Abstract

A free-path-length distribution function (FPDF) of multiply backscattered light is theoretically derived for a fractal aggregate of particles. An effective mean-free path-length l D is newly introduced as a measure of randomness analogous with a homogeneously random medium. We confirm the validity of the FPDF by demonstrating agreement between the dimensions designed for a particle distribution generated by a random walk based on the derived FPDF and estimated by the radius of gyration method. The FPDF is applied to Monte Carlo simulations for copolarized multiply backscattered light from the fractal aggregate of particles. It is shown that a copolarized intensity peak of enhanced backscattering in the far field decreases in accordance with θ2-D and has an angular width of λ/l D. This spatial feature of the backscattering enhancement corresponds to that of the copolarized intensity peak produced from a homogeneously random medium with a dimension of D = 3. As a result, the validity of the model for the fractal structure of particle aggregates and the applicability of the derived FPDF are confirmed by the numerical results.

© 1998 Optical Society of America

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References

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  1. D. W. Schaefer, J. E. Martin, “Fractal geometry of colloidal aggregates,” Phys. Rev. Lett. 52, 2371–2374 (1984).
    [CrossRef]
  2. J. E. Martin, B. J. Ackerson, “Static and dynamic scattering from fractals,” Phys. Rev. A 31, 1180–1182 (1985).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. C. M. Sorensen, J. Cai, N. Lu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
    [CrossRef] [PubMed]
  5. Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
    [CrossRef] [PubMed]
  6. A. Dogariu, J. Uozumi, T. Asakura, “Enhancement of backscattered intensity from fractal aggregates,” Waves Random Media 2, 259–263 (1992).
    [CrossRef]
  7. A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
    [CrossRef]
  8. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. J. Feder, Fractal (Plenum, New York, 1988), pp. 31–40.
  15. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), pp. 22–25.
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    [CrossRef]
  17. M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
    [CrossRef] [PubMed]

1997 (1)

K. Ishii, T. Iwai, T. Asakura, “Angular polarization properties of dynamic light scattering under the influence of enhanced backscattering,” Opt. Commun. 140, 99–109 (1997).
[CrossRef]

1995 (1)

T. Iwai, H. Furukawa, T. Asakura, “Numerical analysis on enhanced backscatterings of light based on Rayleigh–Debye scattering theory,” Opt. Rev. 2, 413–419 (1995).
[CrossRef]

1993 (1)

A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

1992 (2)

1991 (1)

1988 (2)

J. Teixeira, “Small-angle scattering by fractal systems,” J. Appl. Crystallogr. 21, 781–785 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

1987 (4)

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent backscattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

D. Schmeltzer, M. Kaveh, “Back-scattering of electromagnetic waves in random dielectric media,” J. Phys. C 20, L175–L179 (1987).
[CrossRef]

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

1985 (1)

J. E. Martin, B. J. Ackerson, “Static and dynamic scattering from fractals,” Phys. Rev. A 31, 1180–1182 (1985).
[CrossRef] [PubMed]

1984 (1)

D. W. Schaefer, J. E. Martin, “Fractal geometry of colloidal aggregates,” Phys. Rev. Lett. 52, 2371–2374 (1984).
[CrossRef]

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Ackerson, B. J.

J. E. Martin, B. J. Ackerson, “Static and dynamic scattering from fractals,” Phys. Rev. A 31, 1180–1182 (1985).
[CrossRef] [PubMed]

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Asakura, T.

K. Ishii, T. Iwai, T. Asakura, “Angular polarization properties of dynamic light scattering under the influence of enhanced backscattering,” Opt. Commun. 140, 99–109 (1997).
[CrossRef]

T. Iwai, H. Furukawa, T. Asakura, “Numerical analysis on enhanced backscatterings of light based on Rayleigh–Debye scattering theory,” Opt. Rev. 2, 413–419 (1995).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Enhancement of backscattered intensity from fractal aggregates,” Waves Random Media 2, 259–263 (1992).
[CrossRef]

Cai, J.

