Abstract

Light propagation in strongly scattering media can be described by the diffusion approximation to the Boltzmann transport equation. We have derived analytical expressions based on the diffusion approximation that describe the photon density in a uniform, infinite, strongly scattering medium that contains a sinusoidally intensity-modulated point source of light. These expressions predict that the photon density will propagate outward from the light source as a spherical wave of constant phase velocity with an amplitude that attenuates with distance r from the source as exp(−αr)/r. The properties of the photon-density wave are given in terms of the spectral properties of the scattering medium. We have used the Green’s function obtained from the diffusion approximation to the Boltzmann transport equation with a sinusoidally modulated point source to derive analytic expressions describing the diffraction and the reflection of photon-density waves from an absorbing and/or reflecting semi-infinite plane bounded by a straight edge immersed in a strongly scattering medium. The analytic expressions given are in agreement with the results of frequency-domain experiments performed in skim-milk media and with Monte Carlo simulations. These studies provide a basis for the understanding of photon diffusion in strongly scattering media in the presence of absorbing and reflecting objects and allow for a determination of the conditions for obtaining maximum resolution and penetration for applications to optical tomography.

© 1993 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
    [CrossRef]
  3. R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
    [CrossRef] [PubMed]
  4. B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
    [CrossRef] [PubMed]
  5. J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [CrossRef] [PubMed]
  6. 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]
  7. D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
    [CrossRef] [PubMed]
  8. P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
    [CrossRef]
  9. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef] [PubMed]
  10. B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
    [CrossRef] [PubMed]
  11. B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
    [CrossRef] [PubMed]
  12. J. C. Hebden, R. A. Kruger, “A time-of-flight breast imaging system: spatial resolution performance,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).
    [CrossRef]
  13. K. M. Yoo, F. Liu, R. R. Alfano, “Biological materials probed by the temporal and angular profiles of the backscattered ultrafast laser pulses,” J. Opt. Soc. Am. B 7, 1685–1693 (1990).
    [CrossRef]
  14. R. R. Anderson, J. A. Parrish, The Science of Photomedicine, J. D. Regan, J. A. Parrish, eds. (Plenum, New York, 1982), Chap. 6, p. 147.
    [CrossRef]
  15. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  16. H. S. Carslaw, “Some multiform solutions of the partial differential equations of physical mathematics and their applications,” Proc. London Math. Soc. 30, 121–161 (1898).
  17. A. Sommerfeld, “Über verzweigte Potentiale im Raum,” Proc. London Math. Soc. 28, 395–429 (1897);Proc. London Math. Soc. 30, 161–163 (1898).
  18. J. H. Jeans, Mathematical Theory of Electricity and Magnetism (Cambridge U. Press, London, 1933).
  19. B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
    [CrossRef]
  20. B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
    [CrossRef]

1990 (3)

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, “Biological materials probed by the temporal and angular profiles of the backscattered ultrafast laser pulses,” J. Opt. Soc. Am. B 7, 1685–1693 (1990).
[CrossRef]

1989 (4)

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

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

1988 (3)

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

1987 (1)

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

1985 (1)

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

1898 (1)

H. S. Carslaw, “Some multiform solutions of the partial differential equations of physical mathematics and their applications,” Proc. London Math. Soc. 30, 121–161 (1898).

1897 (1)

A. Sommerfeld, “Über verzweigte Potentiale im Raum,” Proc. London Math. Soc. 28, 395–429 (1897);Proc. London Math. Soc. 30, 161–163 (1898).

Alfano, R. R.

Anderson, P. W.

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

Anderson, R. R.

R. R. Anderson, J. A. Parrish, The Science of Photomedicine, J. D. Regan, J. A. Parrish, eds. (Plenum, New York, 1982), Chap. 6, p. 147.
[CrossRef]

Barbieri, B.

B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
[CrossRef]

Berkovits, R.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

Carslaw, H. S.

H. S. Carslaw, “Some multiform solutions of the partial differential equations of physical mathematics and their applications,” Proc. London Math. Soc. 30, 121–161 (1898).

Case, K. M.

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

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Chance, B.

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, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

De Picoli, F.

