Abstract

We develop a numerical algorithm for calculating the light-scattering properties of small particles of arbitrary shape on the basis of a method involving surface integral equations. The calculation error was estimated by performing a comparison between the proposed method and the exact Mie method with regard to the extinction efficiency factor, and the results show that the error is less than 1% when four or more nodes per wavelength are set on the surface of a spherical particle. The accuracy fluctuates in accordance with the distribution of nodal points on the particle surface with respect to the direction of propagation of the incident light. From our examinations, it is shown that the polar incidence alignment yields higher accuracy than equator incidence when a “latitude–longitude” type of mesh generation is adopted. The electric currents on the particle surface and the phase functions of all scattering directions are shown for particles shaped as spheres or hexagonal columns. It is shown that the phase function for a hexagonal column has four or eight cold spots. The phase function of a randomly oriented hexagonal column shows halolike peaks with size parameters of up to 20. This method can be applied to particles with a size parameter of up to about 20 without using the symmetry characteristic of the particle.

© 2009 Optical Society of America

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References

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  1. K. N. Liou and Y. Takano, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818-837 (1995).
    [CrossRef]
  2. A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780-2788 (1993).
    [CrossRef] [PubMed]
  3. J. I. Peltoniemi, K. Lumme, K. Muinonen, and W. M. Irnine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088-4095 (1989).
    [CrossRef] [PubMed]
  4. K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
    [CrossRef]
  5. A. Macke, M. I. Mishchenko, K. Muinonen, and B. E. Carlson, “Scattering of light by large nonspherical particles--approximation versus T-matrix method,” Opt. Lett. 20, 1934-1936(1995).
    [CrossRef] [PubMed]
  6. P. Yang and K. N. Liou, “Light-scattering by hexagonal ice crystals--comparison of finite-difference time-domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162-176(1995).
    [CrossRef]
  7. P. Yang, H. L. Wei, H. L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44, 5512-5523 (2005).
    [CrossRef] [PubMed]
  8. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Leipzig. Ann. Phys. 330, 377-445 (1908).
    [CrossRef]
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    [CrossRef]
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    [PubMed]
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    [CrossRef]
  12. M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
    [CrossRef]
  13. P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072-2085(1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. Y. Mano, “Exact solution of electromagnetic scattering by a three-dimensional hexagonal ice column obtained with the boundary-element method,” Appl. Opt. 39, 5541-5546(2000).
    [CrossRef]
  17. P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
    [CrossRef]
  18. O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
    [CrossRef]
  19. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
  20. J. D. Jackson, Classical Electrodynamics (Wiley, 1962)
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    [CrossRef]

2008 (2)

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693-2703 (2008).
[CrossRef]

2007 (1)

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

2006 (1)

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

2005 (1)

2000 (1)

1999 (1)

1996 (3)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072-2085(1996).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

1995 (3)

1994 (1)

1993 (1)

1989 (1)

1975 (1)

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Leipzig. Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Asano, S.

Baum, B. A.

Carlson, B. E.

Draine, B. T.

Dubovik, O.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Eck, T. F.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Fast, P.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

Flatau, P. J.

Fu, Q.

Holben, B. N.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Hu, Y. X.

Huang, H. L.

Irnine, W. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962)

Kattawar, G. W.

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

P. Yang, H. L. Wei, H. L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44, 5512-5523 (2005).
[CrossRef] [PubMed]

Khlebtsov, N. G.

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

Lapyonok, T.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Leon , J.-F.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Li, C. H.

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

Liou, K. N.

Lumme, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, and W. M. Irnine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088-4095 (1989).
[CrossRef] [PubMed]

Macke, A.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

Mano, Y.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Leipzig. Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Mishchenko, M.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Mishchenko, M. I.

Muinonen, K.

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).

Muñoz, O.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Nousiainen, T.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

Peltoniemi, J. I.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, and W. M. Irnine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088-4095 (1989).
[CrossRef] [PubMed]

Sinyuk, A.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Slutsker, I.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Sorokin, M.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Takano, Y.

K. N. Liou and Y. Takano, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818-837 (1995).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

van der Zande, W. J.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Veihelmann, B.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Videen, G.

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

Volten, H.

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

Wei, H. L.

Wriedt, T.

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

Yamamoto, G.

Yang, P.

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

P. Yang, H. L. Wei, H. L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44, 5512-5523 (2005).
[CrossRef] [PubMed]

P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072-2085(1996).
[CrossRef]

P. Yang and K. N. Liou, “Light-scattering by hexagonal ice crystals--comparison of finite-difference time-domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162-176(1995).
[CrossRef]

Zakharova, N. T.

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

Zhai, P. W.

