Abstract

We propose an effective-medium theory for random aggregates of small spherical particles that accounts for the finite size of the embedding volume. The technique is based on the identification of the first two orders of the Born series within a finite volume for the coherent field and the effective field. Although the convergence of the Born series requires a finite volume, the effective constants that are derived through this identification are shown to admit of a large-scale limit. With this approach we recover successively, and in a simple manner, some classical homogenization formulas: the Maxwell Garnett mixing rule, the effective-field approximation, and a finite-size correction to the quasi-crystalline approximation (QCA). The last formula is shown to coincide with the usual low-frequency QCA in the limit of large volumes, while bringing substantial improvements when the dimension of the embedding medium is of the order of the probing wavelength. An application to composite spheres is discussed.

© 2006 Optical Society of America

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    [CrossRef]
  2. A. Sihvola, Electromagnetic Mixing Formulas and Applications, IEE Electromagnetic Waves Series (The Institute of Electrical Engineers, 1999).
    [CrossRef]
  3. L. Foldy, "The multiple scattering of waves," Phys. Rev. 107, 107-119 (1945).
    [CrossRef]
  4. M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
    [CrossRef]
  5. M. Lax, "Multiple scattering of waves. II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
    [CrossRef]
  6. V. Varadan, V. Bringi, and V. Varadan, "Coherent electromagnetic wave propagation though randomly distributed dielectric scatterers," Phys. Rev. D 19, 2480-2489 (1979).
    [CrossRef]
  7. L. Tsang and J. A. Kong, "Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formalism," J. Appl. Phys. 51, 3465-3485 (1980).
    [CrossRef]
  8. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).
  9. U. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A.T.Barucha-Reid, ed. (Academic, 1953), Vol. 1, pp. 75-198.
  10. A. Doicu and T. Wriedt, "Equivalent refractive index of a sphere with multiple spherical inclusions," J. Opt. A, Pure Appl. Opt. 3, 204-209 (2001).
    [CrossRef]
  11. K. Fuller, "Scattering and absorption cross sections of compounded spheres III. Spheres containing arbitrarily located spherical inhomogeneities," J. Opt. Soc. Am. A 12, 893-904 (1995).
    [CrossRef]
  12. G. Videen, D. Ngo, P. Chýlek, and R. Pinnick, "Light scattering from a sphere with an irregular inclusion," J. Opt. Soc. Am. A 12, 922-928 (1995).
    [CrossRef]
  13. G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).
  14. P. Chýlek and G. Videen, "Scattering by a composite sphere and effective medium approximations," Opt. Commun. 146, 15-20 (1998).
    [CrossRef]
  15. P. Chýlek, G. Videen, and D. Ngo, "Effect of air bubbles on absorption of solar radiation by water droplets," J. Atmos. Sci. 55, 340-343 (1998).
    [CrossRef]
  16. K. H. Ding, L. M. Zurk, and L. Tsang, "Pair distribution functions and attenuation rates for sticky particles in dense media," J. Electromagn. Waves Appl. 8, 1585-1604 (1994).
  17. L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, "Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries," J. Opt. Soc. Am. A 12, 1772-1781 (1995).
    [CrossRef]
  18. L. Tsang, K. H. Ding, S. Shih, and J. Kong, "Scattering of electromagnetic wave from dense distributions of spheroidal particles based on Monte Carlo simulations," J. Opt. Soc. Am. A 15, 2660-2669 (1998).
    [CrossRef]
  19. B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).
  20. C. Ao and J. Kong, "Analytical approximations in multiple scattering of electromagnetic waves by aligned dielectric spheroids," J. Opt. Soc. Am. A 19, 1145-1156 (2002).
    [CrossRef]
  21. A. Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).
    [CrossRef]
  22. G. Smith, "Dielectric constant for mixed media," J. Phys. D 10, L39-42 (1977).
    [CrossRef]
  23. D. Stroud and F. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular theories," Phys. Rev. B 17, 1602 (1978).
    [CrossRef]
  24. G. Niklasson, C. Granqvist, and O. Hunderi, "Effective medium models for the optical properties of inhomogeneous materials," Appl. Opt. 20, 26-30 (1981).
    [CrossRef] [PubMed]
  25. P. Chýlek and V. Srivastava, "Dielectric constant of a composite inhomogeneous medium," Phys. Rev. B 27, 5098 (1983).
    [CrossRef]
  26. P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
    [CrossRef]
  27. C. Tai, Dyadic Green's Functions in Electromagnetic theory (Intext, 1971).
  28. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
    [CrossRef]
  29. L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
    [CrossRef]
  30. J. Percus and G. Yevick, "Analysis of classical statistical mechanics by means of collective coordinates," Phys. Rev. 110, 1-13 (1958).
    [CrossRef]
  31. M. Wertheim, "Exact solution of the Percus-Yevick integral equation for hard spheres," Phys. Rev. Lett. 10, 321-323 (1963).
    [CrossRef]
  32. R. Baxter, "Method of solution of the Percus-Yevick, hypernetted-chain or similar equation," Phys. Rev. 154, 170-175 (1967).
    [CrossRef]
  33. Y. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995).
    [CrossRef] [PubMed]
  34. B. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  35. W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
    [CrossRef]
  36. A. Lakhtakia, "Size-dependent Maxwell-Garnett formula from an integral equation formalism," Optik (Stuttgart) 91, 134-137 (1992).
  37. G. Videen and P. Chýlek, "Scattering by a composite sphere with an absorbing inclusion and effective medium approximations," Opt. Commun. 158, 1-6 (1998).
    [CrossRef]
  38. L. Kolokolova and B. A. Gustafson, "Scattering by inhomogeneous particles: microwave analog experiments and comparison to effective medium theories," J. Quant. Spectrosc. Radiat. Transf. 70, 611-625 (2001).
    [CrossRef]
  39. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  40. R. West, D. Gibbs, L. Tsang, and A. Fung, "Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media," J. Opt. Soc. Am. A 11, 1854-1857 (1994).
    [CrossRef]

