Abstract

Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80–100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, and we suggest a unified partial-current-extrapolated boundary approach.

© 1994 Optical Society of America

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  1. See, for example, the three special journal issues on biomedical optics: Appl. Opt. 28(12), (1989); Appl. Opt. 32(4), (1993); Opt. Eng. 32(2), (1993).
  2. B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
    [CrossRef]
  3. A. Knuttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. 32, 381–389 (1993).
    [CrossRef] [PubMed]
  4. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. (USA) 91, 4887–4891 (1994).
    [CrossRef]
  5. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef] [PubMed]
  6. J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” master’s dissertation (McMaster University, Hamilton, Ont., 1990).
  7. M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.
  8. T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  9. R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
    [CrossRef]
  10. M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  11. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  12. A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
    [CrossRef]
  13. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 1.3.
  14. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  15. B. Davison, Neutron Transport Theory (Oxford, London, 1958).
  16. S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, Princeton, N.J., 1952), Chaps. 5 and 14.
  17. F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2302 (1989).
    [CrossRef] [PubMed]
  18. L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
    [CrossRef]
  19. R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
    [CrossRef] [PubMed]
  20. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  21. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  22. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, London, 1959).
  23. G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).
  24. S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
    [CrossRef] [PubMed]
  25. D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).
  26. V. Allen, A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36, 1621–1638 (1991).
    [CrossRef]
  27. B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
    [CrossRef] [PubMed]
  28. P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
    [PubMed]

1994 (1)

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

1993 (6)

1992 (2)

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

1991 (3)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).

V. Allen, A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36, 1621–1638 (1991).
[CrossRef]

1989 (5)

A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

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

See, for example, the three special journal issues on biomedical optics: Appl. Opt. 28(12), (1989); Appl. Opt. 32(4), (1993); Opt. Eng. 32(2), (1993).

A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
[CrossRef] [PubMed]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2302 (1989).
[CrossRef] [PubMed]

1988 (2)

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

1891 (1)

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

Aarnoudse, J. G.

Allen, V.

V. Allen, A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36, 1621–1638 (1991).
[CrossRef]

Aronson, R.

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
[CrossRef]

Ben-Avraham, D.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).

Berns, M. W.

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

Boas, D. A.

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

Bolin, F. P.

Boretsky, R.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Bryan, G. H.

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, London, 1959).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 1.3.

Chance, B.

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

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

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Cohen, P.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Dassel, A. C. M.

Davison, B.

B. Davison, Neutron Transport Theory (Oxford, London, 1958).

de Mul, F. F. M.

DeVries, P.

A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Duderstadt, J. J.

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

Edlund, M. C.

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, Princeton, N.J., 1952), Chaps. 5 and 14.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Ference, R. J.

Finlander, M.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Fishkin, J. B.

Glasstone, S.

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, Princeton, N.J., 1952), Chaps. 5 and 14.

Graaff, R.

Gratton, E.

Greenfield, R.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Hamilton, L. J.

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

Haskell, R. C.

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

Hefetz, Y.

Ishimaru, A.

Jacques, S. L.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, London, 1959).

Kao, M. H.

Kaplan, P. D.

Kaufmann, K.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Keijzer, M.

Knutson, J. R.

Knuttel, A.

Koelink, M. H.

Lagendijk, A.

A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Leigh, J.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Levy, W.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Madsen, S. J.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.

McKenzie, A. L.

V. Allen, A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36, 1621–1638 (1991).
[CrossRef]

Miyake, H.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Moulton, J. D.

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” master’s dissertation (McMaster University, Hamilton, Ont., 1990).

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.

Nioka, S.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

O’Leary, M. A.

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

Park, Y. D.

Patterson, M. S.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

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

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.

Pine, D. J.

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
[PubMed]

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Preuss, L. E.

Schmitt, J. M.

Smith, D.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Star, W. M.

Storchi, P. R. M.

Svaasand, L. O.

Svasaand, L. O.

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

Taitelbaum, H.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).

Taylor, R. C.

Tromberg, B. J.

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

Tsay, T.-T.

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

Vreeker, R.

A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Weiss, G. H.

