Abstract

Diffuse photon density waves are currently used to probe turbid media for optical anomalies such as tumors in tissue. A basic theory is established for detection of a fixed scatterer embedded in a turbid medium, including the effects of the medium’s boundaries. Several diffuse expressions are obtained for a scattered wave both reflected and transmitted through a turbid layer, as well as within the layer, including important cases of vertically incident light and vertical observation by optical fibers. The importance of the scatterer’s effective cross section when it is embedded in a random medium is emphasized. Specific results are obtained with a simple model of a scatterer.

© 1998 Optical Society of America

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References

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  1. 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]
  2. H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction in turbid media: an experimental study of single-target detectability,” Appl. Opt. 36, 52–63 (1997).
    [CrossRef] [PubMed]
  3. S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
    [CrossRef] [PubMed]
  4. H. Wabnitz, H. Rinneberg, “Imaging in turbid media by photon density waves: spatial resolution and scaling relations,” Appl. Opt. 36, 64–74 (1997).
    [CrossRef] [PubMed]
  5. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
    [CrossRef]
  6. 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. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  7. B. J. Tromberg, L. O. Svaasand, T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
    [CrossRef] [PubMed]
  8. P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  9. 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]
  10. 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 2, 932–944 (1985); K. Furutsu, Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series in Wave Phenomena (Springer-Verlag, New York, 1993), p. 270.
    [CrossRef]
  11. K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
    [CrossRef] [PubMed]
  12. K. Furutsu, “Fixed scatterer in a random medium: shadowing, enhanced backscattering, and the inner structure of the Bethe–Salpeter equation,” Appl. Opt. 32, 2706–2721 (1993).
    [CrossRef] [PubMed]
  13. The diffusion constant D is given by Eqs. (B16) independent of the absorption cross section: K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  14. Equations (B3)–(B5) of Ref. 11.
  15. K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985), Sec. 3 and App. D; Eqs. (5.7)–(5.9) of Ref. 12.
    [CrossRef]
  16. R. A. J. Groenhuis, H. A. Ferwerda, J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  17. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  18. A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
    [CrossRef] [PubMed]
  19. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  20. Equations (5.3)–(5.11) of Ref. 11.
  21. The detailed theory is given in K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985).
    [CrossRef]
  22. Ref. 11, Sec. 4.

1997 (6)

1995 (1)

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

1994 (2)

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

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 (4)

1989 (1)

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

1985 (3)

1983 (1)

1980 (1)

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]

Bosch, J. J. T.

Chance, B.

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]

den Outer, P. N.

Fantini, S.

Feng, T. C.

Ferwerda, H. A.

Fishkin, J. B.

Furutsu, K.

K. Furutsu, “Fixed scatterer in a random medium: shadowing, enhanced backscattering, and the inner structure of the Bethe–Salpeter equation,” Appl. Opt. 32, 2706–2721 (1993).
[CrossRef] [PubMed]

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[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 2, 932–944 (1985); K. Furutsu, Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series in Wave Phenomena (Springer-Verlag, New York, 1993), p. 270.
[CrossRef]

K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985), Sec. 3 and App. D; Eqs. (5.7)–(5.9) of Ref. 12.
[CrossRef]

The detailed theory is given in K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985).
[CrossRef]

The diffusion constant D is given by Eqs. (B16) independent of the absorption cross section: K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
[CrossRef]

Gratton, E.

Groenhuis, R. A. J.

Haskell, R. C.

Hielscher, A. H.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Jacques, S. L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Jiang, H.

Kienle, A.

Lagendijk, A.

McAdams, M. S.

Nieuwenhuizen, Th. M.

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]

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Rinneberg, H.

Schotland, C.

Svaasand, L. O.

Tittle, F. K.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T.

Tsay, T. T.

Wabnitz, H.

Walker, S. A.

