Abstract

Two grid configurations can be employed to implement the finite-difference time-domain (FDTD) technique in a Cartesian system. One configuration defines the electric and magnetic field components at the cell edges and cell-face centers, respectively, whereas the other reverses these definitions. These two grid configurations differ in terms of implication on the electromagnetic boundary conditions if the scatterer in the FDTD computation is a dielectric particle. The permittivity has an abrupt transition at the cell interface if the dielectric properties of two adjacent cells are not identical. Similarly, the discontinuity of permittivity is also observed at the edges of neighboring cells that are different in terms of their dielectric constants. We present two FDTD schemes for light scattering by dielectric particles to overcome the above-mentioned discontinuity on the basis of the electromagnetic boundary conditions for the two Cartesian grid configurations. We also present an empirical approach to accelerate the convergence of the discrete Fourier transform to obtain the field values in the frequency domain. As a new application of the FDTD method, we investigate the scattering properties of multibranched bullet-rosette ice crystals at both visible and thermal infrared wavelengths.

© 2004 Optical Society of America

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  1. K. N. Liou, Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).
  2. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).
  3. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
    [CrossRef]
  4. K. S. Yee, “Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. AP-14, 302–307 (1966).
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  6. A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).
  7. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  8. 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]
  9. P. Yang, K. N. Liou, M. I. Mishchenko, B.-C. Gao, “An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
    [CrossRef]
  10. W. Sun, Q. Fu, 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]
  11. W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
    [CrossRef]
  12. K. Aydin, “Centimeter and millimeter wave scattering from nonspherical hydrometeors,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 451–479.
    [CrossRef]
  13. K. Aydin, C. Tang, “Millimeter wave radar from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sens. 35, 140–146 (1997).
    [CrossRef]
  14. S. C. Hill, G. Videen, W. B. Sun, Q. Fu, “Scattering and internal fields of a microsphere that contains a satiable absorber: finite-difference time domain simulations,” Appl. Opt. 40, 5487–5494 (2001).
    [CrossRef]
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    [CrossRef]
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  19. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
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    [CrossRef]
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    [CrossRef]
  26. A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
    [CrossRef]
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  29. M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
    [CrossRef]
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2003 (3)

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

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

2001 (1)

2000 (3)

1999 (2)

M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
[CrossRef]

W. Sun, Q. Fu, 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]

1998 (1)

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

1997 (1)

K. Aydin, C. Tang, “Millimeter wave radar from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sens. 35, 140–146 (1997).
[CrossRef]

1996 (1)

1995 (2)

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

C. M. Furse, O. P. Gandhi, “Why the DFT is faster than the FFT for FDTD time-to-frequency domain conversions,” IEEE Microwave Guid. Wave Lett. 5, 326–328 (1995).
[CrossRef]

1994 (1)

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

1992 (1)

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

1984 (1)

1966 (1)

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

Aydin, K.

K. Aydin, C. Tang, “Millimeter wave radar from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sens. 35, 140–146 (1997).
[CrossRef]

K. Aydin, “Centimeter and millimeter wave scattering from nonspherical hydrometeors,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 451–479.
[CrossRef]

Baum, B. A.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Berenger, J. P.

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

Bohren, C. F.

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

Ceburnis, D.

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Chamaillard, K.

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Chen, Z.

Chou, M. D.

M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
[CrossRef]

Fu, Q.

Furse, C. M.

C. M. Furse, O. P. Gandhi, “Why the DFT is faster than the FFT for FDTD time-to-frequency domain conversions,” IEEE Microwave Guid. Wave Lett. 5, 326–328 (1995).
[CrossRef]

Gandhi, O. P.

C. M. Furse, O. P. Gandhi, “Why the DFT is faster than the FFT for FDTD time-to-frequency domain conversions,” IEEE Microwave Guid. Wave Lett. 5, 326–328 (1995).
[CrossRef]

Gao, B.-C.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).

Heidinger, A.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Heymsfield, A. J.

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

Hill, S. C.

Hu, Y. X.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Huffman, D. R.

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

Iaquinta, J.

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

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

Isaka, H.

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

Jennings, S. G.

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Jurgens, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

Kahnert, F. M.

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

Kleefeld, C.

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Kratz, D. P.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Kunz, K. S.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Lee, K. T.

