Abstract

Using the principle of energy conservation and laws of geometrical optics, we derive the photon transport equation for turbid biological media with spatially varying isotropic refractive index. We show that when the refractive index is constant, our result reduces to the standard radiative transfer equation and when the medium is lossless and free of scattering to the well known geometrical optics equations in refractive media.

© 2005 Optical Society of America

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References

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  1. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, (SPIE Press, Bellingham, 2000).
  2. J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
    [CrossRef] [PubMed]
  3. X. J. Wang, T. E. Milner, J. F. De Boer, Y. Zhang, D. H. Pashley, and J. S. Nelson “Characterization of dentin and enamel by use of optical coherence tomography,” Appl. Opt. 38, 2092–2096 (1999).
    [CrossRef]
  4. S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
    [CrossRef]
  5. E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
    [CrossRef] [PubMed]
  6. M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Third, Revised Edition, (Springer, New Jersey, 2004).
  7. B. R. Masters (ed.) Selected Papers on Optical Low-Coherence Reflectometry & Tomography, SPIE Milestone Series MS 165 (SPIE Optical Engineering Press, Bellingham, 2001).
  8. A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Lasers, Photonics and Electro-Optics), (Plenum Publishing Corporation, New York, 1995).
  9. L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
    [CrossRef]
  10. S. Chandrasekhar, Radiative Transfer, (Dover Publications, New York, 1960).
  11. L-H. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
    [CrossRef]
  12. E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).
  13. A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
    [CrossRef] [PubMed]
  14. M. Born and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).
  15. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, (Wiley Interscience Publishers, New Jersey, 1965).
  16. H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A: Pure Appl. Opt. 1, L1–L2 (1999).
    [CrossRef]
  17. T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A: Pure Appl. Opt. 5, 137–141 (2003).
    [CrossRef]
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978).
  19. M. Kline, “A note on the expansion coefficient of geometrical optics,” Comm. Pure and Appl. Maths. 14, 473–479 (1961).
    [CrossRef]
  20. S. I. Grossman, Calculus, Fifth Edition, (Harcourt Brace College Publishers, Philadelphia, 1991).
  21. G. Yankovsky, Higher Algebra, (Mir Publishers, Moscow, 1980).
  22. D. J. Griffiths, Introduction to Electrodynamics, Third Edition, (Prentice Hall, New Jersey, 1999).
  23. W. L. Burke, Applied Differential Geometry, (Cambridge University Press, Cambridge, 1985).

2003 (1)

T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A: Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

2002 (2)

E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

1999 (2)

X. J. Wang, T. E. Milner, J. F. De Boer, Y. Zhang, D. H. Pashley, and J. S. Nelson “Characterization of dentin and enamel by use of optical coherence tomography,” Appl. Opt. 38, 2092–2096 (1999).
[CrossRef]

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A: Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

1996 (1)

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

1995 (1)

L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
[CrossRef]

1993 (1)

1987 (1)

A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
[CrossRef] [PubMed]

1961 (1)

M. Kline, “A note on the expansion coefficient of geometrical optics,” Comm. Pure and Appl. Maths. 14, 473–479 (1961).
[CrossRef]

Arridge, S. R.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Austin, T.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Aydin, E. D.

E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).

Baynes, J. W.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Boppart, S. A.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

Bouma, B. E.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

Brezinski, M. E.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

Burke, W. L.

W. L. Burke, Applied Differential Geometry, (Cambridge University Press, Cambridge, 1985).

Chandrasekhar, S.

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

De Boer, J. F.

De Oliveira, C. R. E.

E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).

Deemer, E. K.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Deply, D. T.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Ediger, M. N.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Everdell, N.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Ferwerda, H. A.

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A: Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

Fujimoto, J. G.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

Gibson, A.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Goddard, A. J. H.

E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics, Third Edition, (Prentice Hall, New Jersey, 1999).

Grossman, S. I.

S. I. Grossman, Calculus, Fifth Edition, (Harcourt Brace College Publishers, Philadelphia, 1991).

Hebden, J. C.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Hillman, E. M. C.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Hull, E. L.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Ishimaru, A.

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

Jacques, S. L.

L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
[CrossRef]

L-H. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

Jiang, H.

T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A: Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, (Wiley Interscience Publishers, New Jersey, 1965).

Khan, T.

T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A: Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

Kline, M.

M. Kline, “A note on the expansion coefficient of geometrical optics,” Comm. Pure and Appl. Maths. 14, 473–479 (1961).
[CrossRef]

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, (Wiley Interscience Publishers, New Jersey, 1965).

