Abstract

We have developed a finite-difference time domain (FDTD) method and a novel geometric ray-tracing model for the calculation of light scattering by hexagonal ice crystals. In the FDTD method we use a staggered Cartesian grid with the implementation of an efficient absorbing boundary condition for the truncation of the computation domain. We introduce the Maxwell–Garnett rule to compute the mean values of the dielectric constant at grid points to reduce the inaccuracy produced by the staircasing approximation. The phase matrix elements and the scattering efficiencies for the scattering of visible light by two-dimensional long circular ice cylinders match closely those computed from the exact solution for size parameters as large as 60, with maximum differences less than 5%. In the new ray-tracing model we invoke the principle of geometric optics to evaluate the reflection and the refraction of localized waves, from which the electric and magnetic fields at the particle surface (near field) can be computed. Based on the equivalence theorem, the near field can subsequently be transformed to the far field, in which the phase interferences are fully accounted for. The phase functions and the scattering efficiencies for hexagonal ice crystals computed from the new geometric ray-tracing method compare reasonably well with the FDTD results for size parameters larger than approximately 20. When absorption is involved in geometric ray tracing, the adjusted real and imaginary refractive indices and Fresnel formulas are derived for practical applications based on the fundamental wave theory.

© 1995 Optical Society of America

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  1. A. J. Heymsfield, R. G. Knollenberg, “Properties of cirrus generating cells,”J. Atmos. Sci. 29, 1358–1366 (1972).
    [CrossRef]
  2. K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
    [CrossRef]
  3. K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
    [CrossRef]
  4. G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
    [CrossRef]
  5. H. Jacobowitz, “A method for computing transfer of solar radiation through clouds of hexagonal ice crystals,”J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
    [CrossRef]
  6. P. Wendling, R. Wendling, H. K. Weickmann, “Scattering of solar radiation by hexagonal ice crystals,” Appl. Opt. 18, 2663–2671 (1979).
    [CrossRef] [PubMed]
  7. R. F. Coleman, K. N. Liou, “Light scattering by hexagonal ice crystals,”J. Atmos. Sci. 38, 1260–1271 (1981).
    [CrossRef]
  8. Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  9. Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed for ray optics,” Appl. Opt. 24, 3254–3263 (1985).
    [CrossRef] [PubMed]
  10. 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]
  11. E. Tränkle, R. G. Greenler, “Multiple-scattering effects in halo phenomena,” J. Opt. Soc. Am. A 4, 591–599 (1987).
    [CrossRef]
  12. K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
    [CrossRef] [PubMed]
  13. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,”J. Atmos. Sci. (to be published).
  14. A. Macke, “Scattering of light by irregular ice crystals in the three-dimensional inhomogeneous cirrus clouds,” presented at the Eighth Conference on Atmospheric Radiation, Nashville, Tenn., January 1994.
  15. Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
    [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. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  18. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  19. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1981).
    [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. 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).
  22. A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
    [CrossRef]
  23. K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,”IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
    [CrossRef]
  24. C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,”IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
    [CrossRef]
  25. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,”IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
    [CrossRef]
  26. 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]
  27. R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
    [CrossRef]
  28. Z. Liao, H. L. Wong, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).
  29. B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
    [CrossRef]
  30. G. Mur, “Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations,”IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1982).
    [CrossRef]
  31. J. G. Blaschak, G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,”J. Comput. Phys. 77, 109–139 (1988).
    [CrossRef]
  32. T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
    [CrossRef]
  33. M. Fusco, “FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 38, 76–89 (1990).
    [CrossRef]
  34. T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,”IEEE Trans. Antennas Propag. 40, 357–366 (1992).
    [CrossRef]
  35. K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
    [CrossRef]
  36. M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
    [CrossRef]
  37. J. F. Lee, “Obliquely Cartesian finite difference time domain algorithm,” Proc. Inst. Electr. Eng. Part H, 140, 23–27 (1993).
  38. S. Omick, S. P. Castillo, “A new finite-difference time-domain algorithm for the accurate modeling of wide-band electromagnetic phenomena,”IEEE Trans. Electromagn. Compat. 35, 315–222 (1993).
    [CrossRef]
  39. H. Vinh, H. Duger, C. P. Van Dam, “Finite-difference methods for computational electromagnetics (CEM),” in IEEE AP-S International Symposium Digest, (Institute of Electrical and Electronics Engineers, New York, 1992) Vol. 3, pp. 1682–1683.
  40. D. Steich, R. Luebbers, K. Kunz, “Absorbing boundary condition convergence comparisons,” in IEEE AP-S International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), Vol. 1, pp. 6–9.
  41. K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,”IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
    [CrossRef]
  42. R. H. T. Bates, “Analytic constraints on electromagnetic computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–622 (1975).
    [CrossRef]
  43. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 8.
  44. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 110–113, 377–399, 627–633, and 707–716.
  45. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  46. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), pp. 299–304.
  47. E. A. Hovenac, J. A. Lock, “Assessing the contribution of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]