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

Appl. Opt. (6)

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195(1955).
[CrossRef]

J. Atmos. Sci. (1)

K. N. Liou and Y. Takano, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818-837 (1995).
[CrossRef]

J. Geophys. Res. (1)

O. Dubovik, A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Muñoz, B. Veihelmann, W. J. van der Zande, J.-F. Leon , M. Sorokin, and I. Slutsker, “Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust,” J. Geophys. Res. 111, D11208 (2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transfer (4)

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, and J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577-601 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

M. I. Mishchenko, G. Videen, N. G. Khlebtsov, T. Wriedt, and N. T. Zakharova, “Comprehensive T-matrix reference database: a 2006-07 update,” J. Quant. Spectrosc. Radiat. Transfer 109, 1447-1460 (2008).
[CrossRef]

P. W. Zhai, C. H. Li, G. W. Kattawar, and P. Yang, “FDTD far-field scattering amplitudes: comparison of surface and volume integration methods,” J. Quant. Spectrosc. Radiat. Transfer 106, 590-594 (2007).
[CrossRef]

Leipzig. Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Leipzig. Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Opt. Lett. (1)

Other (2)

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).

J. D. Jackson, Classical Electrodynamics (Wiley, 1962)

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Figures (8)

Fig. 1
Fig. 1

Models of a sphere and a hexagonal column with normal vectors at all matching nodal points. A parametric spline is used for defining the particle size. k 0 denotes the incident wavenumber.

Fig. 2
Fig. 2

Definition of the hexagonal column dimension. Aspect ratio is described by L / D .

Fig. 3
Fig. 3

Phase function of a hexagonal column obtained by SIEM/M. The obtained curve corresponds to that in Fig. 6(a) in Mano [16]. The conditions of the particle are m ˜ = ( 1.3 , 0.0 ) , L / D = 0.866 (corresponding to L / D = 1.0 by the Mano definition), and π L / λ = 3.4 .

Fig. 4
Fig. 4

(a), (b)  Q ext as a function of the size parameter α and (c), (d) relative error of Q ext as obtained by SIEM/M against the exact solution obtained with the Mie theory as a function of the number of matching nodal point for electromagnetic waves with (a), (c) polar incidence and (b), (d) equator incidence.

Fig. 5
Fig. 5

Real part of the obtained electric currents J for the spherical particle in the case of (a) polar incidence (PI) and (b) equator incidence (EI), with complex index of refraction m ˜ = ( 1.395 , 6.99 × 10 3 ) . The particle was rotated in such a way that the incident light propagated toward the target particle from the front-right part in each viewgraph. The total number of matching nodal points is 3136.

Fig. 6
Fig. 6

Phase functions for (a) a sphere and (b) a hexagonal column with an aspect ratio of L / D = 1.0 for all scattering directions. The center and the circumference of each panel denote forward and backward scattering, respectively. The size parameters α are 1, 5, 9, 13, 17, and 21, with a refractive index m ˜ = ( 1.395 , 6.99 × 10 3 ) . The total number of matching nodal points was 2408 ( = 43 × 56 ) for the sphere and 2352 ( = 49 × 48 ) for the hexagonal column.

Fig. 7
Fig. 7

Geometrical alignment of the incident electromagnetic waves and the hexagonal column used in Fig. 6.

Fig. 8
Fig. 8

Phase function of Mie (lower curves) and hexagonal column particles with L / D = 1.0 (upper curves) with a size parameter (a)  α = 1 , (b) 5, (c 9, and (d) 21. Phase functions were normalized by 0.01 (Mie) and 1.0 (SIEM/M). The complex index of refraction is m ˜ = ( 1.313 , 0.0 ) . Phase functions parallel (thin dashed curves) and perpendicular (thin solid curves) to E inc are shown. Azimuthally averaged results of the phase functions are also shown by thick solid curves.

Tables (1)

Tables Icon

Table 1 Average Number of Nodes per Wavelength Along s Parameter of the Spline Function at the Number of Matching Nodal Points

Equations (15)

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

i ( r ) × E inc ( r ) = 1 2 ( m ˜ 2 + 1 ) K ( r ) i ( r ) × s { j k 0 2 J ( r ) ( m ˜ 2 G 1 G 0 ) + k 0 K ( r ) × ( m ˜ 2 G 1 G 0 ) + j ( J ( r ) · ) ( G 1 G 0 ) } d s ,
i ( r ) × H inc ( r ) = J ( r ) i ( r ) × s { j k 0 2 K ( r ) ( m ˜ 2 G 1 G 0 ) k 0 J ( r ) × ( G 1 G 0 ) + j ( K ( r ) · ) ( G 1 G 0 ) } d s ,
G 1 ( r , r ) = e j m ˜ k 0 | r r | 4 π k 0 | r r | ,
G 0 ( r , r ) = e j k 0 | r r | 4 π k 0 | r r | .
J ( r ) = m = 1 M a m f m ( r ) ,
K ( r ) = m = 1 M b m f m ( r ) ,
Δ s = | r ( s , t ) s × r ( s , t ) t | d s d t .
y = Zx .
F ( r ) = j k 0 2 4 π [ i r × ξ i r × s J exp ( j k 0 r · i r ) d s + i r × s K exp ( j k 0 r · i r ) d s ] ,
C s = 1 k 0 2 A 0 2 | F | 2 d Ω ,
C e = 4 π k 0 2 A 0 2 Im { A 0 · F ( r 0 ) } ,
Q ext = C e / ( π r 2 ) ,
Q sca = C s / ( π r 2 ) .
α k 0 r = 2 π r λ .
P normalized = c × P P sin θ d θ ,

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