Abstract

The diffusion equation derived from the space–time transport equation, in contrast to the conventional diffusion equation, should have a diffusion coefficient perfectly independent of the attenuation coefficient, γab, without any constraint on the magnitude of γab and also on its possible spatial inhomogeneity; however, in the case of time-independent diffusion equations, including those for intensity-modulated density waves, such parameters are subjected to constraints including both γab and the modulation frequency. Basic equations for a fixed scatterer embedded in the random medium are presented with reference to previous papers, with some applications. A previous expression for integrated attenuation in a slowly changing medium of γab is improved to be valid to second order of γab for a temporal pulse wave and to first order for an intensity-modulated wave.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Furutsu and T. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [Crossref]
  2. K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980); Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series in Wave Phenomena (Springer-Verlag, New York, 1993), Chap. 6.
    [Crossref]
  3. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993); “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]
  4. A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
    [Crossref]
  5. J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
    [Crossref]
  6. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [Crossref] [PubMed]
  7. J. C. Schotland, J. C. Haselgrove, and J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993). A unit of measurement, photon hitting density, was introduced to describe the deviation of photons from the direct path connecting the source and the detector.
    [Crossref] [PubMed]
  8. K. Furutsu, “Transport theory and boundary-value solutions. I. The Bethe–Salpeter equation and scattering matrices,” J. Opt. Soc. Am. A 2, 913–931 (1985); “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 932–943 (1985), Sec. 3 and App. D; “Fixed scatterer in a random medium: shadowing, enhanced backscattering, and the inner structure of the Bethe–Salpeter equation,” Appl. Opt. APOPAI 32, 2706–2721 (1993), Secs. 4 and 5, Eqs. (4.14), (4.15), and (5.7)–(5.9). In these papers a time-independent wave equation in a random medium with a possible fixed scatterer embedded was considered for a definite wave frequency (with no intensity modulation), starting with the Bethe–Salpeter equation for the mutual coherence function. Here the medium can be divided into two or three layers through rough boundaries that are planar on average, and the theory that was developed treated the random medium and the rough boundaries on an equal basis, so the roles of the medium and the boundaries could be interchanged in the resulting equations, providing a variety of expressions from which to choose. The optical expressions were then introduced with a principle of one-to-one correspondence between the original equations and the optical ones, and the diffusion approximation was finally introduced for a few practical applications. I was exclusively used to stand for the Green’s functions corresponding to S̃ defined by Eq. (2.18) in the present paper. In case of a fixed scatterer in a homogeneously random medium, q, say, the Green’s function, I(a), is obtained as the solution of the Bethe–Salpeter equation in the (coordinate matrix) form I(a)=I+IV(a/q)I. Here V(a/q) is an effective scattering matrix of the scatterer under multiple scattering with the surrounding random medium q.
    [Crossref]
  9. Derivation of a diffusion equation from the transport equation with cross section σ(Ω|Ω′) of general form was made in Ref. 2.
  10. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 203–273.

1995 (1)

A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

1994 (1)

K. Furutsu and T. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

1993 (1)

1992 (1)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

1991 (1)

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Chance, B.

A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Fishkin, J.

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Furutsu, K.

K. Furutsu and T. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

Gratton, E.

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Haselgrove, J. C.

Leigh, J. S.

Mantulin, W. W.

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Schotland, J. C.

van de Ven, M. J.

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Yamada, T.

K. Furutsu and T. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

Yodh, A. G.

A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Appl. Opt. (1)

Phys. Med. Biol. (1)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[Crossref] [PubMed]

Phys. Rev. E (1)

K. Furutsu and T. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

Phys. Today (1)

A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Proc. SPIE (1)

J. Fishkin, E. Gratton, M. J. van de Ven, and W. W. Mantulin, “Diffusion of intensity modulated near-infrared light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, A. Katzir and B. Chance, eds., Proc. SPIE 1431, 122–135 (1991).
[Crossref]

Other (5)

K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980); Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series in Wave Phenomena (Springer-Verlag, New York, 1993), Chap. 6.
[Crossref]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993); “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]

