Abstract

We examine the transport of short light pulses through scattering–absorbing media through different approximate mathematical models. It is demonstrated that the predicted optical signal characteristics are significantly influenced by the various models considered, such as P N expansion, two-flux, and discrete ordinates. The effective propagation speed of the scattered radiation, the predicted magnitudes of the transmitted and backscattered fluxes, and the temporal shape and spread of the optical signals are functions of the models used to represent the intensity distributions. A computationally intensive direct numerical integration scheme that does not utilize approximations is also implemented for comparison. Results of some of the models asymptotically approach those of direct numerical simulation if the order of approximation is increased. In this study therefore we identify the importance of model selection in analyzing short-pulse laser applications such as optical tomography and remote sensing and highlight the parameters, such as wave speed, that must be examined before a model is adopted for analysis.

© 1999 Optical Society of America

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References

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  1. R. L. Fork, J. Y. Brito, C. H. Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  2. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusion light,” Phys. Today 48, 34–40 (Mar.1995).
  3. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  4. S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
    [CrossRef] [PubMed]
  5. M. P. Mengüc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 29, 381–394 (1983).
    [CrossRef]
  6. K. J. Grant, J. A. Piper, D. J. Ramsay, K. L. Williams, “Pulsed lasers in particle detection and sizing,” Appl. Opt. 31, 4515–4520 (1993).
  7. Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” in Annual Review of Heat Transfer, C. L. Tien, ed. (Begell House, New York, 1995), Vol. 6, pp. 1–59.
  8. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
    [CrossRef]
  9. C. D. Mobley, Light and Water: Radiative Transfer in Natural Water (Academic, New York, 1994).
  10. J. H. Churnside, J. J. Wilson, V. V. Tatarskii, “Lidar profiles of fish schools,” Appl. Opt. 36, 6011–6020 (1997).
    [CrossRef] [PubMed]
  11. W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).
  12. M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).
  13. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley Interscience, New York, 1992).
  14. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).
  15. S. Kumar, J. D. Felske, “Radiative transport in a planar media exposed to azimuthally unsymmetric incident radiation,” J. Quant. Spectrosc. Radiat. Transfer 35, 187–212 (1986).
    [CrossRef]
  16. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Academic, New York, 1976).
  17. M. Q. Brewster, C. L. Tien, “Examination of the two-flux model for radiative transfer in participating systems,” Int. J. Heat Mass Transfer 25, 1905–1907 (1982).
    [CrossRef]
  18. Y. Bayazitoglu, J. Higenyi, “The higher order differential equations of radiative transfer: P3 approximation,” AIAA J. 17, 424–431 (1979).
    [CrossRef]
  19. W. A. Fiveland, “Discrete ordinate methods for radiative transfer in isotropically and anisotropically scattering media,” J. Heat Transfer 109, 809–812 (1987).
    [CrossRef]
  20. S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
    [CrossRef]
  21. N. K. Madsen, R. F. Sincovec, “Algorithm 540 pdecol, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
    [CrossRef]
  22. K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef] [PubMed]

1997 (1)

1996 (1)

1995 (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusion light,” Phys. Today 48, 34–40 (Mar.1995).

1994 (1)

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

1993 (1)

1990 (2)

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

1988 (1)

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

1987 (2)

W. A. Fiveland, “Discrete ordinate methods for radiative transfer in isotropically and anisotropically scattering media,” J. Heat Transfer 109, 809–812 (1987).
[CrossRef]

R. L. Fork, J. Y. Brito, C. H. Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

1986 (1)

S. Kumar, J. D. Felske, “Radiative transport in a planar media exposed to azimuthally unsymmetric incident radiation,” J. Quant. Spectrosc. Radiat. Transfer 35, 187–212 (1986).
[CrossRef]

1983 (1)

M. P. Mengüc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 29, 381–394 (1983).
[CrossRef]

1982 (1)

M. Q. Brewster, C. L. Tien, “Examination of the two-flux model for radiative transfer in participating systems,” Int. J. Heat Mass Transfer 25, 1905–1907 (1982).
[CrossRef]

1979 (2)

Y. Bayazitoglu, J. Higenyi, “The higher order differential equations of radiative transfer: P3 approximation,” AIAA J. 17, 424–431 (1979).
[CrossRef]

N. K. Madsen, R. F. Sincovec, “Algorithm 540 pdecol, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

1978 (1)

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Bayazitoglu, Y.

Y. Bayazitoglu, J. Higenyi, “The higher order differential equations of radiative transfer: P3 approximation,” AIAA J. 17, 424–431 (1979).
[CrossRef]

Becker, P. C.

Blakeslee, R. J.

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

Boretsky, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Brewster, M. Q.

