Abstract

We have examined the Maxwell-Garnett, inverted Maxwell-Garnett, and Bruggeman rules for evaluation of the mean permittivity involving partially empty cells at particle surface in conjunction with the finite-difference time-domain (FDTD) computation. Sensitivity studies show that the inverted Maxwell-Garnett rule is the most effective in reducing the staircasing effect. The discontinuity of permittivity at the interface of free space and the particle medium can be minimized by use of an effective permittivity at the cell edges determined by the average of the permittivity values associated with adjacent cells. The efficiency of the FDTD computational program is further improved by use of a perfectly matched layer absorbing boundary condition and the appropriate coding technique. The accuracy of the FDTD method is assessed on the basis of a comparison of the FDTD and the Mie calculations for ice spheres. This program is then applied to light scattering by convex and concave aerosol particles. Comparisons of the scattering phase function for these types of aerosol with those for spheres and spheroids show substantial differences in backscattering directions. Finally, we illustrate that the FDTD method is robust and flexible in computing the scattering properties of particles with complex morphological configurations.

© 2000 Optical Society of America

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References

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  1. S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  2. P. Yang, 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]
  3. P. Yang, 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]
  4. J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A 203, 385–420 (1904).
    [CrossRef]
  5. B. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  6. B. J. Berenger, “Three-dimensional perfect matched layer for the absorption of electromagnetic wave,” J. Comput. Phys. 127, 363–379 (1996).
    [CrossRef]
  7. C. A. G. Bruggeman, “Gerechung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
    [CrossRef]
  8. P. Yang, K. N. Liou, “An efficient algorithm for truncating spatial domain in modeling light scattering by finite-difference technique,” J. Comput. Phys. 140, 346–369 (1998).
    [CrossRef]
  9. D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
    [CrossRef]
  10. W. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
    [CrossRef]
  11. P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 173–221.
  12. G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
    [CrossRef]
  13. S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distribution of soil particles and spheroids,” Appl. Opt. 23, 1025–2430 (1988).
    [CrossRef]
  14. K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).
  15. T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).
  16. M. Wang, H. R. Gordan, “Estimating aerosol optical properties over the oceans with multiangle imaging spectroradiometer: some preliminary results,” Appl. Opt. 33, 4042–4057 (1994).
    [CrossRef] [PubMed]
  17. M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
    [CrossRef]
  18. M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
    [CrossRef]
  19. M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
    [CrossRef] [PubMed]
  20. R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).
  21. J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semi-empirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
    [CrossRef]

1999

1998

P. Yang, K. N. Liou, “An efficient algorithm for truncating spatial domain in modeling light scattering by finite-difference technique,” J. Comput. Phys. 140, 346–369 (1998).
[CrossRef]

1997

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

1996

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

B. J. Berenger, “Three-dimensional perfect matched layer for the absorption of electromagnetic wave,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

P. Yang, 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]

1995

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

P. Yang, 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]

1994

M. Wang, H. R. Gordan, “Estimating aerosol optical properties over the oceans with multiangle imaging spectroradiometer: some preliminary results,” Appl. Opt. 33, 4042–4057 (1994).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

B. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
[CrossRef]

1989

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

1988

1987

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

1980

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semi-empirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

1976

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

1966

S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

1935

C. A. G. Bruggeman, “Gerechung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

1904

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A 203, 385–420 (1904).
[CrossRef]

Arao, K.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Barber, P. W.

Berenger, B. J.

B. J. Berenger, “Three-dimensional perfect matched layer for the absorption of electromagnetic wave,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

B. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bruggeman, C. A. G.

C. A. G. Bruggeman, “Gerechung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

Carlson, B. E.

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

Chen, Z.

Cuzzi, J. N.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semi-empirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

Fu, Q.

Gandhi, O. P.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

Gordan, H. R.

Hill, A. C.

Hill, S. C.

Iwasaka, Y.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

Katz, D. S.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
[CrossRef]

Kobayashi, A.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Lacis, A. A.

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

Lazzi, G.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, “An efficient algorithm for truncating spatial domain in modeling light scattering by finite-difference technique,” J. Comput. Phys. 140, 346–369 (1998).
[CrossRef]

P. Yang, 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, 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]

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 173–221.

