[CrossRef]

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

Bays, R.

Bergh, H. v. d.

Chance, B.

Davison, B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957).

Dögnitz, N.

Essenpreis, M.

Farrell, T. J.

Feng, T.-C.

Gélébart, B.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

Glanzmann, T.

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]

Halvin, S.

Haskell, R. C.

Ishimaru, A.

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

Kienle, A.

Luck, J. M.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

Maynard, R.

S. E. Skipetrov, R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A 217, 181–185 (1996).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

McAdams, M. S.

Mochi, M.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

Nieuwenhuizen, T. M.

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

Ollivier, J.-P.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[CrossRef]

Pacelli, G.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

Patterson, M. S.

Prat, J.

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

Recchioni, M. C.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

Schmitt, J. M.

Skipetrov, S. E.

S. E. Skipetrov, R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A 217, 181–185 (1996).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Svaasand, L. O.

Sykes, J. B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957).

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]

Tinet, E.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[CrossRef]

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

Tromberg, B. J.

Tsay, T.-T.

Tualle, J.-M.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

E. Tinet, S. Avrillier, J.-M. Tualle, J.-P. Ollivier, “Fast semi-analytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[CrossRef]

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

Wagnières, G.

Walker, E. C.

Wall, R. T.

Weiss, G. H.

Wilson, B. C.

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Zhou, G. X.

Zirilli, F.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

Appl. Opt.

IEEE Trans. Biomed. Eng.

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. Opt.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, J.-P. Ollivier, “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Optim. Theory Appl.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[CrossRef]

J. Phys. A

E. Amic, J. M. Luck, T. M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media,” J. Phys. A 29, 4915–4955 (1996).
[CrossRef]

Phys. Lett. A

S. E. Skipetrov, R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A 217, 181–185 (1996).
[CrossRef]

Phys. Rev. Lett.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Other

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

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

J.-M. Tualle, E. Tinet, J. Prat, B. Gélébart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in CLEO/Europe-EQEC Technical Digest, Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 192–194.

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Figures (4)

Fig. 1
Fig. 1

Generalization of the method of images in the case of a flat interface between two scattering media. The virtual source consists of an image point source associated with a more complex source distribution.

Fig. 2
Fig. 2

Case of a scattering medium 1 of finite thickness d having a flat interface with a nonscattering outer medium and placed on a semi-infinite scattering medium 2. The diffuse light interacts alternately with the two interfaces, making n round trips.

Fig. 3
Fig. 3

Infinite-medium problem equivalent to the problem illustrated in Fig. 2. The contribution of each boundary to the signal can clearly be seen. This highlight can be really helpful for the inverse problem.

Fig. 4
Fig. 4

Time-resolved reflectance obtained at ρ=2.5 mm at the free surface of a scattering medium 1 (n1=1.33, μa1=0.026 cm-1, ltr1=1 mm) of thickness d=6 mm, placed on semi-infinite scattering medium 2 (n2=1.33, μa2=0.05 cm-1, ltr2=1/3 mm). The figure shows the effect of the number of round trips n of the light between the two interfaces. These results obtained by the diffusion approximation are compared with Monte-Carlo simulations (noisy curves).

Equations (107)

