Abstract

A novel algorithm for mapping the photon transport equation (PTE) to Maxwell’s equations is presented. Owing to its accuracy, wave propagation through biological tissue is modeled using the PTE. The mapping of the PTE to Maxwell’s equations is required to model wave propagation through foreign structures implanted in biological tissue for sensing and characterization of tissue properties. The PTE solves for only the magnitude of the intensity but Maxwell’s equations require the phase information as well. However, it is possible to construct the phase information approximately by solving the transport of intensity equation (TIE) using the full multigrid algorithm.

© 2008 Optical Society of America

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  1. S. Kumar, K. Mitra, and Y. Yamada, "Hyperbolic damped-wave models for transient light-pulse propagation in scattering media," Appl. Opt. 35, 3372-3378 (1996).
    [CrossRef] [PubMed]
  2. M. Premaratne, E. Premaratne, and A. J. Lowery, "The photon transport equation for turbid biological media with spatially varying isotropic refractive index," Opt. Express 13, 389-399 (2005).
    [CrossRef] [PubMed]
  3. 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).
    [CrossRef]
  4. F. L. Neerhoff and G. Mur, "Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media," Appl. Sci. Res. 28, 73-88 (1973).
  5. S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
    [CrossRef]
  6. S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).
  7. S. C. Chapra and R. P. Canale, Numerical Methods For Engineers, (4th ed., McGraw-Hill, New York, 2002).
  8. D. M. Paganin, Coherent X-Ray Optics, (Oxford University Press, New York, 2006).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, (7th ed., Cambridge University Press, Cambridge, 1999).
  10. M. R. Spiegel, Vector Analysis And An Introduction To Tensor Analysis, (McGraw-Hill, 1959).
  11. M. R. Teague, "Deterministic phase retrieval: a Green’s function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  12. T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
    [CrossRef]
  13. L. J. Allen and M. P. Oxley, "Phase retrieval from series of images obtained by defocus variation," Opt. Commun. 199, 65-75 (2001).
    [CrossRef]
  14. T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
    [CrossRef]
  15. T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
    [CrossRef]
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing, (2nd ed., Cambridge University Press, Cambridge, 1992).
  17. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport of intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
    [CrossRef]
  18. G. E. Thomas and K. Stamnes, Radiative Transfer In The Atmosphere And Ocean, (Cambridge University Press, 1999).
  19. R. Ramamoorthi, and P. Hanrahan, "On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object," J. Opt. Soc. Am. A 18, 2448-2459 (2001).
    [CrossRef]
  20. D. Paganin and K. A. Nugent, "Non-interferometric phase determination," P. Hawkes (editor), in Advances in Imaging and Electron Physics (Harcourt Publishers, Kent, 2001) Vol. 118, pp. 85-127.
  21. D. Paganin and K. A. Nugent, "Non-interferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
    [CrossRef]
  22. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
    [CrossRef]
  23. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
    [CrossRef]
  24. W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
    [CrossRef]
  25. M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, (Springer, Germany, 2004).
  26. S. A. Prahl, J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993).
    [CrossRef] [PubMed]
  27. C. Y. Wu and S. H. Wu, "Integral equation formulation for transient radiative transfer in an anisotropically scattering medium," Int. J. Heat Mass Trans. 43, 2009-2020 (2000).
    [CrossRef]
  28. A. E. Profio, "Light transport in tissue," Appl. Opt. 28, 2216-2222 (1989).
    [CrossRef] [PubMed]
  29. M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
    [CrossRef]
  30. G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
    [CrossRef]
  31. W. D. Burnett, "Evaluation of laser hazards to the eye and the skin," Amer. Ind. Hyg. Assoc. J. 30, 582-587 (1969).

2008

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).
[CrossRef]

2005

2004

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

2001

2000

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

1999

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

1998

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

1997

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

1996

1995

1993

1991

M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
[CrossRef]

1990

W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

1989

1987

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

1983

1973

F. L. Neerhoff and G. Mur, "Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media," Appl. Sci. Res. 28, 73-88 (1973).

1969

W. D. Burnett, "Evaluation of laser hazards to the eye and the skin," Amer. Ind. Hyg. Assoc. J. 30, 582-587 (1969).

Allen, L. J.

L. J. Allen and M. P. Oxley, "Phase retrieval from series of images obtained by defocus variation," Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Burnett, W. D.

W. D. Burnett, "Evaluation of laser hazards to the eye and the skin," Amer. Ind. Hyg. Assoc. J. 30, 582-587 (1969).

Cheong, W.

W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Csapo, L.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[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).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport of intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
[CrossRef]

Handapangoda, C. C.

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).
[CrossRef]

Hanrahan, P.

Janssens, K.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

Kukhlevsky, S. V.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

Kumar, S.

Lowery, A. J.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

Mechler, M.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

Mitra, K.