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

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]

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

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 2, 932–944 (1985); K. Furutsu, Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series in Wave Phenomena (Springer-Verlag, New York, 1993), p. 270.
[CrossRef]

K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985), Sec. 3 and App. D; Eqs. (5.7)–(5.9) of Ref. 12.
[CrossRef]

The detailed theory is given in K. Furutsu, “Transport theory and boundary-value solutions. II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985).
[CrossRef]

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. A 10, 127–140 (1993).
[CrossRef] [PubMed]

P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

Phys. Med. Biol. (1)

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittle, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Phys. Rev. A (1)

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

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 (3)

Equations (B3)–(B5) of Ref. 11.

Equations (5.3)–(5.11) of Ref. 11.

Ref. 11, Sec. 4.

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Figures (5)

Fig. 1
Fig. 1

Examples of exact calculation of a wave distorted by a perfectly absorbing sphere.

Fig. 2
Fig. 2

Phase and amplitude resolution necessary to measure a diffuse photon density wave distorted by a perfect absorber.

Fig. 3
Fig. 3

Geometry and notation of a turbid layer K(q) for Eqs. (3.3) and (3.4). The layer is distributed in the range 0>z>-L with the boundaries σ(12) and σ(23) at z=0 and -L, respectively. The source located in the region z>0 and the fixed scatter V(α) embedded in the layer are represented by a filled square and a hatched circle, respectively. The wave intensity in region a is denoted by Ia1(q+12+23) in this case.

Fig. 4
Fig. 4

Geometry and notation for a semi-infinite turbid layer for Eq. (4.18). The wave is incident and is observed through the apertures A2 and A1, respectively, in vertical directions. The hatched circle represents a fixed embedded scatterer. The vectors Ωˆs and Ωˆs are defined by Eq. (4.20), except for the sign.

Fig. 5
Fig. 5

Geometry and notation for a turbid layer under the same conditions as in Fig. 2. The wave scattered by V(α) into the space z<-L is given by Eq. (4.21) with k3=k1.

Equations (187)