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A 203, 385–420 (1904).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

Nakajima, T.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Nakanishi, Y.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Naruse, H.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Nemoto, O.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Okada, K.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Pollack, J. B.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semi-empirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

Shiobara, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Sun, W.

Taflove, A.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
[CrossRef]

Tanaka, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Tanaka, T.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

Thiele, E. T.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

Wang, M.

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

Yamano, M.

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

Yang, P.

P. Yang, K. N. Liou, “An efficient algorithm for truncating spatial domain in modeling light scattering by finite-difference technique,” J. Comput. Phys. 140, 346–369 (1998).
[CrossRef]

P. Yang, 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, 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]

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 173–221.

Yee, S. K.

S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Zerull, R. H.

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

Ann. Phys. (Leipzig)

C. A. G. Bruggeman, “Gerechung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

Appl. Opt.

Beitr. Phys. Atmos.

R. H. Zerull, “Scattering measurements of dielectric and absorbing nonspherical particles,” Beitr. Phys. Atmos. 49, 168–188 (1976).

Geophys. Res. Lett.

M. I. Mishchenko, A. A. Lacis, B. E. Carlson, L. D. Travis, “Nonsphericity of dust-like tropospherical aerosols: implications for aerosol remote sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995).
[CrossRef]

IEEE Microwave Guided Wave Lett.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 4, 268–270 (1994).
[CrossRef]

IEEE Trans. Antennas Propag.

S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

J. Atmos. Sci.

J. B. Pollack, J. N. Cuzzi, “Scattering by nonspherical particles of size comparable to a wavelength: a new semi-empirical theory and its application to tropospheric aerosols,” J. Atmos. Sci. 37, 868–881 (1980).
[CrossRef]

J. Comput. Phys.

P. Yang, K. N. Liou, “An efficient algorithm for truncating spatial domain in modeling light scattering by finite-difference technique,” J. Comput. Phys. 140, 346–369 (1998).
[CrossRef]

B. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

B. J. Berenger, “Three-dimensional perfect matched layer for the absorption of electromagnetic wave,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

J. Geophys. Res.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

J. Meteorol. Soc. Jpn.

K. Okada, A. Kobayashi, Y. Iwasaka, H. Naruse, T. Tanaka, O. Nemoto, “Features of individual Asian dust-storm particles collected at Nagoya, Japan,” J. Meteorol. Soc. Jpn. 65, 515–521 (1987).

T. Nakajima, M. Tanaka, M. Yamano, M. Shiobara, K. Arao, Y. Nakanishi, “Aerosol optical characteristics in the yellow sand events observed in May 1982 at Nagasaki. II. Models,” J. Meteorol. Soc. Jpn. 67, 279–291 (1989).

J. Opt. Soc. Am. A

Philos. Trans. R. Soc. A

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A 203, 385–420 (1904).
[CrossRef]

Other

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 173–221.

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Figures (10)

Fig. 1
Fig. 1

Mean permittivity evaluated from Maxwell-Garnett, inverted Maxwell-Garnett, and Bruggeman rules for grid cells composed of free space and ice at 0.55 and 15 µm. The refractive indices for the two wavelengths are, respectively, 1.311 + i3.11 × 10-9 and 1.571 + i0.1756.

Fig. 2
Fig. 2

Geometry for a cell edge and four adjacent cells in a Cartesian grid.

Fig. 3
Fig. 3

Comparison of FDTD and Mie results for the phase functions of ice spheres at 15 µm for a size parameter of 1. Three approaches are used for evaluating the mean permittivity for partially empty cells located near the particle surface.

Fig. 4
Fig. 4

Same as Fig. 3 but for a size parameter of 10.

Fig. 5
Fig. 5

Comparison of the performance of three schemes to account for the discontinuity of permittivity at the interface of free space and the particle medium in conjunction with the FDTD computation of phase function for an ice sphere with a size parameter of 1.

Fig. 6
Fig. 6

Same as Fig. 5 but for a size parameter of 10.

Fig. 7
Fig. 7

Comparison of FDTD and Mie results for the phase function of ice spheres by use of grid resolutions λ/Δs = 25, 30, respectively, for optically thin (1.0925 + i0.248) and thick (1.5710 + i0.1756) cases.