M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, M. I. Mishchenko, B.-C. Gao, “An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[CrossRef]

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

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, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp 174–221.

K. N. Liou, Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).

Luebbers, R. J.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Mishchenko, M. I.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

P. Yang, K. N. Liou, M. I. Mishchenko, B.-C. Gao, “An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Mlynczak, M. G.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Moore, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

Pascher, W.

R. Pregla, W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Personne, P.

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

Pregla, R.

R. Pregla, W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Schneider, J. B.

K. L. Shlager, J. B. Schneider, “A survey of the finite-difference time domain literature,” in Advances in Computational Electrodynamics, A. Taflove, ed. (Artech House, Boston, Mass., 1998), pp. 1–62.

Shlager, K. L.

K. L. Shlager, J. B. Schneider, “A survey of the finite-difference time domain literature,” in Advances in Computational Electrodynamics, A. Taflove, ed. (Artech House, Boston, Mass., 1998), pp. 1–62.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Sun, W.

Sun, W. B.

Taflove, A.

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

A. Taflove, Advances in Computational Electromagnetics (Artech House, Boston, Mass., 1998).

A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (Institute of Electrical and Electronics Engineers New York, 1994).

Tang, C.

K. Aydin, C. Tang, “Millimeter wave radar from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sens. 35, 140–146 (1997).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Tsay, S.-C.

M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
[CrossRef]

Umashankar, K.

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

Videen, G.

Warren, S.

Wei, H. L.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Wiscombe, W. J.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

Yang, P.

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

P. Yang, K. N. Liou, M. I. Mishchenko, B.-C. Gao, “An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[CrossRef]

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

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, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp 174–221.

Yee, K. S.

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

Yoon, Y. J.

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Appl. Opt. (5)

Contrib. Atmos. Phys. (1)

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

IEEE Microwave Guid. Wave Lett. (1)

C. M. Furse, O. P. Gandhi, “Why the DFT is faster than the FFT for FDTD time-to-frequency domain conversions,” IEEE Microwave Guid. Wave Lett. 5, 326–328 (1995).
[CrossRef]

IEEE Trans. Antennas Propagat. (2)

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time domain modeling of curved surfaces,” IEEE Trans. Antennas Propagat. 40, 357–366 (1992).
[CrossRef]

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

IEEE Trans. Geosci. Remote Sens. (1)

K. Aydin, C. Tang, “Millimeter wave radar from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sens. 35, 140–146 (1997).
[CrossRef]

J. Atmos. Sci. (2)

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

A. J. Heymsfield, J. Iaquinta, “Cirrus crystal terminal velocities,” J. Atmos. Sci. 57, 916–938 (2000).
[CrossRef]

J. Clim. (1)

M. D. Chou, K. T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmospheric models,” J. Clim. 12, 159–169 (1999).
[CrossRef]

J. Comput. Phys. (1)

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

J. Geophys. Res. (1)

P. Yang, M. G. Mlynczak, H. L. Wei, D. P. Kratz, B. A. Baum, Y. X. Hu, W. J. Wiscombe, A. Heidinger, M. I. Mishchenko, “Spectral signature of cirrus clouds in the far-infrared region: single-scattering calculation and radiative sensitivity study,” J. Geophys. Res. 108D, 4569, 10.1029/2002JD003291 (2003).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (2)

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

K. Chamaillard, S. G. Jennings, C. Kleefeld, D. Ceburnis, Y. J. Yoon, “Light backscattering and scattering by nonspherical sea-salt acrosols,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 577–597 (2003).
[CrossRef]

Other (13)

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp 174–221.

K. L. Shlager, J. B. Schneider, “A survey of the finite-difference time domain literature,” in Advances in Computational Electrodynamics, A. Taflove, ed. (Artech House, Boston, Mass., 1998), pp. 1–62.

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

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K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

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Figures (14)

Fig. 1
Fig. 1

Two Cartesian grid configurations used to implement the FDTD method. The configuration in (a) was originally reported by Yee in 1966.

Fig. 2
Fig. 2

Side view of a cubic particle that is embedded in a Cartesian FDTD grid mesh. The electric field components normal to the particle surface cannot be defined at the locations marked X and O in the diagram because of the discontinuity of the field components at these locations.

