Abstract

A review of Green functions for diffuse light in two semi-infinite scattering and absorbing half-spaces separated by a plane interface is presented. The frequency-domain Green functions for an intensity-modulated point source are derived within the diffusion approximation by the Hankel transform with respect to the variable in the plane of the interface. Green functions for a line source and a plane source parallel to the interface are obtained from the three-dimensional Green functions by the method of descent. Green functions for a steady state are obtained as a limit of zero modulation frequency. Connection of the frequency-domain Green functions with the time-domain Green functions is shown by use of the Fourier transform in time. The influence of the relative optical parameters, namely, the ratios of diffusion coefficients, absorption coefficients, and refractive indices of the two media on the shape of the contour lines of the specific intensity, is shown for the continuous and intensity-modulated point sources.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  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. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. 10, 127–140 (1993).
    [CrossRef]
  4. R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,” IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
    [CrossRef]
  5. J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
    [CrossRef] [PubMed]
  6. I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue,” Appl. Opt. 34, 2973–2982 (1995).
    [CrossRef] [PubMed]
  7. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
  8. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821–830 (2001).
    [CrossRef]
  9. I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
    [CrossRef]
  10. F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002).
    [CrossRef]
  11. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
    [CrossRef]
  12. J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid-turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
    [CrossRef]
  13. J.-M. Tualle, J. Prat, E. Tinet, S. Avrillier, “Real-space Green’s function calculation for the solution of the diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
    [CrossRef]
  14. M. Abramovitz, I. Stegan, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).
  15. M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
    [CrossRef]

2004 (1)

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

2003 (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (2)

1998 (1)

1995 (1)

1993 (1)

1992 (1)

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

1990 (1)

1989 (1)

1968 (1)

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,” IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. Stegan, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).

Anderson, R. R.

Avrillier, S.

Bays, R.

Chance, B.

Culver, J. P.

Dayan, I.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Del Bianco, S.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

Dögnitz, N.

Fishkin, J. B.

Gratton, E.

Havlin, S.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Ishimaru, A.

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

Kienle, A.

Longini, R. L.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,” IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Martelli, F.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002).
[CrossRef]

Nieto-Vesperinas, M.

Ntziachristos, V.

Pattanayak, D. N.

Patterson, M. S.

Prat, J.

Ripoll, J.

Sassaroli, A.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002).
[CrossRef]

Schmitt, J. M.

Shendeleva, M. L.

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

Stegan, I.

M. Abramovitz, I. Stegan, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).

Tinet, E.

Tualle, J.-M.

van den Bergh, H.

Vitkin, I. A.

Wagnières, G.

Walker, E. C.

Wall, R. T.

Weiss, G. H.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Wilson, B. C.

Yamada, Y.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002).
[CrossRef]

Yodh, A. G.

Zaccanti, G.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A 19, 71–80 (2002).
[CrossRef]

Zdrojkowski, R.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,” IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Zhou, G. X.

Appl. Opt. (3)

IEEE Trans. Biomed. Eng. (1)

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,” IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a 2-layer turbid medium—a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

M. L. Shendeleva, “One-dimensional time-domain Green functions for diffuse light in two adjoining turbid media,” Opt. Commun. 235, 233–245 (2004).
[CrossRef]

J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid-turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[CrossRef]

Phys. Rev. E (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by eigenfunction method,” Phys. Rev. E 67, 056623-1–056623-14 (2003).
[CrossRef]

Other (2)

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

M. Abramovitz, I. Stegan, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Contour lines of the specific intensity for a continuous point source located at (0, 0, 1). Parameters of the model are D 1 v 1 = 1 cm2/s, β1 v 1= 1 s-1, N = 1, b = 4. The values of γ are indicated in the figure. Dimensions on the axes are in centimeters. The dashed curve corresponds to the case γ2 b = 1.

Fig. 2
Fig. 2

Contour lines of the specific intensity for a continuous point source located at (0, 0, 1). Parameters of the model are D 1 v 1 = 1 cm2/s, β1 v 1= 1 s-1, N = 1, γ = 0.5. The values of b are indicated in the figure. Dimensions on the axes are in centimeter. The dashed curve corresponds to the case γ2 b = 1.

