Abstract

We describe a method to solve the radiative transfer equation (RTE) in multilayered geometry with index mismatch and demonstrate its potential for modeling light propagation in biological systems. The method is compared to Monte Carlo simulations with high accuracy but is much more efficient in terms of computer time. We illustrate the potential of the method by studying a multilayered system containing a weakly scattering layer surrounded by highly scattering layers, with anisotropic scattering and index mismatched interfaces. The calculation of directional transmitted fluxes has shown that the RTE method can be used to calculate relevant quantities in realistic systems in the presence of nondiffusive behavior.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  2. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, 1995).
  3. P. Sebbah, ed., Waves and Imaging through Complex Media (Kluwer, 2001).
  4. A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
    [CrossRef]
  5. S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
    [CrossRef]
  6. A. Mandelis, "Diffusion waves and their uses," Phys. Today 53, 29-34 (2000).
    [CrossRef]
  7. I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
    [CrossRef] [PubMed]
  8. K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
    [CrossRef] [PubMed]
  9. R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
    [CrossRef]
  10. D. J. Durian and J. Rudnick, "Photon migration at short times and distances and in cases of strong absorption," J. Opt. Soc. Am. A 14, 235-245 (1997).
    [CrossRef]
  11. A. D. Kim and A. Ishimaru, "Optical diffusion of continuous wave, pulsed and density waves in scattering media and comparisons with radiative transfer," Appl. Opt. 37, 5313-5319 (1998).
    [CrossRef]
  12. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
    [CrossRef]
  13. K. Mitra and S. Kumar, "Development and comparison of models for light-pulse transport through scattering-absorbing media," Appl. Opt. 38, 188-196 (1999).
    [CrossRef]
  14. R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
    [CrossRef]
  15. X. Zhang and Z. Q. Zhang, "Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition," Phys. Rev. E 66, 016612 (2002).
    [CrossRef]
  16. R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
    [CrossRef]
  17. A. Periasamy, ed., Methods in Cellular Imaging (Oxford U. Press, 2001).
  18. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  19. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  20. Z. Jin and K. Stamnes, "Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system," Appl. Opt. 33, 431-442 (1994).
    [CrossRef] [PubMed]
  21. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
    [CrossRef]
  22. E. Tinet, S. Avrillier, and J. M. Tualle, "Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media," J. Opt. Soc. Am. A 13, 1903-1915 (1996).
    [CrossRef]
  23. M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
    [CrossRef]
  24. A. D. Kim, "Boundary integral method to compute Green's functions for the radiative transport equation," Waves Random Media 15, 17-42 (2005).
    [CrossRef]

2005 (1)

A. D. Kim, "Boundary integral method to compute Green's functions for the radiative transport equation," Waves Random Media 15, 17-42 (2005).
[CrossRef]

2004 (1)

2002 (3)

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
[CrossRef]

X. Zhang and Z. Q. Zhang, "Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition," Phys. Rev. E 66, 016612 (2002).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

2000 (1)

A. Mandelis, "Diffusion waves and their uses," Phys. Today 53, 29-34 (2000).
[CrossRef]

1999 (2)

K. Mitra and S. Kumar, "Development and comparison of models for light-pulse transport through scattering-absorbing media," Appl. Opt. 38, 188-196 (1999).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

1998 (1)

1997 (2)

D. J. Durian and J. Rudnick, "Photon migration at short times and distances and in cases of strong absorption," J. Opt. Soc. Am. A 14, 235-245 (1997).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

1996 (2)

1995 (1)

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

1994 (1)

1990 (1)

K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

1988 (1)

I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
[CrossRef] [PubMed]

Alfano, R. R.

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
[CrossRef]

K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Avrillier, S.

Cai, W.

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Carminati, R.

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
[CrossRef]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chance, B.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

de Vries, P.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

Durian, D. J.

Elaloufi, R.

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
[CrossRef]

Freund, I.

I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
[CrossRef] [PubMed]

Gayen, S. K.

