Abstract

The finite-difference time domain (FDTD) method for the solution of light scattering by nonspherical particles has been developed for small ice crystals of hexagonal shapes including solid and hollow columns, plates, and bullet rosettes commonly occurring in cirrus clouds. To account for absorption, we have introduced the effective permittivity and conductivity to circumvent the required complex calculations in the direct discretization of the basic Maxwell equations. In the construction of the finite-difference scheme for the time-marching iteration for the near field the mean values of dielectric constants are defined and evaluated by the Maxwell–Garnett rule. In computing the scattered field in the radiation zone (far field) and the absorption cross section, we have applied a new algorithm involving the integration of the electric field over the volume inside the scatterer on the basis of electromagnetic principles. This algorithm removes the high-angular-resolution requirement in integrating the scattered energy for the computation of the scattering cross section. The applicability and the accuracy of the FDTD technique in three-dimensional space are validated by comparison with Mie scattering results for a number of size parameters and wavelengths. We demonstrate that neither the conventional geometric optics method nor the Mie theory can be used to approximate the scattering, absorption, and polarization features for hexagonal ice crystals with size parameters from approximately 5 to 20.

© 1996 Optical Society of America

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  1. G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. M. Kerker, D. Cooke, W. A. Farone, R. T. Jacobsen, “Electromagnetic scattering from an infinite circular cylinder at oblique incidence. I. Radiance functions for m= 1.46,” J. Opt. Soc. Am. 56, 487–491 (1966).
    [CrossRef]
  4. K. N. Liou, “Light scattering by ice clouds in the visible and infrared: a theoretical study,” J. Atmos. Sci. 29, 524–536 (1972).
    [CrossRef]
  5. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  6. S. Asano, “Light scattering properties of spheroidal particle,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  7. Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  8. Y. Takano, K. N. Liou, “Solar radiation transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  9. K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
    [CrossRef] [PubMed]
  10. A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
    [CrossRef] [PubMed]
  11. 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]
  12. Rayleigh, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871) [reprinted in Scientific Papers by Lord Rayleigh, Vol. I: 1869–1881, No. 8(Dover, New York, 1964)].
  13. P. W. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  14. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  15. M. A. Morgan, “Finite element calculation of microwave absorption by the cranial structure,” IEEE Trans. Biomed. Eng. BME-28, 687–695 (1981).
    [CrossRef]
  16. E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
    [CrossRef]
  17. P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [CrossRef]
  18. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  19. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 15, 2431–2437 (1988).
    [CrossRef]
  20. H. Y. Chen, M. F. Iskander, “Light scattering and absorption by fractal agglomerate and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
    [CrossRef]
  21. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  22. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  23. P. Chiappetta, “Multiple scattering approach to light scattering by arbitrarily shaped particles,” J. Phys. A 13, 2101–2108 (1980).
    [CrossRef]
  24. 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).
  25. K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
    [CrossRef]
  26. K. S. Kunz, R. J. Luebbers, The Finite Difference Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  27. A. Taflove, Computational Electrodynamics in the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).
  28. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electro-magn. Compat. EMC-23, 377–382 (1982).
    [CrossRef]
  29. Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).
  30. W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett. 7, 599–604 (1994).
    [CrossRef]
  31. K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
    [CrossRef]
  32. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).
  33. D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
    [CrossRef]
  34. K. S. Kunz, L. Simpson, “A technique for increasing the resolution of finite-difference solution of Maxwell equation,” IEEE Trans. Electromagn. Compat. EMC-23, 419–422 (1981).
    [CrossRef]
  35. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971), Chap. 4, p. 48.
  36. D. S. Saxon, “Lectures on the scattering of light,” in Proceedings of the UCLA International Conference on Radiation and Remote Probing of the Atmosphere (West Periodicals, North Hollywood, Calif., 1973), pp. 227–308.
  37. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  38. M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
    [CrossRef]
  39. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  40. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
    [CrossRef]

1995 (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]

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

1994 (2)

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett. 7, 599–604 (1994).
[CrossRef]

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

1993 (2)

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

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

1990 (3)

M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
[CrossRef]

H. Y. Chen, M. F. Iskander, “Light scattering and absorption by fractal agglomerate and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
[CrossRef]

1989 (2)

Y. Takano, K. N. Liou, “Solar radiation 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]

1988 (1)

G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 15, 2431–2437 (1988).
[CrossRef]

1984 (1)

Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).

