Abstract

This scattering of light by small particles embedded in a continuous transparent medium is influenced not only by the bulk optical properties of the particles and the medium but also by the size, shape, and spatial arrangement of the particles—that is, by the microstructure. If the particles are close together, as in agglomerated coatings or stereolithographic suspensions, interactions between the radiation fields of adjacent particles can lead to variations in the magnitude and spatial arrangement of the scattered light in the near and the far field, which can affect the color and hiding power of a coating, the cure depth and homogeneity in stereolithography, and the threshold intensity for stimulated emission in random lasers. Our calculations of the near- and the far-field scattering distribution for 200-nm TiO2 spheres in pairs of various orientations and in an ordered array of five particles show that, depending on the orientation of the particles with respect to the incident light, these interactions can either increase or decrease the scattering efficiency, the isotropy of the scattering, and the magnitude of the electric field strength within the matrix and the particles. In the mid-visible range, two particles in line increase the backscattering fraction by 28% and the scattering strength by 38% over that of a single particle, whereas if the particles are in the diagonal configuration the backscattering fraction and scattering strength are actually reduced by addition of the second particle. At shorter or longer wavelengths the backscattering fraction is reduced regardless of the location of the second particle, by as much as 60% when five particles are arranged in the zigzag configuration. These results are surprising in that it is generally assumed that multiple scattering enhances backscattering. Simple models of multiple scattering or scattering of two particles as a single, larger particle are inadequate to explain these results.

© 2001 Optical Society of America

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References

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  1. D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).
  2. G. Mie, “Beitrage zur Optik Truber Medien Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
    [CrossRef]
  5. E. S. Thiele, R. H. French, “Computation of light scattering by anisotropic spheres of rutile titania,” Adv. Mater. 10, 1271–1276 (1998).
    [CrossRef]
  6. E. S. Thiele, “Light scattering by high and low contrast particulate systems,” Ph.D. dissertation (Materials Science Department, University of Pennsylvania, Philadelphia, 1998).
  7. L. E. McNeil, R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89, 1898–1906 (2001).
    [CrossRef]
  8. M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of the Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
    [CrossRef]
  9. The MieTab program was obtained from A. Miller, Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003 (1999), amiller@nmsu.edu.

2001 (1)

L. E. McNeil, R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89, 1898–1906 (2001).
[CrossRef]

1998 (2)

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

E. S. Thiele, R. H. French, “Computation of light scattering by anisotropic spheres of rutile titania,” Adv. Mater. 10, 1271–1276 (1998).
[CrossRef]

1993 (1)

D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).

1908 (1)

G. Mie, “Beitrage zur Optik Truber Medien Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Bohren, C. F.

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

Buchacek, R. J.

D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).

Dickinson, J. G.

D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).

Fields, D. P.

D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).

French, R. H.

L. E. McNeil, R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89, 1898–1906 (2001).
[CrossRef]

E. S. Thiele, R. H. French, “Computation of light scattering by anisotropic spheres of rutile titania,” Adv. Mater. 10, 1271–1276 (1998).
[CrossRef]

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

Huffman, D. R.

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

McNeil, L. E.

L. E. McNeil, R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89, 1898–1906 (2001).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik Truber Medien Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Ribarsky, M. W.

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of the Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

Thiele, E. S.

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

E. S. Thiele, R. H. French, “Computation of light scattering by anisotropic spheres of rutile titania,” Adv. Mater. 10, 1271–1276 (1998).
[CrossRef]

E. S. Thiele, “Light scattering by high and low contrast particulate systems,” Ph.D. dissertation (Materials Science Department, University of Pennsylvania, Philadelphia, 1998).

Adv. Mater. (1)

E. S. Thiele, R. H. French, “Computation of light scattering by anisotropic spheres of rutile titania,” Adv. Mater. 10, 1271–1276 (1998).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beitrage zur Optik Truber Medien Speziell Kolloidaler Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

J. Am. Ceram. Soc. (1)

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

J. Appl. Phys. (1)

L. E. McNeil, R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89, 1898–1906 (2001).
[CrossRef]

J. Oil Colour Chem. Assoc. (1)

D. P. Fields, R. J. Buchacek, J. G. Dickinson, “Maximum TiO2 hiding power—the challenge,” J. Oil Colour Chem. Assoc. 2, 87–93 (1993).