Masters, B. R.

B. R. Masters (ed.) Selected Papers on Optical Low-Coherence Reflectometry & Tomography, SPIE Milestone Series MS 165 (SPIE Optical Engineering Press, Bellingham, 2001).

Meek, J. H.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Milner, T. E.

Nelson, J. S.

Niemz, M. H.

M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Third, Revised Edition, (Springer, New Jersey, 2004).

Pashley, D. H.

Stroman, M. L.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Tearney, G. J.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

Tuchin, V.

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, (SPIE Press, Bellingham, 2000).

Unione, A. H. T.

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

Van Gemert, M. J. C.

A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
[CrossRef] [PubMed]

Van-Gemert, M. J. C.

A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Lasers, Photonics and Electro-Optics), (Plenum Publishing Corporation, New York, 1995).

Wang, L-H.

L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
[CrossRef]

L-H. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

Wang, X. J.

Welch, A. J.

A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
[CrossRef] [PubMed]

A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Lasers, Photonics and Electro-Optics), (Plenum Publishing Corporation, New York, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

Wyatt, J. S.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Yankovsky, G.

G. Yankovsky, Higher Algebra, (Mir Publishers, Moscow, 1980).

Yoon, G.

A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
[CrossRef] [PubMed]

Yusof, R. M

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Zhang, Y.

Zheng, L-Q

L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
[CrossRef]

Am. Assoc. Phys. Med. (1)

E. D. Aydin, C. R. E. De Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Am. Assoc. Phys. Med. 29, 2013–2023 (2002).

Appl. Opt. (1)

Comm. Pure and Appl. Maths. (1)

M. Kline, “A note on the expansion coefficient of geometrical optics,” Comm. Pure and Appl. Maths. 14, 473–479 (1961).
[CrossRef]

Comp. Meth. and Prog. in Biomed. (1)

L-H. Wang, S. L. Jacques, and L-Q Zheng, “MCML - Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. and Prog. in Biomed. 47, 131–146 (1995).
[CrossRef]

Devel. Biol. (1)

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, “Investigation of developing embryonic morphology using optical coherence tomography,” Devel. Biol. 177, 54–63 (1996).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A: Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A: Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

Lasers in Surgery and Medicine (1)

A. J. Welch, G. Yoon, and M. J. C. Van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers in Surgery and Medicine 6, 488–493 (1987).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Other (13)

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, (SPIE Press, Bellingham, 2000).

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

E. L. Hull, M. N. Ediger, A. H. T. Unione, E. K. Deemer, M. L. Stroman, and J. W. Baynes, “Noninvasive, optical detection of diabetes: model studies with porcine skin,” Opt. Express12, 4496–4510 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4496
[CrossRef] [PubMed]

M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Third, Revised Edition, (Springer, New Jersey, 2004).

B. R. Masters (ed.) Selected Papers on Optical Low-Coherence Reflectometry & Tomography, SPIE Milestone Series MS 165 (SPIE Optical Engineering Press, Bellingham, 2001).

A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Lasers, Photonics and Electro-Optics), (Plenum Publishing Corporation, New York, 1995).

M. Born and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, (Wiley Interscience Publishers, New Jersey, 1965).

S. I. Grossman, Calculus, Fifth Edition, (Harcourt Brace College Publishers, Philadelphia, 1991).

G. Yankovsky, Higher Algebra, (Mir Publishers, Moscow, 1980).

D. J. Griffiths, Introduction to Electrodynamics, Third Edition, (Prentice Hall, New Jersey, 1999).

W. L. Burke, Applied Differential Geometry, (Cambridge University Press, Cambridge, 1985).

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

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Figures (2)

Fig. 1.
Fig. 1.

Transport of an infinitesimally small photon packet with phase space volume, V S 0 , along an infinitesimally small ray tube surrounding a central light ray.

Fig. 2.
Fig. 2.

