Abstract

An adequate description of the propagation of near-infrared light in highly scattering media is important for imaging hidden luminescent inhomogeneities in tissues. We present an improved approach to modeling of the forward problem based on a telegraph equation. The asymptotic solutions of the telegraph equation are derived and analyzed. The expression for the mean photon path length is also obtained.

© 2003 Optical Society of America

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References

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  1. M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.
  2. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  3. D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, “Analytical solution for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue,” J. Opt. Soc. Am. A 18, 1523–1530 (2001).
    [CrossRef]
  4. B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  5. M. Xu, M. Lax, R. R. Alfano, “Time-resolved Fourier optical diffuse tomography,” J. Opt. Soc. Am. A 18, 1535–1542 (2001).
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  7. S. Fantini, M. A. Franceschini, E. Gratton, “Effective source term in the diffusion equation for transport in turbid media,” Appl. Opt. 36, 156–163 (1997).
    [CrossRef] [PubMed]
  8. D. J. Durian, J. Rundick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. 14, 235–245 (1997).
    [CrossRef]
  9. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
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    [PubMed]
  15. W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
    [CrossRef] [PubMed]
  16. S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. M. Pawlowski, D. F. Wilson, “Monitoring of the oxygen pressure in the blood of live animals using the oxygen dependent quenching of phosphorescence,” Adv. Exp. Med. Biol. 316, 179–183 (1992).
    [PubMed]
  19. R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
    [CrossRef]
  20. S. A. Vinogradov, D. F. Wilson, “Metallotetrabenzoporphyrins. New phosphorescent probes for oxygen measurements,” J. Chem. Soc. Perkin Trans. 2, 103–111 (1994).
  21. S. A. Vinogradov, D. F. Wilson, “Extended porphyrins—new IR phosphors for oxygen measurements,” Adv. Exp. Med. Biol. 411, 597–603 (1997).
  22. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities withing turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef]
  23. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  24. T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
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  28. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  29. N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972).
  30. M. A. Naimark, Lineinye Differentsyal’nye Operatory (Gosudarstvennoe Izdatel’stvo Tekhniko-teoretitcheskoi Literatury, Moscow, 1954; in Russian).
  31. E. C. Titchmarsh, Eigenfunction Expansion Associated with Second-Order Differential Equations (Clarendon, Oxford, 1962).
  32. M. B. Fedoryuk, Asimptotika: Integraly, Summy i Ryady (Nauka, Moscow, 1987; in Russian).
  33. I. N. Minin, Teoria Perenosa Izluchenia v Atmospherah Planet (Nauka, Moscow, 1988; in Russian).

2001 (3)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1998 (1)

1997 (8)

1996 (3)

1994 (2)

S. A. Vinogradov, D. F. Wilson, “Metallotetrabenzoporphyrins. New phosphorescent probes for oxygen measurements,” J. Chem. Soc. Perkin Trans. 2, 103–111 (1994).

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities withing turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

1992 (3)

M. Pawlowski, D. F. Wilson, “Monitoring of the oxygen pressure in the blood of live animals using the oxygen dependent quenching of phosphorescence,” Adv. Exp. Med. Biol. 316, 179–183 (1992).
[PubMed]

R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
[CrossRef]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

1988 (1)

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
[CrossRef] [PubMed]

1987 (1)

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Alfano, R. R.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Boas, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Chance, B.

Chandrasekhar, S.

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

Chernomordik, V.

Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Dugan, B. W.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, D. F. Wilson, “Frequency domain instrument for measuring phosphorescence lifetime distributions in heterogeneous samples,” Rev. Sci. Instrum. 72, 3396–3406 (2001).
[CrossRef]

Durduran, T.

Durian, D. J.

Evans, S. M.

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

Fantini, S.

Fedoryuk, M. B.