2005 (1)

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

2002 (1)

2001 (2)

A. Doicu and T. Wriedt, "Equivalent refractive index of a sphere with multiple spherical inclusions," J. Opt. A, Pure Appl. Opt. 3, 204-209 (2001).
[CrossRef]

L. Kolokolova and B. A. Gustafson, "Scattering by inhomogeneous particles: microwave analog experiments and comparison to effective medium theories," J. Quant. Spectrosc. Radiat. Transf. 70, 611-625 (2001).
[CrossRef]

2000 (1)

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

1998 (4)

L. Tsang, K. H. Ding, S. Shih, and J. Kong, "Scattering of electromagnetic wave from dense distributions of spheroidal particles based on Monte Carlo simulations," J. Opt. Soc. Am. A 15, 2660-2669 (1998).
[CrossRef]

P. Chýlek and G. Videen, "Scattering by a composite sphere and effective medium approximations," Opt. Commun. 146, 15-20 (1998).
[CrossRef]

P. Chýlek, G. Videen, and D. Ngo, "Effect of air bubbles on absorption of solar radiation by water droplets," J. Atmos. Sci. 55, 340-343 (1998).
[CrossRef]

G. Videen and P. Chýlek, "Scattering by a composite sphere with an absorbing inclusion and effective medium approximations," Opt. Commun. 158, 1-6 (1998).
[CrossRef]

1997 (1)

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

1995 (4)

1994 (2)

R. West, D. Gibbs, L. Tsang, and A. Fung, "Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media," J. Opt. Soc. Am. A 11, 1854-1857 (1994).
[CrossRef]

K. H. Ding, L. M. Zurk, and L. Tsang, "Pair distribution functions and attenuation rates for sticky particles in dense media," J. Electromagn. Waves Appl. 8, 1585-1604 (1994).

