Abstract

Scattering of light by an ensemble of nonabsorbing spherical concentric two-layered particles is considered. It has been shown that exponent of the power function describing the wavelength dependence of the extinction coefficient of the medium with subwavelength-sized particles can exceed considerably the value of 4, which takes place for the Rayleigh scattering. Spatial correlation of particles enhances this “anomalous” dependence on the wavelength. Bleaching and darkening effects can be implemented. In the first case transmittance increases, while in the second case transmittance decreases with increased volume concentration. These effects can be used to get a sharp spectral dependence of transmittance. Comparison with the data for spatially correlated homogeneous particles is carried out.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009

Y. Okada and A. A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 902–917 (2009).
[CrossRef]

V. A. Loiko and V. V. Berdnik, “Scattering by a plane-parallel layer with high concentration of optically soft particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1502–1510(2009).
[CrossRef]

2008

M. P. Shepilov, “On light scattering in fluorozirconate glass-ceramics containing BaCl2 nano-crystals,” Opt. Mater. 30, 839–846 (2008).
[CrossRef]

2007

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: Exact 3D results,” Opt. Express 15, 2822–2836 (2007).
[CrossRef] [PubMed]

2006

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles. Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

V. A. Loiko and V. V. Berdnik, “Light scattering in a layer of correlatively arranged optically soft particles,” Opt. Spectrosc. 101, 303–308 (2006).
[CrossRef]

2003

V. A. Babenko, L. G. Astafyeva, and V. N. Kuzmin, “Light scattering and absorption by two- and multilayered spheres,” in Electromagnetic Scattering in Disperse Media (Springer Praxis, 2003), Chap. 3, p. 434.

M. P. Shepilov, “The problem of theoretical description of anomalous light scattering by liquated glasses, coursed by the interparticles interference,” J. Opt. Technol. 70, 882–887 (2003).
[CrossRef]

2001

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

1999

1996

1994

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

1988

A. P. Ivanov, V. A. Loiko, and V. P. Dick, Propagation of Light in Closely Packed Disperse Media (Nauka i Tekhnika, 1988).

1983

1982

1979

J. M. Ziman, Models of Disorder (Cambridge University, 1979).

1978

N. S. Andreev, “Scattering of visible light by glasses undergoing phase separation and homogenization,” J. Non-Cryst. Solids 30, 99–126 (1978).
[CrossRef]

1975

1962

N. A. Voishvillo, “To a question of coherent scattering of light in the glass,” Opt. Spectrosc. 12, 412–418 (1962).

1958

J. K. Percus and G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Andreev, N. S.

N. S. Andreev, “Scattering of visible light by glasses undergoing phase separation and homogenization,” J. Non-Cryst. Solids 30, 99–126 (1978).
[CrossRef]

Astafyeva, L. G.

V. A. Babenko, L. G. Astafyeva, and V. N. Kuzmin, “Light scattering and absorption by two- and multilayered spheres,” in Electromagnetic Scattering in Disperse Media (Springer Praxis, 2003), Chap. 3, p. 434.

Babenko, V. A.

V. A. Babenko, L. G. Astafyeva, and V. N. Kuzmin, “Light scattering and absorption by two- and multilayered spheres,” in Electromagnetic Scattering in Disperse Media (Springer Praxis, 2003), Chap. 3, p. 434.

Berdnik, V. V.

V. A. Loiko and V. V. Berdnik, “Scattering by a plane-parallel layer with high concentration of optically soft particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1502–1510(2009).
[CrossRef]

V. A. Loiko and V. V. Berdnik, “Light scattering in a layer of correlatively arranged optically soft particles,” Opt. Spectrosc. 101, 303–308 (2006).
[CrossRef]

Cairns, B.

Dick, V. P.

Ishimary, A.

Ivanov, A. P.

Kalmykov, A. E.

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

Kokhanovsky, A. A.

Y. Okada and A. A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 902–917 (2009).
[CrossRef]

Kong, J. A.

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

Kuga, Y.

Kuzmin, V. N.