1982 (3)

K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[CrossRef]

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electro-magn. Compat. EMC-23, 377–382 (1982).
[CrossRef]

Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

1981 (2)

M. A. Morgan, “Finite element calculation of microwave absorption by the cranial structure,” IEEE Trans. Biomed. Eng. BME-28, 687–695 (1981).
[CrossRef]

K. S. Kunz, L. Simpson, “A technique for increasing the resolution of finite-difference solution of Maxwell equation,” IEEE Trans. Electromagn. Compat. EMC-23, 419–422 (1981).
[CrossRef]

1980 (3)

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

P. Chiappetta, “Multiple scattering approach to light scattering by arbitrarily shaped particles,” J. Phys. A 13, 2101–2108 (1980).
[CrossRef]

1979 (1)

1975 (2)

1973 (1)

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

1972 (1)

K. N. Liou, “Light scattering by ice clouds in the visible and infrared: a theoretical study,” J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

1966 (2)

M. Kerker, D. Cooke, W. A. Farone, R. T. Jacobsen, “Electromagnetic scattering from an infinite circular cylinder at oblique incidence. I. Radiance functions for m= 1.46,” J. Opt. Soc. Am. 56, 487–491 (1966).
[CrossRef]

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).

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

1871 (1)

Rayleigh, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871) [reprinted in Scientific Papers by Lord Rayleigh, Vol. I: 1869–1881, No. 8(Dover, New York, 1964)].

Asano, S.

Barber, P. W.

P. W. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Cai, Q.

Chen, H. Y.

H. Y. Chen, M. F. Iskander, “Light scattering and absorption by fractal agglomerate and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

Chew, W. C.

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett. 7, 599–604 (1994).
[CrossRef]

Chiappetta, P.

P. Chiappetta, “Multiple scattering approach to light scattering by arbitrarily shaped particles,” J. Phys. A 13, 2101–2108 (1980).
[CrossRef]

Cooke, D.

Draine, B. T.

Farone, W. A.

Fisher, R.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Flatau, P. J.

Furse, M.

M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
[CrossRef]

Gandhi, O. P.

M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
[CrossRef]

Goedecke, G. H.

G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 15, 2431–2437 (1988).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Iskander, M. F.

H. Y. Chen, M. F. Iskander, “Light scattering and absorption by fractal agglomerate and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Jacobsen, R. T.

Kerker, M.

Kunz, K. S.

K. S. Kunz, L. Simpson, “A technique for increasing the resolution of finite-difference solution of Maxwell equation,” IEEE Trans. Electromagn. Compat. EMC-23, 419–422 (1981).
[CrossRef]

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

Liao, Z.

Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).

Liou, K. N.

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]

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

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

Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

K. N. Liou, “Light scattering by ice clouds in the visible and infrared: a theoretical study,” J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

Luebbers, R. J.

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

Macke, A.

Maloney, J. G.

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

Mathur, S. P.

M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
[CrossRef]

Merewether, D. E.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Morgan, M. A.

M. A. Morgan, “Finite element calculation of microwave absorption by the cranial structure,” IEEE Trans. Biomed. Eng. BME-28, 687–695 (1981).
[CrossRef]

Muinonen, K.

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electro-magn. Compat. EMC-23, 377–382 (1982).
[CrossRef]

O’Brien, S. G.

G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 15, 2431–2437 (1988).
[CrossRef]

Pennypacker, C. P.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Peterson, A. F.

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Ray, S. L.

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

Rayleigh,

Rayleigh, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871) [reprinted in Scientific Papers by Lord Rayleigh, Vol. I: 1869–1881, No. 8(Dover, New York, 1964)].

Sato, M.

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” in Proceedings of the UCLA International Conference on Radiation and Remote Probing of the Atmosphere (West Periodicals, North Hollywood, Calif., 1973), pp. 227–308.