1992 (1)

A. Lakhtakia, "Size-dependent Maxwell-Garnett formula from an integral equation formalism," Optik (Stuttgart) 91, 134-137 (1992).

1989 (1)

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

1988 (1)

B. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1983 (1)

P. Chýlek and V. Srivastava, "Dielectric constant of a composite inhomogeneous medium," Phys. Rev. B 27, 5098 (1983).
[CrossRef]

1981 (1)

1980 (2)

A. Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).
[CrossRef]

L. Tsang and J. A. Kong, "Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formalism," J. Appl. Phys. 51, 3465-3485 (1980).
[CrossRef]

1979 (1)

V. Varadan, V. Bringi, and V. Varadan, "Coherent electromagnetic wave propagation though randomly distributed dielectric scatterers," Phys. Rev. D 19, 2480-2489 (1979).
[CrossRef]

1978 (1)

D. Stroud and F. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular theories," Phys. Rev. B 17, 1602 (1978).
[CrossRef]

1977 (1)

G. Smith, "Dielectric constant for mixed media," J. Phys. D 10, L39-42 (1977).
[CrossRef]

1967 (1)

R. Baxter, "Method of solution of the Percus-Yevick, hypernetted-chain or similar equation," Phys. Rev. 154, 170-175 (1967).
[CrossRef]

1963 (1)

M. Wertheim, "Exact solution of the Percus-Yevick integral equation for hard spheres," Phys. Rev. Lett. 10, 321-323 (1963).
[CrossRef]

1958 (1)

J. Percus and G. Yevick, "Analysis of classical statistical mechanics by means of collective coordinates," Phys. Rev. 110, 1-13 (1958).
[CrossRef]

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

1952 (1)

M. Lax, "Multiple scattering of waves. II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
[CrossRef]

1951 (1)

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

1945 (1)

L. Foldy, "The multiple scattering of waves," Phys. Rev. 107, 107-119 (1945).
[CrossRef]

1904 (1)

J. Maxwell Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London 203, 385 (1904).
[CrossRef]

Ao, C.

Ao, C. O.

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Barrowes, B. E.

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

Baxter, R.

R. Baxter, "Method of solution of the Percus-Yevick, hypernetted-chain or similar equation," Phys. Rev. 154, 170-175 (1967).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bringi, V.

V. Varadan, V. Bringi, and V. Varadan, "Coherent electromagnetic wave propagation though randomly distributed dielectric scatterers," Phys. Rev. D 19, 2480-2489 (1979).
[CrossRef]

Chýlek, P.

P. Chýlek and G. Videen, "Scattering by a composite sphere and effective medium approximations," Opt. Commun. 146, 15-20 (1998).
[CrossRef]

P. Chýlek, G. Videen, and D. Ngo, "Effect of air bubbles on absorption of solar radiation by water droplets," J. Atmos. Sci. 55, 340-343 (1998).
[CrossRef]

G. Videen and P. Chýlek, "Scattering by a composite sphere with an absorbing inclusion and effective medium approximations," Opt. Commun. 158, 1-6 (1998).
[CrossRef]

G. Videen, D. Ngo, P. Chýlek, and R. Pinnick, "Light scattering from a sphere with an irregular inclusion," J. Opt. Soc. Am. A 12, 922-928 (1995).
[CrossRef]

P. Chýlek and V. Srivastava, "Dielectric constant of a composite inhomogeneous medium," Phys. Rev. B 27, 5098 (1983).
[CrossRef]

Ding, K.

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Ding, K. H.

Doicu, A.

A. Doicu and T. Wriedt, "Equivalent refractive index of a sphere with multiple spherical inclusions," J. Opt. A, Pure Appl. Opt. 3, 204-209 (2001).
[CrossRef]

Doyle, W. T.

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

Draine, B.

B. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Foldy, L.

L. Foldy, "The multiple scattering of waves," Phys. Rev. 107, 107-119 (1945).
[CrossRef]

Frisch, U.

U. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A.T.Barucha-Reid, ed. (Academic, 1953), Vol. 1, pp. 75-198.

Fuller, K.

Fung, A.

Gibbs, D.

Granqvist, C.

Guérin, C. A.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Gustafson, B. A.

L. Kolokolova and B. A. Gustafson, "Scattering by inhomogeneous particles: microwave analog experiments and comparison to effective medium theories," J. Quant. Spectrosc. Radiat. Transf. 70, 611-625 (2001).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Hunderi, O.

Kolokolova, L.

L. Kolokolova and B. A. Gustafson, "Scattering by inhomogeneous particles: microwave analog experiments and comparison to effective medium theories," J. Quant. Spectrosc. Radiat. Transf. 70, 611-625 (2001).
[CrossRef]

Kong, J.

Kong, J. A.

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

L. Tsang and J. A. Kong, "Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formalism," J. Appl. Phys. 51, 3465-3485 (1980).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, "Size-dependent Maxwell-Garnett formula from an integral equation formalism," Optik (Stuttgart) 91, 134-137 (1992).

Lax, M.

M. Lax, "Multiple scattering of waves. II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
[CrossRef]

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Mallet, P.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Maxwell Garnett, J.

J. Maxwell Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London 203, 385 (1904).
[CrossRef]

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Ngo, D.

P. Chýlek, G. Videen, and D. Ngo, "Effect of air bubbles on absorption of solar radiation by water droplets," J. Atmos. Sci. 55, 340-343 (1998).
[CrossRef]

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

G. Videen, D. Ngo, P. Chýlek, and R. Pinnick, "Light scattering from a sphere with an irregular inclusion," J. Opt. Soc. Am. A 12, 922-928 (1995).
[CrossRef]

Niklasson, G.

Pan, F.

D. Stroud and F. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular theories," Phys. Rev. B 17, 1602 (1978).
[CrossRef]

Pellegrino, P.

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

Percus, J.

J. Percus and G. Yevick, "Analysis of classical statistical mechanics by means of collective coordinates," Phys. Rev. 110, 1-13 (1958).
[CrossRef]

Pinnick, R.

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

G. Videen, D. Ngo, P. Chýlek, and R. Pinnick, "Light scattering from a sphere with an irregular inclusion," J. Opt. Soc. Am. A 12, 922-928 (1995).
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Sentenac, A.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Shih, S.

Sihvola, A.

A. Sihvola, Electromagnetic Mixing Formulas and Applications, IEE Electromagnetic Waves Series (The Institute of Electrical Engineers, 1999).
[CrossRef]

Smith, G.

G. Smith, "Dielectric constant for mixed media," J. Phys. D 10, L39-42 (1977).
[CrossRef]

Srivastava, V.

P. Chýlek and V. Srivastava, "Dielectric constant of a composite inhomogeneous medium," Phys. Rev. B 27, 5098 (1983).
[CrossRef]

Stroud, D.

D. Stroud and F. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular theories," Phys. Rev. B 17, 1602 (1978).
[CrossRef]

Tai, C.

C. Tai, Dyadic Green's Functions in Electromagnetic theory (Intext, 1971).

Teixeira, F. L.

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, "Equation of state calculation by fast computing machines," J. Chem. Phys. 21, 1087-1092 (1953).
[CrossRef]

Tsang, L.

B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L. Tsang, "Monte Carlo simulation of electromagnetic waves propagation in dense random media with dielectric spheroids," IEICE Trans. Electron. E83-C, 1797-1802 (2000).

L. Tsang, K. H. Ding, S. Shih, and J. Kong, "Scattering of electromagnetic wave from dense distributions of spheroidal particles based on Monte Carlo simulations," J. Opt. Soc. Am. A 15, 2660-2669 (1998).
[CrossRef]

L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, "Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries," J. Opt. Soc. Am. A 12, 1772-1781 (1995).
[CrossRef]

R. West, D. Gibbs, L. Tsang, and A. Fung, "Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media," J. Opt. Soc. Am. A 11, 1854-1857 (1994).
[CrossRef]

K. H. Ding, L. M. Zurk, and L. Tsang, "Pair distribution functions and attenuation rates for sticky particles in dense media," J. Electromagn. Waves Appl. 8, 1585-1604 (1994).