Fig. 3
Fig. 3

Contour lines of the amplitude of the specific intensity for the intensity-modulated point source located at (0, 0, 1). Parameters of the model are ω = 10 rad/s, D 1 v 1 = 1 cm2/s, β1 v 1= 1 s-1, N = 1, b = 1. The values of γ are indicated in the figure. Dimensions on the axes are in centimeters. The dashed curve corresponds to the case γ2 N = 1.

Fig. 4
Fig. 4

Contour lines of the amplitude of the specific intensity for the intensity-modulated point source located at (0, 0, 1). Parameters of the model are ω = 1 rad/s, D 1 v 1 = 1 cm2/s, β1 v 1= 1 s-1, b/ N = 1, γ = 0.9. The values of N are (1) N = 1.5, (2) N = 1.23, (3) N = 1, and (4) N = 0.7. Dimensions on the axes are in centimeters. The short-dashed curve corresponds to the case γ2 N = 1; the long-dashed curve corresponds to N = 1.

Equations (58)

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

2x2+2y2+2z2U1-iωD1v1 U1-β1D1 U1=-1D1v1×δxδyδz-z0,
2x2+2y2+2z2U2-iωD2v2 U2-β2D2 U2=0,
U1n12=U2n22,
D1U1z=D2U2z
uj=0 UjJ0λρρdρ
2u1z2-η12u1=12πD1v1 δz-z0,
2u2z2-η22u2=0,
N2u1=u2,
γ2u1z=u2z,
ηj=λ2+iω+βjvjDjvj1/2.
N=n2n1, γ=D1D21/2.
uS=14πD1v1η1exp-η1|z-z0|.
u1=exp-η1|z-z0|4πD1v1η1+-1+2η1η1+μη2×exp-η1z+z04πD1v1η1,
u2=N2η1+μη2exp-η1z0+η2z2πD1v1,
μ=N2γ2.
Uj=0 ujJ0λρλdλ.
US3-D=14πD1v10exp-η1|z-z0|η1 J0λρλdλ=14πD1v1R1exp-R1iω+β1v1D1v11/2,
U13-D=14πD1v1R1exp-R1σ1-14πD1v1R2×exp-R2σ1+12πD1v10×exp-z+z0λ2+σ121/2×J0λρλdλλ2+σ121/2+μλ2+σ221/2.
σj=iω+βjvjDjvj1/2.
U23-D=N22πD1v10×expzλ2+σ221/2-z0λ2+σ121/2×J0λρλdλλ2+σ121/2+μλ2+σ221/2
U¯13-D=14πD1v1R1exp-R1β1D11/2-14πD1v1R2exp-R2β1D11/2+12πD1v10exp-z+z0×λ2+β1D11/2×J0λρλdλλ2+β1D11/2+μλ2+β2D21/2
U¯23-D=N22πD1v10expzλ2+β2D21/2-z0×λ2+β1D11/2J0λρλdλλ2+β1D11/2+μλ2+β2D21/2
ϕ13-D=14πD1v1t3/2exp-R124D1v1t-β1v1t-14πD1v1t3/2exp-R224D1v1t-β1v1t+μν4πD1v1t3/20×exp-ρ24D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tu×G1du1-u+uν2
ϕ23-D=μνN24πD1v1t3/20exp-ρ24D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tu×G2du1-u+uν2
ν=γN=D1v1D2v21/2.
