Abstract

The effect of clustering of small scatterers on optical properties was studied by creation of a Poisson distributed plane-parallel geometry and slow cooling of the particle system in the sense of simulated annealing in an attempt to minimize the assumed total potential energy and sample the spatial distribution during the process. The optical properties were calculated by the volume integral equation method (VIEM). The scattering results for unclustered structures with different size parameters and packing densities were also compared with those given by Monte Carlo simulation for radiative transfer. In particular, measuring the intensity distribution of the VIEM is well suited to the classic radiative transfer approach.

© 2001 Optical Society of America

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References

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  1. L. M. Zurk, L. Tsang, K. H. Ding, 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]
  2. L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
    [Crossref]
  3. L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
    [Crossref]
  4. F. J. Vesely, Computational Physics (Plenum, New York, 1994).
    [Crossref]
  5. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1974).
  6. D. S. Saxon, Lectures on the Scattering of Light (University of California, Los Angeles, Calif., 1955).
  7. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
    [Crossref]
  8. K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
    [Crossref]
  9. J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” licentiate’s thesis (Helsinki University of Technology, Helsinki, 1994).
  10. J. Rahola, “Efficient solution of dense systems of linear equations in electromagnetic scattering calculations,” Ph.D. dissertation (Helsinki University of Technology and Center for Scientific Computing, Espoo, Finland, 1996).
  11. R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–448 (1992).
    [Crossref]
  12. K. Lumme, J. Rahola, “Comparison of light scattering by stochastically rough particles, best-fit spheroids and spheres,” J. Quant. Spectrosc. Radiat. Transfer 60, 439–450 (1998).
    [Crossref]
  13. J. Rahola, “Solution of dense systems of linear equations in the discrete-dipole approximation,” SIAM J. Sci. Comput. 17, 78–89 (1996).
    [Crossref]
  14. J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [Crossref] [PubMed]
  15. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

2001 (1)

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

1998 (2)

K. Lumme, J. Rahola, “Comparison of light scattering by stochastically rough particles, best-fit spheroids and spheres,” J. Quant. Spectrosc. Radiat. Transfer 60, 439–450 (1998).
[Crossref]

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

1996 (1)

J. Rahola, “Solution of dense systems of linear equations in the discrete-dipole approximation,” SIAM J. Sci. Comput. 17, 78–89 (1996).
[Crossref]

1995 (1)

1994 (1)

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[Crossref]

1992 (1)

R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–448 (1992).
[Crossref]

1991 (1)

Balescu, R.

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1974).

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[Crossref]

Ding, K. H.

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

L. M. Zurk, L. Tsang, K. H. Ding, 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]

Draine, B. T.

Flatau, P. J.

Freund, R. W.

R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–448 (1992).
[Crossref]

Goodman, J. J.

Greffet, J.-J.

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[Crossref]

Kong, J. A.

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

Lumme, K.

K. Lumme, J. Rahola, “Comparison of light scattering by stochastically rough particles, best-fit spheroids and spheres,” J. Quant. Spectrosc. Radiat. Transfer 60, 439–450 (1998).
[Crossref]

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[Crossref]

Mareschal, P.

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

Rahola, J.

K. Lumme, J. Rahola, “Comparison of light scattering by stochastically rough particles, best-fit spheroids and spheres,” J. Quant. Spectrosc. Radiat. Transfer 60, 439–450 (1998).
[Crossref]

J. Rahola, “Solution of dense systems of linear equations in the discrete-dipole approximation,” SIAM J. Sci. Comput. 17, 78–89 (1996).
[Crossref]

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[Crossref]

J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” licentiate’s thesis (Helsinki University of Technology, Helsinki, 1994).

J. Rahola, “Efficient solution of dense systems of linear equations in electromagnetic scattering calculations,” Ph.D. dissertation (Helsinki University of Technology and Center for Scientific Computing, Espoo, Finland, 1996).

Roux, L.

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

Saxon, D. S.

D. S. Saxon, Lectures on the Scattering of Light (University of California, Los Angeles, Calif., 1955).

Shih, S. E.

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

Thibaud, J.-B.