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
[CrossRef]

Greffet, J.-J.

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach," J. Opt. Soc. Am. A 21, 1430-1437 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
[CrossRef]

Ishimaru, A.

Jin, Z.

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

Kaveh, M.

I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
[CrossRef] [PubMed]

Kim, A. D.

A. D. Kim, "Boundary integral method to compute Green's functions for the radiative transport equation," Waves Random Media 15, 17-42 (2005).
[CrossRef]

A. D. Kim and A. Ishimaru, "Optical diffusion of continuous wave, pulsed and density waves in scattering media and comparisons with radiative transfer," Appl. Opt. 37, 5313-5319 (1998).
[CrossRef]

Kop, R. H. J.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

Kumar, S.

Lagendijk, A.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

Lax, M.

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Liu, F.

K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Mandelis, A.

A. Mandelis, "Diffusion waves and their uses," Phys. Today 53, 29-34 (2000).
[CrossRef]

Mitra, K.

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

Periasamy, A.

A. Periasamy, ed., Methods in Cellular Imaging (Oxford U. Press, 2001).

Rosenbluh, M.

I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
[CrossRef] [PubMed]

Rudnick, J.

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

Sebbah, P.

P. Sebbah, ed., Waves and Imaging through Complex Media (Kluwer, 2001).

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, 1995).

Sprik, R.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

Stamnes, K.

Thomas, G. E.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

Tinet, E.

Tualle, J. M.

Waitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

Xu, M.

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Yodh, A.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Yoo, K. M.

K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang and Z. Q. Zhang, "Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition," Phys. Rev. E 66, 016612 (2002).
[CrossRef]

Zhang, Z. Q.

X. Zhang and Z. Q. Zhang, "Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition," Phys. Rev. E 66, 016612 (2002).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (3)

J. Opt. A (1)

R. Elaloufi, R. Carminati, and J.-J. Greffet, "Time-dependent transport through scattering media: From radiative transfer to diffusion," J. Opt. A 4, S103-S108 (2002).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Photon. News (1)

S. K. Gayen and R. R. Alfano, "Biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
[CrossRef]

Phys. Rev. E (3)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Waitz, and P. Sheng, "Wave transport in random media: The ballistic to diffusive transition," Phys. Rev. E 60, 4843-4850 (1999).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

X. Zhang and Z. Q. Zhang, "Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition," Phys. Rev. E 66, 016612 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

I. Freund, M. Kaveh, and M. Rosenbluh, "Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation," Phys. Rev. Lett. 60, 1130-1133 (1988).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, and R. R. Alfano, "When does the diffusion approximation fail to describe photon transport in random media?," Phys. Rev. Lett. 64, 2647-2650 (1990).
[CrossRef] [PubMed]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs," Phys. Rev. Lett. 79, 4369-4372 (1997).
[CrossRef]

Phys. Today (2)

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

A. Mandelis, "Diffusion waves and their uses," Phys. Today 53, 29-34 (2000).
[CrossRef]

Waves Random Media (1)

A. D. Kim, "Boundary integral method to compute Green's functions for the radiative transport equation," Waves Random Media 15, 17-42 (2005).
[CrossRef]

Other (7)

A. Periasamy, ed., Methods in Cellular Imaging (Oxford U. Press, 2001).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, 1995).

P. Sebbah, ed., Waves and Imaging through Complex Media (Kluwer, 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Geometry of the multilayered system. The parameters describing layer number k are indicated.

Fig. 2
Fig. 2

Example of the path for ballistic photons in a two-layer medium.

Fig. 3
Fig. 3

Calculation of forward collimated intensity versus optical depth through a slab with three layers for which the refractive indices are n 0 = 1 , n 1 = 1.4 , n 2 = 1.9 , n 3 = 1.3 , and n 4 = 1 . The incident angle of the beam is ϑ 0 = 0 ° ; the optical depth of each layer is L 1 * = 1 ; L 2 * = 1 ; L 3 * = 1 .