1994 (1)

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

1993 (2)

J. F. Lee, “Obliquely Cartesian finite difference time domain algorithm,” Proc. Inst. Electr. Eng. Part H, 140, 23–27 (1993).

S. Omick, S. P. Castillo, “A new finite-difference time-domain algorithm for the accurate modeling of wide-band electromagnetic phenomena,”IEEE Trans. Electromagn. Compat. 35, 315–222 (1993).
[CrossRef]

1992 (5)

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,”IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

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

K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
[CrossRef]

E. A. Hovenac, J. A. Lock, “Assessing the contribution of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
[CrossRef]

1991 (2)

M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
[CrossRef]

R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
[CrossRef]

1990 (4)

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]

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]

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
[CrossRef]

M. Fusco, “FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

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

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,”IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

1988 (3)

J. G. Blaschak, G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,”J. Comput. Phys. 77, 109–139 (1988).
[CrossRef]

T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1987 (1)

1986 (1)

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

1985 (1)

1984 (1)

Z. Liao, H. L. Wong, 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 condition for the finite-difference approximation of the time-domain electromagnetic-field equations,”IEEE Trans. Electromagn. 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)

1980 (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,”IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
[CrossRef]

1979 (1)

1977 (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

1975 (3)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

R. H. T. Bates, “Analytic constraints on electromagnetic computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–622 (1975).
[CrossRef]

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

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)

A. J. Heymsfield, R. G. Knollenberg, “Properties of cirrus generating cells,”J. Atmos. Sci. 29, 1358–1366 (1972).
[CrossRef]

1971 (1)

H. Jacobowitz, “A method for computing transfer of solar radiation through clouds of hexagonal ice crystals,”J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

1966 (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).

1939 (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Barber, P.

Bates, R. H. T.

R. H. T. Bates, “Analytic constraints on electromagnetic computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–622 (1975).
[CrossRef]

Blaschak, J. G.

J. G. Blaschak, G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,”J. Comput. Phys. 77, 109–139 (1988).
[CrossRef]

T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 8.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 110–113, 377–399, 627–633, and 707–716.

Britt, C. L.

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,”IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

Brodwin, M. E.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Cable, V. R.

R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
[CrossRef]

Cai, Q.

Castillo, S. P.

S. Omick, S. P. Castillo, “A new finite-difference time-domain algorithm for the accurate modeling of wide-band electromagnetic phenomena,”IEEE Trans. Electromagn. Compat. 35, 315–222 (1993).
[CrossRef]

Chang, A. H.

K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
[CrossRef]

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]

Chen, J. S.

K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
[CrossRef]

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Coleman, R. F.

R. F. Coleman, K. N. Liou, “Light scattering by hexagonal ice crystals,”J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Duger, H.

H. Vinh, H. Duger, C. P. Van Dam, “Finite-difference methods for computational electromagnetics (CEM),” in IEEE AP-S International Symposium Digest, (Institute of Electrical and Electronics Engineers, New York, 1992) Vol. 3, pp. 1682–1683.

Engquist, B.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

Fang, J.

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,”IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

Flatau, P. J.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
[CrossRef]

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]

Fusco, M.

M. Fusco, “FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

Fusco, M. A.

M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
[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.

Gordon, L. W.

M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
[CrossRef]

Greenler, R. G.

Heymsfield, A. J.