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

Shlager, K. L.

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

Simpson, L.

K. S. Kunz, L. Simpson, “A technique for increasing the resolution of finite-difference solution of Maxwell equation,” IEEE Trans. Electromagn. Compat. EMC-23, 419–422 (1981).
[CrossRef]

Smith, F. W.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Stephens, G. L.

Taflove, A.

K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[CrossRef]

A. Taflove, Computational Electrodynamics in the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971), Chap. 4, p. 48.

Takano, Y.

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

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

Umashankar, K.

K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wang, H. L.

Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Weedon, W. H.

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett. 7, 599–604 (1994).
[CrossRef]

Yamamoto, G.

Yang, B.

Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).

Yang, P.

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).

Yeh, C.

Yuan, Y.

Z. Liao, H. L. Wang, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (8)

Astrophys. J. (1)

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

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).

K. L. Shlager, J. G. Maloney, S. L. Ray, A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propag. 41, 1732–1737 (1993).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

M. A. Morgan, “Finite element calculation of microwave absorption by the cranial structure,” IEEE Trans. Biomed. Eng. BME-28, 687–695 (1981).
[CrossRef]

IEEE Trans. Electro-magn. Compat. (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electro-magn. Compat. EMC-23, 377–382 (1982).
[CrossRef]

IEEE Trans. Electromagn. Compat. (2)

K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[CrossRef]

K. S. Kunz, L. Simpson, “A technique for increasing the resolution of finite-difference solution of Maxwell equation,” IEEE Trans. Electromagn. Compat. EMC-23, 419–422 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Furse, S. P. Mathur, O. P. Gandhi, “Improvements on the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory Tech. 38, 919–927 (1990).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygens’s source in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

J. Atmos. Sci. (3)

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

K. N. Liou, “Light scattering by ice clouds in the visible and infrared: a theoretical study,” J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

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

J. Mod. Opt. (1)

H. Y. Chen, M. F. Iskander, “Light scattering and absorption by fractal agglomerate and coagulations of smoke aerosols,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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Figures (9)

Fig. 1
Fig. 1

Locations of the electric- and magnetic-field components on a cubic cell used in the numerical calculations.

Fig. 2
Fig. 2

Incident and scattering geometries for the transformation of the near field to the far field.

Fig. 3
Fig. 3

Comparison of the phase function computed by the FDTD method and by Mie theory for the size parameter ka of 5 at λ = 0.55 and 10.8 μm in terms of absolute and relative errors. The indices of refraction for ice at these wavelengths are 1.311 + i3.11 × 10−9 and 1.0893 + i0.18216. Three grid sizes are used in the FDTD calculations.

Fig. 4
Fig. 4

Comparison of the phase matrix elements computed by the FDTD method and by Mie theory for the size parameter of 15 at λ = 0.55 μm. The grid size used is λ/20, and errors produced by the FDTD technique are also presented for this larger size parameter (see the text for further discussion).

Fig. 5
Fig. 5

Extinction and absorbing efficiencies for ice spheres as functions of size parameter computed by the FDTD method and by Mie theory at λ = 0.55 and 10.8 μm. Also shown are the absolute and relative differences between the two results.

Fig. 6
Fig. 6

Phase functions and degrees of linear polarization computed by the FDTD method for hexagonal ice columns with two specified orientations at λ = 0.55 μm. L is the length of the ice column, and a is the half-width.

Fig. 7
Fig. 7

Nonzero elements of the phase matrix computed by the FDTD method at two wavelengths, 0.55 and 3.7 μm (m = 1.4005 + i7.1967 × 10−3), for randomly oriented hexagonal columns.

Fig. 8
Fig. 8

Extinction efficiencies of randomly oriented hexagonal ice crystals computed by the FDTD method and by the Mie theory for equivalent-volume and -surface spheres.

Fig. 9
Fig. 9

Comparison of phase functions for hexagonal ice crystals computed by the FDTD method and by a geometric ray-tracing method (GOM1) for solid and hollow columns, plates, and bullet rosettes.

