Abstract

By means of a surface-integral formalism we derive the integral equations for diffuse photon density waves with boundary conditions corresponding to a diffuse–diffuse interface with index mismatch and solve them numerically without any approximation. These numerical results are verified with Monte Carlo simulations for the planar interface case. Since the application of the boundary condition to index-mismatched media is difficult, an approximation that yields very accurate values is found. This approximation can be easily introduced into analytical models, and a study of its limit of validity is shown. We demonstrate with numerical results that the multiple-scattering contribution that is due to surface roughness can be neglected, even when index mismatch is present.

© 1999 Optical Society of America

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References

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1999 (1)

1998 (2)

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

1997 (2)

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

1996 (1)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

1995 (5)

1994 (1)

1993 (3)

1992 (2)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

1991 (1)

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

1989 (2)

1973 (1)

Aarnoudse, J. G.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Aronson, R.

Bai, J.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

Boas, D. A.

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Bolin, F. P.

Bucher, E. A.

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (March1995).
[CrossRef]

S. Feng, F-A. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

Chen, N. G.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

Dassel, A. C. M.

de Haller, E. B.

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

de Mul, F. F. M.

Feng, S.

Feng, T.

Ference, R. J.

Field, M. S.

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Furutsu, K.

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

Gonatas, C. P.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Graaff, R.

Gratton, E.

Haskell, R. C.

Ishii, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Ito, S.

Jacques, S. L.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Koelink, M. H.

Leigh, J. S.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Madrazo, A.

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

McAdams, M. S.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Mendez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Miwa, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Nieto-Vesperinas, M.

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Partovi, F.

Preuss, L. E.

Rava, R. P.

Ripoll, J.

J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A 16, 1453–1465 (1999).
[CrossRef]

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

Sánchez-Gil, J. A.

Schotland, J.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Taylor, R. C.

Tromberg, B. J.

Tsay, T.

Vaasand, L. O.

Walker, S. A.

Wang, L. H.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Wu, J.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (March1995).
[CrossRef]

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Zeng, F-A.

Zheng, L. Q.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zijlstra, W. G.

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (7)

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

Phys. Rev. A (2)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (1)

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Phys. Rev. Lett. (2)

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (March1995).
[CrossRef]

Other (4)

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley-Interscience, New York, 1991).

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Figures (13)

Fig. 1
Fig. 1

Scattering geometry: two semi-infinite homogeneous diffusive media separated by a rough surface with profile z=S(x). The source is located in medium 0. The geometry is constant in the OY axis.

Fig. 2
Fig. 2

Comparison of the normalized quantity |r-rsource||U(r)| calculated by the integral equations (solid curve) and by Monte Carlo simulation (circles) for the following parameters: medium 0, breast parameters μs0=75 cm-1, g=0.8, μa0=0.035 cm-1, n1=1.333; medium 1, breast tumor parameters μs1=50 cm-1, g=0.8, μa0=0.24 cm-1, n1=1.0. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0, 1 cm). The scan was performed in the z direction at x=0.

Fig. 3
Fig. 3

Comparison of the normalized quantity |r-rsource||U(r)| calculated by the integral equations (solid curve) and by Monte Carlo simulation (circles) for the following parameters: medium 0, breast parameters μs0=75 cm-1, g=0.8, μa0=0.035 cm-1, n1=1.333; medium 1, breast tumor parameters μs1=50 cm-1, g=0.8, μa0=0.24 cm-1, n1=2.0. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0, 1 cm). The scan was performed in the z direction at x=0.

Fig. 4
Fig. 4

Comparison of the wave amplitude |U(r)| calculated by the integral equations at zdetect=0.2 cm (solid curves) and zdetect=-0.2 cm (dotted curves) and by Monte Carlo simulation at zdetect=0.2 cm (solid circles) and zdetect=-0.2 cm (open circles) for two values of n1: (a) n0=1.333, n1=1.0; (b) n0=1.333, n1=3.0. Medium 0, breast parameters μs0=75 cm-1, g=0.8, μa0=0.035 cm-1; medium 1, breast tumor parameters μs1=50 cm-1, g=0.8, μa0=0.24 cm-1. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0, 1 cm). The scan was performed in the x direction at a constant z distance from the surface.

Fig. 5
Fig. 5

Values of |Jtotal-|/|Utotalinc| (solid curves) and |Jtotal+|/|Utotalinc| (dotted curves) calculated with the complete boundary conditions, compared with |Jtotal-|/|Utotalinc| (solid circles) and |Jtotal+|/|Utotalinc| (open circles) calculated with the approximate boundary conditions. Media parameters: (a) D0=2D1=0.0666 cm, μs0=4.98 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; (b) D0=D1/2=0.01665 cm, μs0=20.0 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; (c) D0=D1/2=0.0333 cm, μs0=10.0 cm-1, μs1=4.98 cm-1, μa0=μa1=0.02 cm-1, ω=0; (d) breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=0; (e) parameters used in Ref. 24, D0=0.0333, D1=0.1095, μs0=10.0 cm-1, μs1=3.02 cm-1, μa0=μa1=0.02 cm-1, ω=200 MHz; and (f ) breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1; ω=200 MHz. In all cases rsource=(0, 1 cm) and n0=1.333.