Motamedi, M.

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

Mur, G.

F. L. Neerhoff and G. Mur, "Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media," Appl. Sci. Res. 28, 73-88 (1973).

Neerhoff, F. L.

F. L. Neerhoff and G. Mur, "Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media," Appl. Sci. Res. 28, 73-88 (1973).

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport of intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
[CrossRef]

Oxley, M. P.

L. J. Allen and M. P. Oxley, "Phase retrieval from series of images obtained by defocus variation," Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Patterson, M. S.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
[CrossRef]

Prahl, S. A.

S. A. Prahl, J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993).
[CrossRef] [PubMed]

W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Premaratne, E.

Premaratne, M.

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).
[CrossRef]

M. Premaratne, E. Premaratne, and A. J. Lowery, "The photon transport equation for turbid biological media with spatially varying isotropic refractive index," Opt. Express 13, 389-399 (2005).
[CrossRef] [PubMed]

Profio, A. E.

Ramamoorthi, R.

Raven, C.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

Roberts, A.

Samek, O.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

Snigireva, A.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

Teague, M. R.

van Gemert, J. C.

van Gemert, M. C. J.

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

Welch, A. J.

S. A. Prahl, J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993).
[CrossRef] [PubMed]

W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

Wilson, B. C.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
[CrossRef]

Wu, C. Y.

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

Wu, S. H.

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

Wyman, D. R.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
[CrossRef]

Yamada, Y.

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).
[CrossRef]

Yoon, G.

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

Amer. Ind. Hyg. Assoc. J.

W. D. Burnett, "Evaluation of laser hazards to the eye and the skin," Amer. Ind. Hyg. Assoc. J. 30, 582-587 (1969).

Appl. Opt.

Appl. Sci. Res.

F. L. Neerhoff and G. Mur, "Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media," Appl. Sci. Res. 28, 73-88 (1973).

IEEE J. Quantum Electron.

W. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. "Development and application of three-dimensional light distribution model for laser irradiated tissue," IEEE J. Quantum Electron. 23, 1721-1733 (1987).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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).
[CrossRef]

Int. J. Heat Mass Trans.

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

J. Microscopy

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III: The effects of noise," J. Microscopy 214, 51-61 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, "Hard x-ray quantitative non-interferometric phase-contrast microscopy," J. Phys. D: Appl. Phys. 32, 563-567 (1999).
[CrossRef]

Lasers in Medical Science

M. S. Patterson, B. C. Wilson, and D. R. Wyman, "The propagation of optical radiation in tissue 1. Models of radiation transport and their application," Lasers in Medical Science 6, 155-168, (1991).
[CrossRef]

Opt. Commun.

L. J. Allen and M. P. Oxley, "Phase retrieval from series of images obtained by defocus variation," Opt. Commun. 199, 65-75 (2001).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B.

S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, "Enhanced transmission versus localization of a light pulse by a subwavelength metal slit," Phys. Rev. B. 70, 195428 (2004).
[CrossRef]

Phys. Rev. Lett.

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Other

M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, (Springer, Germany, 2004).

D. Paganin and K. A. Nugent, "Non-interferometric phase determination," P. Hawkes (editor), in Advances in Imaging and Electron Physics (Harcourt Publishers, Kent, 2001) Vol. 118, pp. 85-127.

S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).

S. C. Chapra and R. P. Canale, Numerical Methods For Engineers, (4th ed., McGraw-Hill, New York, 2002).

D. M. Paganin, Coherent X-Ray Optics, (Oxford University Press, New York, 2006).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, (7th ed., Cambridge University Press, Cambridge, 1999).

M. R. Spiegel, Vector Analysis And An Introduction To Tensor Analysis, (McGraw-Hill, 1959).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing, (2nd ed., Cambridge University Press, Cambridge, 1992).

G. E. Thomas and K. Stamnes, Radiative Transfer In The Atmosphere And Ocean, (Cambridge University Press, 1999).

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Figures (12)

Fig. 1.
Fig. 1.

(Color online) Metal screen implanted in biological tissue.

Fig. 2.
Fig. 2.

End elevation of the tissue-metal screen model.

Fig. 3.
Fig. 3.

Propagation of an incident wave through a slit in a thick metal screen.

Fig. 4.
Fig. 4.

An illustration of the strategy used for mapping radiance to irradiance.

Fig. 5.
Fig. 5.

The incident radiance profile on the tissue layer (with arbitrary units).

Fig. 6.
Fig. 6.

(Color online) The irradiance profile on a plane just before the tissue-metal screen interface (with arbitrary units).

Fig. 7.
Fig. 7.

(Color online) The electric field distribution on a plane just before the tissue-metal screen interface (with arbitrary units).

Fig. 8.
Fig. 8.

(Color online) The magnetic field distribution on a plane just before the tissue-metal screen interface (with arbitrary units).