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 z ϕ(ρ, 0+, t)=a z ϕ(ρ, 0-, t),
ϕ(ρ, 0+, t)=bϕ(ρ, 0-, t).a=D2/D1,
b=n12/n22.
ϕ=G1(ρ, z-Z, t)+G1SR(z>0),
ϕ=G2ST(z<0),
SR(ρ, z, Z, t)=δ(ρ)δ(z+Z)δ(t)-4αD2θ(-z-Z)×G2(2)(ρ, α(z+Z), t),
ST(ρ, z, Z, t)=-4D2a [G1(1)(ρ, Z, t)δ(z)+α-1θ(z)×G1(2)(ρ, Z+α-1z, t)],
R^mσ=dzSR(ρ, z, z, t) ¯σ(ρ, z, t),
T^mσ=dzST(ρ, z, z, t)¯σ(ρ, z, t),
Sϕ(ρ, z, t)1+(1+R^f)n=1(R^f R^m)n×(1+R^f)S(ρ, z, t).
ϕ(r, t)=G1Sϕ.
ξn(ρ, z, Z, t)(R^fR^m)nS(ρ, z, Z, t),
ϕn(ρ, z, Z, t)=G1ξn.
ϕ(ρ, z, t)=ϕ0(ρ, z, t)+n=1ϕn(ρ, z, Z, t)-ϕn(ρ, 2(d+zb)-z, Z, t)-ϕn(ρ, z, 2(d+zb)-Z, t)+ϕn(ρ, 2(d+zb)-z, 2(d+zb)-Z, t).
ξn(ρ, z, Z, t)=(-1)nδ(ρ)δ(z-zn)δ(t)+k=0n-1pnk(z-zn)k×G2(k+2)(ρ, α(z-zn), t)θ(z-zn),
pn0=(-1)n+14αnD2,
pn+1k=-pnk-2αk pnk-1,ifk<n,
pn+1n=-2αn pnn-1,
ϕn(ρ, z, Z, t)=(-1)nG1(ρ, z-zn, t)+k=0n-1pnkInk(ρ, z, Z, t),
Ink(ρ, z, Z, t)=0dz0tdτψ(ρ, t, τ)zk×G1(z-z-zn, t-τ)G2(k+2)(αz,τ),
ψ(ρ, t, τ)=exp{-ρ2/[4D1c1(t-τ)+4D2c2τ]}4π[D1c1(t-τ)+D2c2τ]×exp[-μa1c1(t-τ)-μa2c2τ],
Gi(z, t)=ciθ(t) exp[-z2/(4Dicit)](4πDicit)1/2.
ψ=l=0ηl(ρ, t)τl,
Ink=l=0ηl(ρ, t)fnkl(t),
fnkl=-dτ0dzτlzkG1(z-z-zn, t-τ)×G2(k+2)(αz, τ).
fnkl=c1i=0l(-1)l(k+i)!4D21κ QkliD1c1κk+1+i-2l×4D1c1tκ2l-1i2l-1 erfczn-z4D1c1t,
Qk00=--1D2c2k+1,
Qk(l+1)i=k+1+i-2l2 Qkli-α2D2c2 Qkli-1,
Qk(l+1)0=k+1-2l2 Qkl0,
Qk(l+1)(l+1)=-α2D2c2 Qkll,fori0, l+1.
R(ρ, t)=Aϕ(ρ, d, t)+BD1z ϕ(ρ, d, t),
14D1c1t i2l-2 erfczn-z4D1c1t.
Vm=Aim erfc(x)+BD12D1c1t im-1 erfc(x),
fnRkl=c1i=0l(-1)l(k+i)!4D21κ QkliD1c1κk+1+i-2l×4D1c1tκ2l-1V2l-1.
ηl(ρ, t)=exp(-μa1c1t)4πD1texp-ρ24D1c1tk=02lβkl(ρ, t),
β00=1,
βk(l+1)=-μa1c1βkl+k D1-D2D1t β(k-1)l-D1-D24D12c1t β(k-2)ll+1,(βkl=0ifk<0ork>l).
ct G-DΔG+μaG=δ(ρ)δ(z)δ(t).
G¯=-G(ρ, z, t)dz,
ct G¯-DΔtG¯+μaG¯=δ(ρ)δ(t),
z G(ρ, 0-, t)-z G(ρ, 0+, t)
=1Dct-DΔt+μaG¯,