Fig. 6
Fig. 6

Values of |U0total|/|Un1=1.333total| (solid curves) and |U1total|/|Un1=1.333total| (dotted curves) calculated with the complete boundary conditions, compared with U0total/Un1=1.333total (solid circles) and U1total/Un1=1.333total (open circles) calculated with the approximate boundary conditions. Media parameters: (a) D0=2D1=0.0666 cm, μs0=4.98 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; (b) D0=D1/2=0.01665 cm, μs0=20.0 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; (c) D0=D1/2=0.0333 cm, μs0=10.0 cm-1, μs1=4.98 cm-1, μa0=μa1=0.02 cm-1, ω=0; (d) breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=0; (e) parameters used in Ref. 24, D0=0.0333, D1=0.1095, μs0=10.0 cm-1, μs1=3.02 cm-1, μa0=μa1=0.02 cm-1, ω=200 MHz; (f ) breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=200 MHz. In all cases rsource=(0, 1 cm) and n0=1.333.

Fig. 7
Fig. 7

Values of |U1total|/|U0total| calculated with the complete boundary conditions for the media parameters: open circles, D0=2D1=0.0666 cm, μs0=4.98 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; open squares, D0=D1/2=0.01665 cm, μs0=20.0 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; stars, D0=D1/2=0.0333 cm, μs0=10.0 cm-1, μs1=4.98 cm-1, μa0=μa1=0.02 cm-1, ω=0; solid curve, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=0; dotted curve, parameters used in Ref. 24, D0=0.0333, D1=0.1095, μs0=10.0 cm-1, μs1=3.02 cm-1, μa0=μa1=0.02 cm-1, ω=200 MHz; solid circles, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=200 MHz. In all cases rsource=(0, 1 cm) and n0=1.333.

Fig. 8
Fig. 8

Error measured as percentage [see expression (32)] when the approximate boundary conditions were used to find the values of (a) |U0total| and (b) |U1total|. Media parameters: open circles, D0=2D1=0.0666 cm, μs0=4.98 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; open squares, D0=D1/2=0.01665 cm, μs0=20.0 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; stars, D0=D1/2=0.0333 cm, μs0=10.0 cm-1, μs1=4.98 cm-1, μa0=μa1=0.02 cm-1, ω=0; solid curves, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=0; dotted curves, parameters used in Ref. 24, D0=0.0333, D1=0.1095, μs0=10.0 cm-1, μs1=3.02 cm-1, μa0=μa1=0.02 cm-1, ω=200 MHz; solid circles, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=200 MHz. In all cases rsource=(0, 1 cm) and n0=1.333.

Fig. 9
Fig. 9

Error measured as percentage [see expression (32)] when index-matched boundary conditions were considered in finding the values of (a) |U0total| and (b) |U1total|. Media parameters: open circles, D0=2D1=0.0666 cm, μs0=4.98 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; open squares, D0=D1/2=0.01665 cm, μs0=20.0 cm-1, μs1=10.0 cm-1, μa0=μa1=0.02 cm-1, ω=0; stars, D0=D1/2=0.0333 cm, μs0=10.0 cm-1, μs1=4.98 cm-1, μa0=μa1=0.02 cm-1, ω=0; solid curves, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=0; dotted curves, parameters used in Ref. 24, D0=0.0333, D1=0.1095, μs0=10.0 cm-1, μs1=3.02 cm-1, μa0=μa1=0.02 cm-1, ω=200 MHz; solid circles, breast and breast tumor parameters, μs0=15.0 cm-1, μs1=10.0 cm-1, μa0=0.035 cm-1, μa1=0.24 cm-1, ω=200 MHz. In all cases rsource=(0, 1 cm) and n0=1.333.

Fig. 10
Fig. 10

Rough surface profiles S(x) with Gaussian statistics: solid curve, T=0.5 cm and σ=0.1 cm; dotted curve, T=1.0 cm and σ=0.2 cm. Note the difference in scale of the x and y axes.

Fig. 11
Fig. 11

Normalized scattered amplitude |U(SC)(r)|/|Un1=n0(SC)|x=0 at zdetect=0.5 cm found by use of the complete boundary conditions for n1=n0=1.333 (solid curves); n0=1.333, n1=1.0 (dotted curve); and n0=1.333, n1=1.5 (dotted curves) and by use of the approximate boundary conditions for n0=1.333, n1=1.0 (solid circles) and n0=1.333, n1=1.5 (open circles). Scattering geometries: (a) plane interface; (b) rough surface with Gaussian statistics T=0.5 cm, σ=0.1 cm; (c) rough surface with Gaussian statistics T=1.0 cm, σ=0.2 cm. In all cases rsource=(0, 1 cm) and ω=200 MHz.