A. J. Heymsfield, R. G. Knollenberg, “Properties of cirrus generating cells,”J. Atmos. Sci. 29, 1358–1366 (1972).
[CrossRef]

Holland, R.

R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
[CrossRef]

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 8.

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 (Wiley, New York, 1962), pp. 299–304.

Jacobowitz, H.

H. Jacobowitz, “A method for computing transfer of solar radiation through clouds of hexagonal ice crystals,”J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

Jayaweera, K.

Jurgens, T. G.

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

Knollenberg, R. G.

A. J. Heymsfield, R. G. Knollenberg, “Properties of cirrus generating cells,”J. Atmos. Sci. 29, 1358–1366 (1972).
[CrossRef]

Kriegsmann, G. A.

T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
[CrossRef]

J. G. Blaschak, G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,”J. Comput. Phys. 77, 109–139 (1988).
[CrossRef]

Kunz, K.

D. Steich, R. Luebbers, K. Kunz, “Absorbing boundary condition convergence comparisons,” in IEEE AP-S International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), Vol. 1, pp. 6–9.

Lee, J. F.

J. F. Lee, “Obliquely Cartesian finite difference time domain algorithm,” Proc. Inst. Electr. Eng. Part H, 140, 23–27 (1993).

Liao, Z.

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

Liou, K. N.

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
[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]

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

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

R. F. Coleman, K. N. Liou, “Light scattering by hexagonal ice crystals,”J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,”J. Atmos. Sci. (to be published).

Lock, J. A.

Luebbers, R.

D. Steich, R. Luebbers, K. Kunz, “Absorbing boundary condition convergence comparisons,” in IEEE AP-S International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), Vol. 1, pp. 6–9.

Macke, A.

A. Macke, “Scattering of light by irregular ice crystals in the three-dimensional inhomogeneous cirrus clouds,” presented at the Eighth Conference on Atmospheric Radiation, Nashville, Tenn., January 1994.

Majda, A.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[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]

Mei, K. K.

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,”IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

Minnis, P.

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Moore, T. G.

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

T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
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O’Brien, S. G.

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S. Omick, S. P. Castillo, “A new finite-difference time-domain algorithm for the accurate modeling of wide-band electromagnetic phenomena,”IEEE Trans. Electromagn. Compat. 35, 315–222 (1993).
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Pennypacker, C. P.

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

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

M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
[CrossRef]

Stackhouse, P. W.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
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Steich, D.

D. Steich, R. Luebbers, K. Kunz, “Absorbing boundary condition convergence comparisons,” in IEEE AP-S International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), Vol. 1, pp. 6–9.

Stephens, G. L.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
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J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Taflove, A.

T. G. Jurgens, A. Taflove, K. Umashankar, T. G. Moore, “Finite-difference time-domain modeling of curved surfaces,”IEEE Trans. Antennas Propag. 40, 357–366 (1992).
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T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
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K. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,”IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
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A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,”IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
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A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
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K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
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Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
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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).
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Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,”J. Atmos. Sci. (to be published).

Tränkle, E.

Tsay, S. C.

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
[CrossRef]

Umashankar, K.

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

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 Dam, C. P.

H. Vinh, H. Duger, C. P. Van Dam, “Finite-difference methods for computational electromagnetics (CEM),” in IEEE AP-S International Symposium Digest, (Institute of Electrical and Electronics Engineers, New York, 1992) Vol. 3, pp. 1682–1683.

Vinh, H.

H. Vinh, H. Duger, C. P. Van Dam, “Finite-difference methods for computational electromagnetics (CEM),” in IEEE AP-S International Symposium Digest, (Institute of Electrical and Electronics Engineers, New York, 1992) Vol. 3, pp. 1682–1683.

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Wendling, P.

Wendling, R.

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R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
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Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 110–113, 377–399, 627–633, and 707–716.

Wong, H. L.

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

Yang, B.

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

Yee, K. S.

K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
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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).

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

Appl. Opt. (6)

Astrophys. J. (2)

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

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Atmos. Res. (1)

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (8)

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).