G(1)(ρ, 0-, t)-G(1)(ρ, 0+, t)=1D δ(ρ)δ(t);
2D z G(ρ, 0-, t)=δ(ρ)δ(t).
ϕ(ρ, z, t)=G1(ρ, z-Z, t)+G1(ρ, z+Z, t)-4aD20dλG1(ρ, z+Z+bλ, t)¯G2(2)(ρ, -aλ, t),
ϕ(ρ, z, t)=G1(ρ, z-Z, t)-2D2×0dλ[bG1(1)(ρ, z+Z+bλ, t)¯G2(1)(ρ, -aλ, t)+aG1(ρ, z+Z+bλ, t)¯G2(2)(ρ, -aλ, t)].
ϕ(ρ, z, t)=-4D2a G1(1)(ρ, Z, t)¯G2(ρ, z, t)-4α-1D20dλG1(2)(ρ, Z+bλ, t)¯G2(ρ, z-aλ, t).
ϕ(ρ, z, t)=-4D20dλ[G1(1)(ρ, Z+bλ, t)
¯G2(1)(ρ, z-aλ, t)].
-2D20dλ λ [G1(ρ, Z+Uλ, t)¯G2(1)(ρ,-Vλ, t)]
=G1(ρ, Z, t)( U, V>0).
ξn(ρ, z, Z, t)(R^f R^m)nS(ρ, z, Z, t),
ξn(ρ, z, Z, t)=(-1)nδ(ρ)δ(z-zn)δ(t)+k=0n-1pnk(z-zn)k×G2(k+2)(ρ, α(z-zn), t)θ(z-zn).
R^f f (z)=-f(2(d+z0)-z)
R^mS=dzSR(ρ, z, z, t)¯S(ρ, z, t),
ξn(ρ, z, Z, t)(R^fR^m)ξn-1(ρ, z, Z, t)=-dzSR(ρ, 2(d+z0)-z, z, t)
¯ξn-1(ρ, z, Z, t).
ξ1(ρ, z, Z, t)=-SR(ρ, 2(d+z0)-z, Z, t);
ξ1(ρ, z, Z, t)=-δ(ρ)δ(Z+2(d+z0)-z)δ(t)+ 4αD2θ(z-Z-2(d+z0))
×G2(2)(ρ, α[Z+2(d+z0)-z], t),
ξn-1(ρ, z, Z, t)=(-1)n-1δ(ρ)δ(z-zn-1)δ(t)+k=0n-2pn-1k(z-zn-1)k×G2(k+2)(ρ, α(z-zn-1), t)×θ(z-zn-1),
I=R^mξn-1(ρ, z, Z, t)=SdzSR(ρ, z, z, t)¯ξn-1(ρ, z, Z, t)=I1+I2+I3+I4,
I1=dz[δ(ρ)δ(z+z)δ(t)]¯[(-1)n-1δ(ρ)δ(z-zn-1)δ(t)],
I2=k=0n-2dz[δ(ρ)δ(z+z)δ(t)]¯[pn-1k(z-zn-1)k×G2(k+2)(ρ, α(z-zn-1), t)θ(z-zn-1)],
I3=dz[-4αD2θ(-z-z)×G2(2)(ρ, α(z+z), t)]¯[(-1)n-1δ(ρ)δ(z-zn-1)δ(t)],
I4=k=0n-2dz[-4αD2θ(-z-z)×G2(2)(ρ, α(z+z), t)]¯[pn-1k(z-zn-1)k×G2(k+2)(ρ, α(z-zn-1), t)θ(z-zn-1)].
I1=(-1)n-1δ(ρ)δ(z+zn-1)δ(t),
I2=k=0n-2pn-1k(-z - zn-1)k×G2(k+2)(ρ, α(-z - zn-1), t)θ(-z-zn-1),
I3=(-1)n4αD2θ(-z - zn-1)×G2(2)(ρ, α(z+zn-1), t).
I4=-4αD2exp(-ρ2/4D2c2t)4πD2c2t
×exp(-μa2c2t)θ(-z-zn-1)
×zn-1-zdzpn-1k(z-zn-1)k
×dτG2(2)(α(z+z), t-τ)
×G2(k+2)(α(z-zn-1), τ).
-12D2G2(k+3)(-α(z+zn-1), t).
zn-1-z(z-zn-1)kdz=(z-zn-1)k+1k+1.
I4=2α (z-zn-1)k+1k+1 θ(-z-zn-1)×G2(k+3)(ρ, -α(z+zn-1), t).
