Abstract

This paper is the second of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for an N-layered finite cylinder. Solutions are given in the steady-state, frequency, and time domains for a point beam incident at an arbitrary position of the first layer and for a circular flat beam incident at the middle of the cylinder top. For special cases the solutions were compared to other solutions of the diffusion equation showing excellent agreement. In addition, the derived solutions were validated by comparison with Monte Carlo simulations. In the time domain we also derived a fast solution (≈ 10ms) for the case of equal reduced scattering coefficients and refractive indices in all layers.

©2010 Optical Society of America

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References

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  1. A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder. I Homogeneous Case,” submitted.
  2. A. Ishimaru, “Wave Propagation and Scattering in Random Media,” Academic Press, New York (1978).
  3. I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
    [Crossref]
  4. 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]
  5. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. 37, 6852–6862 (1998).
    [Crossref]
  6. A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999).
    [Crossref] [PubMed]
  7. J.M. Tualle, H.M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, “Asymptotic Behavior and Inverse Problem in Layered Scattering Media,” J. Opt. Soc. Am. A 21, 24–34 (2004).
    [Crossref]
  8. X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media 16, 121–135 (2006).
    [Crossref]
  9. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted.
    [PubMed]
  10. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted.
    [PubMed]
  11. 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]
  12. S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
    [Crossref]
  13. 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]
  14. M. Das, C. Xu, and Q. Zhu, “Analytical Solution for Light Propagation in a Two-Layer Tissue Structure with a Tilted Interface for Breast Imaging,” Appl. Opt. 45, 5027–5036 (2006).
    [Crossref] [PubMed]
  15. A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. 201, 771–797 (2004).
    [Crossref]
  16. C. Donner and H.W. Jensen, “Rapid Simulations of Steady-State Spatially Resolved Reflectance and Transmittance Profiles of Multilayered Turbid Materials,” J. Opt. Soc. Am. A 23, 1382–1390 (2006).
    [Crossref]
  17. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
    [Crossref]
  18. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
    [Crossref] [PubMed]
  19. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
    [Crossref] [PubMed]
  20. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, “Light Propagation through Biological Tissue and other Diffusive Media,” SPIE Press, Bellingham (2010).
  21. A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005).
    [Crossref]
  22. http://www.uni-ulm.de/ilm/index.php?id=10020200.

2007 (1)

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

2006 (3)

2005 (3)

A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005).
[Crossref]

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
[Crossref] [PubMed]

2004 (2)

A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. 201, 771–797 (2004).
[Crossref]

J.M. Tualle, H.M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, “Asymptotic Behavior and Inverse Problem in Layered Scattering Media,” J. Opt. Soc. Am. A 21, 24–34 (2004).
[Crossref]

2003 (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

2000 (1)

1999 (1)

A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[Crossref] [PubMed]

1998 (3)

1992 (1)

I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[Crossref]

Alexandrakis, G.

Avrillier, S.

Barnett, A.H.

A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. 201, 771–797 (2004).
[Crossref]

Bays, R.

Das, M.

Dayan, I.

I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[Crossref]

Del Bianco, S.

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

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

Dögnitz, N.

Donner, C.

Durkin, A.J.

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

Ettori, D.

Farrell, T.J.

Glanzmann, T.

A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[Crossref] [PubMed]

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[Crossref]

Havlin, S.

I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[Crossref]

Hayakawa, C.

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

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 and other Diffusive Media,” SPIE Press, Bellingham (2010).

Jensen, H.W.

Kienle, A.

A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005).
[Crossref]

A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[Crossref] [PubMed]

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[Crossref]

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]

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted.
[PubMed]

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted.
[PubMed]

A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder. I Homogeneous Case,” submitted.

Liemert, A.

A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder. I Homogeneous Case,” submitted.

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted.
[PubMed]

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted.
[PubMed]

Martelli, F.

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

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

Nghiem, H.M.

Patterson, M.S.

Sablong, R.

Sassaroli, A.

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

Spanier, J.

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

Tinet, E.

Tromberg, B.J.

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

Tseng, S-H.