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2U(r, t)-νμaDU(r, t)-1DU(r, t)t=-1DS(r, t).
(t+Ωˆ·/ρˆ+γt)I(Ωˆ, ρ¯)=dΩˆK(q)(Ωˆ|Ωˆ)I(Ωˆ, ρ¯)+Jc(Ωˆ, ρ¯).
γt=γ+γab,γ=dΩˆK(q)(Ωˆ·Ωˆ),
I(Ωˆ, ρˆ|t=0|Ωˆ, ρˆ)=δ(Ωˆ-Ωˆ)δ(ρˆ-ρˆ).
(t+Ωˆ·/ρˆ+γt)I(Ωˆ, ρˆ|t|Ωˆ, ρˆ)=dΩˆK(q)(Ωˆ|Ωˆ)I(Ωˆ, ρˆ|t|Ωˆ, ρˆ),
I(Ωˆ, ρ¯)=c-tdtdρˆdΩˆI(Ωˆ, ρˆ|t-t|Ωˆ, ρˆ)Jc(Ωˆ, ρ¯),
I(Ωˆ, ρˆ|t|Ωˆ, ρˆ)=exp(-ctγab)I0(Ωˆ, ρˆ|t|Ωˆ, ρˆ).
(t+Ωˆ·/ρˆ+γ)I0(Ωˆ, ρˆ|t|Ωˆ, ρˆ)=dΩˆK(q)(Ωˆ|Ωˆ)I0(Ωˆ, ρˆ|t|Ωˆ, ρˆ).
Jc(Ωˆ, ρ¯)=J0(Ωˆ, ρˆ)+exp(iωt)J(Ωˆ, ρˆ, ω)t>00t<0.
dρˆdΩˆc0tdt exp[-c(t-t)γab+iωt]×I0(Ωˆ, ρˆ|t-t|Ωˆ, ρˆ)J(Ωˆ, ρˆ, ω),
I(Ωˆ, ρˆ, t)=I0(Ωˆ, ρˆ)+exp(iωt)I(Ωˆ, ρˆ, ω).
I(Ωˆ, ρˆ, ω)=dρˆdΩˆI(Ωˆ, ρˆ|Ωˆ, ρˆ)J(Ωˆ, ρˆ, ω),
I(Ωˆ, ρˆ|Ωˆ, ρˆ)=c0dt exp[-ct(γab+iω/c)]×I0(Ωˆ, ρˆ|t|Ωˆ, ρˆ),
(t+Ωˆ·/ρˆ+γ)U(Ωˆ, ρˆ|t|Ωˆ, ρˆ)=0,
U(Ωˆ, ρˆ|t=0|Ωˆ, ρˆ)=δ(ρˆ-ρˆ)δ(Ωˆ-Ωˆ);
U(Ωˆ, ρˆ|Ωˆ, ρˆ)=c0dtU(Ωˆ, ρˆ|t|Ωˆ, ρˆ)
(Ωˆ·/ρˆ+γ)U(Ωˆ, ρˆ|Ωˆ, ρˆ)=δ(ρˆ-ρˆ)δ(Ωˆ-Ωˆ),
I0(t)=U(t)+Id(t),
(t+Ωˆ·/ρˆ+γ)Id(t)=K(q)[U(t)+Id(t)],
(Ut+1)Id(t)=UK(q)[U(t)+Id(t)],
U(Ωˆ, ρˆ|t>0|Ωˆ, ρˆ)=exp(-γct)δ(ρˆ-ρˆ-Ωˆct)δ(Ωˆ-Ωˆ),
c0tdtdΩˆdρˆU(Ωˆ, ρˆ|t-t|Ωˆ, ρˆ)J(Ωˆ, ρˆ, t)dΩˆdρˆU(Ωˆ, ρˆ|Ωˆ, ρˆ)J(Ωˆ, ρˆ, t=t),
U(t)δ(ct+0)U.
(γ-1t+1)Id(t)=UK(q)[Uδ(ct)+Id(t)],
Id(Ωˆ, ρˆ|t|Ωˆ, ρˆ)=ϕA(Ωˆ, iρ)SA(ρˆ|t|ρˆ)ϕ¯A(Ωˆ,-iρ).
ϕA(Ωˆ, iρ)=γ-1(1-3DΩˆ·ρ),
ϕ¯A(Ωˆ,-iρ)=(4π)-1(1+3DΩˆ·ρ),
γ-1(t-Dρ2)SA(ρˆ|t|ρˆ)=δ(ρˆ-ρˆ)δ(ct).
(t-Dρ2)SA(ρˆ|t|ρˆ)=0,t0,