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Wilson, B. C.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

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

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.

Yodh, A. G.

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

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
[PubMed]

Yodshioka, H.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Yound, M.

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Zijlstra, W. G.

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 1.3.

Appl Opt. (1)

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

Appl. Opt. (9)

See, for example, the three special journal issues on biomedical optics: Appl. Opt. 28(12), (1989); Appl. Opt. 32(4), (1993); Opt. Eng. 32(2), (1993).

M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
[CrossRef] [PubMed]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2302 (1989).
[CrossRef] [PubMed]

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy experiments,” Appl. Opt. 32, 3828–3836 (1993).
[PubMed]

A. Knuttel, J. M. Schmitt, J. R. Knutson, “Spatial localization of absorbing bodies by interfering diffusive photon-density waves,” Appl. Opt. 32, 381–389 (1993).
[CrossRef] [PubMed]

R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T.-T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

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

Lasers Life Sci. (1)

D. Ben-Avraham, H. Taitelbaum, G. H. Weiss, “Boundary conditions for a model of photon migration in a turbid medium,” Lasers Life Sci. 4, 29–36 (1991).

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Opt. Eng. (1)

L. O. Svasaand, B. J. Tromberg, R. C. Haskell, T.-T. Tsay, M. W. Berns, “Tissue characterization and imaging using photon density waves,” Opt. Eng. 32, 258–266 (1993).
[CrossRef]

Phys. Lett. A (1)

A. Lagendijk, R. Vreeker, P. DeVries, “Influence of internal reflection on diffusive transport in strongly scattering media,” Phys. Lett. A 136, 81–88 (1989).
[CrossRef]

Phys. Med. Biol. (1)

V. Allen, A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36, 1621–1638 (1991).
[CrossRef]

Phys. Rev. A (1)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Proc. London Math. Soc. (1)

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

Proc. Natl. Acad. Sci. (USA) (2)

B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
[CrossRef]

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

Other (8)

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” master’s dissertation (McMaster University, Hamilton, Ont., 1990).

M. S. Patterson, S. J. Madsen, J. D. Moulton, B. C. Wilson, “Diffusion representation of photon migration in tissue,” in IEEE Microwave Theory and Techniques Symposium Digest, Vol. BB-1 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 905–908.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Sec. 1.3.

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

B. Davison, Neutron Transport Theory (Oxford, London, 1958).

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, Princeton, N.J., 1952), Chaps. 5 and 14.

R. Aronson, “Extrapolation distance for diffusion of light,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 297–305 (1993).
[CrossRef]

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford, London, 1959).

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Figures (13)

Fig. 1
Fig. 1

Infinite-medium geometry. The medium is strongly scattering with scattering coefficient σ, absorption coefficient β, and refractive index n. The detector fiber is oriented perpendicular to the radial flux from the source, so the detector signal is simply proportional to the fluence rate.

Fig. 2
Fig. 2

Semi-infinite-medium geometry. The lower medium is strongly scattering, with scattering coefficient σ, absorption coefficient β, and refractive index n. The upper medium is transparent, with refractive index nout. n ^ is the outward-drawn normal to the boundary.

Fig. 3
Fig. 3

Source, image, and continuous line of sinks that constitute the fluence rate solution in a semi-infinite medium with the partial-current boundary condition expressed in Eq. (2.4.1).

Fig. 4
Fig. 4

Source and image configurations for two different boundary conditions: partial-current at the left and extrapolated boundary at the right. The placement of the images is scaled appropriately for an air–medium interface, with the refractive index of the medium equal to that of water.

Fig. 5
Fig. 5

Simulated phase (A) and modulation (B) data for an infinite medium (solid curves) and for a semi-infinite medium with three boundary conditions: partial-current, extrapolated boundary, and zero boundary. The transport scattering coefficient is 10/cm, the absorption coefficient is 0.05/cm, the refractive index of the turbid medium is 1.40, the effective reflection coefficient is 0.493, and the source–detector separation is 2.0 cm.