B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
[CrossRef]

Feddersen, B. A.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Feng, S.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

Fountain, M.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Gratton, E.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
[CrossRef]

Greenfeld, R.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Hebden, J. C.

J. C. Hebden, R. A. Kruger, “A time-of-flight breast imaging system: spatial resolution performance,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).
[CrossRef]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Holtom, G.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Ishimaru, A.

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

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

Jeans, J. H.

J. H. Jeans, Mathematical Theory of Electricity and Magnetism (Cambridge U. Press, London, 1933).

Kent, J.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Kruger, R. A.

J. C. Hebden, R. A. Kruger, “A time-of-flight breast imaging system: spatial resolution performance,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).
[CrossRef]

Liu, F.

McCully, K.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Nioka, S.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Papanicolaou, G.

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

Parrish, J. A.

R. R. Anderson, J. A. Parrish, The Science of Photomedicine, J. D. Regan, J. A. Parrish, eds. (Plenum, New York, 1982), Chap. 6, p. 147.
[CrossRef]

Patterson, M. S.

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Piston, D. W.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Postel, M.

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

Sheng, P.

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Sommerfeld, A.

A. Sommerfeld, “Über verzweigte Potentiale im Raum,” Proc. London Math. Soc. 28, 395–429 (1897);Proc. London Math. Soc. 30, 161–163 (1898).

Stephen, M. J.

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

vandeVen, M.

B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
[CrossRef]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

White, B.

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

Yoo, K. M.

Zhang, Z. Q.

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Zweifel, P. F.

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

Anal. Biochem. (1)

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time-resolved spectroscopy of hemoglobin and myoglobin in resting and ischemic muscle,” Anal. Biochem. 174, 698–707 (1988).
[CrossRef] [PubMed]

Appl. Opt. (2)

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

Philos. Mag. B (1)

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

Phys. Rev. B (1)

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[CrossRef]

Phys. Rev. Lett. (4)

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

B. White, P. Sheng, M. Postel, G. Papanicolaou, “Probing through cloudiness: theory of statistical inversion for multiply scattered data,” Phys. Rev. Lett. 63, 2228–2231 (1989).
[CrossRef] [PubMed]

B. White, P. Sheng, Z. Q. Zhang, G. Papanicolaou, “Wave localization characteristics in the time domain,” Phys. Rev. Lett. 59, 1918–1921 (1987).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Proc. London Math. Soc. (2)

H. S. Carslaw, “Some multiform solutions of the partial differential equations of physical mathematics and their applications,” Proc. London Math. Soc. 30, 121–161 (1898).

A. Sommerfeld, “Über verzweigte Potentiale im Raum,” Proc. London Math. Soc. 28, 395–429 (1897);Proc. London Math. Soc. 30, 161–163 (1898).

Rev. Sci. Instrum. (1)

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Science (1)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Other (6)

J. C. Hebden, R. A. Kruger, “A time-of-flight breast imaging system: spatial resolution performance,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).
[CrossRef]

R. R. Anderson, J. A. Parrish, The Science of Photomedicine, J. D. Regan, J. A. Parrish, eds. (Plenum, New York, 1982), Chap. 6, p. 147.
[CrossRef]

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

B. Barbieri, F. De Picoli, M. vandeVen, E. Gratton, “What determines the uncertainty of phase and modulation measurements in frequency domain fluorometry?” in Time-Resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1204, 158–170 (1990).
[CrossRef]

J. H. Jeans, Mathematical Theory of Electricity and Magnetism (Cambridge U. Press, London, 1933).

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

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Figures (21)

Fig. 1
Fig. 1

(a) Schematic representation of the time evolution of the light intensity measured in response to a narrow light pulse traversing an arbitrary distance in a scattering and absorbing medium. If the medium is strongly scattering, there are no unscattered components in the transmitted pulse. (b) Time evolution of the intensity from a sinusoidally intensity-modulated source. The transmitted photon wave retains the same frequency as the incoming wave but is delayed owing to the phase velocity of the wave in the medium. The reduced amplitude of the transmitted wave arises from attenuation related to scattering and absorption processes. The demodulation is the ratio ac/dc normalized to the modulation of the source.