M. B. Fedoryuk, Asimptotika: Integraly, Summy i Ryady (Nauka, Moscow, 1987; in Russian).

Fernandez-Seara, M. A.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, D. F. Wilson, “Frequency domain instrument for measuring phosphorescence lifetime distributions in heterogeneous samples,” Rev. Sci. Instrum. 72, 3396–3406 (2001).
[CrossRef]

Foster, T. H.

Franceschini, M. A.

Furutsu, K.

Gaida, G.

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

Gandjbakhche, A.

Gannot, I.

Gratton, E.

Green, T. J.

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

Hattery, D.

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Hull, E. L.

Jenkins, W. T.

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

Jess, H.

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

Kaltenbach, J.

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

Kaschke, M.

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

Koch, C.

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

Lax, M.

Lebedev, N. N.

N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972).

Li, X. D.

Lo, L.-W.

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

Loew, M.

Maniara, G.

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

Minin, I. N.

I. N. Minin, Teoria Perenosa Izluchenia v Atmospherah Planet (Nauka, Moscow, 1988; in Russian).

Naimark, M. A.

M. A. Naimark, Lineinye Differentsyal’nye Operatory (Gosudarstvennoe Izdatel’stvo Tekhniko-teoretitcheskoi Literatury, Moscow, 1954; in Russian).

Nichols, M. G.

O’Leary, M. A.

Pawlowski, M.

M. Pawlowski, D. F. Wilson, “Monitoring of the oxygen pressure in the blood of live animals using the oxygen dependent quenching of phosphorescence,” Adv. Exp. Med. Biol. 316, 179–183 (1992).
[PubMed]

R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
[CrossRef]

Riva, C. E.

R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
[CrossRef]

Rumsey, W. L.

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
[CrossRef] [PubMed]

Rundick, J.

Shonat, R. D.

R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

Titchmarsh, E. C.

E. C. Titchmarsh, Eigenfunction Expansion Associated with Second-Order Differential Equations (Clarendon, Oxford, 1962).

Vanderkooi, J. M.

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
[CrossRef] [PubMed]

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

Vinogradov, S. A.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, D. F. Wilson, “Frequency domain instrument for measuring phosphorescence lifetime distributions in heterogeneous samples,” Rev. Sci. Instrum. 72, 3396–3406 (2001).
[CrossRef]

S. A. Vinogradov, D. F. Wilson, “Extended porphyrins—new IR phosphors for oxygen measurements,” Adv. Exp. Med. Biol. 411, 597–603 (1997).

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

S. A. Vinogradov, D. F. Wilson, “Metallotetrabenzoporphyrins. New phosphorescent probes for oxygen measurements,” J. Chem. Soc. Perkin Trans. 2, 103–111 (1994).

Walker, S. A.

Wilson, B. C.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Wilson, D. F.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, D. F. Wilson, “Frequency domain instrument for measuring phosphorescence lifetime distributions in heterogeneous samples,” Rev. Sci. Instrum. 72, 3396–3406 (2001).
[CrossRef]

S. A. Vinogradov, D. F. Wilson, “Extended porphyrins—new IR phosphors for oxygen measurements,” Adv. Exp. Med. Biol. 411, 597–603 (1997).

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

S. A. Vinogradov, D. F. Wilson, “Metallotetrabenzoporphyrins. New phosphorescent probes for oxygen measurements,” J. Chem. Soc. Perkin Trans. 2, 103–111 (1994).

R. D. Shonat, D. F. Wilson, C. E. Riva, M. Pawlowski, “Oxygen distribution in the retinal and choroidal vessels of the cat as measured by a new phosphorescence imaging method,” Appl. Opt. 33, 3711–3718 (1992).
[CrossRef]

M. Pawlowski, D. F. Wilson, “Monitoring of the oxygen pressure in the blood of live animals using the oxygen dependent quenching of phosphorescence,” Adv. Exp. Med. Biol. 316, 179–183 (1992).
[PubMed]

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
[CrossRef] [PubMed]

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Wrobel, W.

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

Xu, M.

