Abstract

The features of scattered and transmitted light by dilute suspensions of transparent submicron particles are investigated both in the spectral and in the perceived colorimetric domains, as a function of effective particle diameter D, particle-host refractive-index mismatch m, and scattering angle θ. Our results show that the wavelength λ-dependence of the scattering and extinction cross sections remains quite similar well beyond the Rayleigh regime up to particle sizes of a few hundreds nm, but only for specific scattering angles that depend on D and m, and tend to 90° on approaching the Rayleigh regime. Close to this limit (D/λ1), a simple criterion that relates the perceived scattering color at θ=90° and the ratio of the sample extinction coefficients at two properly selected wavelengths is demonstrated. A comparison between computed and measured data is presented.

© 2012 Optical Society of America

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References

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  1. M. G. J. Minnaert, Light and Colors in the Outdoors (Springer-Verlag, 1993).
  2. D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 2001).
  3. C. F. Bohren, “Atmospheric optics,” in The Optics Encyclopedia, T. G. Brown, ed. (Wiley, 2004), pp. 53–91.
  4. “Light and color in the open air,” feature issue, Appl. Opt.42 (2003).
  5. “Light and color in the open air,” feature issue, Appl. Opt.44 (2005).
  6. “Light and color in the open air,” feature issue, Appl. Opt.47 (2008).
  7. J. Cabannes, “Sur la diffusion de la lumière par l’air,” C.R. Acad. Sci. 160, 62–63 (1915).
  8. R. J. Strutt, “Scattering of light by dust-free air, with artificial reproduction of the blue sky. Preliminary note,” Proc. R. Soc. A 94, 453–459 (1918).
    [CrossRef]
  9. L. Rayleigh and J. W. Strutt, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).
  10. C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), pp. 104–108.
  11. M. Kerker, “Founding fathers of light scattering and surface enhanced Raman scattering,” Appl. Opt. 30, 4699–4705 (1991).
    [CrossRef]
  12. S. D. Gedzelman, M. A. Lopez-Alvarez, J. Hernandez-Andres, and R. Greenler, “Quantifying the milky sky experiment,” Appl. Opt. 47, H128–H132 (2008).
    [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  14. M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, 1969).
  15. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).
  16. F. Ferri, A. Bassini, and E. Paganini, “Commercial spectrophotometer for particle sizing,” Appl. Opt. 36, 885–891 (1997).
    [CrossRef]
  17. E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

2008 (1)

1997 (1)

1991 (1)

1918 (1)

R. J. Strutt, “Scattering of light by dust-free air, with artificial reproduction of the blue sky. Preliminary note,” Proc. R. Soc. A 94, 453–459 (1918).
[CrossRef]

1915 (1)

J. Cabannes, “Sur la diffusion de la lumière par l’air,” C.R. Acad. Sci. 160, 62–63 (1915).

1899 (1)

L. Rayleigh and J. W. Strutt, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Bassini, A.

Bohren, C. F.

C. F. Bohren, “Atmospheric optics,” in The Optics Encyclopedia, T. G. Brown, ed. (Wiley, 2004), pp. 53–91.

C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), pp. 104–108.

Cabannes, J.

J. Cabannes, “Sur la diffusion de la lumière par l’air,” C.R. Acad. Sci. 160, 62–63 (1915).

Doxsee, D. D.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Duggal, A. R.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Ferri, F.

Gedzelman, S. D.

Greenler, R.

Hernandez-Andres, J.

Kerker, M.

Levinson, L. M.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Livingston, W.

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 2001).

Lopez-Alvarez, M. A.

Lynch, D. K.

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 2001).

McNulty, T. F.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Minnaert, M. G. J.

M. G. J. Minnaert, Light and Colors in the Outdoors (Springer-Verlag, 1993).

Paganini, E.

Rayleigh, L.

L. Rayleigh and J. W. Strutt, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Srivastava, A. M.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

Stokes, E. B.

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

Strutt, J. W.

L. Rayleigh and J. W. Strutt, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Strutt, R. J.

R. J. Strutt, “Scattering of light by dust-free air, with artificial reproduction of the blue sky. Preliminary note,” Proc. R. Soc. A 94, 453–459 (1918).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

Appl. Opt. (3)

C.R. Acad. Sci. (1)

J. Cabannes, “Sur la diffusion de la lumière par l’air,” C.R. Acad. Sci. 160, 62–63 (1915).

Philos. Mag. (1)

L. Rayleigh and J. W. Strutt, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Proc. R. Soc. A (1)

R. J. Strutt, “Scattering of light by dust-free air, with artificial reproduction of the blue sky. Preliminary note,” Proc. R. Soc. A 94, 453–459 (1918).
[CrossRef]

Other (11)

C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), pp. 104–108.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, 1969).