V. A. Babenko, L. G. Astafyeva, and V. N. Kuzmin, “Light scattering and absorption by two- and multilayered spheres,” in Electromagnetic Scattering in Disperse Media (Springer Praxis, 2003), Chap. 3, p. 434.

Lacis, A. A.

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles. Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Liu, L.

Loiko, V. A.

V. A. Loiko and V. V. Berdnik, “Scattering by a plane-parallel layer with high concentration of optically soft particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1502–1510(2009).
[CrossRef]

V. A. Loiko and V. V. Berdnik, “Light scattering in a layer of correlatively arranged optically soft particles,” Opt. Spectrosc. 101, 303–308 (2006).
[CrossRef]

V. A. Loiko, V. P. Dick, and A. P. Ivanov, “Passage of light through a disperse medium with a high concentration of discrete inhomogeneities: experiment,” Appl. Opt. 38, 2640–2648(1999).
[CrossRef]

A. P. Ivanov, V. A. Loiko, and V. P. Dick, Propagation of Light in Closely Packed Disperse Media (Nauka i Tekhnika, 1988).

Mackowski, D. W.

Mischenko, M. I.

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles. Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Mishchenko, M. I.

Okada, Y.

Y. Okada and A. A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 902–917 (2009).
[CrossRef]

Percus, J. K.

J. K. Percus and G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Shepilov, M. P.

M. P. Shepilov, “On light scattering in fluorozirconate glass-ceramics containing BaCl2 nano-crystals,” Opt. Mater. 30, 839–846 (2008).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. P. Shepilov, “The problem of theoretical description of anomalous light scattering by liquated glasses, coursed by the interparticles interference,” J. Opt. Technol. 70, 882–887 (2003).
[CrossRef]

Sycheva, G. A.

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

Travis, L. D.

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles. Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Tsang, L.

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

Twersky, V.

Videen, G.

Voishvillo, N. A.

N. A. Voishvillo, “To a question of coherent scattering of light in the glass,” Opt. Spectrosc. 12, 412–418 (1962).

Yevick, G. Y.

J. K. Percus and G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Ziman, J. M.

J. M. Ziman, Models of Disorder (Cambridge University, 1979).

Appl. Opt.

J. Non-Cryst. Solids

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

N. S. Andreev, “Scattering of visible light by glasses undergoing phase separation and homogenization,” J. Non-Cryst. Solids 30, 99–126 (1978).
[CrossRef]

M. P. Shepilov, A. E. Kalmykov, and G. A. Sycheva, “Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement,” J. Non-Cryst. Solids 353, 2415–2430 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

J. Quant. Spectrosc. Radiat. Transfer

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

Y. Okada and A. A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 902–917 (2009).
[CrossRef]

V. A. Loiko and V. V. Berdnik, “Scattering by a plane-parallel layer with high concentration of optically soft particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1502–1510(2009).
[CrossRef]

Opt. Express

Opt. Mater.

M. P. Shepilov, “On light scattering in fluorozirconate glass-ceramics containing BaCl2 nano-crystals,” Opt. Mater. 30, 839–846 (2008).
[CrossRef]

Opt. Spectrosc.

N. A. Voishvillo, “To a question of coherent scattering of light in the glass,” Opt. Spectrosc. 12, 412–418 (1962).

V. A. Loiko and V. V. Berdnik, “Light scattering in a layer of correlatively arranged optically soft particles,” Opt. Spectrosc. 101, 303–308 (2006).
[CrossRef]

Phys. Rev.

J. K. Percus and G. Y. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Other

J. M. Ziman, Models of Disorder (Cambridge University, 1979).

V. A. Babenko, L. G. Astafyeva, and V. N. Kuzmin, “Light scattering and absorption by two- and multilayered spheres,” in Electromagnetic Scattering in Disperse Media (Springer Praxis, 2003), Chap. 3, p. 434.

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

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles. Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

A. P. Ivanov, V. A. Loiko, and V. P. Dick, Propagation of Light in Closely Packed Disperse Media (Nauka i Tekhnika, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic presentation of side view of the layer.