Yodh, A. G.

Adv. Exp. Med. Biol. (2)

M. Pawlowski, D. F. Wilson, “Monitoring of the oxygen pressure in the blood of live animals using the oxygen dependent quenching of phosphorescence,” Adv. Exp. Med. Biol. 316, 179–183 (1992).
[PubMed]

S. A. Vinogradov, D. F. Wilson, “Extended porphyrins—new IR phosphors for oxygen measurements,” Adv. Exp. Med. Biol. 411, 597–603 (1997).

Appl. Opt. (6)

Biophys. J. (1)

S. A. Vinogradov, L.-W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, D. F. Wilson, “Noninvasive imaging of the distribution in oxygen in tissue in vivo using near-infrared phosphors,” Biophys. J. 70, 1609–1617 (1996).
[CrossRef] [PubMed]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Biol. Chem. (1)

J. M. Vanderkooi, G. Maniara, T. J. Green, D. F. Wilson, “An optical method for measurement of dioxygen concentration based on quenching of phosphorescence,” J. Biol. Chem. 262, 5476–5482 (1987).
[PubMed]

J. Chem. Soc. Perkin Trans. (1)

S. A. Vinogradov, D. F. Wilson, “Metallotetrabenzoporphyrins. New phosphorescent probes for oxygen measurements,” J. Chem. Soc. Perkin Trans. 2, 103–111 (1994).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Med. Phys. (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (2)

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities withing turbid media: analytical solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Rev. Sci. Instrum. (1)

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, D. F. Wilson, “Frequency domain instrument for measuring phosphorescence lifetime distributions in heterogeneous samples,” Rev. Sci. Instrum. 72, 3396–3406 (2001).
[CrossRef]

Science (1)

W. L. Rumsey, J. M. Vanderkooi, D. F. Wilson, “Imaging of phosphorescence: a novel method for measuring the distribution of oxygen in perfused tissue,” Science 241, 1649–1651 (1988).
[CrossRef] [PubMed]

Other (9)

M. Kaschke, H. Jess, G. Gaida, J. Kaltenbach, W. Wrobel, “Transillumination imaging of tissue by phase modulation techniques,” in Advances in Optical Imaging and Photon Migration, R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 88–92.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972).

M. A. Naimark, Lineinye Differentsyal’nye Operatory (Gosudarstvennoe Izdatel’stvo Tekhniko-teoretitcheskoi Literatury, Moscow, 1954; in Russian).

E. C. Titchmarsh, Eigenfunction Expansion Associated with Second-Order Differential Equations (Clarendon, Oxford, 1962).

M. B. Fedoryuk, Asimptotika: Integraly, Summy i Ryady (Nauka, Moscow, 1987; in Russian).

I. N. Minin, Teoria Perenosa Izluchenia v Atmospherah Planet (Nauka, Moscow, 1988; in Russian).

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Equations (97)