M. Q. Brewster, C. L. Tien, “Examination of the two-flux model for radiative transfer in participating systems,” Int. J. Heat Mass Transfer 25, 1905–1907 (1982).
[CrossRef]

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley Interscience, New York, 1992).

Brito, J. Y.

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusion light,” Phys. Today 48, 34–40 (Mar.1995).

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Churnside, J. H.

Cohen, P.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Cruz, C. H.

Duderstadt, J. J.

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

Felske, J. D.

S. Kumar, J. D. Felske, “Radiative transport in a planar media exposed to azimuthally unsymmetric incident radiation,” J. Quant. Spectrosc. Radiat. Transfer 35, 187–212 (1986).
[CrossRef]

Finander, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Fiveland, W. A.

W. A. Fiveland, “Discrete ordinate methods for radiative transfer in isotropically and anisotropically scattering media,” J. Heat Transfer 109, 809–812 (1987).
[CrossRef]

Fork, R. L.

Grant, K. J.

Greenfeld, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression 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 (Academic, New York, 1976).

Higenyi, J.

Y. Bayazitoglu, J. Higenyi, “The higher order differential equations of radiative transfer: P3 approximation,” AIAA J. 17, 424–431 (1979).
[CrossRef]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Ishimaru, A.

Kaufman, K.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Koshak, W. J.

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

Kumar, S.

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

S. Kumar, J. D. Felske, “Radiative transport in a planar media exposed to azimuthally unsymmetric incident radiation,” J. Quant. Spectrosc. Radiat. Transfer 35, 187–212 (1986).
[CrossRef]

Leigh, J. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression 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. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Madsen, N. K.

N. K. Madsen, R. F. Sincovec, “Algorithm 540 pdecol, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

Majumdar, A.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

Mengüc, M. P.

M. P. Mengüc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 29, 381–394 (1983).
[CrossRef]

Mitra, K.

Miyake, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Water (Academic, New York, 1994).

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).

Nioka, S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Phanord, R. J.

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

Piper, J. A.

Ramsay, D. J.

Shank, C. V.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Sincovec, R. F.

N. K. Madsen, R. F. Sincovec, “Algorithm 540 pdecol, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

Smith, D. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Solakiewicz, R. J.

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

Tatarskii, V. V.

Tien, C. L.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

M. Q. Brewster, C. L. Tien, “Examination of the two-flux model for radiative transfer in participating systems,” Int. J. Heat Mass Transfer 25, 1905–1907 (1982).
[CrossRef]

Viskanta, R.

M. P. Mengüc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 29, 381–394 (1983).
[CrossRef]

Williams, K. L.

Wilson, J. J.

Yamada, Y.

S. Kumar, K. Mitra, Y. Yamada, “Hyperbolic damped-wave models for transient light pulse propagation in scattering media,” Appl. Opt. 35, 3372–3378 (1996).
[CrossRef] [PubMed]

Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” in Annual Review of Heat Transfer, C. L. Tien, ed. (Begell House, New York, 1995), Vol. 6, pp. 1–59.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusion light,” Phys. Today 48, 34–40 (Mar.1995).

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Yoshida, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Young, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

ACM Trans. Math. Softwares (1)

N. K. Madsen, R. F. Sincovec, “Algorithm 540 pdecol, general collocation software for partial differential equations [D3],” ACM Trans. Math. Softwares 5, 326–361 (1979).
[CrossRef]

AIAA J. (1)

Y. Bayazitoglu, J. Higenyi, “The higher order differential equations of radiative transfer: P3 approximation,” AIAA J. 17, 424–431 (1979).
[CrossRef]

Appl. Opt. (3)

Int. J. Heat Mass Transfer (1)

M. Q. Brewster, C. L. Tien, “Examination of the two-flux model for radiative transfer in participating systems,” Int. J. Heat Mass Transfer 25, 1905–1907 (1982).
[CrossRef]

J. Geophys. Res. (1)

W. J. Koshak, R. J. Solakiewicz, R. J. Phanord, R. J. Blakeslee, “Diffusion models for lightning radiative transfer,” J. Geophys. Res. 99, 14,361–14,371 (1994).

J. Heat Transfer (2)

W. A. Fiveland, “Discrete ordinate methods for radiative transfer in isotropically and anisotropically scattering media,” J. Heat Transfer 109, 809–812 (1987).
[CrossRef]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (2)

S. Kumar, J. D. Felske, “Radiative transport in a planar media exposed to azimuthally unsymmetric incident radiation,” J. Quant. Spectrosc. Radiat. Transfer 35, 187–212 (1986).
[CrossRef]

M. P. Mengüc, R. Viskanta, “Comparison of radiative transfer approximations for highly forward scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 29, 381–394 (1983).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusion light,” Phys. Today 48, 34–40 (Mar.1995).

