Abstract

We study theoretically the extinction of collimated light in random systems of highly scattering particles embedded in nonabsorbing media. We aim to provide rough guidelines on the behavior of the extinction coefficient in the so-called dependent-scattering regime. We base our analysis on Keller’s second order perturbative approximation to the effective propagation constant. To gain physical insight, we also analyze a simple model based on the physical notion that particles in a dense system scatter light in an effective medium.

© 2013 Optical Society of America

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  2. Y. Huang and E. M. Sevick-Muraca, “Validating the assumption to the interference approximation by use of measurements of absorption efficiency and hindered scattering in dense suspensions,” Appl. Opt. 43, 814–819 (2004).
    [CrossRef]
  3. G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42, 4023–4030 (2003).
    [CrossRef]
  4. J.-C. Auger, V. Martínez, and B. Stout, “Absorption and scattering properties of dense ensembles of nonspherical particles,” J. Opt. Soc. Am. A 24, 3508–3516 (2007).
    [CrossRef]
  5. V. A. Loiko and G. I. Ruban, “Absorption by a layer of densely packed subwavelength-sized particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 271–278 (2004).
    [CrossRef]
  6. V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
    [CrossRef]
  7. V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
    [CrossRef]
  8. V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
    [CrossRef]
  9. A. Ishimaru and Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  10. V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
    [CrossRef]
  11. R. West, D. Gibbs, L. Tsang, and A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
    [CrossRef]
  12. V. P. Dick, V. A. Loiko, and A. P. Ivanov, “Angular structure of radiation scattered by monolayer of particles: experimental study,” Appl. Opt. 36, 4235–4240 (1997).
    [CrossRef]
  13. V. A. Loiko, V. P. Dick, and A. P. Ivanov, “Passage of light through a dispersion medium with a high concentration of discrete inhomogeneities: experiment,” Appl. Opt. 38, 2640–2646 (1999).
    [CrossRef]
  14. V. P. Dick and A. P. Ivanov, “Extinction of light in dispersive media with high particle concentrations: applicability limits of the interference approximation,” J. Opt. Soc. Am. A 16, 1034–1038 (1999).
    [CrossRef]
  15. L. Tsang and J. A. Kong, “Multiple scattering theory for discrete scatterers,” Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001), Chap. 5, pp. 128–130.
  16. M. Lax, “Multiple scattering of waves II: effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
    [CrossRef]
  17. L. Hespel, S. Mainguy, and J.-J. Greffet, “Theoretical and experimental investigation of the extinction in a dense distribution of particles: nonlocal effects,” J. Opt. Soc. Am. A 18, 3072–3076 (2001).
    [CrossRef]
  18. J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
    [CrossRef]
  19. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2943–2952 (2007).
    [CrossRef]
  20. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2953–2961 (2007).
    [CrossRef]
  21. R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
    [CrossRef]
  22. R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
    [CrossRef]
  23. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  24. K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
    [CrossRef]
  25. H. Zhang, P. Zhu, Y. Xu, H. Zhu, and M. Xu, “Effective medium theory for random media composed of two-layered spheres,” J. Opt. Soc. Am. A 28, 2292–2297 (2011).
    [CrossRef]

2011 (1)

2007 (4)

2004 (2)

V. A. Loiko and G. I. Ruban, “Absorption by a layer of densely packed subwavelength-sized particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 271–278 (2004).
[CrossRef]

Y. Huang and E. M. Sevick-Muraca, “Validating the assumption to the interference approximation by use of measurements of absorption efficiency and hindered scattering in dense suspensions,” Appl. Opt. 43, 814–819 (2004).
[CrossRef]

2003 (2)

2001 (1)

1999 (2)

1997 (1)

1996 (1)

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

1994 (1)

1983 (1)

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

1982 (3)

V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
[CrossRef]

V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

A. Ishimaru and Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

1979 (1)

V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
[CrossRef]

1964 (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

1952 (1)

M. Lax, “Multiple scattering of waves II: effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Auger, J.-C.

Barrera, R. G.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
[CrossRef]

Bohren, C. F.