L. Tsang and J. A. Kong, "Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formalism," J. Appl. Phys. 51, 3465-3485 (1980).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001).

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Varadan, V.

V. Varadan, V. Bringi, and V. Varadan, "Coherent electromagnetic wave propagation though randomly distributed dielectric scatterers," Phys. Rev. D 19, 2480-2489 (1979).
[CrossRef]

V. Varadan, V. Bringi, and V. Varadan, "Coherent electromagnetic wave propagation though randomly distributed dielectric scatterers," Phys. Rev. D 19, 2480-2489 (1979).
[CrossRef]

Videen, G.

P. Chýlek, G. Videen, and D. Ngo, "Effect of air bubbles on absorption of solar radiation by water droplets," J. Atmos. Sci. 55, 340-343 (1998).
[CrossRef]

P. Chýlek and G. Videen, "Scattering by a composite sphere and effective medium approximations," Opt. Commun. 146, 15-20 (1998).
[CrossRef]

G. Videen and P. Chýlek, "Scattering by a composite sphere with an absorbing inclusion and effective medium approximations," Opt. Commun. 158, 1-6 (1998).
[CrossRef]

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

G. Videen, D. Ngo, P. Chýlek, and R. Pinnick, "Light scattering from a sphere with an irregular inclusion," J. Opt. Soc. Am. A 12, 922-928 (1995).
[CrossRef]

Videen, J.

G. Videen, P. Pellegrino, D. Ngo, J. Videen, and R. Pinnick, "Light-scattering intensity fluctuations in microdroplets containing inclusions," Atmos.-Ocean. 36, 6115-6118 (1997).

Wertheim, M.

M. Wertheim, "Exact solution of the Percus-Yevick integral equation for hard spheres," Phys. Rev. Lett. 10, 321-323 (1963).
[CrossRef]

West, R.

Winebrenner, D. P.

Wriedt, T.

A. Doicu and T. Wriedt, "Equivalent refractive index of a sphere with multiple spherical inclusions," J. Opt. A, Pure Appl. Opt. 3, 204-209 (2001).
[CrossRef]

Xu, Y.

Yaghjian, A.

A. Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).
[CrossRef]

Yevick, G.

J. Percus and G. Yevick, "Analysis of classical statistical mechanics by means of collective coordinates," Phys. Rev. 110, 1-13 (1958).
[CrossRef]

Zurk, L. M.

L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, "Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries," J. Opt. Soc. Am. A 12, 1772-1781 (1995).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Random aggregate of particles is obtained by cutting a finite volume inside an infinite distribution.

Fig. 2
Fig. 2

Pair distribution function for two different densities. There is excellent agreement between Monte Carlo simulations (circles and triangles) and Percus–Yevick hard-sphere distribution functions (solid curves).

Fig. 3
Fig. 3

Illustration of the validation procedure. An identification is made between the different cross sections of the random and the homogenized medium.

Fig. 4
Fig. 4

Extinction cross section of the homogeneous medium and average total scattered field of the aggregate for ϵ s = 3.2 .

Fig. 5
Fig. 5

Same as Fig. 4 with ϵ s = 16 .

Fig. 6
Fig. 6

Absorption cross section of the homogeneous medium and incoherent scattering cross section of the aggregate for ϵ s = 3.2 .

Fig. 7
Fig. 7

Same as Fig. 6 with ϵ s = 16 .

Fig. 8
Fig. 8

Imaginary part of the finite-size effective permittivity (FS-QCA) as a function of the radius of the embedding sphere, for otherwise fixed geometry ( K a = 0.1 , ϵ s = 3.2 , f = 0.3 ). QCA is approached in the limit of large volumes.

