Abstract

It is well established that diffusion approximation is valid for light propagation in highly scattering media, but it breaks down in nonscattering regions. The previous methods that manipulate nonscattering regions are essentially boundary-to-boundary coupling (BBC) methods through a nonscattering void region based on the radiosity theory. We present a boundary-to-interior coupling (BIC) method. BIC is based on the fact that the collimated pencil beam incident on the medium can be replaced by an isotropic point source positioned at one reduced scattering length inside the medium from an illuminated point. A similar replacement is possible for the nondiffuse lights that enter the diffuse medium through the void, and it is formulated as the BIC method. We implemented both coupling methods using the finite element method (FEM) and tested for the circle with a void gap and for a four-layer adult head model. For mean time of flight, the BIC shows better agreement with Monte Carlo (MC) simulation results than BBC. For intensity, BIC shows a comparable match with MC data compared with that of BBC. The effect of absorption of the clear layer in the adult head model was investigated. Both mean time and intensity decrease as absorption of the clear layer increases.

© 2004 Optical Society of America

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References

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  1. M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
    [CrossRef] [PubMed]
  2. 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]
  3. S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
    [CrossRef] [PubMed]
  4. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
    [CrossRef]
  5. 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]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  7. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  8. M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef] [PubMed]
  9. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
    [CrossRef] [PubMed]
  10. C. K. Lee, R. E. Hobbs, “Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique,” Comput. Struct. 71, 9–34 (1999).
    [CrossRef]
  11. S. Kim, J. H. Lee, Y. T. Kim, “Near-infrared light propagation in an adult head model with refractive index mismatch,” in Proceedings of the World Congress on Medical Physics and Biomedical Engineering, 24–29 August 2003, Sydney, Australia CD-ROM, ISBN 1877040142.
  12. T. Spott, L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. 39, 6453–6465 (2000).
    [CrossRef]
  13. T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.
  14. J. H. Lee, S. Kim, Y. T. Kim, “Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary,” Med. Phys. (to be published).

2000 (3)

1999 (1)

C. K. Lee, R. E. Hobbs, “Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique,” Comput. Struct. 71, 9–34 (1999).
[CrossRef]

1997 (1)

1996 (1)

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[CrossRef]

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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

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]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Dehghani, H.

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Essenpreis, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Feng, T.-C.

Firbank, 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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Haskell, R. C.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Hobbs, R. E.

C. K. Lee, R. E. Hobbs, “Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique,” Comput. Struct. 71, 9–34 (1999).
[CrossRef]

Ishikawa, T.

T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Kim, S.

J. H. Lee, S. Kim, Y. T. Kim, “Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary,” Med. Phys. (to be published).

S. Kim, J. H. Lee, Y. T. Kim, “Near-infrared light propagation in an adult head model with refractive index mismatch,” in Proceedings of the World Congress on Medical Physics and Biomedical Engineering, 24–29 August 2003, Sydney, Australia CD-ROM, ISBN 1877040142.

Kim, Y. T.

S. Kim, J. H. Lee, Y. T. Kim, “Near-infrared light propagation in an adult head model with refractive index mismatch,” in Proceedings of the World Congress on Medical Physics and Biomedical Engineering, 24–29 August 2003, Sydney, Australia CD-ROM, ISBN 1877040142.

J. H. Lee, S. Kim, Y. T. Kim, “Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary,” Med. Phys. (to be published).

Lee, C. K.

C. K. Lee, R. E. Hobbs, “Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique,” Comput. Struct. 71, 9–34 (1999).
[CrossRef]

Lee, J. H.

J. H. Lee, S. Kim, Y. T. Kim, “Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary,” Med. Phys. (to be published).

S. Kim, J. H. Lee, Y. T. Kim, “Near-infrared light propagation in an adult head model with refractive index mismatch,” in Proceedings of the World Congress on Medical Physics and Biomedical Engineering, 24–29 August 2003, Sydney, Australia CD-ROM, ISBN 1877040142.

McAdams, M. S.

Nieto-Vesperinas, M.

Ohmori, D.

T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.

Ohtomo, T.

T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.

Okada, E.

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[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]

Ripoll, J.

Schweiger, M.

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[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]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Spott, T.

Svaasand, L. O.

Tanifuji, T.

T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.

Tromberg, B. J.