Fig. 4
Fig. 4

Illustration of the boundary conditions. Left, Eq. (21); right, Eq. (22).

Fig. 5
Fig. 5

Ordinate directions when n k < n k + 1 .

Fig. 6
Fig. 6

Ordinate directions when n k > n k + 1 .

Fig. 7
Fig. 7

Transmitted flux versus angle θ between the observation direction and the z axis. Three-layer system with mismatched or matched interfaces and comparison of the RTE solution and a Monte Carlo simulation.

Fig. 8
Fig. 8

Transmitted flux versus angle θ between the observation direction and the z axis. System with three layers with (a) mismatched and (b) matched interfaces. The scattering coefficient of the internal layer changes from 0.01 to 1 mm 1 .

Fig. 9
Fig. 9

Total transmitted flux versus scattering coefficient μ s ( 2 ) of the internal layer for mismatched and matched interfaces.

Fig. 10
Fig. 10

Total transmitted flux versus inverse of optical thickness of the system for mismatched and matched interfaces.

Fig. 11
Fig. 11

Total (angle integrated) transmitted flux versus anisotropy factor g of the internal layer for mismatched and matched interfaces.

Fig. 12
Fig. 12

Illustration of different cycles returning a ray to the same state.

Tables (5)

Tables Icon

Table 1 Definition of Notations Used

Tables Icon

Table 2 Physical Parameters for a System with Three Layers

Tables Icon

Table 3 Physical Parameters for a System with Three Layers a

Tables Icon

Table 4 Physical Parameters for a System with Three Layers a

Tables Icon

Table 5 Physical Parameters for a System with Three Layers a

Equations (58)

