Abstract

We present a boundary-element-method numerical procedure that can be used to solve for the diffusion equation of the field autocorrelation function in any arbitrary geometry with various boundary and source properties. We use this numerical method to study finite-sized effects in a circular slab and the influence of the angle in a cone-plate geometry. The latter is also compared with exact analytical solutions obtained for an equivalent bidimensional geometry. In most cases the deviation from well-known predictions of the correlation function remains small.

© 2001 Optical Society of America

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References

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  1. D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.
  2. G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
    [CrossRef]
  3. P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
    [CrossRef]
  4. R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
    [CrossRef]
  5. D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
    [CrossRef] [PubMed]
  6. D. A. Boas, A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
    [CrossRef]
  7. M. A. Celia, W. G. Gray, Numerical Methods for Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1992).
  8. C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  9. C. A. Brebbia, J. Dominguez, Boundary Elements. An Introductory Course (Computational Mechanics, Southampton, UK, 1989).
  10. I. Stakgold, Boundary Value Problems of Mathematical Physics (Macmillan, New York, 1968), Vol. 2.
  11. P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy,” Appl. Opt. 32, 3828–3836 (1993).
    [PubMed]
  12. SKY/Mesh2, Version 2.34 (Skyblue Systems, Albany, N.Y.).
  13. P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics, Southampton, UK, 1992).

1998

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

1997

1995

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

1994

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

1993

Ackerson, B. J.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Boas, D. A.

D. A. Boas, A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Brebbia, C. A.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984).
[CrossRef]

C. A. Brebbia, J. Dominguez, Boundary Elements. An Introductory Course (Computational Mechanics, Southampton, UK, 1989).

P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics, Southampton, UK, 1992).

Campbell, L. E.

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Celia, M. A.

M. A. Celia, W. G. Gray, Numerical Methods for Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1992).

Dominguez, J.

C. A. Brebbia, J. Dominguez, Boundary Elements. An Introductory Course (Computational Mechanics, Southampton, UK, 1989).

Dorr-Nowkoorani, F.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Dougherty, R. L.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Durian, D. J.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Gray, W. G.

M. A. Celia, W. G. Gray, Numerical Methods for Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1992).

Kao, M. H.

Kaplan, P. D.

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Maret, G.

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

Nobbmann, U.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Partridge, P. W.

P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics, Southampton, UK, 1992).

Pine, D. J.

P. D. Kaplan, M. H. Kao, A. G. Yodh, D. J. Pine, “Geometric constraints for the design of diffusing-wave spectroscopy,” Appl. Opt. 32, 3828–3836 (1993).
[PubMed]

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

Reguigui, N. M.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Stakgold, I.

I. Stakgold, Boundary Value Problems of Mathematical Physics (Macmillan, New York, 1968), Vol. 2.

Telles, J. C. F.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984).
[CrossRef]

Vera, M. U.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Weitz, D. A.

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

Wrobel, L. C.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984).
[CrossRef]

P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics, Southampton, UK, 1992).

Yodh, A. G.

Appl. Opt.

Curr. Opin. Colloid Interface Sci.

G. Maret, “Diffusing-wave spectroscopy,” Curr. Opin. Colloid Interface Sci. 2, 251–257 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorr-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Phys. Rev. E

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Phys. Rev. Lett.

D. A. Boas, L. E. Campbell, A. G. Yodh, “Scattering and imaging with diffusing temporal field correlation functions,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Other

M. A. Celia, W. G. Gray, Numerical Methods for Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1992).

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, Berlin, 1984).
[CrossRef]

C. A. Brebbia, J. Dominguez, Boundary Elements. An Introductory Course (Computational Mechanics, Southampton, UK, 1989).

I. Stakgold, Boundary Value Problems of Mathematical Physics (Macmillan, New York, 1968), Vol. 2.

SKY/Mesh2, Version 2.34 (Skyblue Systems, Albany, N.Y.).

P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics, Southampton, UK, 1992).

D. A. Weitz, D. J. Pine, in Dynamic Light Scattering: The Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

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Figures (4)

Fig. 1
Fig. 1

(a) Correlation functions of the 2-D wedge and the 2-D slab for transmitted light for point-source illumination and point detection as obtained from analytical solutions of the 2-D diffusion equation. θ is the wedge angle, L is the sample thickness at the detection point, and K 2= (1/l* 2)[3l*/l a + k 0 2〈Δr 2(τ)〉], according to Eq. (4). (b) Corresponding effective transport mean free path plotted as a function of wedge angle. Note that, for wedge angles less than 10°, the 2-D slab result holds to within approximately 0.1%.

Fig. 2
Fig. 2

(a) Circular-slab correlation functions for transmitted light for plane-source illumination and point detection as obtained with the BEM. D is the slab diameter, L is the slab thickness, and K 2 = (1/l* 2)[3l*/l a + k 0 2〈Δr 2(τ)〉], according to Eq. (4). The theoretical solution for an infinite slab is also shown [Eq. (2)]. (b) Corresponding effective transport mean free path plotted as a function of the aspect ratio D/ L.

Fig. 3
Fig. 3

Example of the mesh used in the BEM for the cone-plate geometry. Note that the grid spacing is smallest near the illumination and the detection points. The mesh around the side of the sample has been suppressed for clarity.

Fig. 4
Fig. 4

(a) Cone-plate correlation functions for transmitted light for plane-source illumination and point detection as obtained with the BEM. θ is the cone angle, L is the thickness of the sample at the detection point, and K 2 = (1/l* 2)[3l*/l a + k 0 2〈Δr 2(τ)〉], according to Eq. (4). The theoretical solution for an infinite slab is also shown [Eq. (2)]. (b) Corresponding effective transport mean free path plotted as a function of the cone angle. Note that for cone angles less than 10° the planar-slab result holds to within approximately 1%. We also plot the estimate [Eq. (14)] of the effective transport mean free path: l eff*/l eff* = 1/(1 + 1/3 tan2 θ).

Equations (19)

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g1τ=0 Pnasexp-sla-s3l* k02 Δr2τds,
g1x=sinhzpx+zexcoshzpxTzp1+ze2xsinhLx+2zexcoshLx,
2+K2τG1r, τ=-3 Ql* δ3r-rs,
K2τ=1l*23l*la+k02Δr2τ.
zel*n·+1G1r=0,
ΩgHw-wHgdΩ=Γg wn-w gndΓ,
gri=wrirs+Γwrign-g wrindΓ.
wrir=exp-K|r-ri|4π|r-ri|.
gr=j=1p gjΓiϕjΓir,gnr=j=1pgnjΓiϕjΓir.
limε0Γε wrigndΓε=0,limε0Γε g wrindΓε=-12 gri.
12 gi=wrirs+k=1NgnkΓk wridΓk-gkΓkwrindΓk.
zel*gni+gi=0.
1rrr gr+1r22gφ2-K2g=-δr-rsδφ-φsr.
gr, φ=m=12θsinmπφθsinmπφsθ Kmπ/θKr>×Imπ/θKr<,
gx, y=m=1n=14ab×sinmπx/asinmπxs/asinnπy/bsinnπys/bK2+m2π2/a2+n2π2/b2,
Vc=4LΔR21+13tan2 θΔR/L2.
Vs=4LΔR2.
Leff=L1+13tan2 θΔR/L2.
leff*=linf*1+13tan2 θ,

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