I=R^fR^mξn-1(ρ, z, Z, t)=I1+I2+I3+I4,
I1=(-1)nδ(ρ)δ(z-zn)δ(t),
I2=k=0n-2-pn-1k(z-zn)k×G2(k+2)(ρ, α(z-zn), t)θ(z-zn),
I3=(-1)n+14αD2θ(z-zn)×G2(2)(ρ, α(z-zn), t),
I4=-2α (z-zn)k+1k+1 θ(z-zn)
×G2(k+3)(ρ, α(z-zn), t).
fnkl(t)=0dz-dτ τlzkG1(z-z-zn, t-τ)×G2(k+2)(αz, τ).
Gi(z, t)=ciθ(t) exp{-[z2/(4Dicit)]}4πDicit
G˜i(z, ω)=Dici2Diiωexp-iωDici |z|.
G˜i(k+2)(z, ω)=-12Di (sign z)k+2-iωDicik+1
×exp-iωDici |z|.
f˜nkl(ω)=0dzzkG˜1(z-z-zn, ω)(-1)l×dld(iω)lG˜2(k+2)(αz, ω).
dld(iω)lG˜2(k+2)(αz, ω)
=12D2 (sign z)k+2×i=0lQkliiωk+1+i-2l|z|iexp-α iωD2c2 |z|,
dl+1d(iω)l+1G˜2(k+2)(αz, ω)
=12D2 (sign z)k+2×i=0l+1Qk(l+1)iiωk-1+i-2l|z|iexp-α iωD2c2 |z|=12D2 (sign z)k+2i=0lk+1+i-2l2×Qkliiωk-1+i-2l|z|iexp-α iωD2c2 |z|+12D2 (sign z)k+2i=0l-α2D2c2×Qkliiωk+i-2l|z|i+1exp-α iωD2c2 |z|,
Qk00=--1D2c2k+1,
Qk(l+1)i=k+1+i-2l2 Qkli-α2D2c2 Qkli-1,
Qk(l+1)0=k+1-2l2 Qkli,
Qk(l+1)(l+1)=-α2D2c2 Qkll.
f˜nkl(ω)=12D2i=0l(-1)l0dzQklizk+iD1c12D1×iωiωk+1+i-2lexp-iωD1c1 (κz+zn-z),
f˜nkl(ω)=12D2i=0l(-1)l0dzQklizk+i×D1c12D1iω-D1c1κk+1+i-2l×ddzk+1+i-2l×exp-iωD1c1 (κz+zn-z),
f˜nkl(ω)=12D2i=0l(-1)l(k+i)!(2l-1)!×0dzQkliz2l-1D1c12D1iω×D1c1κk+1+i+2l×exp-iωD1c1 (κz+zn-z).
f˜nkl(ω)=i=0l(-1)l(k+i)!2D20dzQkliD1c1κk+1+i-2l×zdz1z1dz2 z2ldz2l-1D1c12D1iω×exp-iωD1c1 (κz2l-1+zn-z).
fnkl(t)=i=0l(-1)l(k+i)!2D20dzQkliD1c1κk+1+i-2l×zdz1z1dz2 z2ldz2l-1c1×exp-(κz2l-1+zn-z)24D1c1t4πD1c1t.
fnkl(t)=i=0l(-1)lc1(k+i)!2D214πD1c1t×QkliD1c1κk+1+i-2l4D1c1tκ2l×0dxzdx1z1dx2z2ldx2l-1×exp-x2l-1+zn-z4D1c1t2,
im erfc(a)=2πadx0x0dx1xm-1dxmexp(-xm2)
=2π0dx0x0dx1xm-1dxm×exp[-(xm+a)2]
fnkl(t)=c1i=0l(-1)l(k+i)!2D214πD1c1t×QkliD1c1κk+1+i-2l×4D1c1tκ2lπ2 i2l-1 erfczn-z4D1c1t,
fnkl(t)=c1i=0l(-1)l(k+i)!4D21κ QkliD1c1κk+1+i-2l×4D1c1tκ2l-1i2l-1 erfczn-z4D1c1t.

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