Abstract

The diffusion approximation proves to be valid for light propagation in highly scattering media, but it breaks down in the presence of nonscattering regions. We present a compact expression of the boundary conditions for diffusive media with nonscattering regions, taking into account small-index mismatch. Results from an integral method based on the extinction theorem boundary condition are contrasted with both Monte Carlo and finite-element-method simulations, and a study of its limit of validity is presented. These procedures are illustrated by considering the case of the cerebro-spinal fluid in the brain, for which we demonstrate that for practical situations in light diffusion, these boundary conditions yield accurate results.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
    [CrossRef] [PubMed]
  2. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, March1995, pp. 38–40.
  3. See related studies in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson, eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).
  4. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
    [CrossRef]
  5. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [CrossRef] [PubMed]
  6. C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
    [CrossRef]
  7. Ch. L. Matson, N. Clark, L. McMackin, J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–220 (1997).
    [CrossRef] [PubMed]
  8. X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
    [CrossRef] [PubMed]
  9. S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
    [CrossRef] [PubMed]
  10. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Simultaneous reconstruction of optical absorption and scattering maps in turbid media from near-infrared frequency-domain data,” Opt. Lett. 20, 2128–2130 (1995).
    [CrossRef] [PubMed]
  11. S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
    [CrossRef]
  12. P. N. den Outer, Th. M. Nieuwenbuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  13. S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
    [CrossRef]
  14. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  15. J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
    [CrossRef]
  16. S. J. Norton, T. Vo-Dinh, “Diffraction tomographic imaging with photon density waves: an explicit solution,” J. Opt. Soc. Am. A 15, 2670–2677 (1998).
    [CrossRef]
  17. S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
    [CrossRef]
  18. S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumor in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
    [CrossRef]
  19. J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
    [CrossRef]
  20. S. R. Arridge, “Optical tomography in medical imaging,” Topical Rev. Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  21. 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]
  22. 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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
    [CrossRef] [PubMed]
  23. S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, Proc. SPIE2979, 130–138 (1997).
  24. S. Takahashi, Y. Yamada, “Simulation of 3D light propagation in a layered head model including a clear CSF layer,” in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).
  25. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  26. N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
    [CrossRef]
  27. C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
    [CrossRef]
  28. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [CrossRef] [PubMed]
  29. K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
    [CrossRef] [PubMed]
  30. J. Wu, F. Partovi, M. S. Field, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
    [CrossRef] [PubMed]
  31. S. Ito, “Diffusion of collimated, narrow beam waves in discrete random media,” Appl. Opt. 34, 7106–7112 (1995).
    [CrossRef] [PubMed]
  32. R. C. Haskell, L. O. Vaasand, T. Tsay, T. 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]
  33. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  34. D. J. Durian, J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source,” J. Opt. Soc. Am. A 16, 837–844 (1999).
    [CrossRef]
  35. A. D. Kim, A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
    [CrossRef]
  36. F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
    [CrossRef]
  37. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. I.
  38. J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A 16, 1453–1465 (1999).
    [CrossRef]
  39. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  40. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  41. S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  42. M. Schweiger, S. R. Arridge, M. Hirakoa, D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef] [PubMed]
  43. S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  44. J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999).
    [CrossRef]
  45. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).
  46. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).
  47. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley-Interscience, New York, 1991).
  48. C. A. Brebbia, J. Dominguez, Boundary Elements, An Introductory Course (Computational Mechanics Publications, McGraw-Hill, New York, 1989).
  49. L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  50. R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
    [CrossRef] [PubMed]

2000 (1)

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

1999 (6)

1998 (5)

1997 (6)

1996 (2)

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]

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

1995 (10)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, March1995, pp. 38–40.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
[CrossRef]

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

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Simultaneous reconstruction of optical absorption and scattering maps in turbid media from near-infrared frequency-domain data,” Opt. Lett. 20, 2128–2130 (1995).
[CrossRef] [PubMed]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

S. Ito, “Diffusion of collimated, narrow beam waves in discrete random media,” Appl. Opt. 34, 7106–7112 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (5)

1992 (1)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

1989 (2)

Aarnoudse, J. G.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Aronson, R.

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Topical Rev. 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]

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. Hirakoa, D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Bai, J.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Boas, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Brebbia, C. A.

C. A. Brebbia, J. Dominguez, Boundary Elements, An Introductory Course (Computational Mechanics Publications, McGraw-Hill, New York, 1989).

Chance, B.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, March1995, pp. 38–40.

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
[CrossRef]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Chen, N. G.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Clark, N.