SA(ρˆ|t=0|ρˆ)=γδ(ρˆ-ρˆ).
Id(Ωˆ, ρˆ|Ωˆ, ρˆ)=ϕA(Ωˆ, iρ)SA(ρˆ|ρˆ)ϕ¯A(Ωˆ,-iρ),
SA(ρˆ|ρˆ)=c0dt exp[-ct(γab+iω/c)]SA(ρˆ|t|ρˆ),
γ-1(γab+iω/c-Dρ2)SA(ρˆ|ρˆ)=δ(ρˆ-ρˆ),
|γab+iω/c|γ
(Ωˆ·/ρˆ+γ+γab+iω/c)I(Ωˆ, ρˆ|Ωˆ, ρˆ)
=dΩˆK(q)(Ωˆ|Ωˆ)I(Ωˆ, ρˆ|Ωˆ, ρˆ)
+δ(Ωˆ-Ωˆ)δ(ρˆ-ρˆ).
I=U(ω)(K(q)I+1)
(Ωˆ·/ρˆ+γ+γab+iω/c)U(ω)(Ωˆ, ρˆ|Ωˆ, ρˆ)=δ(Ωˆ-Ωˆ)δ(ρˆ-ρˆ).
I=U(ω)+I(q),
I(q)=U(ω)K(q)I=U(ω)K(q)[U(ω)+I(q)],
Iab(12)(Ωˆ, z|Ωˆ, z)=Ua(ω)(Ωˆ, z-z)δ(Ωˆ-Ωˆ)δab+Ua(ω)(Ωˆ, z)σab(12)(Ωˆ|Ωˆ)×Ub(ω)(Ωˆ,-z).
Ua(ω)(Ωˆ, z)=|Ωz|-1 exp[-(Ωz)-1i(ω/c-Ω·λ)z],Ωzz>0
Ua(ω)(Ωˆ, ρˆ)=(2π)-2dλ exp(-iλ·ρ)Ua(ω)(Ωˆ, z)=|ρˆ|-2 exp[-i(ω/c)|ρˆ|]δ2(Ωˆ-ρˆ/|ρˆ|),
a=1, 3.
I11(q+12+23)(Ωˆ, z|Ωˆ, z)
=I11(12)(Ωˆ, z|Ωˆ, z)+Ωz>0dΩˆ
×Ωz<0dΩˆU1(ω)(Ωˆ, z)σ12(12)(Ωˆ|Ωˆ)I22(q/12+23)
×(Ωˆ, z=0|Ωˆ, z=0)σ21(12)(Ωˆ|Ωˆ)
×U1(ω)(Ωˆ,-z),
I31(q+12+23)(Ωˆ, z|Ωˆ, z)
=Ωz<0dΩˆΩz<0dΩˆU3(ω)(Ωˆ, z+L)σ32(23)(Ωˆ|Ωˆ)×I22(q/12+23)(Ωˆ, z=-L|Ωˆ, z=0)σ21(12)(Ωˆ|Ωˆ)×U1(ω)(Ωˆ,-z).
σa2(12)(Ωˆ|z)=Ωz>0dΩˆσa2(12)(Ωˆ|Ωˆ)ϕA(Ωˆ, iz),
σ2a(12)(-z|Ωˆ)=Ωz<0dΩˆϕ¯A(Ωˆ,-iz)σ2a(12)(Ωˆ|Ωˆ),
I11(q+12+23)(Ωˆ, z|Ωˆ, z)
=I11(12)(Ωˆ, z|Ωˆ, z)+U1(ω)(Ωˆ, z)σ12(12)(Ωˆ|z)SA(12+23)
×[z(=0)|z(=0)]σ21(12)(-z|Ωˆ)U1(ω)(Ωˆ,-z),
I31(q+12+23)(Ωˆ, z|Ωˆ, z)
=U3(ω)(Ωˆ, z+L)σ32(23)(Ωˆ|z)SA(12+23)×[z(=-L)|z(=0)]σ21(12)(-z|Ωˆ)U1(ω)(Ωˆ,-z)
-DzSA(12+23)(z|z)=ZSA(12+23)(z|z),z=0,
σab(12)(Ωˆ|Ωˆ)=σba(12)(-Ωˆ|-Ωˆ),
γσa2(12)(Ωˆ|z)=4πσ2a(12)(-z|-Ωˆ).
σab(12)(Ωˆ|Ωˆ)=|Ωz(a)|Rab(12)(Ωˆ)2δ(Ωˆ(a)-Ωˆ(a))