Fig. 8
Fig. 8

Comparison of phase function and degree of linear polarization for randomly oriented six-faced convex aerosol particles with aspect ratios of 1, 2, and 1/2.

Fig. 9
Fig. 9

Comparison of the phase functions for oceanic aerosol particles of various shapes. For spherical particles, i.e., a/ b = 1, a power-law size distribution is employed to smooth out the resonant fluctuations. The size parameter used is x max = 10.

Fig. 10
Fig. 10

Phase functions measured by the microwave analog technique and computed from the FDTD method for randomly oriented convex and concave particles with a refractive index m = (1.5 + i0.005) and size parameters ranging from 5.9 to 17.8.

Tables (2)

Tables Icon

Table 1 Extinction Efficiencies Q e and Single-Scattering Albedos ω̃ Corresponding to the Phase Functions Shown in Figs. 3 and 4 for λ/Δs = 30a

Tables Icon

Table 2 Extinction Efficiencies Q e and Single-Scattering Albedos ω̃ Corresponding to the Phase Functions Shown in Figs. 5 and 6a

Equations (22)

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

ε¯=εmε+2εm+2fε+2εmε+2εm-fε+2εm,
ε¯=εεm+2ε+21-fεm+2εεm+2ε-1-fεm+2ε.
1Δv   Δvεr-ε¯εr+2ε¯dv=0,
f ε-ε¯ε+2ε¯+1-fεm-ε¯εm+2ε¯=0.
ε¯n+1=εεm+2fε+1-fεmε¯nfεm+1-fε+2ε¯n.
εr=mr2-mi2,  εi=2mrmi.
εrrcEr, tt+kεirEr, t=×Hr, t,
IΔxI+1ΔxJΔyJ+1Δy×Hr, tzdxdy=ΔxHxrI+1/2, J, K, t-HxrI+1/2, J+1, K, t+ΔyHyrI+1, J+1/2, K, t-HyrI, J+1/2, K, t,
IΔxI+1ΔxJΔyJ+1ΔyεrrcEzr, tt+kεirEzr, tdxdy=ε¯rI+1/2, J+1/2, Kct+kε¯iI+1/2, J+1/2, KIΔxI+1ΔxJΔyJ+1Δy Ezr, tdxdyε¯rI+1/2, J+1/2, Kct+kε¯iI+1/2, J+1/2, KEzrI+1/2, J+1/2, K, tΔxΔy,
ε¯rrI+1/2, J+1/2, KcEzrI+1/2, J+1/2, K, tt+kε¯irI+1/2, J+1/2, KEzrI+1/2, J+1/2, K, t=HxrI+1/2, J, K, t-HxrI+1/2, J+1, K, t/Δy+HyrI+1, J+1/2, K, t-HxrI, J+1/2, K, t/Δx.
ε¯rI+1/2, J+1/2, K=εr1IΔxI+1/2ΔxJΔyJ+1/2Δy+εr2I+1/2ΔxI+1ΔxJΔyJ+1/2Δy+εr3I+1/2ΔxI+1ΔxJ+1/2ΔyJ+1Δy+εr4IΔxI+1/2ΔxJ+1/2ΔyJ+1ΔyEzr, tdxdyIΔxI+1ΔxJΔyJ+1ΔyEzr, tdxdyεr1+εr2+εr3+εr4/4.
ε¯iI+1/2, J+1/2, Kεi1+εi2+εi3+εi4/4.
Ex, Ey, Ez=Exy+Exz, Eyx+Eyz, Ezx+Ezy,
Hx, Hy, Hz=Hxy+Hxz, Hyx+Hyz, Hzx+Hzy,
exp-τyytctexpτyytExy=Hzx+Hzyy,
exp-τzztctexpτzztExz=-Hyx+Hyzz,
exp-τyytctexpτyytHxy=-Ezx+Ezyy,
exp-τzztctexpτzztHxz=Eyx+Eyzz,
τyy=τy,maxy-y0/Dp,
τy,max=-p+12D lnR0°c,
τ¯yJ=1ΔyJ-1/2ΔxJ+1/2Δx τyydy=τy,maxp+1J+1/2p+1-J-1/2p+1LP+1 E field,
τ¯yJ+1/2=1ΔyJΔyJ+1Δy τyydy=τy,maxp+1J+1p+1-Jp+1LP+1 H field.

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