Equations (86)

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ρ 1 ( r ) = 1 V Π V ( r ) ,
ρ n ( r 1 , , r n ) = ρ 1 ( r 1 ) ρ 1 ( r n ) g n ( r 1 , , r n ) ,
g 2 ( r 1 , r 2 ) = 0 , r 1 r 2 < 2 a ,
g 2 ( r 1 , r 2 ) 1 , r 1 r 2 ,
g 2 ( r 1 , r 2 ) = g 2 ( r 1 r 2 ) .
E inc ( r ) = e i K 0 r E 0 .
E ( r ) = E inc ( r ) + K 2 d r [ ϵ ( r ) 1 ] G 0 ( r r ) E ( r ) ,
G 0 ( r ) = ( I + K 2 ) e i K r 4 π r .
{ ( 1 + i K r 1 ( K r ) 2 ) I + [ 3 ( K r ) 2 3 i K r 1 ] r ̂ r ̂ } e i K r 4 π r .
G ( r r ) = 1 K 2 L δ ( r r ) + PV G ( r r ) ,
E ( r ) = 3 ϵ ( r ) + 2 E inc ( r ) + 3 K 2 d r ϵ ( r ) 1 ϵ ( r ) + 2 G 0 ( r r ) E ( r ) ,
d r G ( r r ) lim a 0 r r > a d r G ( r r ) .
E s ( r ) = d r 1 G ( r r 1 ) e i K 0 r φ [ β ] ( r 1 ) ,
G ( r ) = G 0 ( r ) e i K 0 r ,
φ [ β ] = n = 1 φ n [ β ] ,
β ( r ) = ϵ ( r ) 1 ϵ ( r ) + 2 .
φ 1 [ β ] ( r 1 ) = 3 K 2 β ( r 1 ) E 0 ,
φ n [ β ] ( r 1 ) = 3 K 2 β ( r 1 ) d r 2 3 K 2 β ( r 2 ) G ( r 1 r 2 ) d r n 3 K 2 β ( r n ) G ( r n 1 r n ) E 0 , n 2 .
E s ( r ) e i K r r E ̃ s ( K ) , r .
E ̃ s ( K ) = 1 4 π d r 1 ( I K ̂ K ̂ ) e i ( K 0 K ) r 1 φ [ β ] ( r 1 ) ,
φ 1 [ β ] ( r 1 ) = 3 K 2 β 1 ( r 1 ) E 0 ,
φ 2 [ β ] ( r 1 ) = ( 3 K 2 ) 2 d r 2 B 2 ( r 1 , r 2 ) G ( r 2 r 1 ) E 0 ,
( B 1 r ) β ( r ) ,
B 2 ( r , r ) β ( r ) β ( r ) .
( B 1 r ) = β s Proba [ r particle ] = β s f Π V ( r ) ,
f = 4 π a 3 3 N V ,
β s = ϵ s 1 ϵ s + 2 .
B 2 ( r , r ) = β s 2 Proba [ r particle , r particle ] .
B 2 ( r , r ) = f β s 2 Φ ( r r 2 a ) Π V ( r ) Π V ( r ) + f 2 β s 2 Π V ( r ) Π V ( r ) g 2 ( r r ) ,
Φ ( u ) = 1 3 2 u + 1 2 u 3 , if u 1 , Φ ( u ) = 0 otherwise ,
φ 1 + 2 [ β ] ( r 1 ) = 3 K 2 f β s Π V ( r 1 ) E 0 + ( 3 K 2 β s ) 2 f d r 2 Π V ( r 2 ) Φ ( r 12 a ) G ( r 12 ) E 0 + ( 3 K 2 f β s ) 2 Π V ( r 1 ) d r 2 Π V ( r 2 ) g 2 ( r 12 ) G ( r 12 ) E 0 ,
K 2 d r 2 Φ ( r 12 a ) G ( r 12 ) E 0 = 0 a d ( K r ) Φ ( r a ) γ ( K r ) E 0 ,