K. Furutsu, “Transport theory and boundary-value solutions. I. The Bethe–Salpeter equation and scattering matrices,” J. Opt. Soc. Am. A 2, 913–931 (1985); “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 932–943 (1985), Sec. 3 and App. D; “Fixed scatterer in a random medium: shadowing, enhanced backscattering, and the inner structure of the Bethe–Salpeter equation,” Appl. Opt. APOPAI 32, 2706–2721 (1993), Secs. 4 and 5, Eqs. (4.14), (4.15), and (5.7)–(5.9). In these papers a time-independent wave equation in a random medium with a possible fixed scatterer embedded was considered for a definite wave frequency (with no intensity modulation), starting with the Bethe–Salpeter equation for the mutual coherence function. Here the medium can be divided into two or three layers through rough boundaries that are planar on average, and the theory that was developed treated the random medium and the rough boundaries on an equal basis, so the roles of the medium and the boundaries could be interchanged in the resulting equations, providing a variety of expressions from which to choose. The optical expressions were then introduced with a principle of one-to-one correspondence between the original equations and the optical ones, and the diffusion approximation was finally introduced for a few practical applications. I was exclusively used to stand for the Green’s functions corresponding to S̃ defined by Eq. (2.18) in the present paper. In case of a fixed scatterer in a homogeneously random medium, q, say, the Green’s function, I(a), is obtained as the solution of the Bethe–Salpeter equation in the (coordinate matrix) form I(a)=I+IV(a/q)I. Here V(a/q) is an effective scattering matrix of the scatterer under multiple scattering with the surrounding random medium q.
[Crossref]

Derivation of a diffusion equation from the transport equation with cross section σ(Ω|Ω′) of general form was made in Ref. 2.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 203–273.

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

Geometry of the scatterer for Eqs. (4.8) and (4.12).

Fig. 2
Fig. 2

Integration paths for integral (4.1).

Fig. 3
Fig. 3

(a) Temporal changes of the relative intensity A and the normalized power flux toward the absorber center, Ap, defined by Eqs. (4.14) and (4.15), for several values of the scattering angle θ, in degrees (Fig. 1). The dimensionless variables R1, R2, Ra, and T, defined by Eqs. (4.13), are used. (b) Curves of A and Ap as functions of the scattering angle θ for several values of T in the same case as (a).

Fig. 4
Fig. 4

Temporal changes of A and Ap, in picoseconds, for θ = 0°, 90°, 180°. The distances R1, R2, and Ra are given here in millimeters.

Equations (108)