=|Ωz(b)|Rba(12)(-Ωˆ)2δ[Ωˆ(b)-Ωˆ(b)],
σ12(12)(Ωˆ|z)=Ωz(2)R21(12)(-Ωˆ)2ϕA(Ωˆ, iz),
σ21(12)(-z|Ωˆ)=|Ωz(2)|R21(12)(+Ωˆ)2ϕ¯A(Ωˆ,-iz),
SA(12)(z|z)=γ(κλD+Z)-1φ(12)(z>)exp(κλz<),
κλ=[(γab+iω/c)D-1+λ2]1/2,Re(κλ)>0.
φ(12)(z)=cosh(κλz)-(Z/κλD)sinh(κλz)
SA(12+23)(z|z)=CAφ(12)(z>)φ(23)(z<).
φ(23)(z)=cosh[κλ(z+L)]+(Z/κλD)sinh[κλ(z+L)],
CA=γDφ(12) zφ(23)-φ(23) zφ(12)-1
SA(12+23)(z=0|z=-L)
=γ{Zφ(23)(z=0)+D(/z)φ(23)[z(=0)]}-1,
SA(12+23)(z|z)=SA(12+23)(z=0|z=-L)×φ(12)(z>)φ(23)(z<).
I21(q+12+23)(Ωˆ, z|Ωˆ, z)=ϕA(Ωˆ, iz)SA(12+23)[z|z(=0)]σ21(12)(-Z|Ωˆ)×U1(ω)(Ωˆ,-z),z<0,z>0,
I12(q+12+23)(Ωˆ, z|Ωˆ, z)=U1(ω)(Ωˆ, z)σ12(12)(Ωˆ|Z)SA(12+23)(z=0|z)×ϕ¯A(Ωˆ,-iz),z>0,z<0,
Iab()(Ωˆ, ρˆ|Ωˆ, ρˆ)=Iba()(-Ωˆ, ρˆ|-Ωˆ, ρˆ)
V(α)(Ωˆ|ρˆ|Ωˆ)=[σ(α)(Ωˆ|Ωˆ)-γ(α)δ(Ωˆ-Ωˆ)]δ(ρˆ-ρˆα).
dρˆdΩˆV(α)(Ωˆ|ρˆ|ρˆ)=dΩˆσ(α)(Ωˆ|Ωˆ)-γ(α)(Ωˆ)0,
V(α)(Ωˆ|ρˆ|Ωˆ)=σ(α)(Ωˆ|ρˆ|Ωˆ)-γ(α)(Ωˆ, ρˆ)δ(Ωˆ-Ωˆ).
SA(12+α)(ρˆ|ρˆ)=SA(12)(ρˆ|ρˆ)+SA(α/12)(ρˆ|ρˆ).
SA(α/12)(ρˆ|ρˆ)=SA(12)(ρˆ|ρˆα)V(α)(-α|α)ASA(12+α)(ρˆα|ρˆ)
V(α)(-ρ|ρ)A=dΩˆdΩˆϕ¯A(Ωˆ,-iρ)×V(α)(Ωˆ|Ωˆ)ϕA(Ωˆ, iρ).
I11(q+12+α)(Ωˆ, ρˆ|Ωˆ, ρˆ)=[I11(q+12)+I11(α/q+12)]×(Ωˆ, ρˆ|Ωˆ, ρˆ).
I11(α/q+12)(Ωˆ, ρˆ|Ωˆ, ρˆ)
=z=0dρz=0dρU1(ω)(Ωˆ, ρˆ-ρ)σ12(12)(Ωˆ|ρ)×SA(α/12)[ρˆ(z=0)|ρˆ(z=0)]σ21(12)(-ρ|Ωˆ)×U1(ω)(Ωˆ, ρ-ρˆ),
I31(α/q+12+23)(Ωˆ, ρˆ|Ωˆ, ρˆ)=z=-Ldρz=0dρU3(ω)(Ωˆ, ρˆ-ρ)σ32(23)(Ωˆ|ρ)×SA(α/12+23)[ρˆ(z=-L)|ρˆ(z=0)]×σ21(12)(-ρ|Ωˆ)U1(ω)(Ωˆ, ρ-ρˆ),