Tsay, T.-T.

van der Zee, P.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

Appl. Opt. (2)

Comput. Struct. (1)

C. K. Lee, R. E. Hobbs, “Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique,” Comput. Struct. 71, 9–34 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (2)

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Phys. Med. Biol. (2)

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

S. Kim, J. H. Lee, Y. T. Kim, “Near-infrared light propagation in an adult head model with refractive index mismatch,” in Proceedings of the World Congress on Medical Physics and Biomedical Engineering, 24–29 August 2003, Sydney, Australia CD-ROM, ISBN 1877040142.

T. Tanifuji, T. Ohtomo, D. Ohmori, T. Ishikawa, “Time resolved reflectance of an optical pulse from scattering bodies with nonscattering regions utilizing the finite difference time domain analysis,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 1094–1097.

J. H. Lee, S. Kim, Y. T. Kim, “Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary,” Med. Phys. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Manipulation of the void problem in (a) Arridge et al.,3 (b) Ripoll et al.,4 (c) this work. S is a void boundary, V is the region of diffuse medium whose reduced scattering coefficient is μ s ′, and dΩ is the solid angle subtended by dS′ at r.

Fig. 2
Fig. 2

Finite element meshes generated over circular domains with a clear gap with (a) 1-, (b) 2-, (c) 3-mm thicknesses.

Fig. 3
Fig. 3

Exitance and mean time in the circle with a clear gap by use of the various models of diffuse-diffuse coupling by the void region.

Fig. 4
Fig. 4

Finite element meshes for the adult head model: (a) coarse mesh and (b) fine mesh.

Fig. 5
Fig. 5

Mean and partial mean time of flight in the four-layer adult head model by use of the various models of diffuse-diffuse coupling via the clear CSF layer for CSF absorption coefficient μ a,CSF = 0.

Fig. 6
Fig. 6

Mean and partial mean time of flight in the four-layer adult head model by use of the various models of diffuse-diffuse coupling via the clear CSF layer for CSF absorption coefficient μ a,CSF = 0.025 mm-1.

Fig. 7
Fig. 7

Mean and partial mean time of flight in the four-layer adult head model by use of the various models of diffuse-diffuse coupling via the clear CSF layer for CSF absorption coefficient μ a,CSF = 0.05 mm-1.

Fig. 8
Fig. 8

Integrated intensity or exitance in the four-layer adult head model by use of the various models of diffuse-diffuse coupling via the clear CSF layer with respect to CSF absorption coefficient μ a,CSF.

Tables (1)

Tables Icon

Table 1 Optical Properties and Thickness of Each Layer in the Adult Head Modela

Equations (80)