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

[ μ τ ( k ) + 1 ] ϕ ( k ) [ τ ( k ) , μ ] = a ( k ) 2 1 + 1 p ( k ) ( μ , μ ) × ϕ ( k ) ( τ ( k ) , μ ) .
ϕ ( k ) ( τ ( k ) , μ ) = ϕ b + ( k ) ( τ ( k ) ) δ ( μ η k ) + ϕ b ( k ) ( τ ( k ) ) × δ ( μ + η k ) + ϕ d ( k ) ( τ ( k ) , μ ) ,
[ μ τ ( k ) + 1 ] ϕ d ( k ) [ τ ( k ) , μ ] = a ( k ) 2 1 + 1 p ( k ) ( μ , μ ) × ϕ d ( k ) ( τ ( k ) , μ ) + S ( k ) ( τ ( k ) , μ ) .
S ( k ) ( τ ( k ) , μ ) = a ( k ) 2 p ( k ) ( μ , η k ) ϕ b + ( k ) ( τ ( k ) ) + a ( k ) 2 p ( k ) ( μ , η k ) ϕ b ( k ) ( τ ( k ) ) .
γ = s = 1 N i = 1 N + 1 s Γ i ( s ) ,
Γ i ( s ) = [ 1 R i , i 1 ( a = 1 s 1 [ T i , i + a T i + a , i ] ) R i + s 1 , i + s × exp [ 2 a = 0 s 1 ε i + a L i + a * ] ] 1 .
γ = Γ 1 ( 1 ) = 1 1 R 10 R 12 e 2 ε 1 L 1 * = n = 0 ( R 10 R 12 e 2 ε 1 L 1 * ) n .
R 10 = R 12 = 0 γ = 1.
γ = Γ 1 ( 1 ) Γ 2 ( 1 ) Γ 1 ( 2 ) ,
Γ 1 ( 1 ) = 1 1 R 10 R 12 e 2 ε 1 L 1 * = n = 0 ( R 10 R 12 e 2 ε 1 L 1 * ) n ,
Γ 2 ( 1 ) = 1 1 R 21 R 23 e 2 ε 2 L 2 * = n = 0 ( R 12 R 23 e 2 ε 2 L 2 * ) n ,
Γ 1 ( 2 ) = 1 1 R 10 R 12 T 21 R 23 e 2 ( ε 1 L 1 * + ε 2 L 2 * ) = n = 0 ( R 10 T 12 T 21 R 23 e 2 ( ε 1 L 1 * + ε 2 L 2 * ) ) n .
ϕ b + ( k ) ( τ ( k ) ) = A ε k ε 0 γ i = 0 k 1 T i , i + 1 exp [ i = 1 k 1 ε i L i * ] × exp [ ε k τ ( k ) ] ,
ϕ b ( k ) ( τ ( k ) ) = A ε k ε 0 γ i = 0 k 1 T i , i + 1 R k , k + 1 exp [ i = 1 k 1 ε i L i * ] × exp [ ε k [ 2 L k * τ ( k ) ] ] .
ϕ ( k ) = { ϕ ( k ) ( i ) } { ϕ d ( k ) ( τ ( k ) , μ i ( k ) ) } ,
Σ ( k ) = { Σ ( k ) ( i ) } { 1 μ i S ( k ) ( τ ( k ) , μ i ( k ) ) } ,
p ( k ) ( i , j ) p ( k ) ( μ i ( k ) , μ j ( k ) ) ,
M ( k ) = { M i j ( k ) } { 1 μ i ( k ) [ δ i j + a ( k ) 2 ω j ( k ) p ( k ) ( i , j ) ] } .
τ ( k ) ϕ ( k ) ( i ) = j = 1 2 N ( k ) M i j ( k ) ϕ ( k ) ( j ) + Σ ( k ) ( i ) τ ( k ) ϕ ( k ) = M ( k ) ϕ ( k ) + Σ ( k ) .
ϕ ( k ) ( τ ( k ) ) = i = 1 2 N ( k ) C i ( k ) V i ( k ) e λ i ( k ) τ ( k ) + ϕ p ( k ) ( τ ( k ) ) ,
ϕ ( k ) ( τ ( k ) ) = V ( k ) Λ ( k ) ( τ ( k ) ) C ( k ) + ϕ p ( k ) ( τ ( k ) ) ,
Σ ( k ) ( τ ( k ) ) = α ( k ) e ε k τ ( k ) + β ( k ) e ε k ( 2 L k * τ ( k ) ) .
ϕ p ( k ) [ τ ( k ) ] = X ( k ) e ε k τ ( k ) + Y ( k ) e ε k [ 2 L k * τ ( k ) ] ,