Fig. 2
Fig. 2

Cylindrical coordinate system used to describe the configuration of a semi-infinite plane bounded by a straight edge, the point photon source, and the point detector. The edge of the plane lies on the z axis, and the face of the plane is at θ = 0. The coordinates of the point source are (r′,θ′, z′), and the coordinates of the detector are (r,θ, z).

Fig. 3
Fig. 3

Division of the physical space into three regions about the semi-infinite plane bounded by a straight edge and the corresponding values of U1 and U2 used for U(θ′) and U(−θ′) in each region of the physical space.

Fig. 4
Fig. 4

Schematic of the experimental setup used to test the validity of the diffusion approximation result of Eq. (10) above.

Fig. 5
Fig. 5

(a) Setup for a diffraction measurement: the ends of the source and detector optical fibers are set a fixed distance apart (3 cm) on opposite sides of the absorbing plane, the face of the absorbing plane is oriented perpendicularly to the line joining the ends of the source and detector optical fibers, and each point of the face of the plane is equidistant from the end of each optical fiber. Measurements are performed as a function of the position of the edge of the plane relative to the line joining the source and the detector optical fibers. The 0-cm position of the plane edge is defined to be the point where the edge of the plane crosses the line joining the ends of the source and detector optical fibers. (b) Setup for a reflection measurement: same as that in (a), except that now the ends of the source and detector optical fibers are set a fixed distance (1.4 cm) apart on the same side of the absorbing plane, with the ends of the optical fibers pointing in a direction that is perpendicular to the face of the plane. The 0-cm position of the plane edge is defined to be the point where the edge of the plane crosses the line coming from the end of the source optical fiber.

Fig. 6
Fig. 6

Phase lag versus source/detector separation r, where the source and the detector are immersed in 3.78 L of skim milk mixed with three amounts of black India ink. The phase data were plotted relative to the phase measured with the shortest source/detector separation. The data were collected at a 120-MHz modulation frequency. The correlation coefficients for the lines fitting data of each milk/ink mixture are equal to 1.000. (b) Frequency-domain Monte Carlo simulation of phase lag versus source/detector separation r for four values of μa. The data were plotted relative to the phase at 2.5 cm from the source.

Fig. 7
Fig. 7

Frequency-domain Monte Carlo simulation of phase lag versus the square root of modulation frequency. Each set of data was obtained at a fixed source/detector separation.

Fig. 8
Fig. 8

(a) Natural logarithm of the source/detector separation r multiplying the dc intensity obtained at r, versus r, where the source and the detector are immersed in 3.78 L of skim milk mixed with three amounts of black India ink. The data are normalized to the r dc value at r = 2.5 cm. The correlation coefficients for the lines fitting these data are equal to 1.000. (b) Monte Carlo simulation of ln(r dc) versus r for four values of μa. The data are normalized to the r dc value at r = 2.5 cm.

Fig. 9
Fig. 9

(a) Natural logarithm of the source/detector separation r multiplying the ac amplitude obtained at r, versus r, where the source and the detector are immersed in 3.78 L of skim milk mixed with three amounts of black India ink. The data are normalized to the r ac value at r = 2.5 cm. The correlation coefficients for the lines fitting these data are equal to 1.000. (b) Frequency-domain Monte Carlo simulation of ln(r ac) versus r for three values of μa. The data are normalized to the r ac value at r = 2.5 cm.

Fig. 10
Fig. 10

(a) Plots calculated from diffusion theory of dc intensity at four medium absorptions versus the position of an absorbing edge. (b) Measurement in skim milk mixed with four amounts of black India ink of dc light intensity versus the position of an absorbing edge.

Fig. 11
Fig. 11

Inverse sharpness of an edge (the sharpness is the inverse of the distance moved by the edge, relative to the source/detector position, that causes the intensity to decrease from 90% to 10% of the initial value) versus the square root of the volume of black India ink in the skim milk. This figure was derived from the plots in Fig. 10(b).

Fig. 12
Fig. 12

Monte Carlo simulation of the time-gated intensity versus the position of an absorbing edge.

Fig. 13
Fig. 13

(a) Plots calculated from diffusion theory of ac amplitude at four modulation frequencies versus the position of an absorbing edge. (b) Frequency-domain Monte Carlo simulation of ac amplitude at three modulation frequencies versus the position of an absorbing edge.