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

Tsang, L.

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

L. M. Zurk, L. Tsang, K. H. Ding, 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]

Vesely, F. J.

F. J. Vesely, Computational Physics (Plenum, New York, 1994).
[Crossref]

Vukadinovic, N.

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

Winebrenner, D. P.

Zurk, L. M.

Astrophys. J. (1)

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[Crossref]

J. Opt. Soc. Am. A (3)

L. Tsang, K. H. Ding, S. E. Shih, J. A. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 10, 2660–2669 (1998).
[Crossref]

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 2, 374–384 (2001).
[Crossref]

L. M. Zurk, L. Tsang, K. H. Ding, 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]

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

K. Lumme, J. Rahola, “Comparison of light scattering by stochastically rough particles, best-fit spheroids and spheres,” J. Quant. Spectrosc. Radiat. Transfer 60, 439–450 (1998).
[Crossref]

Opt. Lett. (1)

SIAM J. Sci. Comput. (1)

J. Rahola, “Solution of dense systems of linear equations in the discrete-dipole approximation,” SIAM J. Sci. Comput. 17, 78–89 (1996).
[Crossref]

SIAM J. Sci. Stat. Comput. (1)

R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–448 (1992).
[Crossref]

Other (7)

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” licentiate’s thesis (Helsinki University of Technology, Helsinki, 1994).

J. Rahola, “Efficient solution of dense systems of linear equations in electromagnetic scattering calculations,” Ph.D. dissertation (Helsinki University of Technology and Center for Scientific Computing, Espoo, Finland, 1996).

F. J. Vesely, Computational Physics (Plenum, New York, 1994).
[Crossref]

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1974).

D. S. Saxon, Lectures on the Scattering of Light (University of California, Los Angeles, Calif., 1955).

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Pigment structure with 150 Poisson-distributed particles: size parameter 1.2; midpoints in a cylinder with radius 10.0 and height 3.0; effective packing density, 7%; optical thickness (over 10 realizations), 0.40.

Fig. 2
Fig. 2

Same structure as in Fig. 1 clustered by simulated annealing down to a virtual temperature T = 0.14 (dT = 0.05%). Optical thickness (over 10 realizations), 0.35.

Fig. 3
Fig. 3

Pigment structure with 300 Poisson-distributed particles; size parameter 1.2; midpoints in a cylinder with radius 10.0 and height 3.0. Thus the packing density has been doubled, to 14%, with respect to that in Figs. 1 and 2. Optical thickness (over 10 realizations), 0.86.

Fig. 4
Fig. 4

Same structure as in Fig. 3 clustered by simulated annealing down to a virtual temperature T = 0.14. Optical thickness (over 10 realizations), 0.70.

Fig. 5
Fig. 5

Calculated relative intensities (log10 of matrix element P 11). Unclustered structures in the left column; moderately clustered structures in the right column. Lower size parameter (1.2) in the two upper rows; the higher (1.9) in the two lower rows. Lower packing density (7%) in the first and third rows, higher density (14%) in the third and fourth rows. Three random structure realizations are represented by the dashed curves; the average, by the solid curves.

Fig. 6
Fig. 6

Calculated polarizations (-P 12/P 11 × 100%) that correspond to the relative intensities shown in Fig. 5.

Fig. 7
Fig. 7

Scattering coefficient as a function of packing density and clustering. Size parameter, 1.9.

Fig. 8
Fig. 8

Relative intensities as functions of scattering angle. Solid curves, results of the Monte Carlo simulation; dotted curves, solutions given by the integral equation. Optical thickness, τ0; packing density, p; size parameter, x 1.

Fig. 9
Fig. 9

Polarizations as functions of scattering angle. Solid curves, results of Monte Carlo simulation; dotted curves, solution given by the integral equation. Optical thickness, τ0, packing density, p; size parameter, x 1.

Tables (1)

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Table 1 Scattering Coefficients for the Main Particle Configurations Treated in This Paper

Equations (2)

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Csca=1Iinc4πdnscaIscansca,
Isca=Z11Iinc+Z12Qinc+Z13Uinc+Z14Vinc,

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