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

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufman, W. Levy, M. Young, P. Cohen, H. Yoshida, R. Boretsky, “Compression of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef]

Other (6)

C. D. Mobley, Light and Water: Radiative Transfer in Natural Water (Academic, New York, 1994).

Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” in Annual Review of Heat Transfer, C. L. Tien, ed. (Begell House, New York, 1995), Vol. 6, pp. 1–59.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley Interscience, New York, 1992).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

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

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Figures (8)

Fig. 1
Fig. 1

Schematic of the boundary conditions for the scattered intensity field: (a) Propagating pulse that decays spatially owing to absorption and scattering. Outscattering from the pulse is the source for the scattered intensity field in the medium. (b) Boundary-driven problem.

Fig. 2
Fig. 2

Transmittance through a medium having D = 10.0, ω = 0.998, t p = 1 ps as a function of time for hyperbolic P 1, P 3, P 5, two-flux, and 12-quadrature-discrete-ordinates models.

Fig. 3
Fig. 3

Transmittance through a medium having D = 30.0, ω = 0.998, t p = 1 ps as a function of time for hyperbolic P 1, P 3, P 5, two-flux, 12-quadrature-discrete-ordinates models, and parabolic and quasi-steady methods.

Fig. 4
Fig. 4

Reflectivity through a medium having D = 30.0, ω = 0.998, t p = 1 ps as a function of time for hyperbolic P 1, P 3, P 5, two-flux, and 12-quadrature-discrete-ordinates models.

Fig. 5
Fig. 5

Transmittance as a function of time for different values of optical depth for 12 discrete ordinates. The albedo is kept constant for a boundary-driven problem.

Fig. 6
Fig. 6

Transmittance through a medium having D = 30.0, ω = 0.998, as a function of nondimensional time for hyperbolic P 1, P 3, P 5, two-flux, and 12-quadrature-discrete-ordinates models for the case of a boundary-driven problem.

Fig. 7
Fig. 7

Comparison of transmittance between 12 discrete ordinates and the numerical integration method by using a different number of nodes for a medium having D = 1.0, ω = 0.998 for a boundary-driven problem.

Fig. 8
Fig. 8

Transmittance through a medium having D = 30.0, ω = 0.998 for a boundary source on for 1 ps for hyperbolic, parabolic, and a quasi-steady P 1 approximation.

Tables (1)

Tables Icon

Table 1 Ratio of Predicted Effective Propagation Speed of the Scattered Flux to Actual Speed of Light

Equations (24)

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

1cIx, μ, tt+μ Ix, μ, tx=-σeIx, μ, t+σs2-11 Ix, μ,tpμμdμ+Sx,μ,t,
pμμ=n=0K akPkμPkμ,
Ix, μ, t=ux, t+34π qx, tμ,
ux, t=14π4π Ix, μ, tdΩ=12-11 Ix, μ, tdμ,
qx, t=4π Ix, μ, tμdΩ=2π -11 Ix, μ, tμdμ,
4πcux, tt+qx, tx+4πσaux, t=2π -11 Sx, μ, tdμ,
1cqx, tt+4π3ux, tx+σeqx, t-σsp¯qx, t=2π -11 Sx, μ, tμdμ,
3c22ut2-2ux2+3cσa+σe-σsp¯ut+3σe-σsp¯σau=σe-σsp¯32-11 Sdμ-32-11Sx μdμ+32-111cStdμ.
3cσa+σe-σsp¯ut-2ux2+3σe-σsp¯σau=σe-σsp¯32-11 Sdμ-32-11Sx μdμ+32-111cStdμ.
Ix, t, μ=I+x, t, -π2<θ<π2μ>0,
Ix, t, μ=I-x, t, π2<θ<3π2μ<0,
1cI+t+12I+x+σa+σsBI+-σsBI-=01 Sdμ,
1cI-t-12I-x+σa+σsBI--σsBI+=-10 Sdμ,
B=1201-10 pμμdμdμ.
Ix, μ, t=m=0N Imx, tPmμ,
1cIkt+k+12k+3Ik+1x+k2k-1Ik-1x+σe-σsak2k+1Ik=2k+12-11 SPkdμ.
1cIix, tt+μiIix, tx+σeIix, t=σs2j=-MM wjIjx, tpμjμi+Sx, μi, t, j0, i=-M,, -1, 1,, M.
Icx, μ, t=Iincident exp-σexHt-x/c-Ht-tp-x/cδμ-1,
Sx, μ, t=σs2-11 Icx, μ, tpμμdμ
=σs2 Iincident exp-σexHt-x/c-Ht-tp-x/cp1μ.
Ix=0, μ>0, t=Ix=L, μ<0, t=0.
Ix=L, μ<0, t=0,
Ix=0, μ>0, t=IbHt, step input,
Ix=0, μ>0, t=IbHt-Ht-tp, onoff input,

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