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982).

E. B. Stokes, T. F. McNulty, D. D. Doxsee, A. M. Srivastava, L. M. Levinson, and A. R. Duggal, “Solid state illumination system containing a light emitting diode, a light scattering material and a luminescent material,” U.S. patent 6,791,259 (14September2004).

M. G. J. Minnaert, Light and Colors in the Outdoors (Springer-Verlag, 1993).

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 2001).

C. F. Bohren, “Atmospheric optics,” in The Optics Encyclopedia, T. G. Brown, ed. (Wiley, 2004), pp. 53–91.

“Light and color in the open air,” feature issue, Appl. Opt.42 (2003).

“Light and color in the open air,” feature issue, Appl. Opt.44 (2005).

“Light and color in the open air,” feature issue, Appl. Opt.47 (2008).

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Figures (7)

Fig. 1.
Fig. 1.

Comparison between the computed scattering differential cross sections dσ/dΩ(θ,λ) (continuous lines) and the total extinction cross section σ(λ) (dotted lines) as a function of λ for different scattering angles θ. All curves were computed according to Mie theory with the same optical mismatch m=1.4, and two effective diameters D=100nm (a) and D=200nm (b).

Fig. 2.
Fig. 2.

Shape mismatch χ(θ) between the spectra of the scattering differential cross sections and that of the extinction cross section plotted as a function of the scattering angle for fixed m and different effective diameters D (a), and for different m and fixed D (b). Behavior of the angle of minimum mismatch θmin as a function of D for different m values (c). Independently of m, θmin tends to 90° as the particle diameter D0.

Fig. 3.
Fig. 3.

Computed scattering-color maps in the Dm plane obtained for spherical particles illuminated with ideal white light. The three frames correspond to different scattering angles: 60°, 90°, and 120°.

Fig. 4.
Fig. 4.

Computed distance ρ=Δx2+Δy2 in the colorimetric xy plane between the color points recovered by using dσ/dΩ(θ,λ) and σ(λ) plotted as a function of effective particle diameter D for two optical mismatch values m=1.40 (a) and m=1.70 (b).

Fig. 5.
Fig. 5.

Set of iso-γ contour lines represented in the D–M space (a) and in the x–y space (b). In the D–M plane, the curves are plotted with the correspondent true scattering colors, while in the x–y space, the lines are with false colors and the plane is painted with true colors.

Fig. 6.
Fig. 6.

Schematic diagram of the setup. The light emitted by the LED source is collected and weakly focused with a condenser lens (C), and the beam is then reduced with a diaphragm (D) and a 10X microscope objective. The final beam, with a diameter of about 1.5 mm, impinges on the cell containing the scattering particles. Both the transmitted and the scattered light are collected by two collimator-fiber systems and guided to the spectrometers S1 and S2. The photo shows a side view of the beam emerging from the objective, propagating through the cell and scattered by a submicron particle solution.

Fig. 7.
Fig. 7.

Experimental results taken on four dilute suspensions of scattering particles and reported in the x–y color plane: white lines represent the iso-γ curves obtained from the transmission measurements, while triangles and circles indicate the color points obtained from the scattering measured at 90°. The values of γ indicated in the figure refer to samples Silica B (γ=2.13), Silica A, and Latex 152 nm (γ=1.89), Latex 200 nm (γ=1.66). The white and Rayleigh points are represented with open square symbols. Note that color points derived from scattering measurements lie accurately on the iso-γ curves.

Equations (11)

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

T(λ)PT(λ)P0(λ)=eνσ(λ)L,
σ(λ)=4πdσdΩ(λ,θ)dΩ,
dPSdΩ(θ,λ)P0(λ)LνdσdΩ(θ,λ),
D=n1d,
m=n2n1,
σ(λ)=π4D2n12Qsca(λ,D,m),
dσdΩ(θ,λ)=(λ2πn1)2F(θ,λ,D,m),
σ(λ)=23π5D6n12λ4(m21m2+2)2,
dσdΩ(θ,λ)=π44D6n12λ4(m21m2+2)2(1+cos2θ)2.
χ(θ)=s.d.[dσ/dΩσ]dσ/dΩσ,
γσ(λ1)σ(λ2)=log[T(λ1)]log[T(λ2)],

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