Equations (49)

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× H ( r , t ) = c E ( r , t ) t ,
× E ( r , t ) = - 1 c H ( r , t ) t ,
= r + i i ,
r = m r 2 - m i 2 ,             i = 2 m r m i ,
× H ( r , t ) = c E ( r , t ) t + 4 π c J ( r , t ) ,
J ( r , t ) = σ E ( r , t ) .
× H ( r ) = - i k E ( r ) ,
× H ( r ) = - i k ( + i 4 π σ / k c ) E ( r ) ,
r = ,
i = 4 π σ / k c .
× H ( r , t ) = r c [ E ( r , t ) t + τ E ( r , t ) ] ,
τ = k c i / r .
[ exp ( τ t ) E ( r , t ) ] t = exp ( τ t ) c r × H ( r , t ) .
exp [ τ ( n + 1 ) t ] E n + 1 ( r ) - exp ( τ n Δ t ) E n ( r ) = n Δ t ( n + 1 ) Δ t exp ( τ t ) c r × H ( r , t ) d t Δ t exp [ τ ( n + 1 / 2 ) Δ t ] c r × H n + 1 / 2 ( r ) .
E n + 1 ( r ) = exp ( - τ Δ t ) E n ( r ) + exp ( - τ Δ / 2 ) × c Δ t r × H n + 1 / 2 ( r ) .
H n + 1 / 2 ( r ) = H n - 1 / 2 ( r ) - C Δ t × E n ( r ) .
n ^ · ( × f ) d s = f · d 2 l             ( Stokes theorem ) ,
top n ^ · H n ± 1 / 2 ( r ) d 2 s ( Δ s ) 2 H z n ± 1 / 2 ( I , J , K + 1 / 2 ) ,
top n ^ · [ × E n ( r ) ] d 2 s = top E n ( r ) · d 1 Δ s [ E y n ( I + 1 / 2 , J , K + 1 / 2 ) - E y n ( I - 1 / 2 , J , K ) + E x n ( I , J - 1 / 2 , K + 1 / 2 ) - E x n ( I , J + 1 / 2 , K + 1 / 2 ) ] ,
E x n + 1 ( I , J + 1 / 2 , K + 1 / 2 ) = exp [ - Δ t τ ¯ ( I , J + 1 / 2 , K + 1 / 2 ) ] E x n ( I , J + 1 / 2 , K + 1 / 2 ) + exp [ - Δ t τ ¯ ( I , J + 1 / 2 , K + 1 / 2 ) / 2 ] c Δ t Δ s ¯ r ( I , J + 1 / 2 , K + 1 / 2 ) [ H z n + 1 / 2 ( I , J + 1 , K + 1 / 2 ) - H z n + 1 / 2 ( I , J , K + 1 / 2 ) + H y n + 1 / 2 ( I , J + 1 / 2 , K ) - H y n + 1 / 2 ( I , J + 1 / 2 , K + 1 ) ] ,
H x n + 1 / 2 ( I + 1 / 2 , J , K ) = H x n - 1 / 2 ( I + 1 / 2 , J , K ) + c Δ t Δ s [ E z n ( I + 1 / 2 , J - 1 / 2 , K ) - E z n ( I + 1 / 2 , J + 1 / 2 , K ) + E y n ( I + 1 / 2 , J , K + 1 / 2 ) - E y n ( I + 1 / 2 , J , K - 1 / 2 ) ] ,
E ˜ x n + 1 ( I , J A - 1 / 2 , K + 1 / 2 ) = E x n + 1 ( I , J A - 1 / 2 , K + 1 / 2 ) - c Δ t Δ s H o , z n + 1 / 2 ( I , J A - 1 , K + 1 / 2 ) , E ˜ x n + 1 ( I , J B + 1 / 2 , K + 1 / 2 ) = E x n + 1 ( I , J B + 1 / 2 , K + 1 / 2 ) + c Δ t Δ s H o , z n + 1 / 2 ( I , J B + 1 , K + 1 / 2 ) , K [ K A - 1 , K B ] ,