Fig. 3
Fig. 3

Electric and magnetic fields associated with two adjacent cells that are not identical in terms of their dielectric constants.

Fig. 4
Fig. 4

Top panel: the variation of a pulse as a function of time, which is observed at the third grid point in a 1-D FDTD grid; middle panel: the real parts of F N (ω) in Eq. (24) and F c (ω) in Eq. (26); bottom panel: the imaginary parts of F N (ω) and F c (ω).

Fig. 5
Fig. 5

Variation of the y component of the electric field at the center of a sphere versus the time step used for the Fourier transform to derive the field signals in the frequency domain. The sphere is illuminated by a y-polarized incident pulse that propagates along the z axis. The complex refractive index for the scattering particle is m = 1.0925 + i0.248.

Fig. 6
Fig. 6

Phase functions computed from the FDTD technique with the modified Fourier-transform method given by Eq. (27) and the conventional Fourier-transform method by Eq. (24), which are compared with their Lorenz-Mie counterparts. The complex refractive index for the scattering particle is m = 1.0925 + i0.248.

Fig. 7
Fig. 7

Phase functions computed by use of the two grid configurations shown in Figs. 1(a) and 1(b). Also shown are the relative errors in comparison with Lorenz-Mie theory. The complex refractive index for the scattering particle is m = 1.5015 + i0.067.

Fig. 8
Fig. 8

Same as Fig. 7, except for a size parameter of x = 10.

Fig. 9
Fig. 9

Comparison of the two FDTD schemes for a case with a large refractive index (m = 8.2252 + i1.6808).

Fig. 10
Fig. 10

Comparison of the two FDTD schemes for light-scattering computations involving cubic particles. Note that the solid and dotted curves in the upper panels are essentially overlapped. The refractive index of sea-salt aerosols, m = 1.5 + i10-8, is used in the computation.

Fig. 11
Fig. 11

Morphological geometry of bullet-rosette ice crystals defined for scattering calculations.

Fig. 12
Fig. 12

Phase functions of bullet-rosette ice crystals with 1–12 branches. The wavelength is 11 μm. The size parameter for the particle is defined as x = 5, where x = 2πD/λ and D is the length of a bullet element. The complex refractive index for the scattering particle is m = 1.0925 + i0.248. The term Lorenz-Mie 1 refers to the results for the spheres that have the same volume as the bullet-rosette ice crystals with 12 branches, whereas the term Lorenz-Mie 2 refers to the results for the spheres that have the same diameters as the bullet-rosette ice crystals.

Fig. 13
Fig. 13

Nonzero phase-matrix elements for the bullet-rosette ice crystals with 1, 6, and 12 elements for a wavelength of 0.66 μm at which the refractive index of ice is 1.3078 + i1.66 × 10-8. The size parameter is x = 5. The phase-matrix elements of a sphere that has the same volume as the 12-branched bullet rosettes are also shown.

Fig. 14
Fig. 14

Same as Fig. 13, except for x = 10.

Tables (1)

Tables Icon

Table 1 Extinction Efficiency (Qe ), Single-Scattering Albedo (ω̃), and Asymmetry Factor (g) for Bullet-Rosette Ice Crystals Associated with the Phase Functions Shown in Fig. 12, a

Equations (37)

