Abstract

We consider the problem of optical tomographic imaging in a weakly scattering medium in the presence of highly scattering inclusions. The approach is based on the assumption that the transport coefficient of the scattering media differs by an order of magnitude for weakly and highly scattering regions. This situation is common for optical imaging of live objects such an embryo. We present an approximation to the radiative transfer equation, which can be applied to this type of scattering case. Our approach was verified by reconstruction of two optical parameters from numerically simulated datasets.

© 2011 OSA

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2010 (3)

2009 (3)

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036607 (2009).
[CrossRef] [PubMed]

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009).
[CrossRef]

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17(22), 20178–20190 (2009).
[CrossRef] [PubMed]

2008 (3)

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss-Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008).
[CrossRef] [PubMed]

2006 (2)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. 65(3), 383–405 (2006).
[CrossRef]

G. Bal, “Radiative transfer equations with varying refractive index: a mathematical perspective,” J. Opt. Soc. Am. A 23(7), 1639–1644 (2006).
[CrossRef] [PubMed]

2005 (2)

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

M. Fauver, E. J. Seibel, J. R. Rahn, M. G. Meyer, F. W. Patten, T. Neumann, and A. C. Nelson, “Three-dimensional imaging of single isolated cell nuclei using optical projection tomography,” Opt. Express 13(11), 4210–4223 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

D. V. Finch, “The attenuated X-Ray transforms: recent developments,” Inverse Probl. 47, 47–66 (2003).

2002 (2)

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

2001 (1)

F. Natterer, “Inversion of the attenuated Radon transform,” Inverse Probl. 17(1), 113–119 (2001).
[CrossRef]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41 (1999).
[CrossRef]

1998 (1)

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998).
[CrossRef]

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[CrossRef] [PubMed]

1992 (1)

1989 (1)

P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989).
[CrossRef] [PubMed]

1985 (1)

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12(2), 252–255 (1985).
[CrossRef] [PubMed]

1980 (1)

O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39(2), 341–354 (1980).
[CrossRef]

1979 (1)

F. Natterer, “On the inversion of the attenuated Radon transform,” Numer. Math. 32(4), 431–438 (1979).
[CrossRef]

Abdoulaev, G. S.

Agard, D. A.

P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989).
[CrossRef] [PubMed]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Alerstam, E.

Andersson-Engels, S.

Arridge, S. R.

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009).
[CrossRef]

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss-Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41 (1999).
[CrossRef]

Bal, G.

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Bassi, A.

Beuthan, J.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

Boas, D. A.

Brida, D.

Brown, C. S.

Burns, D. H.

Cerullo, G.

Cubeddu, R.

D’Andrea, C.

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

De Silvestri, S.

Dorn, O.

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998).
[CrossRef]

Dunsby, C.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Fang, Q.

Fauver, M.

Finch, D. V.

D. V. Finch, “The attenuated X-Ray transforms: recent developments,” Inverse Probl. 47, 47–66 (2003).

Florescu, L.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(1), 016602 (2010).
[CrossRef] [PubMed]

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036607 (2009).
[CrossRef] [PubMed]

French, P. M. W.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Han, T. D.

Hecksher-Sørensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29(6), 578–580 (2004).
[CrossRef] [PubMed]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Hiraoka, Y.

P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989).
[CrossRef] [PubMed]

Jacques, S. 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(2), 131–146 (1995).
[CrossRef] [PubMed]

Kaipio, J. P.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. 65(3), 383–405 (2006).
[CrossRef]

Klose, A. D.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

Kolehmainen, V.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. 65(3), 383–405 (2006).
[CrossRef]

Laine, R.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Lilge, L.

Markel, V. A.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(1), 016602 (2010).
[CrossRef] [PubMed]

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036607 (2009).
[CrossRef] [PubMed]

McGinty, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Metz, C.

O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39(2), 341–354 (1980).
[CrossRef]

Meyer, M. G.

Natterer, F.

F. Natterer, “Inversion of the attenuated Radon transform,” Inverse Probl. 17(1), 113–119 (2001).
[CrossRef]

F. Natterer, “On the inversion of the attenuated Radon transform,” Numer. Math. 32(4), 431–438 (1979).
[CrossRef]

Neil, M. A. A.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Nelson, A. C.

Netz, U.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002).
[CrossRef]

Neumann, T.

Ntziachristos, V.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008).
[CrossRef] [PubMed]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[CrossRef]

Patten, F. W.

Perrimon, N.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008).
[CrossRef] [PubMed]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Pitsouli, C.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008).
[CrossRef] [PubMed]

Quintana, L.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Rahn, J. R.

Razansky, D.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008).
[CrossRef] [PubMed]

Ren, K.

Rose, J.

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Schotland, J.

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009).
[CrossRef]

Schotland, J. C.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(1), 016602 (2010).
[CrossRef] [PubMed]

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036607 (2009).
[CrossRef] [PubMed]

Sedat, J. W.

P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989).
[CrossRef] [PubMed]

Seibel, E. J.

Sharpe, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[CrossRef] [PubMed]

Shaw, P. J.

P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989).
[CrossRef] [PubMed]

Siddon, R. L.

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12(2), 252–255 (1985).
[CrossRef] [PubMed]

Spelman, F. A.