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

Bringi, V. N.

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
[CrossRef]

V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
[CrossRef]

Busch, K.

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

Calvo-Perez, O.

Del Bianco, S.

Dick, V. P.

Durant, S.

Fund, A. K.

García-Valenzuela, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
[CrossRef]

Gibbs, D.

Greffet, J.-J.

Hespel, L.

Huang, Y.

Huffman, D. R.

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

Ishimaru, A.

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

A. Ishimaru and Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Ivanov, A. P.

Keller, J. B.

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Kong, J. A.

L. Tsang and J. A. Kong, “Multiple scattering theory for discrete scatterers,” Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001), Chap. 5, pp. 128–130.

Kuga, Y.

Lax, M.

M. Lax, “Multiple scattering of waves II: effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Loiko, V. A.

Mainguy, S.

Martelli, F.

Martínez, V.

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Ruban, G. I.

V. A. Loiko and G. I. Ruban, “Absorption by a layer of densely packed subwavelength-sized particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 271–278 (2004).
[CrossRef]

Sevick-Muraca, E. M.

Soukoulis, C. M.

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

Stout, B.

Tsang, L.

R. West, D. Gibbs, L. Tsang, and A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

L. Tsang and J. A. Kong, “Multiple scattering theory for discrete scatterers,” Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001), Chap. 5, pp. 128–130.

Varadan, V. K.

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
[CrossRef]

Varadan, V. V.

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
[CrossRef]

Varandan, V. K.

V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
[CrossRef]

Varandan, V. V.

V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
[CrossRef]

Vukadinovic, N.

West, R.

Xu, M.

Xu, Y.

Zaccanti, G.

Zhang, H.

Zhu, H.

Zhu, P.

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

V. N. Bringi, V. V. Varandan, and V. K. Varandan, “The effects of pair correlation function on coherent wave attenuation in discrete random media,” IEEE Trans. Antennas Propag. 30, 805–808 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

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

L. Hespel, S. Mainguy, and J.-J. Greffet, “Theoretical and experimental investigation of the extinction in a dense distribution of particles: nonlocal effects,” J. Opt. Soc. Am. A 18, 3072–3076 (2001).
[CrossRef]

R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
[CrossRef]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2943–2952 (2007).
[CrossRef]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. A 24, 2953–2961 (2007).
[CrossRef]

J.-C. Auger, V. Martínez, and B. Stout, “Absorption and scattering properties of dense ensembles of nonspherical particles,” J. Opt. Soc. Am. A 24, 3508–3516 (2007).
[CrossRef]

H. Zhang, P. Zhu, Y. Xu, H. Zhu, and M. Xu, “Effective medium theory for random media composed of two-layered spheres,” J. Opt. Soc. Am. A 28, 2292–2297 (2011).
[CrossRef]

R. West, D. Gibbs, L. Tsang, and A. K. Fund, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

V. P. Dick and A. P. Ivanov, “Extinction of light in dispersive media with high particle concentrations: applicability limits of the interference approximation,” J. Opt. Soc. Am. A 16, 1034–1038 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

V. A. Loiko and G. I. Ruban, “Absorption by a layer of densely packed subwavelength-sized particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 271–278 (2004).
[CrossRef]

Phys. Rev. (1)

M. Lax, “Multiple scattering of waves II: effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Phys. Rev. B (2)

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

Phys. Rev. D (1)

V. K. Varadan, V. N. Bringi, and V. V. Varadan, “Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers,” Phys. Rev. D 19, 2480–2489 (1979).
[CrossRef]

Proc. Symp. Appl. Math. (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Radio Sci. (2)

V. N. Bringi, V. K. Varandan, and V. V. Varandan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. 17, 946–952 (1982).
[CrossRef]

V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiments,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Other (3)

L. Tsang and J. A. Kong, “Multiple scattering theory for discrete scatterers,” Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2001), Chap. 5, pp. 128–130.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

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

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Figures (7)

Fig. 1.
Fig. 1.