Details of the stationary and moving coordinate systems

Equations (54)

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

Ω ̂ = d r d s = 1 μ 2 cos ( φ ) x ̂ + 1 μ 2 sin ( φ ) y ̂ + μ z ̂
V 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s ( μ a + μ s ) 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
+ Δ s δ V s μ s ( 1 1 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ
+ Δ s δ V s ε ( x , y , z , μ , φ , t ) d x d y d z μ d φ + O ( Δ s 2 )
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= δ V s 0 [ ( 𝓛 J ) s + Δ s , t + Δ t ( 𝓛 J ) s , t ] d x 0 d y 0 d z 0 d μ 0 d φ 0
( 𝓛 J ) s + Δ s , t + Δ t ( 𝓛 J ) s , t = J n c 𝓛 t Δ s + J 𝓛 s Δ s + L J s Δ s + O ( Δ s 2 )
J = x x 0 x y 0 x z 0 x μ 0 x φ 0 y x 0 y y 0 y z 0 y μ 0 y φ 0 z x 0 z y 0 z z 0 z μ 0 z φ 0 μ x 0 μ y 0 μ z 0 μ μ 0 μ φ 0 φ x 0 φ y 0 φ z 0 φ μ 0 φ φ 0
J = σ sgn ( σ ) J 1 , σ ( 1 ) J 2 , σ ( 2 ) J 3 , σ ( 3 ) J 4 , σ ( 4 ) J 5 , σ ( 5 )
J = σ sgn ( σ ) j = 1 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
J s = v { x 0 , y 0 , z 0 } σ sgn ( σ ) ( k = 1 5 Ω v ϑ ( k ) ϑ ( k ) v ) j = 1 ϑ 0 ( j ) v 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
+ v { μ 0 , φ 0 } σ sgn ( σ ) ( k = 1 5 ϑ ( k ) v ϑ ( k ) ( v s ) ) j = 1 ϑ 0 ( j ) v 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
J s = ( Ω x x + Ω y y + Ω z z + μ ( μ s ) + φ ( φ s ) ) J
r · V = V x x + V y y + V z z
J s = J r · Ω ̂ + ( μ ( μ s ) + φ ( φ s ) ) J
r · Ω ̂ = 1 R 1 ( s ) + 1 R 2 ( s )
( n Ω ̂ ) s = r n
Ω ̂ s = r n n 1 n d n d s Ω ̂
μ ̂ = μ cos ( φ ) x ̂ μ sin ( φ ) y ̂ + 1 μ 2 z ̂
φ ̂ = sin ( φ ) x ̂ + cos ( φ ) y ̂
Ω ̂ s = 1 1 μ 2 μ s μ ̂ + 1 μ 2 φ s φ ̂
μ s = 1 μ 2 r n · μ ̂ n
φ s = r n · φ ̂ n 1 μ 2
μ ̂ μ = Ω ̂ 1 μ 2
φ ̂ φ = μ μ ̂ 1 μ 2 Ω ̂
μ ( μ s ) = μ 1 μ 2 r n · μ ̂ n r n · Ω ̂ n
φ ( φ s ) = μ 1 μ 2 r n · μ ̂ n r n · Ω ̂ n
J s = ( 1 R 1 ( s ) + 1 R 2 ( s ) 2 r n · Ω ̂ n ) J
J s = ( 1 R 1 ( s ) + 1 R 2 ( s ) 2 n d n d s ) J
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s 0 [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] J d x 0 d y 0 d z 0 d μ 0 d φ 0 + O ( Δ s 2 )
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] d x d y d z d μ d φ + O ( Δ s 2 )
δ V s [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] d x d y d z d μ d φ
δ V s ( ( μ a + μ s ) 𝓛 + ε ) d x d y d z d μ d φ
δ V s μ s ( 1 1 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ = 0
n ( r ) c 𝓛 ( r , Ω ̂ , t ) t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) + n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
r 𝒮 = n ( r ) Ω ̂
n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= n 2 ( r ) r s · r ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + n 2 ( r ) Ω ̂ s · Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) = μ ̂ 1 μ 2 μ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + φ ̂ 1 μ 2 φ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= Ω ̂ · r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) r n ( r ) · Ω ̂ 𝓛 ( r , Ω ̂ , t ) 2 n ( r ) Ω ̂ · r n ( r )
n ( r ) c 𝓛 ( r , Ω ̂ , t ) t + Ω ̂ · r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) r n ( r ) · Ω ̂ 𝓛 ( r , Ω ̂ , t )
+ ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) 2 n ( r ) Ω ̂ · r n ( r )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
n 0 c 𝓛 ( r , Ω ̂ , t ) t + Ω ̂ · r 𝓛 ( r , Ω ̂ , t )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ ) + n 2 ( r ) d d s ( 𝓛 ( r , Ω ̂ ) n 2 ( r ) ) = 0
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0 exp ( s 0 s ( 1 R 1 ( s ) + 1 R 2 ( s ) ) d s )
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0

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