Fig. 2
Fig. 2

Dependence of asymmetry parameter g on λ r and w. (a) Layer of homogeneous spherical particles with radius R = 0.3 mkm and refractive index n = 1.108 , (b) layer of two-layered spherical particles with core radius R c = 0.18 mkm and refractive index of core n c = 1.108 , shell radius R s = R = 0.3 mkm and refractive index of shell n s = 0.96 .

Fig. 3
Fig. 3

Dependence of u on λ r and w. (a) Homogeneous spherical particles with R = 0.3 mkm and relative refractive index n = 1.108 , (b) two-layered spherical particles with R c = 0.18 mkm , n c = 1.108 , R s = 0.3 mkm , n s = 0.96 .

Fig. 4
Fig. 4

Angular dependence of scattered light intensity: phase function for homogeneous particles with R = 0.3 mkm and n = 1.108 (curve 1) and two-layered particles with R c = 0.18 mkm , n c = 1.108 , n s = 0.96 , R s = 0.3 mkm (curve 2). Structure factor S 3 for layer with particles R = 0.3 mkm and volume concentration w = 0.1 (curve 3), 0.4 (curve 4), 0.5 (curve 5), 0.55 (curve 6), and 0.6 (curve 7).

Fig. 5
Fig. 5

Dependence of the u value on n c and n s for two-layered spherical particles with R s = 0.3 mkm , w = 0.6 , R c = 0.18 mkm (a), R c = 0.20 mkm (b) at λ = 0.9 mkm .

Fig. 6
Fig. 6

Dependence of p on λ / R s and w for two-layered spherical particles with n c = 1.2 , R s = 0.3 mkm , n s = 0.9 . (a)  R c = 0.05 mkm , (b)  R c = 0.2 mkm .

Fig. 7
Fig. 7

Dependence of parameter p on λ / R s and R c for two-layered particles at n c = 1.1084 , n s = 0.9636 , R s = 0.3 mkm , w = 0.5 .

Fig. 8
Fig. 8

Dependence of transmittance T on the wavelength for a layer with homogenous particles at w = 0.01 (curve 1), w = 0.4 (curve 2), and w = 0.6 (curve 3). R = 0.3 mkm , n = 1.1 , n m = 1 , z 0 = 20 mkm .

Fig. 9
Fig. 9

Dependence of transmittance T on the wavelength for a layer with two-layered particles at w = 0.01 (curve 1), w = 0.4 (curve 2), and w = 0.6 (curve 3). R c = 0.2 mkm , n c = 1.2 , R s = 0.3 mkm , n s = 0.9 , n m = 1.0 , z 0 = 20 mkm .

Equations (18)

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σ h ( γ ) = w σ 0 l p l ( γ ) S 3 ( γ , w ) ,
σ h = w σ 0 l u ,
ε h = w ( ε 0 l σ 0 l + σ 0 l u ) ,
u = 0 π p l ( γ ) S 3 ( γ , w ) sin γ d γ .
S 3 ( γ , w ) = 1 + 4 π n 0 [ g ( r , w ) 1 ] sin z r z r r 2 d r .
S 3 ( γ , w ) = ( 1 24 n 0 1 g ( r , w ) sin z r z r r 2 d r ) 1 .
g ( r , w ) = { a b r 2 R s c ( r 2 R s ) 3 , r 2 R s 0 , r > 2 R s ,
a = ( 1 + 2 w ) 2 ( 1 w ) 4 ,
b = 6 w ( 1 + 0.5 w ) 2 ( 1 w ) 4 ,
c = 0.5 w ( 1 + 2 w ) 2 ( 1 w ) 4 .
τ 0 = τ 0 l ( 1 Λ l ( 1 u ( R s , w ) ) ) ,
Λ = Λ l u ( R s , w ) 1 Λ l ( 1 u ( R p , w ) ) ,
p h ( γ ) = p l ( γ ) S 3 ( γ , w ) u ( R s , w ) .
α h = ε h σ h = w α 0 l
g = 0 π p h ( γ ) cos γ sin γ d γ .
ε h ( λ ) = C λ p ( λ ) .
p ( λ , w ) = d ln ε h ( λ ) d ln λ .
T = exp ( τ 0 ( λ ) )

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