SA(12+23+α)(ρˆ|ρˆ)=SA(12+23)(ρˆ|ρˆ)+SA(α/12+23)(ρˆ|ρˆ),
SA(α/12+23)(ρˆ|ρˆ)=SA(12+23)(ρˆ|ρˆα)V(α)(-α|α)A×SA(12+23+α)(ρˆα|ρˆ),
Ua(ω)[Ωˆ(Ω=0)|ρˆ-ρˆ]=δ(ρ-ρ)exp[-i(ω/c)|z-z|],
a=1, 3,Ωz(z-z)>0,
γσ12(12)(Ωˆ|ρ)=4πσ21(12)(-ρ|Ωˆ)
=R21(12)2(1+3Z),
V(α)(Ωˆ|Ωˆ)=(4π)-1γ(α)-fγ(α)δ(Ωˆ-Ωˆ).
4πdΩˆV(α)(Ωˆ|Ωˆ)=4πdΩˆV(α)(Ωˆ|Ωˆ)=(1-f)γ(α)0.
V(α)(-ρ|ρ)A=γ-1γ(α)[1-f+3fD2ρ·ρ],
I11(α/q+12)(A1|A2)
=(4πγ)-1R21(12)4(1+3Z)2A1dρ
×A2dρSA(12)[ρˆ(z=0)|ρˆ(=ρˆα)]V(α)(-ρ|ρ)A
×SA(12+α)[ρˆ(=ρˆα)|ρˆ(z=0)],
ρ·ρ=Ωˆs·Ωˆs,
Ωˆs=ρˆαlog[SA(12)(ρˆ(z=0)|ρˆα)]
I31(α/q+12+23)(A1|A2)
=(4πγ)-1R21(12)2R23(23)2(1+3Z)2A1dρA2dρ×SA(12+23)[ρˆ(z=-L)|ρˆ(=ρˆα)]V(α)(-ρ|ρ)A×SA(12+23+α)[ρˆ(=ρˆα)|ρˆ(z=0)],
V(α)(-ρ|ρ)A=γ-1γ(α)3D2ρ·ρ+0perfectscatterer-1perfectabsorber.
V(α)ASA(12+α)=V(α/q)ASA(12)
V(α)(-ρ|ρˆ|ρ)AdΩˆdΩˆϕ¯A(Ωˆ,-iρ)×V(α)(Ωˆ|ρˆ|Ωˆ)ϕA(Ωˆ, iρ).
SA(12+α)=SA(12)+SA(12)V(α/q)ASa(12);
V(α)A(1+SA(12)V(α/q)A)=V(α/q)A,
V(α/q)A=[1-V(α)ASA(12)]-1V(α)A
=V(α)A+V(α)ASA(12)V(α)A+,
IdI(q)=U(ω)S(q)U(ω),
I=U(ω)+U(ω)S(q)U(ω).
S(q)=K(q)[1+U(ω)S(q)],
S(q)U(ω)=K(q)I,U(ω)S(q)=IK(q),
S(q)=K(q)+K(q)IK(q).
I(12)=U(ω)+U(ω)σ(12)U(ω)
Iab(12)(Ωˆ, z|Ωˆ, z)
=Ua(ω)(Ωˆ, z-z)δ(Ωˆ-Ωˆ)δab+Ua(ω)(Ωˆ, z)
×σab(12)(Ωˆ|Ωˆ)Ub(ω)(Ωˆ,-z),
a=12kadΩˆσab(12)(Ωˆ|Ωˆ)=kb|Ωz(b)|.
I(q+12)=I(12)+I(12)S(q/12)I(12).
S(q/12)=K(q)[1+I(12)S(q/12)].
S(q/12)=S(q)[1+U(ω)σ(12)U(ω)S(q/12)].
I(q/12)=U(ω)S(q/12)U(ω),
I(q/12)=I(q)[1+σ(12)I(q/2)]