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,”IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

T. G. Moore, J. G. Blaschak, A. Taflove, G. A. Kriegsmann, “Theory and application of radiation boundary operators,”IEEE Trans. Antennas Propag. 36, 1797–1812 (1988).
[CrossRef]

M. Fusco, “FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 38, 76–89 (1990).
[CrossRef]

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

K. S. Yee, J. S. Chen, A. H. Chang, “Conformal finite difference time domain (FDTD) with overlapping grids,”IEEE Trans. Antennas Propag. 40, 1068–1075 (1992).
[CrossRef]

M. A. Fusco, M. V. Smith, L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,”IEEE Trans. Antennas Propag. 39, 1463–1471 (1991).
[CrossRef]

K. K. Mei, J. Fang, “Superabsorption—a method to improve absorbing boundary conditions,”IEEE Trans. Antennas Propag. 40, 1001–1010 (1992).
[CrossRef]

IEEE Trans. Electromagn. Compat. (5)

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

S. Omick, S. P. Castillo, “A new finite-difference time-domain algorithm for the accurate modeling of wide-band electromagnetic phenomena,”IEEE Trans. Electromagn. Compat. 35, 315–222 (1993).
[CrossRef]

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,”IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
[CrossRef]

R. Holland, V. R. Cable, L. C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,”IEEE Trans. Electromagn. Compat. 33, 281–293 (1991).
[CrossRef]

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

IEEE Trans. Microwave Theory Tech. (3)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

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]

R. H. T. Bates, “Analytic constraints on electromagnetic computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–622 (1975).
[CrossRef]

J. Atmos. Sci. (5)

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiation properties,”J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

G. L. Stephens, S. C. Tsay, P. W. Stackhouse, P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback,”J. Atmos. Sci. 47, 1742–1753 (1990).
[CrossRef]

A. J. Heymsfield, R. G. Knollenberg, “Properties of cirrus generating cells,”J. Atmos. Sci. 29, 1358–1366 (1972).
[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]

R. F. Coleman, K. N. Liou, “Light scattering by hexagonal ice crystals,”J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

J. Comput. Phys. (1)

J. G. Blaschak, G. A. Kriegsmann, “A comparative study of absorbing boundary conditions,”J. Comput. Phys. 77, 109–139 (1988).
[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. A (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

H. Jacobowitz, “A method for computing transfer of solar radiation through clouds of hexagonal ice crystals,”J. Quant. Spectrosc. Radiat. Transfer 11, 691–695 (1971).
[CrossRef]

Math. Comput. (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

Mon. Weather Rev. (1)

K. N. Liou, “Influence of cirrus clouds on weather and climate process: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

J. F. Lee, “Obliquely Cartesian finite difference time domain algorithm,” Proc. Inst. Electr. Eng. Part H, 140, 23–27 (1993).

Sci. Sin. (1)

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

Other (7)

H. Vinh, H. Duger, C. P. Van Dam, “Finite-difference methods for computational electromagnetics (CEM),” in IEEE AP-S International Symposium Digest, (Institute of Electrical and Electronics Engineers, New York, 1992) Vol. 3, pp. 1682–1683.

D. Steich, R. Luebbers, K. Kunz, “Absorbing boundary condition convergence comparisons,” in IEEE AP-S International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), Vol. 1, pp. 6–9.

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,”J. Atmos. Sci. (to be published).

A. Macke, “Scattering of light by irregular ice crystals in the three-dimensional inhomogeneous cirrus clouds,” presented at the Eighth Conference on Atmospheric Radiation, Nashville, Tenn., January 1994.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), pp. 299–304.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 8.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 110–113, 377–399, 627–633, and 707–716.

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Figures (8)

Fig. 1
Fig. 1

Geometry for the conventional ray-tracing technique.

Fig. 2
Fig. 2

Geometry for the new geometric optics model: E&M, electric and magnetic.

Fig. 3
Fig. 3

Geometry for the inhomogeneity effect of the refracted wave in the absorptive case.

Fig. 4
Fig. 4

Nonzero elements of the phase matrix for light scattering by a circular cylinder with a size parameter of 20.

Fig. 5
Fig. 5

Same as Fig. 4, except for a size parameter of 60.

Fig. 6
Fig. 6

Scattering efficiencies for the circular cylinders computed by the FDTD and the exact methods. Also shown are differences between the two.