Fig. 12
Fig. 12

Scattered phase ϕ(r), of U(SC)=|U(SC)|exp[iϕ], at zdetect=0.5 cm found by use of the complete boundary conditions for n1=n0=1.333 (solid curves); n0=1.333, n1=1.0 (dotted curve); and n0=1.333, n1=1.5 (dashed curves) and by use of the approximate boundary conditions for n0=1.333, n1=1.0 (solid circles) and n0=1.333, n1=1.5 (open circles). Scattering geometries: (a) plane interface; (b) rough surface with Gaussian statistics T=0.5 cm, σ=0.1 cm; (c) rough surface with Gaussian statistics T=1.0 cm, σ=0.2 cm. In all cases rsource=(0, 1 cm) and ω=200 MHz.

Fig. 13
Fig. 13

(a) Discretized curved surface studied to find the minimum value of Δ required for the expressions of RU,Jjk to remain valid when a numerical calculation is performed when a surface with curvature radius R by elements dS is sampled. nˆ is the surface normal. (b) Detail of (a), where α=Δ/2. J-,+ represent the downward and upward flux densities, respectively.

Equations (39)

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1c U(r, t)t+J(r, t)+μa(r)U(r, t)=S0(r, t),
J(r, t)=-13[μa+(1-g)μs] U(r, t),
U(r)=4πI(r, sˆ)dΩ,
J+(r, nˆ)=(2π)+I(r, sˆ)sˆnˆdΩ,
J-(r, nˆ)=(2π)-I(r, sˆ)sˆ(-nˆ)dΩ.
J(r)=4πI(r, sˆ)sˆdΩ,
J(r)nˆ=J+(r, nˆ)-J-(r, nˆ),
I(r, sˆ)14π U(r)+34π J(r)sˆ,
J+(r, nˆ)=U12 01[1-R10(μi)]μidμi+3J1n2 01[1-R10(μi)]μi2dμi,
J-(r, nˆ)=U02 01[1-R01(μi)]μidμi-3J0n2 01[1-R01(μi)]μi2dμi.
J(r)nˆ=U12 RU10+J1n2 RJ10-U02 RU01+J0n2 RJ01.
RUjk=01[1-Rjk(μ)]μdμ,
RJjk=301[1-Rjk(μ)]μ2dμ.
U1(r)-(n1/n0)2U0(r)=C Jn(r),
C=2-RJ10-RJ01RU10.
2U0(r)+κ02U0(r)=-S0(r)D0,
2G(κ0|r-r|)+κ02G(κ0|r-r|)=-4πδ(r-r),
κ=-μa0D0+i ωn0cD01/2,
2U1(r)+κ12U1(r)=0,
2G(κ1|r-r|)+κ12G(κ1|r-r|)=-4πδ(r-r),
κ=-μa1D1+i ωn1cD11/2.
U1(r)|S-n1n02U0(r)|S=-CD0U0(r)nS,
-D0U0(r)nS=-D1U1(r)nS,
v(U2G-G2U)d3r=S(UrG-GrU)ds,
U0(r>)=U(inc)(r>)+14π S(x)U0(r) G(κ0|r>-r|)n-G(κ0|r>-r|) U0(r)ndS.
0=U(inc)(r<)+14π S(x)U0(r) G(κ0|r<-r|)n-G(κ0|r<-r|) U0(r)ndS.
U1(r<)=-14π S(x)n1n02 G(κ1|r<-r|)n U0(r)-D0D1 G(κ1|r<-r|)+CD0 G(κ1|r<-r|)n×U0(r)ndS.
0=-14π S(x)n1n02 G(κ1|r>-r|)n U0(r)-D0D1 G(κ1|r>-r|)+CD0 G(κ1|r>-r|)n×U0(r)ndS.
U(inc)=14π V˜G(κ0|r-r|) S0(r)D0 d3r.
U1(r)|Sn1n02U0(r)|S,
-D0 U0(r)nS=-D1 U1(r)nS.
|Jtotal+,-||Utotal(inc)|=S|J+,-(r, nˆ)|dSS|U(inc)(r)|dS.
|U0,1total|=S|U0,1(r)|dS.
|U1total||U0total|=n1n02-CD0U0nS/|U0total|.
Errori(%)=|Uitotal|approx-|Uitotal|complete|Uitotal|complete×100.
J+(r, nˆ)=02πdψ0π/2-αI(r, sˆ)sˆnˆ sin θdθ,
J-(r, nˆ)=02πdψπ/2+απI(r, sˆ)sˆ(-nˆ)sin θdθ,
RUjk=sin Δ/21[1-Rjk(μ)]μdμ,
RJjk=3sin Δ/21[1-Rjk(μ)]μ2dμ.

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