Colak, S. B.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Dassel, A. C. M.

de Haller, E. B.

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

de Mul, F. F. M.

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Delpy, D. T.

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
[CrossRef]

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. Hirakoa, D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

den Outer, P. N.

Dominguez, J.

C. A. Brebbia, J. Dominguez, Boundary Elements, An Introductory Course (Computational Mechanics Publications, McGraw-Hill, New York, 1989).

Durduran, T.

Durian, D. J.

Fantini, S.

Fender, J. S.

Feng, S.

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
[CrossRef]

Feng, T.

Field, M. S.

Firbank, M.

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]

Franceschini, M. A.

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Fry, M. E.

Furutsu, K.

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

Gonatas, C. P.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Graaff, R.

Gratton, E.

Haskell, R. C.

Hebden, J. C.

Hillman, E. M. C.

Hirakoa, M.

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

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Hooft, G. W.’t

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Imai, D.

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, Proc. SPIE2979, 130–138 (1997).

Ishii, M.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Ishimaru, A.

Ito, S.

Jacques, S. L.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Jiang, H.

Kaschke, M.

Kim, A. D.

Koelink, M. H.

Lagendijk, A.

Leigh, J. S.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Li, X. D.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Martinelli, F.

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

Matson, Ch. L.

McAdams, M. S.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

McMackin, L.

Melissen, J. B. M.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Miwa, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Moesta, K. T.

Nieto-Vesperinas, M.

Nieuwenbuizen, Th. M.

Norton, S. J.

O’Leary, M. A.

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Osterberg, U. L.

Paasschens, J. C. J.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Papaioannou, D. G.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Partovi, F.

Pattanayak, D. N.

Patterson, M. S.

Paulsen, K. D.

Pogue, B. W.

Rava, R. P.

Ripoll, J.

Rudnick, J.

Sánchez-Gil, J. A.

Sassaroli, A.

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

Schlag, P. M.

Schmidt, F. E. W.

Schomberg, H.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

Schotland, J.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Schotland, J. C.

J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
[CrossRef]

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. Hirakoa, D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Takahashi, S.

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, Proc. SPIE2979, 130–138 (1997).

S. Takahashi, Y. Yamada, “Simulation of 3D light propagation in a layered head model including a clear CSF layer,” in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).

Tromberg, B. J.

Tsay, T.

Vaasand, L. O.

van Asten, N. A. A. J.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

van der Mark, M. B.

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Vo-Dinh, T.

Walker, S. A.

Wang, L. H.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Wilson, B. C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Wu, J.

Yamada, Y.

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, Proc. SPIE2979, 130–138 (1997).

S. Takahashi, Y. Yamada, “Simulation of 3D light propagation in a layered head model including a clear CSF layer,” in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, March1995, pp. 38–40.

Yodh, A. G.

Zaccanti, G.

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

Zeng, F.

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
[CrossRef]

Zheng, L. Q.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zijlstra, W. G.

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (12)

R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
[CrossRef] [PubMed]

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

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

Ch. L. Matson, N. Clark, L. McMackin, J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–220 (1997).
[CrossRef] [PubMed]

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
[CrossRef] [PubMed]

S. B. Colak, D. G. Papaioannou, G. W.’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 181–213 (1997).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3836 (1995).
[CrossRef]

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumor in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

J. Wu, F. Partovi, M. S. Field, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef] [PubMed]

S. Ito, “Diffusion of collimated, narrow beam waves in discrete random media,” Appl. Opt. 34, 7106–7112 (1995).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (9)

J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

S. J. Norton, T. Vo-Dinh, “Diffraction tomographic imaging with photon density waves: an explicit solution,” J. Opt. Soc. Am. A 15, 2670–2677 (1998).
[CrossRef]

P. N. den Outer, Th. M. Nieuwenbuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

R. C. Haskell, L. O. Vaasand, T. Tsay, T. 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]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

D. J. Durian, J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source,” J. Opt. Soc. Am. A 16, 837–844 (1999).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A 16, 1453–1465 (1999).
[CrossRef]

J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

Med. Phys. (3)

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

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

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 non-scattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Med. Biol. (3)

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

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

Phys. Rev. A (2)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (2)

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–5324 (1998).
[CrossRef]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today, March1995, pp. 38–40.

Topical Rev. Inverse Probl. (1)

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

Other (9)

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, Proc. SPIE2979, 130–138 (1997).

S. Takahashi, Y. Yamada, “Simulation of 3D light propagation in a layered head model including a clear CSF layer,” in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).