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

1cϕr, tt- · Drϕr, t+μarϕr, t=S0r, t, Jr, t=-Drϕr, t,
 · Drϕr, ω+-μar+iω/cϕr, ω=-S0r, ω,
Jnr=ζrϕr, ζr=121-Rϕr1+Rjr,
R101 2μRFμdμ, R101 3μ2RFμdμ.
Gr, rνr, rcos θ cos θ|r-r|2×exp-μa+i ωc|r-r|,
q0r=Sζrϕrπ Gr, rdS,
J-r=S I0r, ûr-rGr, rdS
I0r, ŝ=J+r/π,
J+r=1-Rϕr4 ϕr+1-Rjr2 Jnr.
Jnr=J+r-J-r.
Jnr=ζrϕr-2π1+Rjr×S1-Rϕr4 ϕr+1-Rjr2 JnrGr, rdS.
dPr, ŝ=I0r, ŝcos θdSdΩ.
dS0rs=dPr, ŝ1|rs-r|2 δ|rs-r|-1μs×δŝ-rs-r|rs-r|,
S0rs=S2πdSdΩI0r, ŝcos θ1|rs-r|2×δ|rs-r|-1μsδŝ-rs-r|rs-r|.
S0rs=SSdSdSI0r, ûr-rGr, r×1|rs-r|2 δ|rs-r|-1μs×δûr-r-rs-r|rs-r|,
S0rs=SSdSdS ζrϕrπ Gr, r×1|rs-r|2 δ|rs-r|-1μs×δûr-r-rs-r|rs-r|,
Vdνφir · Drϕr+-μar+iω/cϕr=-VdνφirS0r.
F0Φ+Λϕ, Jn=q0,
F0=K+C-iωB,
Λiϕ, JnSdSφirJnr.
Kij=VdνDrφirφjr,
Cij=Vdνμarφirφjr,
Bij=Vdνnrc0 φirφjr,
q0,i=VdνφirS0r.
Λϕ, Jn=AΦ.
Aij=SdSζrφirφjr.
q1=0Φ.
Σij0=S1S1dSdS ζrπ Gr, rφirφjr.
FΦ=q0,
F=K+C-iωB+A-Σ0.
Λϕ, Jn=EJ.
Eij=SdSφirφjr,
JJn,1, Jn,2,T.
Λiφ, Jn=j AijΦj-jΞijΦΦ+ΞijJJn,j,
ΞijΦ=SSdSdS2π φirφjr1-Rϕr1+RjrGr, r,
ΞijJ=SSdSdSπ φirφjr1-Rjr1+RjrGr, r.
J=E+ΞJ-1A-ΞΦΦ.
Σ1=EE+ΞJ-1A-ΞΦ.
F=K+C-iωB+Σ1.
F˜0Φ=q0+q1,
q1,i=VdνsφirsS0rs=VdνsϕirsSSdSdSπ ζrφrGr, r×δ|rs-r|-1/μs|rs-r|2 δûr-r-rs-r|rs-r|.
q1,i=SSdSdSπ φir+1μsûr-r×ζrϕrGr, r.
q1=Σ2Φ,
Σij2=SSdSdSπ ζrφir+1μs×r-r|r-r|φjrGr, r.
F=K+C-iωB+A-Σ2.
S0r=δr-rp+1/μsnˆp.
Ir=- Γr, tdt,
tr=- tΓr, tdt - Γr, tdt.
Ir=- Γr, texpiωtdtω=0=Γ˜r, ωω=0,
tr=-iω- Γr, texpiωtdtω=0Γ˜r, ωω=0=-i1Γ˜r, ωΓ˜r, ωωω=0=-iωlnΓ˜r, ωω=0.
-1IrIrμa=ctr.
ctr=-1ΓdcrΓdcrμa.
citir=-1ΓdcrΓdcrμa,i,
tr=i tir,
Lr=i citir.
Γdc=ζΦdc
tk=-iΦk/ωω=0/Φdc,k
Lmk=-Φdc,k/μa,m/Φdc,k,
F-iΦ/ω=iF/ωΦ.
iF/ω=B-iΣ0/ω
iF/ω=B+iΣ1/ω
iF/ω=B-iΣ2/ω
-i Σij0ω=S1S1dSdS ζrπ φirφjrνr, r×cos θ cos θc|r-r|exp-μa+iω/c|r-r|.
-iΣ1/ω=-EE+ΞJ-1-iΞJ/ωE-1Σ1+-iΞΦ/ω.
-i Σij2ω=SSdSdS ζrπ×φir+1μsr-r|r-r|×φjrνr, rcos θ cos θc|r-r|exp-μa+iω/c|r-r|.
F-Φ/μa,m=F/μa,mΦ,
F/μa,m=C/μa,m,
C/μa,mij=Vmdνφirφjr,
F/μa=-Σ0/μa
F/μa=Σ1/μa
F/μa=-Σ2/μa
-Σij0μa=S1S1dSdS ζrπ φirφjrνr, r×cos θ cos θ|r-r|exp-μa+iω/c|r-r|,
Σ1/μa=-EE+ΞJ-1ΞJ/μaE-1Σ1+ΞΦ/μa,
-Σij2μa=SSdSdS ζrπ×φir+1μsr-r|r-r|φjr×νr, rcos θ cos θ|r-r|×exp-μa+iω/c|r-r|.
Σ2-D,ij0=C1C1dRdR ζRπ G2-DR, RφiRφjR
Ξ2-D,ijΦ=CCdRdR2π φiRφjR×1-RϕR1+RjRG2-DR, R,
Ξ2-D,ijJ=CCdRdRπ φiRφjR×1-RjR1+RjRG2-DR, R
Σ2-D,ij2=CCdRdRπνR, RζRφjR×cos Θ cos Θ|R-R|-π/2π/2dβ cos2 β×φiR+1μsR-R|R-R|sec β×exp-μa+iω/c|R-R|sec β
G2-DR, R=νR, Rcos Θ cos Θ|R-R|-π/2π/2dβ cos2 β×exp-μa+iω/c|R-R|sec β.
G2-DaR, RνR, Rπ2cos Θ cos Θ|R-R|×exp-μa+iω/c|R-R|.

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