Abstract

The pseudospectral time-domain (PSTD) method is a powerful approach for computing the single-scattering properties of arbitrarily shaped particles with small-to-moderate-sized parameters. In the PSTD method, the spatial derivative approximation based on the spectral method is more accurate than its counterpart based on the finite-difference technique. Additionally, the PSTD method can substantially diminish accumulated errors that increase with the spatial scale and temporal duration of simulation. We report on the application of the PSTD method to the scattering of light by nonspherical ice particles. The applicability of the PSTD method is validated against the Lorenz–Mie theory and the T-matrix method. The phase functions computed from the PSTD method and the Lorenz–Mie theory agree well for size parameters as large as 80. Furthermore, the PSTD code is also applied to the scattering of light by nonspherical ice crystals, namely, hollow hexagonal columns and aggregates, which are frequently observed in cirrus clouds. The phase functions computed from the PSTD method are compared with the counterparts computed from the finite-difference time-domain (FDTD) method for a size parameter of 20 and an incident wavelength of 3.7μm. The comparisons show good agreement between the two methods.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. Q. Fu, W. Sun, and P. Yang, “On model of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelength,” J. Atmos. Sci. 56, 2937-2947 (1999).
    [CrossRef]
  18. P. Yang, G. W. Kattawar, K. N. Liou, and J. Q. Lu, “Comparison of Cartesian grid configurations for applying the finite-difference time domain method to electromagnetic scattering by dielectric particles,” Appl. Opt. 43, 4611-4624 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  22. Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158-165 (1997).
    [CrossRef]
  23. Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1044-1055 (1998).
    [CrossRef]
  24. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  25. Q. H. Liu, “PML and PSTD algorithm for arbitrary lossy anisotropic media,” IEEE Microw. Guid. Wave Lett. 9, 48-50 (1999).
    [CrossRef]
  26. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  27. B. Tian and Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithm,” Prog. Electromagn. Res. 28, 259-279 (2000).
    [CrossRef]
  28. B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216-230 (1997).
    [CrossRef]
  29. B. Yang and J. S. Hesthaven, “Multidomain pseudospectral computation of Maxwell's equations in 3-D general curvilinear coordinates,” Appl. Numer. Math. 33, 281-289 (2000).
    [CrossRef]
  30. J. W. Cooley and J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
    [CrossRef]
  31. X. Gao, M. S. Mirotznik, and D. W. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE Trans. Antennas Propag. 52, 1665-1671 (2004).
    [CrossRef]
  32. G. X. Fan and Q. H. Liu, “Pseudospectral time-domain algorithm applied to electromagnetic scattering from electrically large objects,” Microwave Opt. Technol. Lett. 29, 123-125 (2001).
    [CrossRef]
  33. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
    [CrossRef]
  34. W. Sun, N. G. Loeb, and Q. Fu, “Finite-difference time domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41, 5728-5743 (2002).
    [CrossRef] [PubMed]
  35. P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, and S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532-1547 (2001).
    [CrossRef]

2005

2004

2003

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 775-824 (2003).
[CrossRef]

2002

2001

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, and S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532-1547 (2001).
[CrossRef]

G. X. Fan and Q. H. Liu, “Pseudospectral time-domain algorithm applied to electromagnetic scattering from electrically large objects,” Microwave Opt. Technol. Lett. 29, 123-125 (2001).
[CrossRef]

A. J. Baran, S. Haveman, P. N. Francis, and P. Yang, “A study of the absorption and extinction properties of hexagonal ice columns and plates in random and preferred orientation, using exact T-matrix theory and aircraft observations of cirrus,” J. Quant. Spectrosc. Radiat. Transfer 70, 505-518 (2001).
[CrossRef]

2000

M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026-1031 (2000).
[CrossRef]

B. Tian and Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithm,” Prog. Electromagn. Res. 28, 259-279 (2000).
[CrossRef]

B. Yang and J. S. Hesthaven, “Multidomain pseudospectral computation of Maxwell's equations in 3-D general curvilinear coordinates,” Appl. Numer. Math. 33, 281-289 (2000).
[CrossRef]

A. J. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 5, 916-938 (2000).
[CrossRef]

1999

W. Sun, Q. Fu, and Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions,” Appl. Opt. 38, 3141-3151 (1999).
[CrossRef]

Q. Fu, W. Sun, and P. Yang, “On model of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelength,” J. Atmos. Sci. 56, 2937-2947 (1999).
[CrossRef]

Q. H. Liu, “PML and PSTD algorithm for arbitrary lossy anisotropic media,” IEEE Microw. Guid. Wave Lett. 9, 48-50 (1999).
[CrossRef]

1998

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1044-1055 (1998).
[CrossRef]

P. Yang and K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223-248 (1998).

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

1997

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216-230 (1997).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158-165 (1997).
[CrossRef]

1996

1995

J. Iaquinta, H. Isaka, and P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401-1413 (1995).
[CrossRef]

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

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

1994

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

1993

1989

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3-19 (1989).
[CrossRef]

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044-3050 (1989).
[CrossRef] [PubMed]

1966

S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1965

J. W. Cooley and J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
[CrossRef]

Appl. Numer. Math.