Chen, Z.-Y.

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

Dogariu, A.

A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Enhancement of backscattered intensity from fractal aggregates,” Waves Random Media 2, 259–263 (1992).
[CrossRef]

Edrei, I.

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

Feder, J.

J. Feder, Fractal (Plenum, New York, 1988), pp. 31–40.

Freund, I.

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

Furukawa, H.

T. Iwai, H. Furukawa, T. Asakura, “Numerical analysis on enhanced backscatterings of light based on Rayleigh–Debye scattering theory,” Opt. Rev. 2, 413–419 (1995).
[CrossRef]

Gelbart, W. M.

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

Hasegawa, Y.

Ishii, K.

K. Ishii, T. Iwai, T. Asakura, “Angular polarization properties of dynamic light scattering under the influence of enhanced backscattering,” Opt. Commun. 140, 99–109 (1997).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), pp. 22–25.

Iwai, T.

K. Ishii, T. Iwai, T. Asakura, “Angular polarization properties of dynamic light scattering under the influence of enhanced backscattering,” Opt. Commun. 140, 99–109 (1997).
[CrossRef]

T. Iwai, H. Furukawa, T. Asakura, “Numerical analysis on enhanced backscatterings of light based on Rayleigh–Debye scattering theory,” Opt. Rev. 2, 413–419 (1995).
[CrossRef]

Kaveh, M.

D. Schmeltzer, M. Kaveh, “Back-scattering of electromagnetic waves in random dielectric media,” J. Phys. C 20, L175–L179 (1987).
[CrossRef]

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

Lagendijk, A.

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent backscattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

Lu, N.

Maret, G.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Martin, J. E.

J. E. Martin, B. J. Ackerson, “Static and dynamic scattering from fractals,” Phys. Rev. A 31, 1180–1182 (1985).
[CrossRef] [PubMed]

D. W. Schaefer, J. E. Martin, “Fractal geometry of colloidal aggregates,” Phys. Rev. Lett. 52, 2371–2374 (1984).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Meakin, P.

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

Nomura, Y.

Rosenbluh, M.

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

Schaefer, D. W.

D. W. Schaefer, J. E. Martin, “Fractal geometry of colloidal aggregates,” Phys. Rev. Lett. 52, 2371–2374 (1984).
[CrossRef]

Schmeltzer, D.

D. Schmeltzer, M. Kaveh, “Back-scattering of electromagnetic waves in random dielectric media,” J. Phys. C 20, L175–L179 (1987).
[CrossRef]

Sorensen, C. M.

Tamura, M.

Teixeira, J.

J. Teixeira, “Small-angle scattering by fractal systems,” J. Appl. Crystallogr. 21, 781–785 (1988).
[CrossRef]

Uozumi, J.

A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Enhancement of backscattered intensity from fractal aggregates,” Waves Random Media 2, 259–263 (1992).
[CrossRef]

van Albada, M. P.

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent backscattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

Weaklien, P.

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Yamada, Y.

Appl. Opt. (2)

J. Appl. Crystallogr. (1)

J. Teixeira, “Small-angle scattering by fractal systems,” J. Appl. Crystallogr. 21, 781–785 (1988).
[CrossRef]

J. Phys. (Paris) (1)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

J. Phys. C (1)

D. Schmeltzer, M. Kaveh, “Back-scattering of electromagnetic waves in random dielectric media,” J. Phys. C 20, L175–L179 (1987).
[CrossRef]

Med. Phys. (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Opt. Commun. (1)

K. Ishii, T. Iwai, T. Asakura, “Angular polarization properties of dynamic light scattering under the influence of enhanced backscattering,” Opt. Commun. 140, 99–109 (1997).
[CrossRef]

Opt. Rev. (1)

T. Iwai, H. Furukawa, T. Asakura, “Numerical analysis on enhanced backscatterings of light based on Rayleigh–Debye scattering theory,” Opt. Rev. 2, 413–419 (1995).
[CrossRef]

Phys. Rev. A (2)