I(q+12)=I(12)+[1+U(ω)σ(12)]I(q/12)[σ(12)U(ω)+1].
I11(q+12)=I11(12)+U1(ω)σ12(12)I22(q/12)σ21(12)U1(ω),
I22(q/12)(Ωˆ, z|Ωˆ, z)=ϕA(Ωˆ, iz)SA(12)(z|z)ϕ¯A(Ωˆ,-iz).
-DzSA(12)[z(=0)|z]=ZSA(12)[z(=0)|z].
-ϕA-(z)SA(12)[z(=0)|z]=σ22(12)(z)SA(12)[z(=0)|z],
ϕA±(z)=(4π)-1Ωz0dΩˆΩzϕA(Ωˆ, iz)
=(2γ)-1(±2-1-Dz),
σ22(12)(z)=(4π)-1Ωz<0dΩˆΩz>0dΩˆ×σ22(12)(Ωˆ|Ωˆ)ϕA(Ωˆ, iz).
Z=12-12πΩz<0dΩˆΩz>0dΩˆσ22(12)(Ωˆ|Ωˆ)×1+32πΩz<0dΩˆΩz>0dΩˆσ22(12)(Ωˆ|Ωˆ)Ωz-1.
Z=2-11-(2π)-1dΩˆ2Ωz(2)R22(12)(Ωˆ)21+(2π)-1dΩˆ3(Ωz(2))2R22(12)(Ωˆ)2.
-ϕA-(z)SA(12)[z(=0)|z]=ReffϕA+(z)SA(12)[z(=0)|z]
Z=2-1(1-Reff)/(1+Reff).
Iab(12+23)=Ua(ω)δab+Ua(ω)σab(12+23)Ub(ω),
σ(12+23)σ(12)+σ(23),
I(q+12+23)=I(12+23)+[1+U(ω)σ(12+23)]×I(q/12+23)[σ(12+23)U(ω)+1].
I(q/12+23)=I(q){1+[σ(12)+σ(23)]I(q/12+23)},
I22(q/12+23)(Ωˆ, z|Ωˆ, z)=ϕA(Ωˆ, iz)SA(12+23)(z|z)×ϕ¯A(Ωˆ,-iz),
I11(q+12+23)=I11(12)+U1(ω)σ12(12)I22(q/12+23)σ21(12)U1(ω),
I31(q+12+23)=U3(ω)σ32(23)I22(q/12+23)σ21(12)U1(ω),
I(q/12+23)=I(q/12)[1+σ(23)I(q/12+23)].
SA(12+23)(z|z)=SA(12)(z|z)+SA(12)(z|z)×σ22(23)(-z|z)ASA(12+23)(z|z)|z=-L.
σ22(23)(-z|z)A=dΩˆdΩˆϕ¯A(Ωˆ,-iz)×σ22(23)(Ωˆ|Ωˆ)ϕA(Ωˆ, iz),
I(q/12)=I(q)+I(q)σ(12/q)I(q),
σ(12/q)=[1-σ(12)I(q)]-1σ(12).
I˜d(Ωˆ, λˆ|t|Ωˆ, λˆ)=dρˆdρˆ exp[i(λˆ·ρˆ-λˆ·ρˆ)]Id(Ωˆ, ρˆ|t|Ωˆ, ρˆ),
(γ-1t+1)I˜d(Ωˆ, λˆ|t|Ωˆ, λˆ)
=U˜(Ωˆ, λˆ)[K(q)(Ωˆ|Ωˆ)U˜(Ωˆ, λˆ)(2π)3δ(λˆ-λˆ)δ(ct)+dΩˆK(q)(Ωˆ|Ωˆ)I˜d(Ωˆ, λˆ|t|Ωˆ, λˆ)].