Fig. 6
Fig. 6

FDPM phase (A, C) and modulation (B, D) data for an infinite medium (circles) and for a semi-infinite medium (squares). The tissue phantom was a 1% agar gel with 40% milk. The source–detector separation was 1.5 cm for the data in A and B and 2.5 cm for the data in C and D. The solid curves represent nonlinear least-squares fits to Eqs. (2.2.6) and (2.2.7) for the infinite-medium data and to Eqs. (2.4.7) and (2.4.8) or (2.7.1) and (2.7.2) for the surface data. The fits were performed with n = 1.33 and Reff = 0.431.

Fig. 7
Fig. 7

Values for absorption coefficient β derived from fits to phase and modulation data at nine different values of the source–detector separation. The tissue phantom was a 1% agar gel with 40% milk (as in Fig. 6). The circles are fits to infinite-medium data with Eqs. (2.2.6) and (2.2.7), and the squares are fits to surface data with Eqs. (2.4.7) and (2.4.8) or (2.7.1) and (2.7.2). The triangles are fits to surface data with infinite-medium equations (2.2.6) and (2.2.7), i.e., with no account taken of the presence of the boundary. B.C., boundary condition.

Fig. 8
Fig. 8

Values for the parameter f derived from fits to phase and modulation data at nine different values of the source–detector separation. The tissue phantom was a 1% agar gel with 40% milk (as in Figs. 6 and 7). The circles are fits to infinite-medium data with Eqs. (2.2.6) and (2.2.7), and the squares are fits to surface data with Eqs. (2.4.7) and (2.4.8) or (2.7.1) and (2.7.2). The curves represent linear least-squares fits to the infinite-medium f values (solid lines) (with r > 1.0 cm) and to the surface f values (dashed curves) (with ρ > 1.0 cm). From the slopes of the fits we find that σtr = 7.6 ± 1.8/cm from the infinite-medium data and σtr = 6.4 ± 0.5/cm from the surface data.

Fig. 9
Fig. 9

Phase versus fiber separation for three different modulation frequencies: 50, 100, and 150 MHz. The source and the detector fibers were either pushed deep into the gel to simulate an infinite-medium geometry (A) or placed on the surface of the gel (B). The gel was the same one used in Figs. 6 and 8. For both infinite-medium and surface data the x intercepts are approximately −0.5 cm.

Fig. 10
Fig. 10

Fitted values for parameter f versus source–detector separation. The source and detector fibers were facing each other in an effectively infinite medium of 10% Intralipid. These FDPM measurements were made with just the argon-pump laser (514 nm) and have been described in more detail by Tromberg et al.27 From the slope of the fitted line (solid line), the transport scattering coefficient is found to be σtr = 140/cm. The fitted value of the absorption coefficient is β = 0.021/cm (at 514 nm). The x intercept of the f-versus-r line is −0.07 cm, considerably less than that of Fig. 8.

Fig. 11
Fig. 11

Phase-versus-fiber (r) separation for three different modulation frequencies: 50, 100, and 150 MHz. The source and detector fibers were facing each other in an effectively infinite medium of 10% Intralipid (as in Fig. 10). The x intercepts of the fitted lines are approximately −0.08 cm, considerably less than those of Fig. 9.

Fig. 12
Fig. 12

Phase (A) and modulation (B) data collected under three sets of conditions: infinite-medium geometry (circles), semi-infinite-medium geometry (squares), and semi-infinite-medium geometry with the surface of the gel covered by aluminum foil (triangles). The gel was as for Fig. 6 (40% milk), and the source–detector separation was 1.5 cm. The solid curves are fits to Eqs. (2.2.6) and (2.2.7) for the infinite-medium data, to Eqs. (2.7.1) and (2.7.2) with Reff = 0.431 for the semi-infinite data, and to Eqs. (2.7.1) and (2.7.2) with Reff = 0.8 for the aluminum foil data.

Fig. 13
Fig. 13

Simulated phase data for an infinite medium (top curve) and for a semi-infinite medium with four different surface reflectivities: Reff = 0.96, 0.90, 0.80, 0.431. The absorption coefficient is β = 0.05/cm, and the transport scattering coefficient is σtr = 10/cm. The refractive index of the medium is n = 1.33 for all cases, and the source–detector separation is 2.0 cm.

