Abstract

The discrete-ordinates method is formulated to solve transient radiative transfer with the incorporation of a transient term in the transfer equation in two-dimensional rectangular enclosures containing absorbing, emitting, and anisotropically scattering media subject to diffuse and/or collimated laser irradiation. The governing equations resulting from the discrete-ordinates discretization of the angular directions are further discretized in the spatial and the temporal domains by the finite-volume approach. The current formulation is suitable for solving transient laser transport in turbid media as well as for steady-state radiative transfer in many engineering problems. The method is applied to several example problems and compared with existing steady-state solutions and Monte Carlo transient solutions. Good agreement is found in all cases. Short-pulsed laser interaction and propagation in a turbid medium with high scattering albedo are studied. The imaging of an inhomogeneous zone inside a turbid medium is demonstrated.

© 2001 Optical Society of America

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References

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  1. S. Kumar, K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transfer 33, 187–294 (1998).
    [CrossRef]
  2. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  3. 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]
  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. C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).
  6. S. A. Prahl, M. J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Appl. Opt. 32, 559–568 (1993).
    [CrossRef] [PubMed]
  7. K. Mitra, S. Kumar, “Development and comparison of models for light-pulse transport through scattering–absorbing media,” Appl. Opt. 38, 188–196 (1999).
    [CrossRef]
  8. Z. M. Tan, P.-F. Hsu, “An integral formulation of transient radiative transfer—theoretical investigation,” in Proceedings of 34th National Heat Transfer Conference (American Society of Mechanical Engineers, Pittsburgh, Pa., 2000), Paper NHTC2000–12077.
  9. Z. Guo, S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane–parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
    [CrossRef]
  10. Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in biological tissues,” JSME Int. J. B 39, 754–761 (1996).
    [CrossRef]
  11. K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
    [CrossRef]
  12. C.-Y. Wu, S.-H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43, 2009–2020 (2000).
    [CrossRef]
  13. B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  14. S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
    [CrossRef] [PubMed]
  15. M. Q. Brewster, Y. Yamada, “Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements,” Int. J. Heat Mass Transfer 38, 2569–2581 (1995).
    [CrossRef]
  16. Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).
  17. W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
    [CrossRef]
  18. W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” ASME Heat Transfer Div. 72, 89–96 (1991).
  19. T. K. Kim, H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
    [CrossRef]
  20. K. D. Lathrop, “Spatial differencing of the transport equation: positive vs accuracy,” J. Comput. Phys. 4, 475–498 (1968).
    [CrossRef]
  21. M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993).
  22. N. Shah, “New method of computation of radiation heat transfer in combustion chambers,” Ph.D. dissertation (Department of Mechanical Engineering, Imperial College of Science and Technology, London, 1979).
  23. Z. Guo, S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
    [CrossRef]

2001 (1)

Z. Guo, S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane–parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
[CrossRef]

2000 (3)

C.-Y. Wu, S.-H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43, 2009–2020 (2000).
[CrossRef]

Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).

Z. Guo, S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
[CrossRef]

1999 (1)

1998 (1)

S. Kumar, K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transfer 33, 187–294 (1998).
[CrossRef]

1997 (1)

K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
[CrossRef]

1996 (2)

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, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in biological tissues,” JSME Int. J. B 39, 754–761 (1996).
[CrossRef]

1995 (1)

M. Q. Brewster, Y. Yamada, “Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements,” Int. J. Heat Mass Transfer 38, 2569–2581 (1995).
[CrossRef]

1993 (1)

1991 (1)

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” ASME Heat Transfer Div. 72, 89–96 (1991).

1990 (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]

1989 (2)

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

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

1988 (2)

W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
[CrossRef]

T. K. Kim, H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

1983 (2)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

1968 (1)

K. D. Lathrop, “Spatial differencing of the transport equation: positive vs accuracy,” J. Comput. Phys. 4, 475–498 (1968).
[CrossRef]

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

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]

Brewster, M. Q.

M. Q. Brewster, Y. Yamada, “Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements,” Int. J. Heat Mass Transfer 38, 2569–2581 (1995).
[CrossRef]

Buckius, R. O.

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

Chance, B.

Fiveland, W. A.

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” ASME Heat Transfer Div. 72, 89–96 (1991).

W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
[CrossRef]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Guo, Z.