Fig. 9.
Fig. 9.

(Color online) The magnetic field distribution on a plane just after the tissue-metal screen interface (with arbitrary units).

Fig. 10.
Fig. 10.

(Color online) The electric field component in the x-direction on a plane just after the tissue-metal screen interface (with arbitrary units).

Fig. 11.
Fig. 11.

(Color online) The electric field component in the z-direction on a plane just after the tissue-metal screen interface (with arbitrary units).

Fig. 12.
Fig. 12.

(Color online) The irradiance profile on a plane just after the tissue-metal screen interface (with arbitrary units).

Equations (40)

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1 v t I PTE ( z , u , ϕ , t ) + u z I PTE ( z , u , ϕ , t ) + σ t I PTE ( z , u , ϕ , t )
σ s 4 π 0 2 π 1 1 P ( u , ϕ ; u , ϕ ) I PTE ( z , u , ϕ , t ) d u d ϕ = F ( z , u , ϕ , t ) ,
[ ε ( x , y , z ) μ 0 2 t 2 2 ] E ( x , y , z , t ) = 0 ,
[ ε ( x , y , z ) μ 0 2 t 2 2 ] H ( x , y , z , t ) = 0 ,
[ ε ( x , y , z ) μ 0 2 t 2 2 ] Ψ ( x , y , z , t ) = 0 .
Ψ ( x , y , z , t ) = 1 2 π 0 ψ ω ( x , y , z ) e j ω t d ω ,
[ 2 + ε ω ( x , y , z ) μ 0 c 2 k 0 2 ] ψ ω ( x , y , z ) = 0 ,
[ 2 + k 0 2 n ω 2 ( x , y , z ) ] ψ ω ( x , y , z ) = 0 ,
ψ ω ( x , y , z ) = ψ ˜ ω ( x , y , z ) e jkz ,
2 ( ψ ˜ ω ( x , y , z ) e jkz ) + k 0 2 n ω 2 ( x , y , z ) ψ ˜ ω ( x , y , z ) e jkz = 0 .
2 [ A ( x , y , z ) B ( x , y , z ) ] A ( x , y , z ) 2 B ( x , y , z )
+ B ( x , y , z ) 2 A ( x , y , z ) + 2 A ( x , y , z ) . B ( x , y , z ) ,
[ xy 2 + 2 z 2 + 2 j k 0 z + k 0 2 ( n ω 2 ( x , y , z ) 1 ) ] ψ ˜ ω ( x , y , z ) = 0 ,
[ xy 2 + 2 j k 0 z + k 0 2 ( n ω 2 ( x , y , z ) 1 ) ] ψ ˜ ω ( x , y , z ) = 0 ,
ψ ˜ ω ( x , y , z ) = I e j ϕ .
xy · ( I ( x , y , z ) xy ϕ ( x , y , z ) ) = k I ( x , y , z ) z .
I ( x , y , z ) xy 2 ϕ ( x , y , z ) + I ( x , y , z ) x ϕ ( x , y , z ) x + I ( x , y , z ) y ϕ ( x , y , z ) y = k I ( x , y , z ) z .
Γ u = f ,
( I i + 1 , j I i 1 , j 2 Δ ) ( ϕ i + 1 , j ϕ i 1 , j 2 Δ ) + ( I i , j + 1 I i , j 1 2 Δ ) ( ϕ i , j + 1 ϕ i , j 1 2 Δ )
+ I i , j ( ϕ i 1 , j + ϕ i , j 1 + ϕ i + 1 , j + ϕ i , j + 1 4 ϕ i , j Δ 2 ) = f i , j ,
I ( x , y , z ) I ( x , y , z + δ z ) + I ( x , y , z ) 2 ,
I ( x , y , z ) z I ( x , y , z + δ z ) I ( x , y , z ) δ z .
I = 2 π I PTE cos θ d ω ˜ ,
Γ h u h = f h .
v h = u h u ˜ h ,
d h = Γ u ˜ h f h .
Γ h v h = d h .
u ˜ h new = u ˜ h + v h .
Γ H v H = d H .
I I A + I A δ z 2 ,
I z I A I A δ z δ z .
E A = I A e j ϕ A e j ( kz ω t ) E 0 .
H A = j 1 ω × E A .
H y ( x , y , z , t ) = U ( x , z , t ) e j ω t e y ,
( 2 + k j 2 ) U j ( x , z , t ) = 0 ,
U 1 ( x , z , t ) = U i ( x , z , t ) + U r ( x , z , t ) + U d ( x , z , t ) ,
U i ( x , z , t ) = e j k 1 z ,
U r ( x , z , t ) = U i ( x , 2 b z , t ) .
I = 1 2 v ε E 2 ,
f ( t ) = I 0 e ( ( t t 0 ) T ) 2 ,

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