See related studies in Advances in Optical Imaging and Photon Migration, J. Fujimoto, M. S. Patterson, eds., Vol. 21 of Trends in Optics and Photonic Series (Optical Society of America, Washington, D.C., 1998).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley-Interscience, New York, 1991).

C. A. Brebbia, J. Dominguez, Boundary Elements, An Introductory Course (Computational Mechanics Publications, McGraw-Hill, New York, 1989).

Cited By

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

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Schematic view of the geometry with the upward (J+) and downward (J-) density flux at an interface.

Fig. 2
Fig. 2

Scheme for the solid-angle relationship between dS=nˆdS and dS=n^dS.

Fig. 3
Fig. 3

Scheme for the specific intensities inside the diffusive medium and emerging into the nonscattering medium. Note the angular dependence to first order of I1 in the diffusive medium, whereas in the nonscattering medium I0 is angle independent. The light distribution radiated into the nonscattering medium, i.e., I0(r), gives rise to Lambert’s power law: dp(r)=I0(r)cos θ dS.

Fig. 4
Fig. 4

Scattering geometry of a nonscattering region embedded in a diffusive medium.

Fig. 5
Fig. 5

Cases considered: (a) nonscattering cylinder of radius Rin embedded in a diffusive cylinder of radius Rout, (b) nonscattering gap of inner radius Rin and outer radius Rgap embedded in a diffusive cylinder of radius Rout.

Fig. 6
Fig. 6

Average intensity measured on the outer surface Rout=2.5 cm versus detector angle separation, for the configuration depicted in Fig. 5(a). Values of Rin=0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0 cm. Results are presented for simulations performed with MC (solid curve), FEM (○), and ET (●). In all cases a dc source (ω=0) was located at rsource=2.45 cm at θ=0. μa0=0.05 cm-1, μa1=0.1 cm-1, μs1=20 cm-1, g=0.

Fig. 7
Fig. 7

Average intensity measured on the outer surface Rout=2.5 cm versus detector angle separation, for the configuration depicted in Fig. 5(b). Values of (a) Rin=1.5 cm, Rgap=1.6 cm (W=0.1 cm), 1.8 cm (W=0.3 cm), and 2.0 cm (W=0.5 cm). (b) Rin=1.0 cm, Rgap=1.1 cm (W=0.1 cm), 1.3 cm (W=0.3 cm), and 1.5 cm (W=0.5 cm). Results are presented for simulations performed with MC (solid curve), FEM (○), and ET (●). In all cases a dc source (ω=0) was located at rsource=2.45 cm at θ=0. μa0=0.05 cm-1, μa1=0.1 cm-1, μs1=20 cm-1, g=0.

Equations (70)