Z. Guo, S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane–parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
[CrossRef]

Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).

Z. Guo, S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
[CrossRef]

Hasegawa, Y.

Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in biological tissues,” JSME Int. J. B 39, 754–761 (1996).
[CrossRef]

Hsu, P.-F.

Z. M. Tan, P.-F. Hsu, “An integral formulation of transient radiative transfer—theoretical investigation,” in Proceedings of 34th National Heat Transfer Conference (American Society of Mechanical Engineers, Pittsburgh, Pa., 2000), Paper NHTC2000–12077.

Kim, T. K.

T. K. Kim, H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

Kumar, S.

Z. Guo, S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane–parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
[CrossRef]

Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).

Z. Guo, S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
[CrossRef]

K. Mitra, S. Kumar, “Development and comparison of models for light-pulse transport through scattering–absorbing media,” Appl. Opt. 38, 188–196 (1999).
[CrossRef]

S. Kumar, K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transfer 33, 187–294 (1998).
[CrossRef]

K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
[CrossRef]

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]

Lai, M.-S.

K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
[CrossRef]

Lathrop, K. D.

K. D. Lathrop, “Spatial differencing of the transport equation: positive vs accuracy,” J. Comput. Phys. 4, 475–498 (1968).
[CrossRef]

Lee, H.

T. K. Kim, H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (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]

Mitra, K.

K. Mitra, S. Kumar, “Development and comparison of models for light-pulse transport through scattering–absorbing media,” Appl. Opt. 38, 188–196 (1999).
[CrossRef]

S. Kumar, K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transfer 33, 187–294 (1998).
[CrossRef]

K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
[CrossRef]

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]

Modest, M. F.

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

Patterson, M. S.

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

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Prahl, S. A.

Rackmil, C. I.

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

San, K.-C.

Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).

Shah, N.

N. Shah, “New method of computation of radiation heat transfer in combustion chambers,” Ph.D. dissertation (Department of Mechanical Engineering, Imperial College of Science and Technology, London, 1979).

Tan, Z. M.

Z. M. Tan, P.-F. Hsu, “An integral formulation of transient radiative transfer—theoretical investigation,” in Proceedings of 34th National Heat Transfer Conference (American Society of Mechanical Engineers, Pittsburgh, Pa., 2000), Paper NHTC2000–12077.

van Gemert, M. J. C.

Welch, A. J.

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

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

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Wu, C.-Y.

C.-Y. Wu, S.-H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43, 2009–2020 (2000).
[CrossRef]

Wu, S.-H.

C.-Y. Wu, S.-H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43, 2009–2020 (2000).
[CrossRef]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Yamada, Y.

Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in biological tissues,” JSME Int. J. B 39, 754–761 (1996).
[CrossRef]

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]

M. Q. Brewster, Y. Yamada, “Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements,” Int. J. Heat Mass Transfer 38, 2569–2581 (1995).
[CrossRef]

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]

Adv. Heat Transfer (1)

S. Kumar, K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transfer 33, 187–294 (1998).
[CrossRef]

Appl. Opt. (5)

ASME Heat Transfer Div. (1)

W. A. Fiveland, “The selection of discrete ordinate quadrature sets for anisotropic scattering,” ASME Heat Transfer Div. 72, 89–96 (1991).

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Int. J. Heat Mass Transfer (3)

M. Q. Brewster, Y. Yamada, “Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements,” Int. J. Heat Mass Transfer 38, 2569–2581 (1995).
[CrossRef]

C.-Y. Wu, S.-H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43, 2009–2020 (2000).
[CrossRef]

T. K. Kim, H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

J. Thermophys. Heat Transfer (1)

W. A. Fiveland, “Three-dimensional radiative heat transfer solutions by the discrete-ordinates method,” J. Thermophys. Heat Transfer 2, 309–316 (1988).
[CrossRef]

J. Comput. Phys. (1)

K. D. Lathrop, “Spatial differencing of the transport equation: positive vs accuracy,” J. Comput. Phys. 4, 475–498 (1968).
[CrossRef]

J. Thermophys. Heat Transfer (2)

Z. Guo, S. Kumar, K.-C. San, “Multi-dimensional Monte Carlo simulation of short pulse laser radiation transport in scattering media,” J. Thermophys. Heat Transfer 14, 504–511 (2000).