Fig. 7
Fig. 7

Normalized phase functions computed by the FDTD, GOM1, and GOM2 methods for light scattering by randomly oriented 2-D hexagonal ice crystals.

Fig. 8
Fig. 8

Scattering efficiencies for randomly oriented 2-D hexagonal ice crystals and differences between GOM2 (GOM1) and FDTD results.

Tables (1)

Tables Icon

Table 1 Effect of Inhomogeneity of Refraction on the Single-Scattering Parameters for Size Parameters 40 and 100a

Equations (88)

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( x , y ) c E z ( x , y , t ) t = H y ( x , y , t ) x - H x ( x , y , t ) y ,
1 c H x ( x , y , t ) t = - E z ( x , y , t ) y ,
1 c H y ( x , y , t ) t = E z ( x , y , t ) x ,
1 c H ˜ z ( x , y , t ) t = E y ( x , y , t ) x - E x ( x , y , t ) y ,
( x , y ) c E x ( x , y , t ) t = - H ˜ z ( x , y , t ) x ,
( x , y ) c E y ( x , y , t ) t = H ˜ z ( x , y , t ) x ,
1 c E z o ( x , t ) t = H y o ( x , t ) x ,
1 c H y o ( x , t ) t = E z o ( x , t ) x ,
( x , y ) c E z s ( x , y , t ) t = H y s ( x , y , t ) x - H x s ( x , y , t ) y - [ ( x , y ) - 1 ] H y o ( x , t ) x ,
1 c H x s ( x , y , t ) t = - E z s ( x , y , t ) y ,
1 c H y s ( x , y , t ) t = E z s ( x , y , t ) y .
cell ( i , j ) ¯ i j c E z s ( i , j , t ) t Δ s 2 ( x , y ) c E z s ( x , y , t ) t d x d y ¯ i j c E z s ( i , j , t ) t Δ s 2 ,
cell ( i , j ) [ H y s ( x , y , t ) x - H x s ( x , y , t ) y ] d x d y - cell ( i , j ) [ H y o ( x , t ) x ] [ ( x , y ) - 1 ] d x d y Δ s [ H x s ( i , j - 1 / 2 , t ) - H x s ( i , j + 1 / 2 , t ) + H y s ( i + 1 / 2 , j , t ) - H y s ( i - 1 / 2 , j , t ) ] - Δ s ( ¯ i j - 1 ) [ H y o ( i + 1 / 2 , t ) - H y o ( i - 1 / 2 , t ) ] .
¯ i j - 1 ¯ i j + 2 = 1 Δ s 2 cell ( i , j ) ( x , y ) - 1 ( x , y ) + 2 d x d y .
E z s ( n + 1 ) ( i , j ) = E z s ( n ) ( i , j ) + c Δ t Δ s ¯ i j { [ H y s ( n + 1 / 2 ) ( i + 1 / 2 , j ) - H y s ( n + 1 / 2 ) ( i - 1 / 2 , j ) ] + [ H x s ( n + 1 / 2 ) ( 1 , j - 1 / 2 ) - H x s ( n + 1 / 2 ) ( i , j + 1 / 2 ) ] } + ( 1 - 1 ¯ i j ) c Δ t Δ s × [ H y o ( n + 1 / 2 ) ( i - 1 / 2 ) - H y o ( n + 1 / 2 ) ( i + 1 / 2 ) ] ,
H x s ( n + 1 / 2 ) ( i , j + 1 / 2 ) = H x s ( n - 1 / 2 ) ( i , j + 1 / 2 ) + c Δ t Δ s [ E z s ( n ) ( i , j ) - E z s ( n ) ( i , j + 1 ) ] ,