Other (4)

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

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of the Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

The MieTab program was obtained from A. Miller, Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003 (1999), amiller@nmsu.edu.

E. S. Thiele, “Light scattering by high and low contrast particulate systems,” Ph.D. dissertation (Materials Science Department, University of Pennsylvania, Philadelphia, 1998).

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Figures (5)

Fig. 1
Fig. 1

Near-field values of the square of the electric field magnitude of the scattered wave for 200-nm TiO2 particles illuminated by 250-nm light. The images show a cross-section in the plane containing the incident direction and the particle center. The light is incident from the top edge of each image. All values are relative to the incident wave. (a) Single particle; spatial dimension 1 µm × 1 µm; scale 1.50 (red) to 5.47 × 10-4 (blue). (b) In-line configuration; spatial dimension 1 µm × 2 µm; scale 1.78 (red) to 5.94 × 10-4 (blue). (c) Side-by-side configuration; spatial dimension 1.25 µm × 1 µm; scale 1.39 (red) to 5.74 × 10-4 (blue). (d) Diagonal configuration; spatial dimension 1.25 µm × 2 µm; scale 1.50 (red) to 2.11 × 10-5 (blue). (e) Zigzag configuration; spatial dimension 1.5 µm × 2 µm; scale 1.40 (red) to 3.76 × 10-4 (blue).

Fig. 2
Fig. 2

Near-field values of the square of the electric field magnitude of the scattered wave for 200-nm TiO2 particles illuminated by 550-nm light. The images show a cross section in the plane containing the incident direction and the particle center. The light is incident from the top edge of each image. All values are relative to the incident wave. (a) Single particle; spatial dimension 1 µm × 1 µm; scale 4.19 (red) to 1.05 × 10-3 (blue). (b) In-line configuration; spatial dimension 1 µm × 2 µm; scale 8.91 (red) to 5.19 × 10-4 (blue). (c) Side-by-side configuration; spatial dimension 1.25 µm × 1 µm; scale 4.21 (red) to 3.32 × 10-3 (blue). (d) Diagonal configuration; spatial dimension 1.25 µm × 2 µm; scale 4.22 (red) to 6.50 × 10-4 (blue). (e) Zigzag configuration; spatial dimension 1.5 µm × 2 µm; scale 9.44 (red) to 2.69 × 10-4 (blue).

Fig. 3
Fig. 3

Near-field values of the square of the electric field magnitude of the scattered wave for 200-nm TiO2 particles illuminated by 850-nm light. The images show a cross-section in the plane containing the incident direction and the particle center. The light is incident from the top edge of each image. All values are relative to the incident wave. (a) Single particle; spatial dimension 1 µm × 1 µm; scale 0.931 (red) to 1.82 × 10-3 (blue). (b) In-line configuration; spatial dimension 1 µm × 2 µm; 1.53 (red) to 1.32 × 10-3 (blue). (c) Side-by-side configuration; spatial dimension 1.25 µm × 1 µm; 1.73 (red) to 3.70 × 10-4 (blue). (d) Diagonal configuration; spatial dimension 1.25 µm × 2 µm; scale 1.58 (red) to 5.11 × 10-4 (blue). (e) Zigzag configuration; spatial dimension 1.5 µm × 2 µm; scale 3.81 (red) to 3.59 × 10-4 (blue).

Fig. 4
Fig. 4

Volume-normalized scattering parameter S versus particle diameter for single TiO2 particles at various wavelengths, calculated with Mie theory. Solid curve, λ = 250 nm; dotted curve; λ = 550 nm; dashed curve, λ = 850 nm.

Fig. 5
Fig. 5

Backscattering fraction B versus particle diameter for single TiO2 particles at various wavelengths, calculated with Mie theory. Solid curve, λ = 250 nm; dotted curve, λ = 550 nm; dashed curve, λ = 850 nm.

Tables (2)

Tables Icon

Table 1 Scattering Parameters for Single and Multiple 200-nm TiO2 Particles from emflex Calculationsa

Tables Icon

Table 2 Volume-Normalized Scattering Parameters S and Backscattering Fraction B for Single TiO2 Particles, Calculated from Mie Theorya

Equations (3)

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2E-μ 2Et2=0.
Csca=1I04πIscaθdΩ.
S=Csca/V.

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