Tables (3)

Tables Icon

Table 1 Values for the Ratio of the Fluence Rate to the Flux in an Infinite Mediuma

Tables Icon

Table 2 Values for the Ratio of the Fluence Rate to the Flux at the Surface for Typical Mismatches in Refractive Indices

Tables Icon

Table 3 Results of Nonlinear Least-Squares Fits to Simulated Semi-Infinite-Medium Dataa

Equations (47)

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1 c L ( r , s ^ , t ) t + · L ( r , s ^ , t ) s ^ = - ( σ + β ) L ( r , s ^ , t ) + σ 4 π L ( r , s ^ , t ) f ( s ^ · s ^ ) d Ω + Q ( r , s ^ , t ) ,
4 π f ( s ^ · s ^ ) d Ω = 1 .
1 c ϕ ( r , t ) t + · j ( r , t ) = - β ϕ ( r , t ) + S ( r , t ) ,
S ( r , t ) 4 π Q ( r , s ^ , t ) d Ω , ϕ ( r , t ) 4 π L ( r , s ^ , t ) d Ω , j ( r , t ) 4 π L ( r , s ^ , t ) s ^ d Ω .
L ( r , s ^ , t ) = 1 4 π ϕ ( r , t ) + 3 4 π j ( r , t ) · s ^ .
1 c j ( r , t ) t = - 1 3 ϕ ( r , t ) - 1 3 D j ( r , t ) ,
D 1 3 [ ( 1 - g ) σ + β ] 1 3 σ tr l tr 3 ,
j ( r , t ) = - D ϕ ( r , t ) .
D 2 ϕ ( r , t ) - β ϕ ( r , t ) = ( 1 + 3 D β ) 1 c ϕ ( r , t ) t - S ( r , t ) + 3 D c 2 2 ϕ ( r , t ) t 2 - 3 D c S ( r , t ) t .
D 2 ϕ ( r , t ) - β ϕ ( r , t ) = 1 c ϕ ( r , t ) t - S ( r , t ) .
ϕ G ( r , t - t ) = c [ 4 π D c ( t - t ) ] 3 / 2 × exp [ - r 2 4 D c ( t - t ) - β c ( t - t ) ] .
ϕ ω ( r , t ) = - t ϕ G ( r , t - t ) P exp ( i ω t ) d t = P     exp ( i ω t ) 4 π D exp ( - k r ) r = P 4 π D exp ( - k real r ) exp [ - i ( k imag r - ω t ) ] r ,
k = k real + i k imag = β c + i ω D c ,             τ 1 β c ,
k real = 3 2 β σ tr [ 1 + ( ω τ ) 2 + 1 ] 1 / 2 , k imag = 3 2 β σ tr [ 1 + ( ω τ ) 2 - 1 ] 1 / 2 .
ϕ ( r , t ) = A dc exp ( - r / δ ) r + A ac exp ( - k real r ) r × exp [ - i ( k imag r - ω t ) ] ,
phase lag = k imag r = 3 2 β σ tr [ 1 + ( ω τ ) 2 - 1 ] 1 / 2 r .
modulation = [ A ac exp ( - k real r ) r / A dc exp ( - r / δ ) r ] [ A ac / A dc ] = exp [ - ( k real - 1 δ ) r ] = exp { - 3 2 β σ tr × [ [ 1 + ( ω τ ) 2 + 1 ] 1 / 2 - 2 ] r } .
E irrad = s ^ · n ^ > 0 R Fresnel ( s ^ ) L ( s ^ ) s ^ · n ^ d Ω ,
R Fresnel ( θ ) = 1 2 ( n     cos θ - n out cos θ n cos θ + n out cos θ ) 2 + 1 2 ( n cos θ - n out cos θ n cos θ + n out cos θ ) 2 when 0 θ θ c , = 1             when θ c θ π / 2 ,
E irrad = s ^ · n ^ < 0 L ( s ^ ) s ^ · ( - n ^ ) d Ω = ϕ 4 + j z 2 ,
s ^ · n ^ > 0 R Fresnel ( s ^ ) L ( s ^ ) s ^ · n ^ d Ω = R ϕ ϕ 4 - R j j z 2 ,
R ϕ 0 π / 2 2 sin θ cos θ R Fresnel ( θ ) d θ , R j 0 π / 2 3 sin θ cos 2 θ R Fresnel ( θ ) d θ .
ϕ 4 + j z 2 = R ϕ ϕ 4 - R j j z 2             or ϕ = 1 + R j 1 - R ϕ ( - 2 j z ) .
