Abstract

We study light propagation in biological tissue containing an absorbing obstacle. In particular, we solve the infinite-domain problem in which an absorbing plate of negligible thickness prevents a portion of the light from the source from reaching the detector plane. Inasmuch as scattering in the medium is sharply peaked in the forward direction, we replace the governing radiative transport equation with the Fokker-Planck equation. The problem is solved first by application of the Kirchhoff approximation to determine the secondary source distribution over the surface of the plate. That result is propagated to the detector plane by use of Green’s function. The Green’s function is given as an expansion of plane-wave modes that are calculated numerically. The radiance is shown to obey Babinet’s principle. Results from numerical computations that demonstrate this theory are shown.

© 2004 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1996).
  2. K. Furutsu, “Theory of a fixed scatterer embedded in a turbid medium,” J. Opt. Soc. Am. A 15, 1371–1382 (1998).
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  4. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef] [PubMed]
  5. 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).
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  7. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  11. K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium on Applied Mathematics, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), Vol. 1, pp. 17–36.
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).
  13. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).
  14. J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
    [CrossRef]
  15. S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
    [CrossRef]
  16. J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

2003 (2)

S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
[CrossRef]

A. D. Kim, J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

2001 (1)

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

1998 (1)

1997 (2)

1996 (1)

1995 (1)

1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

1993 (1)

1985 (1)

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

Carminati, R.

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

Case, K. M.

K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium on Applied Mathematics, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), Vol. 1, pp. 17–36.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chance, B.

Clark, N.

den Outer, P. N.

Fender, J. S.

Feng, S.

Feng, S. C.

Furutsu, K.

Ishimaru, A.

S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1996).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

Jaruwatanadilok, S.

S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
[CrossRef]

Keller, J. B.

Kim, A. D.

Kuga, Y.

S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
[CrossRef]

Matson, C. L.

McMackin, L.

Morel, J. E.

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Nieto-Vesperinas, M.

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

Nieuwenhuizen, Th. M.

Ntziachristos, V.

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Ripoll, J.

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

Wei, S.-P.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Zeng, F.-A.

Zhu, X. D.

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens. (1)

S. Jaruwatanadilok, A. Ishimaru, Y. Kuga, “Optical imaging through clouds and fog,” IEEE Trans. Geosci. Remote Sens. 41, 1834–1843 (2003).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nucl. Sci. Eng. (1)

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Phys. Rev. E (1)

J. Ripoll, V. Ntziachristos, R. Carminati, M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917(2001).
[CrossRef]

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Other (5)

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium on Applied Mathematics, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), Vol. 1, pp. 17–36.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1996).

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Figures (10)

Fig. 1
Fig. 1

Diagram of the absorbing-plate problem. An absorbing plate of vanishing thickness on the plane z = z 1 lies between source plane z = z 0 and detector plane z = z 2. It prevents light from the source from reaching the detector plane.

Fig. 2
Fig. 2

Diagram of the compliment to the absorbing-plate problem shown in Fig. 1. An absorbing screen contains an aperture opening with the same shape as that of the absorbing plate in Fig. 1.

Fig. 3
Fig. 3

Radiance Ψ D at the detector plane evaluated in direction (μ, φ) = (0.9894, π) as a function of x and y. The absorbing plate is the rectangular region |x| ≤ 0.625l s and |y| ≤ 1.875l s . The point source with direction (μ0, φ0) = (0.9894, 0) is located at (x, y) = (0, 0). The optical properties of the medium are ∑ a /∑ s = 0.01 and g = 0.95.

Fig. 4
Fig. 4

Radiance Ψ D at location (x, y) = (0, 0) on the detector plane as a function of ω. All other parameters are the same as for Fig. 3.

Fig. 5
Fig. 5

Same as Fig. 4, except that the location on the detector plane is (x, y) = (5l s , 5l s ).

Fig. 6
Fig. 6

Same as Fig. 3, except that the absorbing plate occupies |x| ≤ 0.625l s and |y| ≤ 2.5l s .

Fig. 7
Fig. 7

Absorbing plate used for studying Babinet’s principle.

Fig. 8
Fig. 8

Integrated flux F D as a function of x and y that is due to the absorbing plate shown in Fig. 7.

Fig. 9
Fig. 9

Integrated flux F D c for the solution to the complimentary problem in which the absorbing plate shown in Fig. 7 is an aperture opening.

Fig. 10
Fig. 10

(a) Sum of the absorbing-plate problem and its compliment shown in Figs. 8 and 9 and (b) the homogeneous solution to the problem with no absorbing plate.

