Abstract

The time-dependent one-dimensional photon transport (radiative transfer) equation is widely used to model light propagation through turbid media with a slab geometry, in a vast number of disciplines. Several numerical and semi-analytical techniques are available to accurately solve this equation. In this work we propose a novel efficient solution technique based on eigen decomposition of the vectorized version of the photon transport equation. Using clever transformations, the four variable integro-differential equation is reduced to a set of first order ordinary differential equations using a combination of a spectral method and the discrete ordinates method. An eigen decomposition approach is then utilized to obtain the closed-form solution of this reduced set of ordinary differential equations.

© 2011 Optical Society of America

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  1. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
    [CrossRef]
  2. N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” N. Astron. 6, 437–455 (2001).
    [CrossRef]
  3. D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992).
    [CrossRef]
  4. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinatemethod radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef]
  5. J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
    [CrossRef]
  6. C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985).
    [CrossRef]
  7. C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994).
    [CrossRef]
  8. E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975).
    [CrossRef]
  9. G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979).
    [CrossRef]
  10. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001).
    [CrossRef]
  11. J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).
  12. A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999).
    [CrossRef]
  13. A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
    [CrossRef]
  14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).
  15. A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
    [CrossRef]
  16. A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication (McGraw-Hill, 1986).
  17. S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).
  18. K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
    [CrossRef]
  19. W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + +, 2nd ed. (Cambridge University Press, 2003).
  20. C. H. Edwards and D. E. Penney, Differential Equations: Computing andModeling, 3rd ed. (Prentice Hall, 2004).
  21. C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009).
    [CrossRef]

2009 (1)

2008 (1)

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

2005 (1)

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

2002 (2)

A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

2001 (2)

N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” N. Astron. 6, 437–455 (2001).
[CrossRef]

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001).
[CrossRef]

1999 (1)

A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999).
[CrossRef]

1994 (1)

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994).
[CrossRef]

1992 (1)

D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992).
[CrossRef]

1989 (1)

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
[CrossRef]

1988 (1)

1985 (1)

C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985).
[CrossRef]

1981 (1)

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

1979 (1)

G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979).
[CrossRef]

1975 (1)

E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975).
[CrossRef]

1963 (1)

J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).

Deuz, J. L.

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
[CrossRef]

Fleck, J. A.

J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).

Foglesong, G. M.

G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979).
[CrossRef]

Friend, J.

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

Gnedin, N. Y.

N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” N. Astron. 6, 437–455 (2001).
[CrossRef]

Handapangoda, C. C.

C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009).
[CrossRef]

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

Herman, M.

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
[CrossRef]

Hielscher, A. H.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

Hsu, P. F.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001).
[CrossRef]

Ishimaru, A.

A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999).
[CrossRef]

Jayaweera, K.

Kim, A. D.

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999).
[CrossRef]

Klose, A. D.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

Larsen, E. W.

E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975).
[CrossRef]

Moscoso, M.

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

Ntziachristos, V.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

O’Brien, D. M.

D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979).
[CrossRef]

Premaratne, M.

C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009).
[CrossRef]

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

Santer, R.

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994).
[CrossRef]

C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985).
[CrossRef]

Stamnes, K.

K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinatemethod radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef]

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Swanson, R. A.

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Tan, Z. M.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001).
[CrossRef]

Thomas, J. R.

C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985).
[CrossRef]

Tsay, S. C.

Wiscombe, W.

Yeo, L.

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

Appl. Opt. (1)

ASME J. Heat Transfer (1)

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008).

J. Atmos. Sci. (1)

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

J. Comput. Phys. (3)

A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005).
[CrossRef]

G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (5)

D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992).
[CrossRef]

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989).
[CrossRef]

C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985).
[CrossRef]

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994).
[CrossRef]

E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975).
[CrossRef]

Methods Comput. Phys. (1)

J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).

N. Astron. (1)

N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” N. Astron. 6, 437–455 (2001).
[CrossRef]

Opt. Express (1)

SIAM J. Sci. Comput. (USA) (2)

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002).
[CrossRef]

Other (4)

A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication (McGraw-Hill, 1986).

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + +, 2nd ed. (Cambridge University Press, 2003).

C. H. Edwards and D. E. Penney, Differential Equations: Computing andModeling, 3rd ed. (Prentice Hall, 2004).

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