Plot of the extinction coefficient αext for λ=500nm versus the volume filling fraction for particles of refractive index (a) np=1.5 and (b) np=2.0 suspended in water (nm=1.33), for various particle radii a (indicated by the plots). Full lines are for the independent-scattering approximation, and the dashed lines are for Keller’s self-consistent approximation.

Fig. 2.
Fig. 2.

Plots of the EEF for λ=500nm and for particles of different radii and refractive index np=1.7 immersed in water and calculated with the quadratic (full lines) and self-consistent (dashed lines) Keller’s approximations for different particle radii a (indicated by the plots).

Fig. 3.
Fig. 3.

Plots of the EEF for λ=500nm for particles immersed in water (nm=1.33) versus the particle radius for various values of the refractive index of the particles np=1.4, 1.6, 1.7, 1.8, 2.0, 2.2, 2.4, and 2.7 (indicated by the plots). The plots were generated using the self-consistent Keller’s approximation. We assume the volume fraction of the particles is 10%. The horizontal dotted line indicates where EEF=1.

Fig. 4.
Fig. 4.

Plots of Δn˜ (full lines) and Δn˜ (dashed lines) versus the particle radius for particles of refractive index np=1.6, 1.8, and 2.2. The plots are for spherical particles suspended in water (nm=1.33) and a wavelength of λ=500nm. The labels a and a indicate Δn˜ and Δn˜, respectively, for np=1.6, b and b indicate Δn˜ and Δn˜, respectively, for np=1.8, and c and c indicate Δn˜ and Δn˜, respectively, for np=1.8.

Fig. 5.
Fig. 5.

Plots of the EEF for particles of different radii and refractive index np=1.7 immersed in water for λ=500nm. The curves were calculated by the effective medium scattering approximation discussed in the text (SEMM).

Fig. 6.
Fig. 6.

Plots of the EEF for particles immersed in water (nm=1.33) versus the particle radius for λ=500nm and for various values of the refractive index of the particles np=1.4, 1.6, 1.7, 1.8, 2.0, 2.2, 2.4, and 2.7 (indicated by each plot). The plots were calculated with the SEMM. We assume the volume fraction of the particles is 10%. The horizontal dotted line indicates where EEF=1.

Fig. 7.
Fig. 7.

Plots of Im(A/nm) versus the particle radius (full lines) for particles of refractive index np=1.6 (a), 1.8 (b), and 2.2 (c), and plots of Im(A/nm) versus the particle radius (dashed lines) for particles of refractive index np=1.6 (a), 1.8 (b), and 2.2 (c). All plots are for spherical particles suspended in water (nm=1.33) and for λ=500nm.

Equations (23)

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

αext=ρCext,
αextfC/scaVp.
αext=2Im(keff).
keff=k(1+Af+Bf2+),
keffind=k[1+2πiρk3S(0)],
A=i32S(0)x3.
Im(A)=Cext/2kVp.
γαextαextind(1+BAf),
keff2=k2+4πiρkS(0)(4π)2ρ2S2(0)k20du[g(u)1]sin(keffu)keffexp(iku),
keff=k{1+2πik3ρS(0)+(2π)2k6ρ2S2(0)[1+I(k)]},
I(k)=2k20du[g(u)1]sin(ku)kexp(iku).
B=A2[1+I(k)].
A=ΔneffindnmfΔn˜
γ=(12Δn˜Δn˜(1+I)+[(Δn˜)2(Δn˜)2]IΔn˜f),
g(u)={0ifu<2a1ifu>2a.
I(k)=sin2(2ka)+2i[a14sin(4ka)].
γ1+2Δn˜If,
γ=[1(12+3χ)f],
A(f)A(0)+Af|f=0f,
Af=Akeffkefff+Akeffkefff.
Akeff|f=01kAnmandAkeff|f=01kAnm,
γ=1+[Im(Anm)Δn˜Δn˜+Im(Anm)]f.
γ1+Im(Anm)f,

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