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Er, tt=cε ×Hr, t,
Hr, tt=-c×Er, t,
expωtεi/εrEr, tt=expωtεi/εrcεr ×H,
Hn+1/2r=Hn-1/2r-cΔt×Enr,
En+1r=exp-τrΔtEnr+exp-τrΔt/2cΔtεr ×Hn+1/2r,
Ezn+1i, j, k+1/2=exp-τi,j,k+1/2Δt×Ezni, j, k+1/2+exp-τi, j, k+1/2Δt/2×cΔtεri, j, k+1/2Δs×Hxn+1/2i, j-1/2, k+1/2-Hxn+1/2i, j+1/2, k+1/2+Hyn+1/2i+1/2, j, k+1/2-Hyn+1/2i-1/2, j, k+1/2,
n×E+-E-=0,
n×H+-H-=K,
n·D+-D-=ρs,
Ez+i, j, k+1/2, tt=cεi,j,k+1ΔsHx+i,j-1/2, k+1/2, t-Hx+i, j+1/2, k+1/2, t+Hy+i+1/2, j, k+1/2, t-Hy+i-1/2, j, k+1/2, t,
Ez-i, j, k+1/2, tt=cεi,j,kΔsHx-i, j-1/2, k+1/2, t-Hx-i, j+1/2, k+1/2, t+Hy-i+1/2, j, k+1/2, t-Hy-i-1/2, j, k+1/2, t,
Hx+i, j±1/2, k+1/2, t=Hx-i, j±1/2, k+1/2, t,
Hy+i±1/2, j, k+1/2, t=Hy-i±1/2, j, k+1/2, t.
Ez+i, j, k+1/2, tEz-i, j, k+1/2, t.
Ezi, j, k+1/2, t=Ez+i, j, k+1/2, t+Ez-i, j, k+1/2, t/2,
1εi, j, k+1/2=121εi,j,k+1εi,j,k+1.
Ezi, j, k+1/2, tt=cεi, j, k+1/2ΔsHxi, j-1/2, k+1/2, t-Hxi, j+1/2, k+1/2, t+Hyi+1/2, j, k+1/2, t-Hyi-1/2, j, k+1/2, t.
expiωεii, j, k+1/2/εri, j, k+1/2Ezi, j, k+1/2, tt=expiωεii, j, k+1/2/εri, j, k+1/2cεri, j, k+1/2ΔsHxi, j-1/2, k+1/2, t-Hxi, j+1/2, k+1/2, t+Hyi+1/2, j, k+1/2, t-Hyi-1/2, j, k+1/2, t,
Hxi, j+1/2, k+1/2t=-cΔs2×AB Eyx, y, z, tdy+Bc Ezx, y, z, tdz+cD Eyx, y, z, tdy+DA Ezx, y, z, tdz.
1Δs2DA Ezx, y, z, tdz=1Δs2Ez-i, j, k+1/2×Δs/2+Ez+i, j, k+1/2Δs/2=-1Δs Ezi, j, k+1/2.
Hxn+1/2i, j+1/2, k+1/2=Hxn-1/2i, j+1/2, k+1/2-cΔtΔsEyni, j+1/2, k-Eyni, j+1/2, k+1+Ezni+1, j, k+1/2-Ezni, j, k+1/2.
εi,j,k+1Ez+i, j, k+1/2=εi,j,kEz-i, j, k+1/2.
Ez+i, j, k+1/2=2εi,j,kεi,j,k+εi,j,k+1 Ezi, j, k+1/2,
Ez-i, j, k+1/2=2εi,j,kεi,j,k+εi,j,k+1 Ezi, j, k+1/2.
Ezi, j, k=εi,j,k+1εi,j,k+εi,j,k+1 Ezi, j, k+1/2+εi,j,k-1εi,j,k+εi,j,k-1 Ezi, j, k-1/2.
Ezi+1/2, j+1/2, k, tt=cεi+1/2,j+1/2,kΔsHxi,j-1/2,k+1/2,t-Hxi,j+1/2,k+1/2,t +Hyi+1/2,j,k+1/2,t -Hyi-1/2,j,k+1/2,t,
εi+1/2, j+1/2, k=εi,j,k+εi+1,j,k+εi+1,j+1,k+εi,j+1,k/4.
Ezi, j, k=Ezi-1/2, j-1/2,k+Ezi+1/2, j-1/2, k+Ezi+1/2, j+1/2, k+Ezi-1/2,j+1/2, k/4.
FNω=n=0N fn expiωnΔt,
FN0ω=n=0N0 fn expiωnΔt,
FN0+1ω=n=0N0+1 fn expiωnΔt,
FN0+Lω=n=0N0+L fn expiωnΔt.
Fcω=FN0ω+FN0+1ω++FN0+Lω/L+1.
Fcω=n=0N0 fn expiωnΔt+LL+1 fN0+1×expiωN0+1Δt+L-1L+1 fN0+2 expiωN0+2Δt++1L+1 fN0+L expiωN0+LΔt =n=0N0 fn expiωnΔt+j=1LL+1-jL+1 fN0+j×expiωN0+jΔt.
fn=10.001n-10002+1.
fn=exp-n/30-52.
1-P22/P11=P44/P11-P33/P11.

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