[ M ( k ) + ε k l ] X ( k ) = α ( k ) ,
[ M ( k ) ε k l ] Y ( k ) = β ( k ) .
ϕ ( 1 ) ( 0 , μ ( 1 ) > 0 ) = R 10 ( μ ( 1 ) ) ϕ ( 1 ) ( 0 , μ ( 1 ) < 0 ) ,
ϕ ( k ) ( L k * , μ ( k ) < 0 ) = R k , k + 1 ( μ ( k ) ) ϕ ( k ) ( L k * , μ ( k ) > 0 ) + ( n k n k + 1 ) 2 T k + 1 , k ( μ ( k + 1 ) ) ϕ ( k + 1 ) ( 0 , μ ( k + 1 ) < 0 ) ,
ϕ ( k + 1 ) ( 0 , μ ( k + 1 ) > 0 ) = R k + 1 , k ( μ ( k + 1 ) ) ϕ ( k + 1 ) ( 0 , μ ( k + 1 ) < 0 ) + ( n k + 1 n k ) 2 T k , k + 1 ( μ ( k ) ) ϕ ( k ) ( L k * , μ ( k ) > 0 ) ,
ϕ ( N ) ( L N * , μ ( N ) < 0 ) = R N , N + 1 ( μ ( N ) ) ϕ ( N ) ( L N * , μ ( N ) > 0 ) ,
E j i ( k ) = V i ( k ) ( j ) R k , k 1 ( μ j ( k ) ) V i ( k ) ( 2 N ( k ) + 1 j ) .
F j i ( k ) = ( V i ( k ) ( N ( k ) + j ) R k , k + 1 ( μ N ( k ) + 1 j ( k ) ) V i ( k ) × ( N ( k ) + 1 j ) ) e λ i ( k ) L k * .
G j i ( k + 1 ) = { 0 j = 1 . . . r ( n k n k + 1 ) 2 T k + 1 , k ( μ N ( k ) + 1 j ( k + 1 ) ) V i ( k + 1 ) ( 2 N ( k + 1 ) N ( k ) + j ) j = r + 1 . . . N ( k ) i = 1 . . . 2 N ( k + 1 ) , r = max ( N ( k ) N ( k + 1 ) , 0 ) .
H j i ( k ) = { ( n k + 1 n k ) 2 T k , k + 1 ( μ j ( k ) ) V i ( k ) ( j ) e λ i ( k ) L k * j = 1 . . . s 0 j = s + 1 . . . N ( k + 1 ) i = 1 . . . 2 N ( k ) , s = min ( N ( k ) , N ( k + 1 ) ) .
T k + 1 , k ( μ N ( k ) + 1 j ( k + 1 ) ) = 0 , V i ( k + 1 ) ( N ( k ) + j ) = 0 , for   j = 1 . . . N ( k ) N ( k + 1 ) .
P ( k ) ( j ) = R k , k 1 ( μ j ( k ) ) ϕ p ( k ) ( 0 , 2 N ( k ) + 1 j ) ϕ p ( k ) ( 0 , j ) ( n k n k 1 ) 2 T k 1 , k ( μ j ( k 1 ) ) ϕ p ( k 1 ) ( L k 1 * , j ) j = 1 . . . N ( k )
P ( 1 ) ( j ) = R 1 , 0 ( μ j ( 1 ) ) ϕ p ( 1 ) ( 0 , 2 N ( 1 ) + 1 j ) ϕ p ( 1 ) ( 0 , j ) j = 1 . . . N ( 1 ) .
Q ( k ) ( j ) = R k , k + 1 ( μ j ( k ) ) ϕ p ( k ) ( L k * , j ) ϕ p ( k ) ( L k * , 2 N ( k ) + 1 j ) ( n k n k + 1 ) 2 T k + 1 , k ( μ j ( k + 1 ) ) ϕ p ( k + 1 ) ( 0 , 2 N ( k ) + 1 j ) j = 1 . . . N ( k )
Q ( N ) ( j ) = R N , N + 1 ( μ j ( N ) ) ϕ p ( N ) ( L N * , j ) ϕ p ( N ) ( L N * , 2 N ( N ) + 1 j ) j = 1 . . . N ( N ) .