K S r d S r G ( r ) E 0 = γ ( K r ) E 0 ,
S r d S r e i K r = 4 π r 2 sin ( K r ) K r ,
S r d S r e i K r r ̂ r ̂ = 4 π r 2 { [ sin ( K r ) ( K r ) 3 cos ( K r ) ( K r ) 2 ] I + [ sin ( K r ) K r + 3 cos ( K r ) ( K r ) 2 3 sin ( K r ) ( K r ) 3 ] K ̂ K ̂ } ,
γ ( u ) = e i u u { ( u 2 + i u 1 ) sinc ( u ) + 3 3 i u u 2 u 2 [ sinc ( u ) cos ( u ) ] } .
γ ( u ) 11 15 u + 2 3 i u 2 , u 0 .
φ 1 + 2 [ β ] ( r 1 ) = 3 K 2 f β s Π V ( r 1 ) [ 1 + 11 i 10 ( K a ) 2 β s + 2 i 3 ( K a ) 3 β s + 3 K 2 f β s d r 2 Π V ( r 2 ) g 2 ( r 12 ) G ( r 12 ) ] E 0 ,
β s = β s + i β s .
β 0 = f β s [ 1 + 11 i 10 ( K a ) 2 β s + 2 i 3 β s ( K a ) 3 ] ,
φ 1 + 2 [ β ] = 3 K 2 β 0 [ 1 + 3 K 2 β 0 Π V ( g 2 G ) ] Π V E 0 ,
ϵ e = ϵ MG 1 + 3 f β s 1 β s f ,
β e ( r 1 ) [ 1 + 3 K 2 ( Π V β e ) G ( r 1 ) ] E 0 = β 0 [ 1 + 3 K 2 β 0 Π V ( g 2 G ) ( r 1 ) ] E 0 .
ϵ e = 1 + 3 f β s 1 β s f { 1 + [ 11 i 10 ( K a ) 2 β s + 2 i 3 ( K a ) 3 β s ] 1 1 β s f } .
ϵ EFA = 1 + 3 f β s [ 1 + 2 i 3 ( K a ) 3 β s ] .
β e ( r ) = [ 1 + η ( r ) ] β 0 .
η = 3 K 2 β 0 { [ Π V ( g 2 G ) ] [ 1 + η ] [ Π V ( 1 + η ) G ] } ,
η = 3 K 2 β 0 { Π V [ ( g 2 1 ) G ] } .
ϵ e ( r 1 ) = 1 + 2 β e ( r 1 ) 1 β e ( r 1 ) .
β e ( r 1 ) = β 0 ( 1 + 3 K 2 β 0 { Π V [ ( g 2 1 ) G ] } ( r 1 ) ) .
β e = β 0 [ 1 + 3 K 2 β 0 d r ( g 2 1 ) ( r ) G ( r ) ] .
β e = β 0 { 1 + 3 β 0 0 d ( K r ) γ ( K r ) [ g 2 ( r ) 1 ] } .
β e β 0 [ 1 + 11 i 5 β 0 ( K a ) 2 M 1 + 2 i β 0 ( K a ) 3 M 2 ] ,
M n = 0 d u [ g ̃ 2 ( u ) 1 ] u n , n = 1 , 2 ,
ϵ QCA = 1 + 3 f β s 1 β s f [ 1 + 2 i 3 ( K a ) 3 β s 1 β s f ( 1 + 3 f M 2 ) + 11 i 10 ( K a ) 2 β s 1 β s f ( 1 + 2 f M 1 ) ] ,
β e ¯ = 1 V d r 1 β e ( r 1 ) Π V ( r 1 ) .
β e ¯ = β 0 { 1 + 3 K 2 β 0 d r Φ ( r L ) [ g 2 ( r ) 1 ] G ( r ) } ,
β e ¯ = β 0 { 1 + 3 β 0 0 L d ( K r ) Φ ( r L ) [ g 2 ( r ) 1 ] γ ( K r ) } ,
ϵ e = 1 + 2 β e ¯ 1 β e ¯ ,