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

|γab+iω/c|γ.
(t+Ω·+γt)I(Ω, ρ¯)= dΩσ(Ω|Ω)I(Ω, ρ¯)+Jc(Ω, ρ¯).
γt=γ+γab,γ= dΩσ(Ω·Ω),
S(Ω, ρ|t-t=0|Ω, ρ)=δ(Ω-Ω)δ(ρ-ρ).
(t+Ω·+γt)S(Ω, ρ|t-t|Ω, ρ)
= dΩσ(Ω|Ω)S(Ω, ρ|t-t|Ω, ρ),
I(Ω, ρ¯)=c -t dt  dρdΩ×S(Ω, ρ|t-t|Ω, ρ)Jc(Ω, ρ¯),
S(Ω, ρ|t-t|Ω, ρ)=exp[-c(t-t)γab]×S0(Ω, ρ|t-t|Ω, ρ).
(t+Ω·+γ)S0(Ω, ρ|t-t|Ω, ρ)
= dΩσ(Ω|Ω)S0(Ω, ρ|t-t|Ω, ρ),
S0(ρ|t-t|ρ)= dΩS0(Ω, ρ|t-t|Ω, t),
(t-D2)S0(ρ|t-t|ρ)=0
a1=γ-1  dΩΩ·Ωσ(Ω·Ω)
S0(ρ|t-t=0|ρ)=δ(ρ-ρ).
(t+γab-D2)S(ρ|t-t|ρ)=0,
Jc(Ω, ρ¯)=J0(Ω, ρ)+exp(iωt)J(Ω, ρ, ω),0,t>0t<0.
 dρdΩc 0t dt exp[-c(t-t)γab+iωt]
×S0(Ω, ρ|t-t|Ω, ρ)J(Ω, ρ, ω),
I(Ω, ρ, t)=I0(Ω, ρ)+exp(iωt)I(Ω, ρ, ω).
I(Ω, ρ, ω)= dρdΩS˜(Ω, ρ|ω|Ω, ρ)J(Ω, ρ, ω),
S˜(Ω, ρ|ω|Ω, ρ)=c 0 dt exp[-ct(γab+iω/c)]×S0(Ω, ρ|t|Ω, ρ).
(Ω·+γab+iω/c+γ)S˜(Ω, ρ|ω|Ω, ρ)
= dΩσ(Ω|Ω)S˜(Ω, ρ|ω|Ω, ρ)
+δ(Ω-Ω)δ(ρ-ρ),
I(ρ, ω)= dΩI(Ω, ρ, ω)= dρS˜(ρ|ω|ρ)J(ρ, ω).
S˜(ρ|ω|ρ)= dΩS˜(Ω, ρ|ω|Ω, ρ),
J(ρ, ω)= dΩJ(Ω, ρ, ω),
S˜(ρ|ω|ρ)=c 0 dt exp[-ct(γab+iω/c)]S0(ρ|t|ρ),
(γab+iω/c-D2)S˜(ρ|ω|ρ)=δ(ρ-ρ),
|γab+iω/c|γ
γ(ρ)= dΩσ(Ω·Ω, ρ),
D(ρ)=3-1(1-a1)-1γ-1(ρ),
[t+γab(ρ)-jD(ρ)j]S(ρ|t-t|ρ)=0,
V(α)(Ω|ρ|Ω)=σ(α)(Ω|ρ|Ω)-γ(α)(Ω, ρ)δ(Ω-Ω).
V(α)(Ω|ρ|Ω)=[σ(α)(Ω|Ω)-γ(α)(Ω)δ(Ω-Ω)]×δ(ρ-ρα).
-γab(α)(Ω, ρ) dΩV(α)(Ω|ρ|Ω)= dΩσ(α)(Ω|ρ|Ω)-γ(α)(Ω, ρ)0.
[t+γab+γab(α)(ρ)-D2]S(α)(ρ|t-t|ρ)=0.
[γab+γab(α)(ρ)+iω/c-D·2]S˜(α)(ρ|ω|ρ)
=δ(ρ-ρ),
S˜(α)(ρ|ω|ρ)=c 0 dt exp[-ct(γab+iω/c)]×S0(α)(ρ|t|ρ),
[t+γab(α)(ρ)-D2]S0(α)(ρ|t|ρ)=0,
S0(α)(ρ|t|ρ)=(2πc)-1 --i0-i0 dω exp(iωt)S˜0(α)(ρ|ω|ρ),
S˜0(ρ|ω|ρ)=(4πD|ρ-ρ|)-1 exp(-k|ρ-ρ|),
k=(iω/cD)1/2,π/4arg(k)-π/4,
[γab(α)+iω/c-D2]S˜0(α)(ω)=1.
S˜0(α)(ω)=S˜0(ω)[1-γab(α)S˜0(α)(ω)],
S˜0(α)(ω)=S˜0(ω)-S˜0(ω)T˜ab(ω)S˜0(ω)
T˜ab(ω)=[1+γab(α)S˜0(ω)]-1γab(α)
=γab(α)-γab(α)S˜0(ω)γab(α)+.
S˜0(ρ|ω|ρ)=(4πD)-1k n=0(2/π)(2n+1)×Pn(cos θ)in+1/2(kρ<)kn+1/2(kρ>).
in+1/2(x)=(π/2x)1/2In+1/2(x),