( E ( 1 ) 0 0 0 0 F ( 1 ) G ( 2 ) 0 0 0 H ( 1 ) E ( 2 ) 0 0 0 0 F ( 2 ) G ( 3 ) 0 0 0 H ( 2 ) E ( 3 ) 0 0 0 F ( k ) G ( k + 1 ) 0 0 H ( k ) E ( k + 1 ) 0 0 0 F ( k + 1 ) G ( k + 2 ) 0 0 0 H ( k + 1 ) E ( k + 2 ) 0 0 0 0 F ( N 1 ) G ( N ) 0 0 0 H ( N 1 ) E ( N ) 0 0 0 0 F ( N ) ) ( C ( 1 ) C ( 2 ) C ( k ) C ( N 1 ) C ( N ) ) = ( P ( 1 ) Q ( 1 ) P ( k ) Q ( k ) P ( N ) Q ( N ) ) .
C i ( k ) = C ˜ i ( k ) i = 1 . . . N ( k ) , C i ( k ) = C ˜ i ( k ) e λ i ( k ) L k * i = N ( k ) + 1 . . . 2 N ( k ) .
C ( k ) = ( 1 0 0 L ( k ) ) C ˜ ( k ) ,
ϕ ( k ) ( τ ( k ) ) = i = 1 N ( k ) C ˜ i ( k ) V i ( k ) e λ i ( k ) τ ( k ) + i = N ( k ) + 1 2 N ( k ) C ˜ i ( k ) V i ( k ) e λ i ( k ) ( τ ( k ) L k * ) + ϕ p ( k ) ( τ ( k ) ) ,
ϕ ( k ) ( τ ( k ) ) = V ( k ) Λ ˜ ( k ) ( τ ( k ) ) C ˜ ( k ) + ϕ p ( k ) ( τ ( k ) ) ,
Λ ˜ ( k ) ( τ ( k ) ) = diag { e λ 1 ( k ) τ ( k ) , . . . , e λ i ( k ) τ ( k ) , . . . , e λ N ( k ) ( k ) τ ( k ) , e λ 1 ( k ) ( τ ( k ) L k * ) , . . . , e λ i ( k ) ( τ ( k ) L k * ) , . . . , e λ N ( k ) ( k ) ( τ ( k ) L k * ) } .
R k , k + 1 R k , k 1 e 2 ε k L k * .
T k , k + 1 R k + 1 , k + 2 T k + 1 , k R k , k 1 e 2 ( ε k L k * + ε k + 1 L k + 1 * ) .
Γ k ( 1 ) = 1 1 R k , k + 1 R k , k 1 e 2 ε k L k * .
Γ k ( 2 ) = 1 1 T k , k + 1 R k + 1 , k + 2 T k + 1 , k R k , k 1 e 2 ( ε k L k * + ε k + 1 L k + 1 * ) .
1 + 1 f ( μ ( 1 ) ) ( 1 ) = i = 1 2 N ( 1 ) ω i ( 1 ) f ( μ i ( 1 ) ) .
μ L ( 2 ) = 1 ( n 2 n 1 ) 2 ,
1 + 1 f ( μ ( 2 ) ) ( 2 ) = ( 1 μ L ( 2 ) + μ L ( 2 ) + μ L ( 2 ) + L ( 2 ) +1 ) × f ( μ ( 2 ) ) ( 2 ) .
μ L ( 2 ) + μ L ( 2 ) f ( μ ( 2 ) ) ( 2 ) = μ L ( 2 ) 1 + 1 f ( μ L ( 2 ) u ) d u = μ L ( 2 ) 1 + 1 g ( u ) d u = j = 1 2 ( N ( 2 ) N ( 1 ) ) μ L ( 2 ) ϖ j g ( u j ) ,
μ N ( 1 ) + j ( 2 ) = μ L ( 2 ) u j j = 1 ( N ( 2 ) N ( 1 ) ) ω N ( 1 ) + j ( 2 ) = μ L ( 2 ) ϖ j ,
( 1 μ L ( 2 ) + + μ L ( 2 ) + 1 ) f ( μ ( 2 ) ) ( 2 ) .
μ ( 2 ) = h ( μ ( 1 ) ) = 1 ( n 1 n 2 ) 2 ( 1 ( μ ( 1 ) ) 2 ) ,
( 1 0 + 0 + 1 ) f ( h ( μ ( 1 ) ) ) d h ( 1 ) ( 1 ) = 1 + 1 f ( h ( μ ( 1 ) ) ) × d h ( 1 ) ( 1 ) ,
1 + 1 f ( h ( μ ( 1 ) ) ) d h ( 1 ) ( 1 ) = j = 1 2 N ( 1 ) f ( h ( μ ( 1 ) ) ) × d h ( 1 ) | μ j ( 1 ) ω j ( 1 ) ,
{ μ j ( 2 ) = 1 ( n 1 n 2 ) 2 ( 1 μ j ( 1 ) 2 ) i = 1 . . . N ( 1 ) ω j ( 2 ) = ( n 1 n 2 ) 2 μ j ( 1 ) μ j ( 2 ) ω j ( 1 )

Metrics