Fig. 14
Fig. 14

(a) Plots calculated from diffusion theory of the relative phase at three media absorptions versus the position of an absorbing edge. (b) Measurement in skim milk mixed with three amounts of black India ink of the relative phase versus the position of an absorbing edge.

Fig. 15
Fig. 15

(a) Plots calculated from diffusion theory of the demodulation at three media absorptions versus the position of an absorbing edge. (b) Measurement in skim milk mixed with three amounts of black India ink of signal demodulation versus the position of an absorbing edge.

Fig. 16
Fig. 16

(a) Plots calculated from diffusion theory of the relative phase at three modulation frequencies versus the position of an absorbing edge. (b) Measurement in skim milk of the relative phase at three modulation frequencies versus the position of an absorbing edge.

Fig. 17
Fig. 17

(a) Plots calculated from diffusion theory of the demodulation at three modulation frequencies versus the position of an absorbing edge. (b) Measurement in skim milk of the signal demodulation at three modulation frequencies versus the position of an absorbing edge.

Fig. 18
Fig. 18

(a) Plots calculated from diffusion theory of the relative phase at four modulation frequencies versus the position of an absorbing edge. (b) Frequency-domain Monte Carlo simulation of the relative phase at five modulation frequencies versus the position of an absorbing edge.

Fig. 19
Fig. 19

(a) Plots calculated from diffusion theory of the demodulation at four modulation frequencies versus the position of an absorbing edge. (b) Frequency-domain Monte Carlo simulation of the demodulation at five modulation frequencies versus the position of an absorbing edge.

Fig. 20
Fig. 20

(a) Plots calculated from diffusion theory of the relative phase at three modulation frequencies versus the position of an absorbing edge. (b) Measurement in skim milk of the relative phase at three modulation frequencies versus the position of an absorbing edge. The light source that was used to generate these data was a synchronously pumped Rhodamine 6G single-jet dye laser that was cavity dumped at 2 MHz. The dye laser was tuned to a wavelength of 690 nm. Apart from the light source, the experimental setup is as described in Section 5.

Fig. 21
Fig. 21

(a) Plots calculated from diffusion theory of the demodulation at three modulation frequencies versus the position of an absorbing edge. (b) Measurement in skim milk of the signal demodulation at three modulation frequencies versus the position of an absorbing edge. The light source described in the caption to Fig. 20(b) was used in the acquisition of these data, but, apart from the light source, the experimental setup is as described in Section 5.

Tables (1)

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Table 1 Wavelength and Phase Velocity of a Photon Density Wave in 3.78 L of Skim Milk Mixed with Black India Inka

Equations (47)