ϕ 3 j z = 2 3 1 + R j 1 - R ϕ 2 3 1 + R eff 1 - R eff
R eff R ϕ + R j 2 - R ϕ + R j .
E irrad = R eff E emitt = R eff s ^ · n ^ > 0 L ( s ^ ) s ^ · n ^ d Ω = R eff ( ϕ 4 - j z 2 ) .
ϕ = l s ϕ z             at z = 0 ,
l s 1 + R eff 1 - R eff 2 D = 1 + R eff 1 - R eff 2 3 l tr .
ϕ G ( ρ , z , t - t ) = c [ 4 π D c ( t - t ) ] 3 / 2 exp [ - ρ 2 4 D c ( t - t ) - β c ( t - t ) ] × { exp [ - ( z - l tr ) 2 4 D c ( t - t ) ] + exp [ - ( z + l tr ) 2 4 D c ( t - t ) ] - 2 l s 0 d l exp ( - l / l s ) exp [ - ( z + l tr + l ) 2 4 D c ( t - t ) ] } .
ϕ ω ( ρ , z , t ) = - t ϕ G ( ρ , z , t - t ) P exp ( i ω t ) d t = P exp ( i ω t ) 4 π D ( exp ( - k r 1 ) r 1 + exp ( - k r 2 ) r 2 - 2 l s × 0 d l exp ( - l / l s ) exp { - k [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 } [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 ) ,
r 1 = [ ( z - l tr ) 2 + ρ 2 ] 1 / 2 ,             r 2 = [ ( z + l tr ) 2 + ρ 2 ] 1 / 2 ,
ϕ ( ρ , z , t ) = A dc ( exp ( - r 1 / δ ) r 1 + exp ( - r 2 / δ ) r 2 - 2 l s × 0 d l exp ( - l / l s ) exp { - [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 / δ } [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 ) + A ac exp ( i ω t ) ( exp ( - k r 1 ) r 1 + exp ( - k r 2 ) r 2 - 2 l s × 0 d l exp ( - l / l s ) exp { - k [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 } [ ( z + l tr + l ) 2 + ρ 2 ] 1 / 2 ) .
signal = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) × L ( x , y , z = 0 , s ^ ) ( s ^ · n ^ ) = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) 1 4 π × [ ϕ ( x , y , z = 0 ) + 3 D ϕ z × ( x , y , z = 0 ) cos θ ] cos θ = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) 1 4 π × [ ϕ ( x , y , z = 0 ) + 3 2 ( 1 - R eff ) ( 1 + R eff ) × ϕ ( x , y , z = 0 ) cos θ ] cos θ ϕ ( x , y , z = 0 ) ,
phase lag = k imag r 0 - arctan ( IMAG / REAL ) , modulation = ( REAL 2 + IMAG 2 ) 1 / 2 / dc ,
REAL = exp ( - k real r 0 ) r 0 - 1 l s 0 d l exp ( - l / l s ) × exp ( - k real r 0 l ) r 0 l cos [ k imag ( r 0 l - r 0 ) ] , IMAG = 1 l s 0 d l exp ( - l / l s ) × exp ( - k real r 0 l ) r 0 l sin [ k imag ( r 0 l - r 0 ) ] , dc = exp ( - r 0 / δ ) r 0 - 1 l s 0 d l exp ( - l / l s ) exp ( - r 0 l / δ ) r 0 l , r 0 = ( l tr 2 + ρ 2 ) 1 / 2 ,             r 0 l = [ ( l tr + l ) 2 + ρ 2 ] 1 / 2 .
ϕ ( ρ , z , t ) = A dc [ exp ( - r 1 / δ ) r 1 - exp ( - r 2 / δ ) r 2 ] + A ac exp ( i ω t ) [ exp ( - k r 1 ) r 1 - exp ( - k r 2 ) r 2 ] ,
r 1 = [ ( z - l tr ) 2 + ρ 2 ] 1 / 2 ,             r 2 = [ ( z + l tr ) 2 + ρ 2 ] 1 / 2 .