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

Tualle, J.M.

van den Bergh, H.

Wagnières, G.

Wang, S.M.

X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media 16, 121–135 (2006).
[Crossref]

Wang, X.C.

X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media 16, 121–135 (2006).
[Crossref]

Weiss, G.H.

I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[Crossref]

Xu, C.

Yamada Y, Y.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

Zaccanti, G.

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

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

Zhu, Q.

Appl. Opt. (5)

J. Comp. Phys. (1)

A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. 201, 771–797 (2004).
[Crossref]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, and 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. A (3)

Opt. Lett. (1)

S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005).
[Crossref]

Phys. Med. Biol. (3)

F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007).
[Crossref] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. 50, 2159–2166 (2005).
[Crossref] [PubMed]

A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[Crossref] [PubMed]

Phys. Rev. E (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E 67, 056623 (2003).
[Crossref]

Waves Rand. Compl. Media (1)

X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media 16, 121–135 (2006).
[Crossref]

Other (6)

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted.
[PubMed]

A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted.
[PubMed]

A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder. I Homogeneous Case,” submitted.

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 and other Diffusive Media,” SPIE Press, Bellingham (2010).

http://www.uni-ulm.de/ilm/index.php?id=10020200.

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Figures (9)

Fig. 1.
Fig. 1. Scheme of the N-layered cylinder used in the calculations. In the figure the beam is incident onto the cylinder top, but it can also be incident onto the cylinder barrel of the first layer.

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Fig. 2.
Fig. 2. Comparison of the reflectance from a two-layered cylinder with radius a = 22mm calculated with Eq. 26 and by the eigenvalue method [17,18] (solid curves). The parameters are μ′ s1 = 1.1mm-1, μ a1 = 0.02mm-1, μ′ s2 = 1.3mm-1, μ a2 = 0.008mm-1, l 1 = 7mm, l 2 = 8mm, n 1 = 1.4, and n 2 = 1.4. For the non-scattering surrounding the refractive index is 1.0. The distance between the incident beam and detection is ρ = 17mm. The dashed curve shows the reflectance for a laterally infinitely extended medium.

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Fig. 3.
Fig. 3. Relative difference (|Martelli et al. - Eq. 26|/Martelli et al.) of the time resolved reflectance calculated with the two methods for the cylinder shown in Fig. 2.

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Fig. 4.
Fig. 4. Comparison of the time resolved transmittance from a finite 7-layered turbid medium. The distances between the center of the cylinder bottom and the detection positions are ρ= 50mm (black curve), ρ= 60mm (green curve) and ρ= 70mm (red curve). The radius of the cylinder is a = 150mm.

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Fig. 5.
Fig. 5. Relative difference of the two solutions (|cylinder - laterally infinite|/cylinder) shown in Fig. 4.

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Fig. 6.
Fig. 6. Comparison of the time resolved transmission from the cylinder barrel of a homogeneous cylinder. The parameters are μ′ s = 1.5mm-1, μa = 0.02mm-1, l 1 = 4mm, l 2 = 6mm, n = 1.4 and the two radii are a = 18mm (red curve) and a = 40mm (green curve). The dashed lines show the transmittance calculated with the solution for a homogeneous parallelepiped.

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Fig. 7.
Fig. 7. Relative difference between time resolved transmittance from a two-layered and a homogeneous cylinder for a = 18mm (red curve) and a = 40mm (green curve) shown in Fig. 6.

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Fig. 8.
Fig. 8. Comparison of the solution of the 7-layered diffusion equation (solid curves) with Monte Carlo simulations (circles) in the steady-state domain. The three radii are a = 4mm (red curve), a = 5mm (green curve), and a = 10mm (blue curve).

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Fig. 9.
Fig. 9. Comparison of the reflectance from a 7-layered turbid medium calculated with the solution of the diffusion equation (solid curves) and with the Monte Carlo method (points).