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I0/I=τ0/τ=1+KSVO2.
Uer, r0, t-t0=C Vd3rs --dtdtUirs, r0, t-t0Upr, rs, t-tqτrs, t-t,
s, Idiff+1vIdifft=-αIdiff+αλ4π 02πdφ -11 pϑ×Idiffη, φdη+αλpϑ0P0exp-α|r-r0|4π|r-r0|2×δt-t0-|r-r0|v.
s, Idiff=-γIdiff+αλ4π 02πdφ -11 pϑIdiffη, φdη+αλpϑ0P0exp-ikvt0-γ|r-r0|4π|r-r0|2,
γ=α+ik,
k=ω/v.
Idiff=m=0cos mφ n=m InmPnmη+m=1sin mφ n=m InmPnmη.
div F=-3γ-αλU-3αλP0 exp-iωt0exp-γ|r-r0|4π|r-r0|2,
U=-γF.
ΔU-χ2U=-q,
χ2=3γγ-αλ,
q=3P0αλγ exp-iωt0exp-γ|r-r0|4π|r-r0|2.
ΔU-3aU-3b1vtU-31v22t2U=-q,
a=α21-λ, b=α2-λ, q=3P0αλα+1vtexp-α|r-r0|4π|r-r0|2 δt-t0-|r-r0|v.
ΔΨ-1w22t2Ψ=-3b24-aΨ-q×exp32bwt-t0,
Idiffη>0=0, z=0,
Idiffη<0=0, z=zd,
Idiff=U+s, F.
s, Idiff=-γIdiff+αλ4π 02πdφ -10 Idiffη, φdη+αλP0exp-iωt0-γ|r-r0|z=04π|r-r0|z=02.
U=-Fz, z=0.
Uz+Fzz=34αλU+αλP0exp-iωt0-γ|r-r0|z=04π|r-r0|z=02,
-Uz+Fzz=-2γU+¾ αλU+αλP0exp-iωt0-γ|r-r0|z=04π|r-r0|z=02.
Uz=γU, z=0.
Uz=-γU, z=zd.
|γ|  |χ|,
|γ||r-r0|  1.
q3λP0γ exp-ikvt0δr-r0.
U=3λP0γ exp-ikvt0G,
ΔG-χ2G=-δr-r0,
δφ-φ0=12πn=-+expinφ-φ0, G=12πn=-+ ψnr, z expinφ-φ0,
Lrψn+Lzψn=-1rδr-r0δz-z0,
Lr=1rrrr-n2r2, r0, 
Lz=2z2-χ2, z0, zd
LrRnμ, r=-μ2Rnμ, r,
limr|r Rnμ, r|=0,
limr0|Rnμ, r|<.
ψnr, z=0 Φnμ, zJnμrμdμ.
1rδr-r0=0 Jnμr0Jnμrμdμ
2z2Φnμ, z-μ2+χ2Φnμ, z=-Jnμr0δz-z0,
zΦnμ, z-γΦnμ, z=0, z=0,
zΦnμ, z+γΦnμ, z=0, z=zd.
Φnμ, z=Φn1μ, z+Φn2μ, z,
Φn1μ, z=-12Jnμr0μ2+χ2 c1zexp-z-zμ2+χ2δz-z0dz+12Jnμr0μ2+χ2 c2zexp-z-zμ2+χ2δz-z0dz,
Φn2μ, z=a expzμ2+χ2+b×exp-zμ2+χ2.
Reμ2 + χ2 = 0,
Reμ2 + χ2  0
exp-z - z0μ2 + χ2,
exp-z0 - zμ2 + χ2,
G=12π 0 Kμ, z, z0J0μRμdμ,
Kμ, z, z0=12μ2+χ2exp-|z-z0|μ2+χ2+1Wμμ2+χ2-γ2 exp-zd-z+z0μ2+χ2+1Wμμ2+χ2-γ2exp-zd-z-z0μ2+χ2+1Wμμ2+χ2-γ2 exp-zd+z-z0μ2+χ2+1Wμμ2+χ2-γ2expzd-z-z0μ2+χ2,
Wμ=μ2+χ2+γ2 expzdμ2+χ2-μ2+χ2-γ2 exp-zdμ2+χ2,
R=r2+r02-2rr0 cosφ-φ01/2.
exp-χR2+z-z021/2R2+z-z021/2=0exp-|z-z0|χ2+μ2χ2+μ2J0μRμdμ,
G0=14πexp-χR2+z-z021/2R2+z-z02.