E ( r ) = E Mie ( r ) + K 2 d r [ ϵ ( r ) ϵ Mie ( r ) ] G Mie ( r , r ) E ( r ) .
E Mie ( r ) 3 2 + ϵ h E inc ( r ) .
G Mie ( r , r ) G h ( r r ) , if r r λ ,
ϵ e = ϵ h + 3 ϵ h f β s h ϵ h β s f ,
β s h = ϵ s ϵ h ϵ s + 2 ϵ h ;
ϵ EFA = ϵ h + 3 ϵ h f β s h [ 1 + 2 i 3 ( K h a ) 3 β s h ] ;
ϵ e ( r 1 ) = ϵ h + 2 β e ( r 1 ) ϵ h β e ( r 1 )
β e ( r 1 ) = β 0 h ( 1 + 3 K h 2 β 0 h { Π V [ ( g 2 1 ) G h ] } ( r 1 ) ) ,
E s ( r ) = α s K 2 j = 1 N G 0 ( r r j ) E j ,
E i = E inc ( r i ) + α s K 2 j i G 0 i j E j .
α s = 4 π a 3 β s [ 1 + 2 3 i β s ( K a ) 3 ] .
E ̃ s ( K ) = α s K 2 4 π j = 1 N ( I K ̂ K ̂ ) E j e i K r j .
σ e = 4 π K I { [ E s ˜ ( K ) . E 0 ] } K = K 0 ,
σ s = 4 π E s ˜ 2 d Ω ,
σ a = σ e σ s .
E ̃ s = E ̃ s + δ E ̃ s .
4 π E ̃ s + δ E ̃ s 2 d Ω = 4 π K I [ ( E ̃ s + δ E ̃ s ) . E 0 ] K = K 0 .
4 π E ̃ s 2 d Ω + 4 π δ E ̃ s 2 d Ω = 4 π K I ( E ̃ s . E 0 ) K = K 0 .
σ s [ V , ϵ e ] = σ s coh [ Aggregate ] ,
σ a [ V , ϵ e ] = σ s incoh [ Aggregate ] ,
σ e [ V , ϵ e ] = ( σ s coh + σ s incoh ) [ Aggregate ] .
M 2 = 1 3 f [ ( 1 f ) 4 ( 1 + 2 f ) 2 1 ] ,
ϵ e = 1 + 3 f ϵ e ( ϵ s 1 ) 3 ϵ e + ( ϵ s 1 ) ( 1 f ) [ 1 + 2 i 3 ( K a ) 3 β s ϵ e 3 2 3 ϵ e + ( ϵ s 1 ) ( 1 f ) ( 1 + 3 f M 2 ) ] .
Proba [ r particle , r particle ] = N Proba [ r , r 1 ] + N ( N 1 ) Proba [ r 1 , r 2 ] .
N Proba [ r , r 1 ] = Π V ( r ) Π V ( r ) N Proba [ r r 1 < a , r r 1 < a ] = Π V ( r ) Π V ( r ) N d u 1 ρ 1 ( u 1 ) Π a ( r u 1 ) Π a ( r u 1 ) = f Π V ( r ) Π V ( r ) Φ ( r r 2 a ) ,
Proba [ r 1 , r 2 ] = Proba [ r r 1 < a , r r 2 < a ] = Π V ( r ) Π V ( r ) d u 1 d u 2 ρ 2 ( u 1 , u 2 ) Π a ( r u 1 ) Π a ( r u 2 ) = Π V ( r ) Π V ( r ) d u 1 d u 2 ρ 1 ( u 1 ) ρ 1 ( u 2 ) g 2 ( u 1 u 2 ) Π a ( r u 1 ) Π a ( r u 2 ) .
N ( N 1 ) Proba [ r 1 , r 2 ] = f 2 Π V ( r ) Π V ( r ) g 2 ( r r ) ,

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