kn+1/2(x)=(π/2x)1/2Kn+1/2(x),
|ρ-ρ|=(ρ2+ρ2-2ρρ cos θ)1/2.
S˜0(α)(ρ|ω|ρ)=(4πD)-1k n=0(2/π)(2n+1)Pn(cos θ)×in+1/2(kρ<)-kn+1/2(kρ<)×in+1/2(ka)kn+1/2(ka)kn+1/2(kρ>),
T=ct/D,Ra=a(Dct)-1/2,
Rj=ρj(Dct)-1/2,j=1, 2,
S(α)(ρ|t|ρ)=AS0(ρ|t|ρ),
Sp(α)(ρ|t|ρ)=-D ρS(ρ|t|ρ)=ApS0(ρ|t|ρ).
[γ˜ab(ρ)+k2-2]S˜(ρ|ω|ρ)=D-1δ(ρ-ρ).
γ˜ab(ρ)=γab(ρ)D-1,
k2=i(ω/c)D-1,Re(k)>0.
S˜(ρ|ω|ρ)=exp(-ψ(ρ|ω|ρ)),
γ˜ab(ρ)+k2-(ψ)2+2ψ=eψD-1δ(ρ-ρ).
k2-(ψ0)2+2ψ0=eψ0D-1δ(ρ-ρ),
ψ0(ρ|ω|ρ)=k|ρ-ρ|+ln(4πD|ρ-ρ|);
ψ0=k(ρ-ρ)/|ρ-ρ|+(ρ-ρ)/|ρ-ρ|2,
ψ0(ρ|ω|ρ)kk(ρ-ρ)/|ρ-ρ|.
ψ(ρ|ω|ρ)=(ψ0+ψ1)(ρ|ω|ρ),
γ˜ab(ρ)-(ψ1)2+2ψ1-2(ψ1)·(ψ0)=0,
γ˜ab(ρ)+2ψ1-2k·ψ1=0.
γ˜ab(x, z)+[(/x)2-2k/z]ψ1(x, z)=0.
ψ1(x, z)=(2k)-1 0z dz exp[(2k)-1×(z-z)x2]γ˜ab(x, z),
ψ1(x, z)(2k)-1 0z dz[1+(2k)-1×(z-z)x2]γ˜ab(x, z).
ψ1(x, z)=0z dz  dxS(x-x, z-z)×(2k)-1γ˜ab(x, z),
S(x, z-z)=k[2π(z-z)]-1 exp[-kx2/2(z-z)],
 dxS(x, z-z)=1.
S(ρ|t|ρ)=exp[-w(ρ|t|ρ)],
S0(ρ|t|ρ)=exp[-w0(ρ|t|ρ)];
w(ρ|t|ρ)=w0(ρ|t|ρ)+ct 01 dτγab[ρ(τ)]-D(ct)3 01 dτ 01 dτG(τ|τ)
×γab,ρ[ρ(τ)]·γab,ρ[ρ(τ)],
ρ(τ)=ρ+τ(ρ-ρ),
γab,ρ(ρ)=ργab(ρ),
G(τ|τ)=(1-τ)ττ(1-τ)τ>ττ<τ,
τ2G(τ|τ)=-δ(τ-τ),
G(τ=1|τ)=G(τ=0|τ)=0.
γab(ρ)=γab0+γab,ρ·ρ,
ct 01 dτγab[ρ(τ)]=ct2-1[γab(ρ)+γab(ρ)],
(w-w0)(ρ|t|ρ)=2-1ct[γab(ρ)+γab(ρ)]-12-1D(ct)3(γab,ρ)2
01 dτ 01 dτG(τ|τ)=12-1.
tSλ(t)=HλSλ(t),Sλ(t=0)=1,
Hλ=H0-λγab,H0=D2.
Sλ(t)=exp(ctHλ).
Hλ+δλ=Hλ-δλγab,
Sλ+δλ(t)=Sλ(t)+δSλ(t).
δSλ(t)=Sλ(t)δAλ(t),
Sλ(t)tδAλ(t)=-δλγabSλ(t)
tδAλ(t)=-δλγˆab(t)
γˆab(t)=Sλ-1(t)γab(ρ)Sλ(t)=γab[ρˆ(t)],
ρˆ(t)=Sλ-1(t)ρSλ(t).
δAλ(t)=-δλc 0t dtγˆab(t).
Sλ(ρ|t|ρ)ρ|Sλ(t)|ρ=exp[-wλ(ρ|t|ρ)],
δSλ(ρ|t|ρ)ρ|δSλ(t)|ρ=ρ|SλδAλ(t)|ρ,
δwλ(ρ|t|ρ)=-ρ|Sλ(t)δAλ(t)|ρρ|Sλ(t)|ρ-δAλ(t)λ
Qλ=ρ|Sλ(t)Q|ρρ|Sλ(t)|ρ,
δwλ(ρ|t|ρ)=δλc 0t dtγab[ρˆ(t)]λ,
w(ρ|t|ρ)-w0(ρ|t|ρ)=c 0t dt 01 dλγab[ρˆ(t)]λ,
w0(ρ|t|ρ)=(ρ-ρ)2/4ctD+ln(4πctD)3/2.
ρˆ(τ)λ=ρ(τ)-2Dλ(ct)2 01 dτG(τ|τ)×γab,ρ[ρˆ(τ)]λ.

Metrics