B. Yang and J. S. Hesthaven, “Multidomain pseudospectral computation of Maxwell's equations in 3-D general curvilinear coordinates,” Appl. Numer. Math. 33, 281-289 (2000).
[CrossRef]

Appl. Opt.

W. Sun, N. G. Loeb, and Q. Fu, “Finite-difference time domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41, 5728-5743 (2002).
[CrossRef] [PubMed]

P. Yang, B.-C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, M. I. Mishchenko, D. M. Winker, and S. L. Nasiri, “Asymptotic solutions for optical properties of large particles with strong absorption,” Appl. Opt. 40, 1532-1547 (2001).
[CrossRef]

P. Yang, G. W. Kattawar, K. N. Liou, and J. Q. Lu, “Comparison of Cartesian grid configurations for applying the finite-difference time domain method to electromagnetic scattering by dielectric particles,” Appl. Opt. 43, 4611-4624 (2004).
[CrossRef] [PubMed]

M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39, 1026-1031 (2000).
[CrossRef]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780-2788 (1993).
[CrossRef] [PubMed]

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044-3050 (1989).
[CrossRef] [PubMed]

P. Yang and K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568-6584 (1996).
[CrossRef] [PubMed]

W. Sun, Q. Fu, and Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions,” Appl. Opt. 38, 3141-3151 (1999).
[CrossRef]

Contrib. Atmos. Phys.

P. Yang and K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223-248 (1998).

IEEE Microw. Guid. Wave Lett.

Q. H. Liu, “PML and PSTD algorithm for arbitrary lossy anisotropic media,” IEEE Microw. Guid. Wave Lett. 9, 48-50 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

X. Gao, M. S. Mirotznik, and D. W. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE Trans. Antennas Propag. 52, 1665-1671 (2004).
[CrossRef]

S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1044-1055 (1998).
[CrossRef]

J. Atmos. Sci.

Q. Fu, W. Sun, and P. Yang, “On model of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelength,” J. Atmos. Sci. 56, 2937-2947 (1999).
[CrossRef]

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystal,” J. Atmos. Sci. 53, 2813-2825 (1996).
[CrossRef]

J. Iaquinta, H. Isaka, and P. Personne, “Scattering phase function of bullet rosette ice crystals,” J. Atmos. Sci. 52, 1401-1413 (1995).
[CrossRef]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3-19 (1989).
[CrossRef]

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

A. J. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 5, 916-938 (2000).
[CrossRef]

J. Comput. Phys.

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

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216-230 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 775-824 (2003).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

A. J. Baran, S. Haveman, P. N. Francis, and P. Yang, “A study of the absorption and extinction properties of hexagonal ice columns and plates in random and preferred orientation, using exact T-matrix theory and aircraft observations of cirrus,” J. Quant. Spectrosc. Radiat. Transfer 70, 505-518 (2001).
[CrossRef]

Math. Comput.

J. W. Cooley and J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
[CrossRef]

Microwave Opt. Technol. Lett.

G. X. Fan and Q. H. Liu, “Pseudospectral time-domain algorithm applied to electromagnetic scattering from electrically large objects,” Microwave Opt. Technol. Lett. 29, 123-125 (2001).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158-165 (1997).
[CrossRef]

Opt. Express

Part. Part. Syst. Charact.

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

Prog. Electromagn. Res.

B. Tian and Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithm,” Prog. Electromagn. Res. 28, 259-279 (2000).
[CrossRef]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29-60.
[CrossRef]

P. Yang and K. N. Liou, “Light scattering and absorption by nonspherical ice crystals,” in Light Scattering Reviews: Single and Multiple Light Scattering, A.Kokhanovsky, ed. (Springer-Praxis, 2006), pp. 31-71.

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Figures (6)

Fig. 1
Fig. 1

Comparison of the phase functions computed from the PSTD method and the Lorenz–Mie theory for a sphere with four grid spatial resolutions at a refractive index of 1.2 + 0 i and a size parameter of 20.

Fig. 2
Fig. 2

Comparison of the Lorenz–Mie and PSTD solutions for the phase matrix elements associated with polarization for a sphere with a grid spatial resolution of 16 grid points per wavelength at a refractive index of 1.2 + 0 i and a size parameter of 20.

Fig. 3
Fig. 3

Comparison of the phase functions computed from the PSTD method and the Lorenz–Mie theory for spheres with size parameters of 30 and 50.

Fig. 4
Fig. 4

Comparison of the phase functions computed from the PSTD method and the Lorenz–Mie theory for spheres with a size parameter of 80. The spatial resolution is 8. The refractive index is 1.05 + i 0 .

Fig. 5
Fig. 5

Comparison of the PSTD and T-matrix solutions for the phase functions for spheroids and circular cylinders.

Fig. 6
Fig. 6

Comparison of the PSTD and T-matrix solutions for the phase functions for hollow hexagonal and aggregate ice crystals.

Tables (1)

Tables Icon

Table 1 Extinction Efficiencies Corresponding to the Phase Functions Shown in Figs. 3, 4

Metrics