K. Mitra, M.-S. Lai, S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” J. Thermophys. Heat Transfer 11, 409–414 (1997).
[CrossRef]

JSME Int. J. B (1)

Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in biological tissues,” JSME Int. J. B 39, 754–761 (1996).
[CrossRef]

Med. Phys. (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Numer. Heat Transfer (1)

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

Numer. Heat Transfer B (1)

Z. Guo, S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane–parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
[CrossRef]

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]

Other (3)

Z. M. Tan, P.-F. Hsu, “An integral formulation of transient radiative transfer—theoretical investigation,” in Proceedings of 34th National Heat Transfer Conference (American Society of Mechanical Engineers, Pittsburgh, Pa., 2000), Paper NHTC2000–12077.

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

N. Shah, “New method of computation of radiation heat transfer in combustion chambers,” Ph.D. dissertation (Department of Mechanical Engineering, Imperial College of Science and Technology, London, 1979).

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Figures (9)

Fig. 1
Fig. 1

(a) Sketch of the system, (b) control volume.

Fig. 2
Fig. 2

Transient DO prediction of surface heat flux for a purely absorbing medium and comparison with exact solution.22

Fig. 3
Fig. 3

Transient DO predictions of incident radiation and radiative heat flux for a purely anisotropically scattering medium and comparison with S–14 steady-state solution.19

Fig. 4
Fig. 4

Comparison of temporal transmittance profiles between DO and MC methods in a square isotropically scattering medium with one hot wall.

Fig. 5
Fig. 5

Influence of time step and comparison of equivalent isotropic scattering results with the direct anisotropic scattering simulations in transient radiation transport.

Fig. 6
Fig. 6

Nondimensional incident radiation profiles along the centerline at various time instants for a medium subject to ultra-short-pulsed laser irradiation.

Fig. 7
Fig. 7

Nondimensional incident radiation profiles along the y direction near the laser incident surface at various time instants.

Fig. 8
Fig. 8

Influences of absorption coefficient and detector position on the temporal reflectance profiles.

Fig. 9
Fig. 9

Comparison of the temporal transmittance profiles at different positions between the homogeneous medium and the medium with a small inhomogeneous zone.

Tables (1)

Tables Icon

Table 1 Ck , Expansion Coefficient Values for the Phase Functions

Equations (17)

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

1 c I i t + ξ i I i x + η i I i y + β I i = β S i ,   i = 1 ,   2 ,     ,   n ,
S i = 1 - ω I b + ω 4 π j = 1 n w j Φ ij I j + S c ,   i = 1 ,   2 ,     ,   n ,
Φ ij = k = 0 M   C k P k cos   ϕ ,
cos   ϕ = s ˆ i · s ˆ j = ξ i ξ j + η i η j + μ i μ j .
I w = w I bw + 1 - w π ξ j < 0 n / 2 w j I j | ξ j | .
I c x ,   ξ c ,   t = I 0   exp - β x H t - x / c - H t - t p - x / c δ ξ c - 1 ,
S c = ω / 4 π I c Φ ξ c ξ i + η c η i + μ c μ i ,
G = j = 1 n   w j I j + I c ,
Q x = j = 1 n   ξ j w j I j + I c ,
Q y = j = 1 n   η j w j I j .
· q = κ 4 E b - G ,
T y ,   t = Q x x = L ,   y ,   t I 0 ,   R y ,   t = Q x x = 0 ,   y ,   t - I c x = 0 ,   y ,   t I 0 .
V / c Δ t I Pi - I Pi 0 + ξ i A E I Ei - A W I Wi + η i A N I Ni - A S I Si = β V - I Pi + S Pi ,
I Pi = γ y I Ni + 1 - γ y I Si = γ x I Ei + 1 - γ x I Wi .
I Pi = 1 β c Δ t   I Pi 0 + S Pi + | ξ i | γ x β Δ x   I xi + | η i | γ y β Δ y   I yi 1 β c Δ t + 1 + | ξ i | γ x β Δ x + | η i | γ y β Δ y ,
Δ x < | ξ i | min / β 1 - γ x ,     Δ y < | η i | min / β 1 - γ y .
Δ t * < Min | ξ i | 1 - γ x ,   | η i | 1 - γ y .

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