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Equations (40)

Equations on this page are rendered with MathJax. Learn more.

z bk = 1 + R eff , k 1 − R eff , k 2 D k ,
ΔΦ ( r → , ω ) − ( μ a D + i ω Dc ) Φ ( r → , ω ) = − 1 D S ( r → , ω ) ,
∂ 2 ∂ ρ 2 Φ + 1 ρ ∂ ∂ ρ Φ + 1 ρ 2 ∂ 2 ∂ ϕ 2 Φ + ∂ 2 ∂ z 2 Φ − ( μ a D + i ω Dc ) Φ = − 1 D ρ δ ( ρ − ρ 0 ) δ ( ϕ − ϕ 0 ) δ ( z − z 0 ) .
Φ ( s n , ϕ , m ) = ∫ 0 2 π ∫ 0 a ′ ρΦ ( ρ , ϕ ′ ) J m ( s n ρ ) cos ⁡ ( m ( ϕ − ϕ ′ ) ) d ρ d ϕ ′ ,
J m ( a ′ s n ) = 0 , n = 1,2 , … ,
∂ 2 ∂ z 2 Φ − ( μ a D + s n 2 + i ω Dc ) Φ = − 1 D J m ( s n ρ 0 ) cos ⁡ ( m ( ϕ − ϕ 0 ) ) δ ( z − z 0 ) .
∂ 2 ∂ z 2 G ( s n , z , ω ) − α k 2 G ( s n , z , ω ) = − 1 D δ ( z − z 0 ) ,
Φ ( s n , ϕ , m , z , ω ) = G ( s n , z , ω ) J m ( s n ρ 0 ) cos ⁡ ( m ( ϕ − ϕ 0 ) ) .
G 1 ( p ) ( s n , k , ω ) = ∫ − ∞ ∞ G 1 ( p ) ( s n , z , ω ) e − ikz d z
G 1 ( p ) ( s n , k , ω ) = 1 D 1 1 k 2 + α 1 2 e − ik z 0 .
G 1 ( p ) ( s n , z , ω ) = 1 2 π ∫ − ∞ ∞ G 1 ( p ) ( s n , k , ω ) e ikz d k
G 1 ( p ) ( s n , z , ω ) = e − α 1 ∣ z − z 0 ∣ 2 D 1 α 1 .
G 1 ( s n , z , ω ) = A 1 e α 1 z + B 1 e − α 1 z + e − α 1 ∣ z − z 0 ∣ 2 D 1 α 1 , k = 1
G k ( s n , z , ω ) = A k e α k z + B k e − α k z , k = 2 , … , N .
G 1 ( s n , z = − z b 1 , ω ) = 0
n k + 1 2 G k ( s n , z = L k , ω ) = n k 2 G k + 1 ( s n , z = L k , ω )
D k ∂ G k ( s n , z = L k , ω ) / ∂ z = D k + 1 ∂ G k + 1 ( s n , z = L k , ω ) / ∂ z
G N ( s nN , z = L N + z bN , ω ) = 0 ,
G 1 ( s n , z , ω ) = e − α 1 ∣ z − z 0 ∣ − e − α 1 ( z + z 0 + 2 z b 1 ) 2 D 1 α 1 + sinh ⁡ [ α 1 ( z 0 + z b 1 ) ] sinh ⁡ [ α 1 ( z + z b 1 ) ] D 1 α 1 e α 1 ( l 1 + z b 1 )
× D 1 α 1 n 1 2 β 3 − D 2 α 2 n 2 2 γ 3 D 1 α 1 n 1 2 β 3 cosh ⁡ [ α 1 ( l 1 + z b 1 ) ] + D 2 α 2 n 2 2 γ 3 sinh ⁡ [ α 1 ( l 1 + z b 1 ) ]
G N ( s n , z , ω ) = n N 2 ∏ i = 2 N − 1 ( D i α i n i 2 ) sinh ⁡ [ α 1 ( z 0 + z b 1 ) ] sinh ⁡ [ α N ( L N + z bN − z ) ] D 1 α 1 n 1 2 β 3 cosh ⁡ [ α 1 ( l 1 + z b 1 ) ] + D 2 α 2 n 2 2 γ 3 sinh ⁡ [ α 1 ( l 1 + z b 1 ) ] .
β N = D N − 1 α N − 1 n N − 1 2 cosh ⁡ ( α N − 1 l N − 1 ) sinh ⁡ [ α N ( l N + z b 2 ) ]