U˜(Ωˆ, λˆ)=(γ-iΩˆ·λˆ)-1.
dΩˆK(q)(Ωˆ|Ωˆ)U˜(Ωˆ, λˆ)fA(Ωˆ, λˆ)=A(λˆ)fA(Ωˆ, λˆ),
dΩˆf¯A(Ωˆ, λˆ)U˜(Ωˆ, λˆ)K(q)(Ωˆ|Ωˆ)=A(λˆ)f¯A(Ωˆ, λˆ),
dΩˆf¯A(Ωˆ, λˆ)U˜(Ωˆ, λˆ)fB(Ωˆ, λˆ)=δAB.
K(q)(Ωˆ|Ωˆ)=AA(λˆ)fA(Ωˆ, λˆ)f¯A(Ωˆ, λˆ).
ϕA(Ωˆ, λˆ)=U˜(Ωˆ, λˆ)fA(Ωˆ, λˆ),
ϕ¯A(Ωˆ, λˆ)=f¯A(Ωˆ, λˆ)U˜(Ωˆ, λˆ),
dΩˆϕ¯A(Ωˆ, λˆ)fB(Ωˆ, λˆ)=dΩˆf¯A(Ωˆ, λˆ)ϕB(Ωˆ, λˆ)=δAB,
I˜d(Ωˆ, λˆ|t|Ωˆ, λˆ)=AϕA(Ωˆ, λˆ)S˜A(λˆ|t|λˆ)ϕ¯A(Ωˆ, λˆ).
[γ-1t+1-A(λˆ)]S˜A(λˆ|t|λˆ)
=A(λˆ)(2π)3δ(λˆ-λˆ)δ(ct)
[γ-1t+1-A(i/ρˆ)]SA(ρˆ|t|ρˆ)
=A(i/ρˆ)δ(ρˆ-ρˆ)δ(ct).
U˜(Ωˆ, λˆ)δ(Ωˆ-Ωˆ)=AϕA(Ωˆ, λˆ)ϕ¯A(Ωˆ, λˆ),
1-A(λˆ)=γ-1Dλˆ2,
D=(3γ)-1(1-a1)-1,
a1=γ-1dΩˆ(Ωˆ·Ωˆ)K(q)(Ωˆ·Ωˆ).
γ-1[t-D(/ρˆ)2]SA(ρˆ|t|ρˆ)=δ(ρˆ-ρˆ)δ(ct).
ϕA(Ωˆ, λˆ)=γ-1(1+i3DΩˆ·λˆ),
ϕ¯A(Ωˆ, λˆ)=(4π)-1(1+i3DΩˆ·λˆ),
Id(Ωˆ, ρˆ|t|Ωˆ, ρˆ)
=ϕA(Ωˆ, i/ρˆ)SA(ρˆ|t|ρˆ)ϕ¯A(Ωˆ,-i/ρˆ).
I22(q/12)(Ωˆ, ρˆ|Ωˆ, ρˆ)
=ϕA(Ωˆ, i/ρˆ)SA(12)(ρˆ|ρˆ)ϕ¯A(Ωˆ,-i/ρˆ),
SA(12)(ρˆ|ρˆ)=(2π)-2dλ exp[-iλ·(ρ-ρ)]×γ(κλD+Z)-1φ(12)(z>)exp(κλz<).
κλ=(κ2+λ2)1/2,κ=[(γab+iω/c)D-1]1/2.
SA(ρˆ)=(2π)-2dλγ(2Dκλ)-1 exp(-iλ·ρ-κλ|z|)
=(γ/D)|4πρˆ|-1 exp(-κ|ρˆ|)
γ(κλD+Z)-1=(γ/D)[κλ-1-Zκλ-1(Z+κλD)-1].
SA(12)(ρˆ|ρˆ)=2{×SA(ρˆ-ρˆ)-Zγ(2π)-2×dλ(2Dκλ)-1(Z+κλD)-1×exp[-iλ·(ρ-ρ)+κλz<]}.
(Z+κλD)-1=0dx exp[-(Z+κλD)x],
SA(12)(ρˆ|ρˆ)=2[SA(ρˆ-ρˆ)-Z×0dx exp(-Zx)SA(ρ-ρ, Dx-z<)],
SA(12)(ρˆ|ρˆ)=-20dx exp(-Zx) x×SA(ρ-ρ, Dx-z<)=2D0dx exp(-Zx) z<×SA(ρ-ρ, Dx-z<),

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