M. Rosenbluh, I. Edrei, M. Kaveh, I. Freund, “Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories,” Phys. Rev. A 35, 4458–4460 (1987).
[CrossRef] [PubMed]

J. E. Martin, B. J. Ackerson, “Static and dynamic scattering from fractals,” Phys. Rev. A 31, 1180–1182 (1985).
[CrossRef] [PubMed]

Phys. Rev. B (1)

M. P. van Albada, A. Lagendijk, “Vector character of light in weak localization: spatial anisotropy in coherent backscattering from a random medium,” Phys. Rev. B 36, 2353–2356 (1987).
[CrossRef]

Phys. Rev. Lett. (2)

D. W. Schaefer, J. E. Martin, “Fractal geometry of colloidal aggregates,” Phys. Rev. Lett. 52, 2371–2374 (1984).
[CrossRef]

Z.-Y. Chen, P. Weaklien, W. M. Gelbart, P. Meakin, “Second-order light scattering and local anisotropy of diffusion-limited aggregates and bond-percolation clusters,” Phys. Rev. Lett. 58, 1996–1999 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

A. Dogariu, J. Uozumi, T. Asakura, “Source of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Waves Random Media (1)

A. Dogariu, J. Uozumi, T. Asakura, “Enhancement of backscattered intensity from fractal aggregates,” Waves Random Media 2, 259–263 (1992).
[CrossRef]

Other (2)

J. Feder, Fractal (Plenum, New York, 1988), pp. 31–40.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), pp. 22–25.

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Figures (6)

Fig. 1
Fig. 1

FPDF’s for the fractal aggregates of particles with dimensions of D = 2.5 and 2.8 and a homogeneously random medium with D = 3.0. The horizontal and vertical axes are plotted on a logarithmic scale and the horizontal axis is normalized by the effective mean-free path length l D .

Fig. 2
Fig. 2

Particle distributions for (a) a homogeneously random medium with D = 3.0 and (b) a fractal medium with D = 2.5. The plots are from the results obtained by a random walk whose free-path length is based on Eq. (5).

Fig. 3
Fig. 3

Logarithmic plot of the particle number in a sphere as a function of radius r also given on a logarithmic scale. The solid lines with slopes of 2.5 and 3.0 are for reference. ▲ and ■ denote the results for the media with dimensions of 2.5 and 3.0, respectively.

Fig. 4
Fig. 4

Copolarized angular intensity peak produced in the far field for three dimensions of D = 2.5, 2.8, and 3.0, which were scanned along the Y direction perpendicular to the incident polarization. The vertical and horizontal axes denote the logarithms of the coherent intensity I - 1 and the backscattered angle θ, respectively. The curves with slopes of -0.5, -0.8, -1, and -2 are for reference.

Fig. 5
Fig. 5

Angular widths of the copolarized intensity peak of multiply backscattered light as a function of expected width λ/l D for five different aggregates of particles with dimensions of D = 2.6, 2.7, 2.8, 2.9, and 3.0.

Fig. 6
Fig. 6

Slope of the copolarized intensity peak of multiply backscattered light from five different aggregates of particles with dimensions of D = 2.6, 2.7, 2.8, 2.9, and 3.0. The slopes fit the angular intensity distribution in the region around the angle at which the intensity reduces to one fourth of the peak intensity. The solid line denotes the line that follows 2 - D.

Equations (8)

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

p r d r = 1 - 0 r   p r d r C sca ρ r d r .
N D r = K r a D ,
K = N 3 R R / a D ,
ρ r = 1 4 π r 2 d d r   N D r = KD 4 π a 3 r a D - 3 .
p r = D - 2 l D r l D D - 3 exp - r l D D - 2
l D = a 4 D - 2 KDQ sca 1 / D - 2
I X ex = n = 1   I nX ex ,
I nX = p 1 p 1 + 2   n = 2   p n F nX n = 1 2 p n F nX 1 + cos   Φ n p 1 + 2   n = 2   p n F nX n 1 ,

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