+ D N α N n N 2 sinh ⁡ ( α N − 1 l N − 1 ) cosh ⁡ [ α 1 ( l N + z b 2 ) ]
γ N = D N − 1 α N − 1 n N − 1 2 sinh ⁡ ( α N − 1 l N − 1 ) sinh ⁡ [ α N ( l N + z b 2 ) ]
+ D N α N n N 2 cosh ⁡ ( α N − 1 l N − 1 ) cosh ⁡ [ α N ( l N + z b 2 ) ]
β k − 1 = D k − 2 α k − 2 n k − 2 2 cosh ⁡ ( α k − 2 l k − 2 ) β k + D k − 1 α k − 1 n k − 1 2 sinh ⁡ ( α k − 2 l k − 2 ) γ k
γ k − 1 = D k − 2 α k − 2 n k − 2 2 sinh ⁡ ( α k − 2 l k − 2 ) β k + D k − 1 α k − 1 n k − 1 2 cosh ⁡ ( α k − 2 l k − 2 ) γ k .
Φ ( ρ , ϕ ) = 1 π a ′ 2 ∑ m = − ∞ ∞ ( 2 − δ m , 0 ) ∑ n = 1 ∞ Φ ( s n , ϕ , m ) J m ( s n ρ ) J m + 1 − 2 ( a ′ s n ) .
J m + 1 ( a ′ s n ) = 2 m a ′ s n J m ( a ′ s n ) − J m − 1 ( a ′ s n ) = − J m − 1 ( a ′ s n )
Φ k ( r → , ω ) = 1 π a ′ 2 ∑ m = − ∞ ∞ cos ⁡ ( mφ ) ∑ n = 1 ∞ G k ( s n , z , ω ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 − 2 ( a ′ s n ) k = 1,2 … N .
Φ k ( r → , ω ) = 1 π a ′ 2 ∑ n = 1 ∞ G k ( s n , z , ω ) J 0 ( s n ρ ) J 1 − 2 ( a ′ s n ) .
Φ k ( r → , ω ) = 2 π a ′ 2 ρ w ∑ n = 1 ∞ G k ( s n , z , ω ) J 0 ( s n ρ ) J 1 ( s n ρ w ) s n J 1 2 ( s n a ′ ) .
( 1 c ∂ ∂ t + μ a − D Δ ) Φ ( r → , t ) = S ( r → , t ) ,
Φ ( r → , t ) = 1 2 π ∫ − ∞ ∞ Φ ( r → , ω ) e iωt dω .
Φ k ( r → , t ) = 1 2 π 2 a ′ 2 ∑ m = − ∞ ∞ cos ⁡ ( mφ ) ∑ n = 1 ∞ ∫ − ∞ ∞ G k ( s n , z , ω ) e iωt dω J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 − 2 ( a ′ s n ) .
α k = s n 2 + μ ak D + i ω Dc = μ ak c + Dc s n 2 + iω Dc .
X ( σ + iω ) = ∫ − ∞ ∞ x ( t ) e − ( σ + iω ) t dt , ROC : Re { s } > − σ ,
X ( μ ak c + Dc s n 2 + iω ) = ∫ − ∞ ∞ x ( t ) e − ( μ ak c + Dc s n 2 + iω ) t dt = ∫ − ∞ ∞ x ( t ) e − Dc s n 2 t e − ( μ ak c + iω ) t dt .
Φ k ( r → , t ) = 1 2 π 2 a ′ 2 ∫ − ∞ ∞ G k ( z , ω ) e iωt dω ∑ m = − ∞ ∞ cos ⁡ ( mφ ) ∑ n = 1 ∞ e − Dc s n 2 t J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 − 2 ( s n a ′ ) .
Φ k ( r → , t ) = 1 2 π 2 a ′ 2 ∫ − ∞ ∞ G k ( z , ω ) e iωt dω ∑ n = 1 ∞ e − Dc s n 2 t J 0 ( s n ρ ) J 1 − 2 ( s n a ′ ) .

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