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U ( r , t ) t + υ μ a U ( r , t ) + · J ( r , t ) = q 0 ( r , t ) ,
U ( r , t ) + 3 J ( r , t ) υ 2 t + J ( r , t ) υ D = 0 ,
D = { 3 [ μ a + μ s ( 1 g ) ] } 1 ,
q 0 ( r , t ) = δ ( r ) S { 1 + A exp [ i ( ω t + ) ] } ,
U ( r , t ) = [ U ( r ) ] dc + [ U ( r ) ] ac exp [ i ( ω t + ) ] ,
J ( r , t ) = [ J ( r ) ] dc + [ J ( r ) ] ac exp [ i ( ω t + ) ] ,
υ μ a [ U ( r ) ] dc + · [ J ( r ) ] dc = S δ ( r ) ,
[ J ( r ) ] dc = υ D [ U ( r ) ] dc
( υ μ a i ω ) [ U ( r ) ] ac + · [ J ( r ) ] ac = S A δ ( r ) ,
[ J ( r ) ] ac = υ D [ 1 + i 3 ω D / υ 1 + ( 3 ω D / υ ) 2 ] [ U ( r ) ] ac .
[ J ( r , t ) ] ac υ D [ U ( r ) ] ac .
2 [ U ( r ) ] dc ( μ a / D ) [ U ( r ) ] dc = ( S / υ D ) δ ( r ) .
2 [ U ( r ) ] ac ( υ μ a i ω υ D ) [ U ( r ) ] ac = S A υ D δ ( r ) .
U ( r , t ) = S 4 π υ D r exp [ r ( μ a D ) 1 / 2 ] + S A 4 π υ D r × exp { r ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 cos [ 1 2 tan 1 ( ω υ μ a ) ] } × exp { ir ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 sin [ 1 2 tan 1 ( ω υ μ a ) ] i ( ω t + ) } ·
U ( r , t ) = S 4 π υ D r + S A 4 π υ D r exp [ r ( ω 2 υ D ) 1 / 2 ] × exp [ ir ( ω 2 υ D ) 1 / 2 i ( ω t + ) ] .
ρ ( r , t ) = 1 ( 4 π υ D t ) 3 / 2 exp ( r 2 4 υ D t μ a υ t ) ,
λ = 2 π ( 2 υ D / ω ) 1 / 2 ,
V = ( 2 υ D ω ) 1 / 2 .
Φ = r ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 sin [ 1 2 tan 1 ( ω υ μ a ) ] ,
ln [ ( r ) ( dc ) ] = r ( μ a D ) 1 / 2 + ln ( S 4 π υ D ) ,
ln [ ( r ) ( ac ) ] = r ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 × cos [ 1 2 tan 1 ( ω υ μ a ) ] + ln ( S A 4 π υ D ) ·
2 U + k 2 U = 4 π δ ( R )
R = [ r 2 + r 2 + ( z z ) 2 2 r r cos ( θ θ ) ] 1 / 2
U 1 ( θ ) = exp ( ikR ) R 1 π cos [ 1 2 ( θ θ ) ] 0 exp { i k [ r 2 + r 2 + ( z z ) 2 + 2 r r cosh β ] 1 / 2 } cosh ( β / 2 ) d β [ r 2 + r 2 + ( z z ) 2 + 2 r r cosh β ] 1 / 2 cos ( θ θ ) + cosh β ] ,
U 2 ( θ ) = 1 π cos [ 1 2 ( θ θ ) ] 0 exp { ik [ r 2 + r 2 + ( z z ) 2 + 2 r r cosh β ] 1 / 2 } cosh ( β / 2 ) d β [ r 2 + r 2 + ( z z ) 2 + 2 r r cosh β ] 1 / 2 [ cos ( θ θ ) + cosh β ] ,
k dc i ( μ a / D ) 1 / 2 ,
k ac i ( υ μ a i ω υ D ) 1 / 2
i k ac = ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 cos [ 1 2 tan 1 ( ω υ μ a ) ] + i ( υ 2 μ a 2 + ω 2 υ 2 D 2 ) 1 / 4 sin [ 1 2 tan 1 ( ω υ μ a ) ]
0 < θ < 2 π ,
2 π < θ < 0 .
Ū = U ( θ ) U ( θ )
Ū dc = U dc 1 ( θ ) U dc 1 ( θ ) ,
Ū a = U ac 1 ( θ ) U ac 1 ( θ ) ;
Ū dc = U dc 1 ( θ ) U dc 2 ( θ ) ,
Ū ac = U ac 1 ( θ ) U ac 2 ( θ ) ;
Ū dc = U dc 2 ( θ ) U dc 2 ( θ ) ,
Ū ac = U ac 2 ( θ ) U ac 2 ( θ ) ;
Ū dc = U dc 1 ( θ ) 1 2 S 4 π υ D exp [ R ( μ a / D ) 1 / 2 ) ] R ,
Ū ac = U ac 1 ( θ ) 1 2 S A 4 π υ D exp ( i k ac R ) R ;
Ū dc = 1 2 S 4 π υ D exp [ R ( μ a / D ) 1 / 2 ] R U dc 2 ( θ ) ,
Ū ac = 1 2 S A 4 π υ D exp ( i k ac R ) ) R U ac 2 ( θ ) .
Ū = U ( θ ) + U ( θ )
Ū = U ( θ ) p abs U ( θ ) + ( 1 p abs ) U ( θ ) .
Φ = tan 1 [ Im ( Ū ac ) Re ( Ū ac ) ] ,
dc = Ū dc ,
ac = [ Re ( Ū ac ) 2 + Im ( Ū ac ) 2 ] 1 / 2 ,
demodulation = ac / dc ,

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