G1=1u+μ2ν21-u3/2×exp-μ2ν2z+z024D1v1tu+μ2ν21-u×erfcz+z0u2D1v1t1-u1/2u+μ2ν21-u1/2×1-μ2ν2z+z022D1v1tu+μ2ν21-u+μ2ν2z+z0u1-u2πD1v1t1/2u+μ2ν21-u2×exp-z+z024D1v1t1-u,
G2=1u+μ2ν21-u3/2×1-ν2μz0+z22D1v1tu+μ2ν21-u×exp-ν2μz0+z24D1v1tu+μ2ν21-u×erfcz0u-μν2z1-u2D1v1tu1-u1/2u+μ2ν21-u1/2+-zu+z0μ3ν21-uμD1v1tu1-u1/2u+μ2ν21-u2×exp-z2ν21-u+z02u4D1v1tu1-u,
Uj2-D=- Uj3-Ddy.
US2-D=14πD1v1-exp-R1σ1R1dx=12πD1v1 K0r1σ1,
0exp-ax2+11/2x2+11/2dx=K0a
U12-D=12πD1v1 K0r1σ1-12πD1v1 K0r2σ1+1πD1v10exp-z+z0×λ2+σ121/2×cosλxdλλ2+σ121/2+μλ2+σ221/2
U22-D=N2πD1v10expzλ2+σ221/2-z0λ2+σ121/2×cosλxdλλ2+σ121/2+μλ2+σ221/2
U¯12-D=12πD1v1 K0r1β1D11/2-12πD1v1 K0r2β1D11/2+1πD1v10exp-z+z0×λ2+β1D11/2×cosλxdλλ2+β1D11/2+μλ2+β2D21/2
U¯22-D=N2πD1v10expzλ2+β2D21/2-z0×λ2+β1D11/2cosλxdλλ2+β1D11/2+μλ2+β2D21/2
ϕ12-D=14πD1v1texp-r124D1v1t-β1v1t-14πD1v1texp-r224D1v1t-β1v1t+μν4πD1v1t0exp-x24D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tuG1du1-u+uν21/2
ϕ22-D=μνN24πD1v1t0exp-x24D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tuG2du1-u+uν21/2
Uj1-D=- Uj2-Ddx,
US1-D=12πD1v1- K0r1σ1dx=12D1v1σ1exp-z-z0σ1.
1K0ayydyy2-11/2=πe-a2a
U11-D=12D1v1σ1exp-|z-z0|σ1+12D1v1σ1σ1-μσ2σ1+μσ2exp-z+z0σ1
U21-D=N2D1v1σ1+μσ2expzσ2-z0σ1
U¯11-D=12v1β1D11/2exp-|z-z0|β1D11/2+12v1β1D11/2β1D11/2-N2β2D21/2β1D11/2+N2β2D21/2×exp-z+z0β1D11/2
U¯21-D=N2v1β1D11/2+N2β2D21/2expzβ2D21/2-z0β1D11/2
ϕ11-D=12πD1v1t1/2exp-z-z024D1v1t-β1v1t-12πD1v1t1/2exp-z+z024D1v1t-β1v1t+μν2πD1v1t1/20exp-β1v1t1-u-β2v2tuG1du
ϕ21-D=μνN22πD1v1t1/20exp-β1v1t1-u-β2v2tuG2du
b=β2β1.
U¯13-D=14πD1v1R1exp-R1β1D11/2+14πD1v1R21-μ1+μexp-R2β1D11/2,
U¯23-D=N22πD1v1R111+μexp-R1β1D11/2.
U13-D=14πD1v1R1exp-R1σ1+14πD1v1R21-μ1+μexp-R2σ1,
U23-D=N22πD1v1R111+μexp-R1σ1,
Uj=0 ϕj exp-iωtdt.
0exp-at-bt dtt3/2=πaexp-2ab.
C1=μν4πD1v1t3/201exp-ρ24D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tuG1du1-u+uν2.
0 J0λρexp-aλ2λdλ=exp-ρ2/4a2a,
C1=μν4π3D1v1t1/20dλ 01 λJ0λρ×exp-λ2D1v1t1-u+uν2×exp-β1v1t1-u-β2v2tuG1du1-u+uν2.
G1=12πD1v1tu1-u3/20 ww+z+z0×exp-w+z+z024D1v1t1-u-μ2ν2w24D1v1tudw.
FC1=0 C1 exp-iωtdt,
0 C1 exp-iωtdt=12πD1v10λJ0λρη1+μη2×exp-z+z0η1dλ,

Metrics