signal = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) × L ( x , y , z = 0 , s ^ ) ( s ^ · n ^ ) = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) × 3 4 π D ϕ z ( x , y , z = 0 ) cos 2 θ ϕ z ( x , y , z = 0 ) .
phase lag = k imag r 0 - arctan ( k imag k real + 1 / r 0 ) , modulation = [ ( k real + 1 / r 0 ) 2 + k imag 2 ] 1 / 2 ( 1 / δ + 1 / r 0 ) × exp [ - ( k real - 1 / δ ) r 0 ] .
z b = l s = 1 + R eff 1 - R eff 2 3 l tr .
ϕ ( ρ , z , t ) = A dc [ exp ( - r 1 / δ ) r 1 - exp ( - r b / δ ) r b ] + A ac exp ( i ω t ) [ exp ( - k r 1 ) r 1 - exp ( - k r b ) r b ] ,
r 1 = [ ( z - l tr ) 2 + ρ 2 ] 1 / 2 ,             r b = [ ( z + 2 z b + l tr ) 2 + ρ 2 ] 1 / 2 .
signal = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) × L ( x , y , z = 0 , s ^ ) ( s ^ · n ^ ) = A fiber d x d y Ω fiber d Ω T Fresnel ( s ^ ) × 1 4 π [ ϕ ( x , y , z = 0 ) + 3 D ϕ z ( x , y , z = 0 ) cos θ ] cos θ .
phase lag = k imag r 0 - arctan ( IMAG / REAL ) , modulation = ( REAL 2 + IMAG 2 ) 1 / 2 / dc ,
REAL = exp ( - k real r 0 ) r 0 - cos [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b + ( k real + 1 r 0 ) l tr 2 r 0 exp ( - k real r 0 ) r 0 + ( k real + 1 r 0 b ) × ( 2 z b + l t r ) r 0 b cos [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b + k imag ( 2 z b + l tr ) l tr r 0 b sin [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b , IMAG = sin [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b + k imag l tr 2 r 0 exp ( - k real r 0 ) r 0 + k imag ( 2 z b + l tr ) l tr r 0 b cos [ k imag ( r 0 b - r 0 ) ] × exp ( - k real r 0 b ) r 0 b - ( k real + 1 r 0 b ) ( 2 z b + l tr ) l tr r 0 b sin [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b , dc = exp ( - r 0 / δ ) r 0 - exp ( - r 0 b / δ ) r 0 b + ( 1 δ + 1 r 0 ) l tr 2 r 0 exp ( - r 0 / δ ) r 0 + ( 1 δ + 1 r 0 b ) ( 2 z b + l tr ) l tr r 0 b exp ( - r 0 b / δ ) r 0 b , r 0 = ( l tr 2 + ρ 2 ) 1 / 2 ,             r 0 b = ( 2 z b + l tr ) 2 + ρ 2 ] 1 / 2 .
phase lag = k imag r 0 - arctan ( IMAG / REAL ) , modulation = ( REAL 2 + IMAG 2 ) 1 / 2 / dc ,
REAL = exp ( - k real r 0 ) r 0 - cos [ k imag ( r 0 b - r 0 ) ] × exp ( - k real r 0 b ) r 0 b , IMAG = sin [ k imag ( r 0 b - r 0 ) ] exp ( - k real r 0 b ) r 0 b , dc = exp ( - r 0 / δ ) r 0 - exp ( - r 0 b / δ ) r 0 b , r 0 = ( l tr 2 + ρ 2 ) 1 / 2 ,             r 0 b = [ ( 2 z b + l tr ) 2 + ρ 2 ] 1 / 2 , z b = l s = 1 + R eff 1 - R eff 2 3 l tr .

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