E ˜ x n + 1 ( I , J + 1 / 2 , K A - 1 / 2 ) = E x n + 1 ( I , J + 1 / 2 , K A - 1 / 2 ) + c Δ t Δ s H o , y n + 1 / 2 ( I , J + 1 , K A - 1 ) , E ˜ x n + 1 ( I , J + 1 / 2 , K B + 1 / 2 ) = E x n + 1 ( I , J + 1 / 2 , K B + 1 / 2 ) - c Δ t Δ s H o , y n + 1 / 2 ( I , J + 1 / 2 , K B + 1 ) , J [ J A - 1 , J B ] ,
σ e = σ s + σ a .
( 2 + k 2 ) E ( r ) = - 4 π ( k 2 I + ) · P ( r ) ,
P ( r ) = ( r ) - 1 4 π E ( r ) .
E ( r ) = E o ( r ) + 4 π v G ( r , ξ ) ( k 2 I + ξ ξ ) · P ( ξ ) d 3 ξ ,
G ( r , ξ ) = exp ( i k r - ξ ) 4 π r - ξ .
E s ( r ) k r = k 2 exp ( i k r ) 4 π r v [ ( ξ ) - 1 ] { E ( ξ ) - r ^ [ r ^ · E ( ξ ) ] } exp ( - i k r ^ · ξ ) d 3 ξ ,
E s ( r ) = α ^ E s , α ( r ) + β E s , β ( r ) ,
r ^ = β ^ × α ^ .
( E s , α ( r ) E s , β ( r ) ) = k 2 exp ( i k r ) 4 π r v [ ( ξ ) - 1 ] × ( α ^ · E ( ξ ) β ^ · E ( ξ ) ) exp ( - i k r · ξ ) d 3 ξ = exp ( i k r ) - i k r [ s 2 s 3 s 4 s 1 ] ( E o , α E o , β ) ,
( E o , α E o , β ) = [ β ^ · x ^ - β ^ · y ^ β ^ · y ^ β ^ · x ^ ] ( E o , y E o , x ) ,
( F α , x F β , x ) = i k 3 4 π v [ 1 - ( ξ ) ] ( α ^ · E ( ξ ) β ^ · E ( ξ ) ) × exp ( - i k r ^ · ξ ) d 3 ξ | E o , x = 1 , E o , y = 0 ,
( F α , y F β , y ) = i k 3 4 π v [ 1 - ( ξ ) ] ( α ^ · E ( ξ ) β ^ · E ( ξ ) ) × exp ( - i k r · ξ ) d 3 ξ | E o , x = 0 , E o , y = 1 .
[ s 2 s 3 s 4 s 1 ] = [ F α , y F α , x F β , y F β , x ] [ β ^ · x ^ β ^ · y ^ - β ^ · y ^ β · x ^ ] .
c × H = - i ω ( r + i i ) E ,
c × E = i ω H .
- · s = i ω 4 π ( r E · E * - H · H * ) + ω i 4 π E · E * ,
s = c 4 π E × H * ,
- Re [ v · s ( ξ ) d 3 ξ ] = - Re [ n ^ · s ( ξ ) d 2 ξ ] = ω 4 π v i E ( ξ ) · E * ( ξ ) d 3 ξ ,
F o = c 4 π E o · E o * = c 4 π E o 2 .
σ a = - Re [ n ^ · s ( ξ ) d 2 ξ ] / F o = k E o 2 v i ( ξ ) E ( ξ ) · E * ( ξ ) d 3 ξ .
s = s e + s s + s i .
s e = c 4 π ( E o × H * + E * × H o ) .
- Re [ n ^ · s e ( ξ ) d 2 ξ ] = ω 4 π Im { v [ ( ξ ) - 1 ] E ( ξ ) · E * ( ξ ) d 3 ξ } .
σ e = ω 4 π Im { v [ ( ξ ) - 1 ] E ( ξ ) · E * ( ξ ) d 3 ξ } / F o = Im { k E o 2 v [ ( r ) - 1 ] E ( ξ ) · E o * ( ξ ) d 3 ξ } .
σ ¯ e = ( σ e , + σ e , ) / 2 = 2 π k 2 Re [ s 1 ( z ^ ) + s 2 ( z ^ ) ] .
P ( θ ) = 1 2 π 0 2 π P ( θ , ϕ ) d ϕ .

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