Swoger, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Tahir, K. B.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Talbot, C. B.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008).
[CrossRef] [PubMed]

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss-Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. 65(3), 383–405 (2006).
[CrossRef]

Tretiak, O.

O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39(2), 341–354 (1980).
[CrossRef]

Valentini, G.

Vauhkonen, M.

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Supplementary Material (2)

» Media 1: AVI (3964 KB)     
» Media 2: AVI (3390 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) (Media 1) Three twisted spirals built from scattering balls are embedded in a weakly scattering cylinder with background transport coefficient μ =0.1 mm −1 and albedo λ = 0.999. Value of the transport coefficient for each scattering ball is set to μ = 0.75 mm −1. Two spirals have the background value of the albedo and one absorbing spiral has the value of albedo λ = 0.25. The direct light enters the domain along the direction s 0 = 2−1/2 (1,0,−1) T . Camera was rotated around the weakly scattering cylinder, whose axis is aligned along z-axis, by 153° with respect to the initial position n = (1,0,0) T , where n is the camera normal. (b) (Media 2) Two highly scattering cylinders are embedded in a weakly scattering cylinder with the same optical properties as in (a). Both highly scattering cylinders have μ = 0.75 mm −1, one of them has a low value of the albedo, λ = 0.25. The direct light enters the domain along the same direction as in (a).The camera was rotated by 117° from its initial position around z-axis in the positive direction.

Fig. 2.
Fig. 2.

Reconstruction results showing middle slices at z = 10. (a) Reconstructed transport coefficient μ (b) Reconstructed albedo λ

Fig. 3.
Fig. 3.

(a) Isosurface of the transport coefficient μ. (b) Isosurface of the albedo λ.

Fig. 4.
Fig. 4.

Slices showing reconstruction results of the triple helix at two different heights (a–b) Reconstructed transport coefficient μ albedo λ at z = 9mm. (c–d) μ and λ at z = 11mm.

Fig. 5.
Fig. 5.

(a) Isosurface of the transport coefficient μ. (b) Isosurface of the albedo λ.

Equations (30)

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s · I + μ ˜ I = λ μ B ,
B ( r , s ) = ( 4 π ) p ( s · s ) I ( r , s ) d 2 s + p ( s · s 0 ) I 0 ( r , s 0 ) .
p ( s · s ) = ( 1 4 π ) ( 1 + ε s · s ) ,
s · I 0 + μ ˜ I 0 = Q 0 δ ( r r 0 ) δ ( s s 0 ) ,
I 0 ( r , s 0 ) = Q 0 exp ( 0 l μ ˜ ( r 0 + s 0 l ) d l ) .
I ( r , s ) = 0 l max λ ( r s l ) μ ( r s l ) B ( r s l , s ) exp ( 0 l μ ˜ ( r s l ) d l ) d l .
I = u 3 κ s · u ,
u = 1 4 π ( 4 π ) I ( s ) d 2 s ,
κ = ( 1 3 ) [ μ ( 1 λ ε 3 ) + i ω c ] 1 .
Λ u = λ μ ρ ,
Λ = · κ + ( 1 λ ) μ + i ω c ,
B ( r , s ) = u ε κ s · u + p ( s · s 0 ) I 0 .
ε 0 l λ μ κ [ l u ( r s l ) ] exp ( 0 l μ ˜ ( r s l ) d l ) d l .
l u ( r s l ) = Σ i [ u ] l i δ ( l l i ) ,
ε Σ i λ l i μ l i κ l i [ u ] l i exp { Σ j = 0 i μ ˜ Δ l j } ,
= ς ( ω ) ( 𝓔 + ) d ω + ϒ ,
𝓔 = s = 1 ξ ( s ) d 2 s V χ ( r ) I E I 2 d 3 r .
ξ ( s ) = Σ 0 n < N δ ( s s n ) ,
χ ( r ) = Σ 0 m < M σ m δ ( r r m ) , ς ( ω ) = Σ 0 l < L δ ( ω ω l ) ,
= Re s = 1 ξ ( s ) J , s · I + μ ˜ I λ μ B d 2 s ,
ϒ ( μ , λ ) = 1 2 [ α μ Δ μ 2 + α λ Δ λ 2 ] ,
s n · J * + μ ˜ J * = 2 χ ( r ) ( I E * I * ) ,
{ μ k + 1 = μ k + α μ 1 f μ , λ k + 1 = λ k + α λ 1 f λ ,
f μ = Re s = 1 ξ ( s ) [ λ ( p ( s · s 0 ) I 0 + u ) I ] J * d 2 s
3 ε λ μ Re { κ 2 ( i ω μ c ) s = 1 ξ ( s ) J * s · u d 2 s } ,
f λ = μ Re s = 1 ξ ( s ) [ p ( s · s 0 ) I 0 + u ] J * d 2 s
3 ε Re { ( μ κ ) 2 ( 1 + i ω μ c ) s = 1 ξ ( s ) J * s · u d 2 s } .
α μ 1 = Δ μ f μ 1 .
α μ 1 = C μ ( k ) 𝓔 1 2 f μ 1 ,
α λ 1 = C λ ( k ) 𝓔 1 2 f λ 1 ,

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