G=G0+14πexp-χR2+z+z021/2R2+z+z021/2-γ2π 0exp-z+z0μ2+χ2μ2+χ2+γμ2+χ2J0μRμdμ.
|χ| |r-r0|  1
J0μR=½H01μR+H02μR, H01-μR=-H02μR,
G=n=04 Gn,
G0=14πexp-χR2+z-z021/2R2+z-z021/2, Gn=χ8πm0Γ Fnmξ×exp-χZnm cosh ξH01χR sinh ξsinh ξdξ, n=1, 2, 3, 4,
Z1m=2m+2zd-z+z0,Z2m=2m+2zd-z-z0,Z3m=2m+2zd+z-z0,Z4m=2mzd+z+z0,F1mξ=F3mξ=χ cosh ξ-γχ cosh ξ+γ2m+2,F2mξ=F4mξ=χ cosh ξ-γχ cosh ξ+γ2m+1.
cos ζ=Re χ|χ|, sin ζ=Im χ|χ|,
H01χR sinh ξ=2πχR sinh ξ1/2×expiχR sinh ξ-iπ4.
Gn=14πχ2πR exp-iπ4m0×Γ Fnmξsinh ξ exp-nmΨnmξdξ,
nm=R2+Znm21/2, Ψnmξ=χ cosiξ+θnm, cos θnm=Znmnm, sin θnm=R/nm.
ddξΨnmξnm=0, d2dξ2Ψnmξnm=Ψξnm,
argξ-ξnm=-ζ/2, ξΓnm.
Ψnmξχ+½ χξ-ξnm2,
Fnmξsinh ξexpiπ4Rnm Fnmiθnm.
GG0+14πn,m0 Fnmexp-χnmnm, n=1, 2, 3, 4,
Fnm=1-2γnmχZnm+γnm2m+2, n=1, 3, Fnm=1-2γnmχZnm+γnm2m+1, n=2, 4.
U3λP0γ exp-ikvt0G,
G=G0+14πexp-χR2+z+z021/2R2+z+z021/2-14π2γ exp-χR2+z+z021/2χz+z0+γR2+z+z021/2.
Ut=3λαv8π2P0n=041n - γkFnkexpΨnkdk,
Ψnk=-α3nk0+ik2-σ21/2+iαvt-t0k, σ=½ λ, k0=1-σ, Rek0+ik2-σ21/20.
ksn=ik0-iσ vt-t0v2t-t02-3n21/2.
Ψnksn=-αk0vt-t0+ασ3 Ln,
Ln=w2t-t02-n2,
w=v3.
UtvP04π23αλ3/2n=041+iksnLn3/2×Fnksnexp-αk0vt-t0+ασ3Ln.
dE=1vIdΩd3r,
dϱ=1vUtd3r,
1vUt=3λP08π2v|r-r0| - γ exp-χ|r-r0|+iωt-t0dω.
ϱ=λP02πi -expiωt-t0ω-iαv1-λdω.
ϱ=0, t<t0, ϱ=λP0 exp-αv1-λt-t0, t>t0,
pt=1vN0hνV Utd3r=t-t0 Wtdt, t>t0,
Wt=1vN0hνV-tUtd3r.
t=- tWtdt, l=vt.
Uω|ω=0=- Utdt,
t=1vN0hνV Uω|ω=0d3r.
t=1αvλ1-λ, l=1αλ1-λ.
t=j=04 tj,t0=t1-exp-χ0zd2 coshχ0zd2-z0,t1=tα-χ02W0exp-χ032zd+z0×sinhχ0zd2,t2=-tα2-χ02W0exp-χ032zd-z0×sinhχ0zd2,t3=tα-χ02W0exp-χ052zd-z0×sinhχ0zd2,t4=-tα2-χ02W0exp-χ0zd2+z0×sinhχ0zd2,
W0=α+χ02-α-χ02 exp-2χzd, χ0=α31-λ1/2.
ϱ=ϱr,
ϱr=12πv -dω expiωt02πdφ×-11dη 0ϱ Uω2d.
ϱ-ϱr=3λP02π - γ exp-χϱ+ikvt-t01+χϱχ2dk=0.
χ3ασw2t-t02-21/2, ϱwt-t0.
U0=38παλγP0 exp-ikvt0×lnγ+χγ-χexp-χ|r-r0|χ|r-r0|+2|r-r0| γdμχ2-μ2exp-μ|r-r0|.

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