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

U(r)=4πI(r, sˆ)dΩ,
J(r)=4πI(r, sˆ)sˆdΩ,
Jn(r)=J(r)nˆ=4πI(r, sˆ)nˆsˆdΩ,
dp(r, sˆ)=I(r, sˆ)nˆsˆdSdΩ,
dP(r)=4π=I(r, sˆ)nˆsˆdSdΩ=Jn(r)dS,
J+(r)=(2π)+I10t(r, sˆ)sˆnˆdΩ,
J-(r)=(2π)-I01t(r, -sˆ)(-sˆnˆ)dΩ,
dpi(r, θi)=dpr(r, θr)+dpt(r, θt),
dpr(r, θt)=Rp(θi)dpi(r, θi),
dpt(r, θt)=[1-Rp(θi)]dpi(r, θi).
J+(r)=(2π)-[1-|R10(θi)|2]I1(r, sˆ)sˆnˆdΩi,
J-(r)=(2π)+[1-|R01(θi)|2]I0(r, -sˆ)(-sˆnˆ)dΩi,
ncI(r, sˆ)t+sˆI(r, sˆ)+μaI(r, sˆ)=0,
I(r, u^r-r)=I(r, u^r-r)exp[-ua|r-r|],
dpi(r, θ)=I0(r, u^r-r)exp[-μa0|r-r|]cos θdSdΩ,
dPt(r)=(2π)+(1-|R01(θ)|2)I0(r, u^r-r)×exp[-μa0|r-r|]nˆu^r-rdSdΩ.
J-(r)=SI0(r, u^r-r)G(r-r)dS,rS,
G(r-r)=[1-|R01(θ)|2] exp[-μa0|r-r|]|r-r|2×V(r-r)cos θcos θ,
cos θ=nˆ (r-r)|r-r|,cos θ=n^ (r-r)|r-r|.
U0(r)=SI0(r, u^r-r) nu^r-r|r-r|2dS,rV,
U0(r)=SI0(r, u^r-r)Γ(r-r)dS,rV.
Γ(r-r)=exp[-μa0|r-r|]|r-r|2cos θ.
I1(r, sˆ)=αU1(r)+βJ1(r)sˆ,rV˜,
J1(r)=-13(μs1+μa1) [U1(r)-Q1(r)],rV˜,
Q1(r)=4π(r, sˆ)sˆdΩ,
J1=Jn1nˆ+Jt1tˆ,
J+(r)=(2π)-U1(r)4π [1-|R10(θ)|2]cos θdΩ+(2π)-3Jn1(r)4π [1-|R10(θ)|2]cos2 θdΩ,
Jn(r)=Jn1(r)=Jn0(r),
Jn1(r)=-D1[nˆU1(r)-nˆQ1(r)]=-D1U1(r)nˆ+D1nˆQ1(r).
J+(r)=RUU1(r)2+RJJn(r)2,rS,
RU=01[1-|R01(θ)|2]cos θd(cos θ),
RJ=301[1-|R01(θ)|2]cos2 θd(cos θ).
I0(r, sˆ)=I0(r)=αJ+(r),
dp(r, θ)=I0(r)cos θdSdΩ=J+(r)πcos θdSdΩ,
dP(r)=1π(2π)+J+(r)cos θdSdΩ=J+(r)dS.
J-(r)=1πSJ+(r)G(r-r)dS,rS.
Jn(r)=RUU1(r)2+RJJn(r)2-1πSRUU1(r)2+RJJn(r)2×G(r-r)dS,rS.
U1(r)=CnJn(r)+1πSU1(r)+RJRU Jn(r)×G(r-r)dS,rS,
U1(r)=CnJn(r),rS.
U0(r)=1πSJ+(r)Γ(r-r)dS,rV.
U(sec)(r)=1πSJn(0)(r)G(r-r)dS,rS,
Jn(0)(r)J+(r)=RUU1(r)2+RJJn(0)(r)2,rS,
2U1(r)+κ12U1(r)=-S0(r)D1-Q1(r),rV˜.
S0(r)=4π(r, sˆ)dΩ
κ12=-μa1D1+i ωn1cD1,
2G(κ1|r-r|)+κ12G(κ1|r-r|)=-4πδ(r-r),
r, rV˜,
G(κ1|r-r|)=exp[iκ1|r-r|]/|r-r|].
I0(r, u^r-r)=I0(r, u^r-r)×exp-μa0+i ωn0c|r-r|,
r, rV,
U0(r)=1πSRU2 U1(r)+RJ2 Jn(r)Γω(r-r)dS,rV,
Γω(r-r)=exp{[-μa0+i(ωn0/c)]|r-r|}|r-r|2cos θ.
v˜(U12G-G2U1)d3r=s˜(U1G-GU1)ds,
V˜(U12G-G2U1)d3r
=SU1Gnˆ+GJnD1+nˆQ1dS.
r, rV˜:
U1(r)=U(inc)(r)+ΣQ(r)+14π×SU1(r) G(κ1|r-r|)n+G(κ1|r-r|) Jn(r)D1dS,
rV,rV˜:
0=U(inc)(r)+ΣQ(r)+14πSU1(r) G(κ1|r-r|)n+G(κ1|r-r|) Jn(r)D1dS,
U(inc)(r)=-14πV˜S0(r)D1+Q1(r)×G(κ1|r-r|)dr
ΣQ(r)=14πSn^Q1(r)D1G(κ1|r-r|)dS.
U1(r)=CnJn(r)+1πSU1(r)+RJRU Jn(r)×Gω(r-r)dS,rS,
Gω(r-r)=[1-|R01(θ)|2]Γω(r-r)cos θ.
U0(R)=1πAJ+(R)Γω2D(R-R)dA,
Γω2D(R-R)=-+Γω(r-r)dz.
Γω2D(R-R)=u^R-RNˆ|R-R|×-π/2π/2exp-μa0+i ωn0cR-Rcos β×cos βdβ.
U1(R)=CnJn(R)+1πAU1(R)+RJRU Jn(R)×Gω2D(R-R)dA,RA,
Gω2D(R-R)=-+Gω(r-r)dz.
Gω2D(R-R)=(u^R-RNˆ)(u^R-RN^)|R-R|-π/2π/2[1-|R01(θ(i))|2]×exp-μa0+i ωn0c|R-R|cos β×cos2 βdβ,
Gω2D(R-R)π2cos Θ cos Θ|R-R|×exp-μa0+i ωn0c|R-R|.

Metrics