Equations (39)

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ω·Ψ+aΨ=-sΨ+sΩ fω·ωΨω, rdω.
ω · Ψ+aΨ=½s1-gΔΨ.
g=2π -1+1ω · ωfω · ωdω · ω.
ω · G+aG-½s1-gΔG=δr-r ×δω-ω
Ψω, r=DΩ Gω, r; ω, rQω, rdωdr+SΩ Gω, r; ω, rs×ω · nˆrsΨω, rsdωdrs.
Ψω, r=12π22 Vω; κexpλκz+iκ · ρdκ.
λμV+iν · κV+aV=½s1-gΔV.
λjκ-λkκΩ Vjω; κVkω; κμdω=0.
Ω Vjω; κVjω; κμdω=-Ω V-jω; κ×V-jω; κμdω.
Ω Vjω; κVjω; κμdω=-1j>0+1j<0.
Gω, r; ω, r=12π22 Ĝω, z; ω, z, κ×expiκ · ρ-ρdκ.
μzĜ+iν · κĜ+aĜ-½s1-gΔĜ=×δω-ωδz-z.
μĜω, z+0; ω, z, κ-μĜω, z-0; ω, z, κ=δω-ω.
Ĝω, z; ω, z, κ=j gjz; ω, z, κVjω.
jzgj-λjgjμVjω; κ=δω-ωδz-z.
zgj-λjgj=-sgnjVjω; κδz-z,
zCj-λjCj=δz-z.
Cjz+0; z, κ-Cjz-0; z, κ=1.
Cjz; z, κ=-expλjκz-zz<z, j>00z<z, j<00z>z, j>0+expλjκz-zz>z, j<0.
Ĝω, z; ω, z, κ=j>0 expλjκz-zVjω; κVjω; κz<zj<0 expλjκz-zVjω; κVjω; κz>z.
ΨDhω, ρ=3Ω Gω, ρ, z2; ω, ρ, z×δω-ω0δρ-ρ0δz-z0dωdr =Gω, ρ, z2; ω0, ρ0, z0.
ΨˆDhω; κ=z=z2 ΨDhω, ρexp-iκ · ρdρ=Ĝω, z2; ω0, z0; κexp-iρ0 · κ=j<0 expλjκz2-z0-iρ0 · κVjω; κVjω0; κ.
Ψ1hω, ρ=Gω, ρ, z1; ω0, ρ0, z0.
Ψω, ρ, z2=z=z1Ω Gω, ρ, z2; ω, ρ, z1×Ψ1hω, ρμdωdρ.
Ψˆω, z2; κ=Ω Ĝω, z2; ω, z1, κΨˆ1hω; κμdω=j<0 expλjκz2-z1Vjω; κ×k<0 expλkκz1-z0-iρ0 · κVkω0; κ×Ω Vjω; κVkω; κμdω=j<0 expλjκz2-z1Vjω; κ×k<0 expλkκz1-z0-iρ0 · κVkω0; κδj,k=j<0 expλjκz2-z0-iρ0 · κVjω; κVjω0; κ.
ΨDhω, ρ=z=z1Ω Gω, ρ, z2; ω, ρ, z1×Ψ1hω, ρμdωdρ.
ΨDω, ρ=ΨDhω, ρ+PΩ Gω, ρ, z2; ω, ρ, z1×ΨPω, ρμdωdρ,
ΨPω, ρ=-Ψ1hω, ρ, ρ in P.
ΨDω, ρ=P¯Ω Gω, ρ, z2; ω, ρ, z1×Ψ1hω, ρμdωdρ.
ΨDcω, ρ=PΩ Gω, ρ, z2; ω, ρ, z1×Ψ1hω, ρμdωdρ.
ΨD+ΨDc=ΨDh.
FDρ=Ω ΨDω, ρμdω
ΔV=μ1-μ2μV+1-μ2-1φ2V.
αm+1/2=αm-1/2-2μmwm, m=1,, M.
ΔVμm, φnwm-1αm+1/2Vμm+1, φn-Vμm, φnμm+1-μm-αm-1/2Vμm, φn-Vμm-1, φnμm-μm-1+1-μm2-1Vμm, φn+1-2Vμm, φn+Vμm, φn-1Δφ2.
Vμ, φ=Vμ, φ+2π.
λμmVμm, φn+i1-μm21/2κx cos φn+κy sin φnVμm, φn+aVμm, φn=½s1-gΔVμm, φn.
γj=m=1Mn=1N Vjμm, φnVjμm, φnμmwmΔφ.
Reλ-MN/2<<Reλ-1<Reλ+1<<Reλ+MN/2.

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