H y s ( n + 1 / 2 ) ( i + 1 / 2 , j ) = H y s ( n - 1 / 2 ) ( i + 1 / 2 , j ) + c Δ t Δ s [ E z s ( n ) ( i + 1 , j ) - E z s ( n ) ( i , j ) ] ,
U ( t + Δ t , x 1 ) = j = 1 N ( - 1 ) j + 1 C j N T ¯ j U ¯ j ,
T ¯ j = T ¯ 1 [ T j - 1 , 1 T j - 1 , 2 T j - 1 , 2 j - 1 0 0 0 T j - 1 , 1 T j - 1 , 2 T j - 1 , 2 j - 1 0 0 0 T j - 1 , 1 T j - 1 , 2 T j - 1 , 2 j - 1 ] ,             j 2 ,
T ¯ 1 = [ ( 2 - β ) ( 1 - β ) / 2 , β ( 2 - β ) , β ( β - 1 ) / 2 ] ,
U ( t + Δ t , x ) = j = 1 N ( - 1 ) ( j + 1 ) C j N × U [ t - ( j - 1 ) Δ t , x - j α c Δ t ] .
c Δ t Δ s 2 2 0.707 ,             2 - D case ,
c Δ t Δ s 3 3 0.577 ,             3 - D case .
U ( n + 1 ) ( 1 , j ) = 2 U ( n - 1 ) ( 2 , j ) - U ( n - 3 ) ( 3 , j ) ,
U ( n + 1 ) ( 1 , j ) = 3 U ( n - 1 ) ( 2 , j ) - 3 U ( n - 3 ) ( 3 , j ) + U ( n - 5 ) ( 4 , j ) .
U ( n + 1 ) ( 1 , j ) = ( 3 - γ ) U ( n - 1 ) ( 2 , j ) - ( 3 - 2 γ ) U ( n - 3 ) ( 3 , j ) + ( 1 - γ ) U ( n - 5 ) ( 4 , j ) .
f ( t ) = n = 0 N f n δ ( t - n Δ t ) ,
F ( k ) = - [ n = 0 N f n δ ( t - n Δ t ) ] exp ( i k c t ) d t = n = 0 N f n exp ( i k c n Δ t ) ,
F ( k ) = n = 0 N f n exp ( i n k / 2 p k max ) ,
F ( k ) = n = 0 N f n exp ( i n q / 2 p ) .
f n = A exp [ - ( n / w - 5 ) 2 ] ,
E z ( r ) = n ^ · [ E z ( r ) G ( r , r ) - G ( r , r ) E z ( r ) ] d r ,
G ( r , r ) = i 4 H 0 ( 1 ) ( r - r ) ,             i = - 1 .
E z s ( r ) = ( 2 π k r ) 1 / 2 exp [ i ( k r + 3 π / 4 ) ] i 4 × [ i k ( n ^ · r ^ ) E z s ( r ) + E z s ( r ) n ] × exp ( - i k r ^ · r ) d r ,
E z s n ( i Δ s , j Δ s ) = E z s x ( i Δ s , j Δ s ) 4 3 E z s ( i + 1 , j ) - E z s ( i - 1 , j ) 2 Δ s - 1 3 E z s ( i + 2 , j ) - E z s ( i - 2 , j ) 4 Δ s .
[ E TM s E TE s ] = ( 2 π k r ) 1 / 2 exp [ i ( k r + 3 π / 4 ) - i k x ] × [ F TM ( s ^ ) 0 0 F TE ( s ^ ) ] [ E TM i E TE i ] ,
F TM ( s ^ ) = i 4 1 E TM i [ E z s ( r ) n + i k E z s ( r ) ( n ^ · s ^ ) ] × exp ( - i k s ^ · r ) d r ,
F TE ( s ^ ) = i 4 1 H ˜ TE i [ H ˜ z s ( r ) n + i k H ˜ z s ( r ) ( n ^ · s ^ ) ] × exp ( - i k s ^ · r ) d r ,
P 11 ( s ^ ) = ½ [ F TM ( s ^ ) 2 + F TE ( s ^ ) 2 ] ,
P 12 ( s ^ ) = ½ [ F TM ( s ^ ) 2 - F TE ( s ^ ) 2 ] ,
P 33 ( s ^ ) = Re { F TM ( s ^ ) · [ F TE ( s ^ ) ] * } ,
P 34 ( s ^ ) = Im { F TM ( s ^ ) · [ F TE ( s ^ ) ] * } ,
σ TM = 4 k Re [ F TM ( s ^ ) ] s ^ = x ^ ,
σ TE = 4 k Re [ F TE ( s ^ ) ] s ^ = x ^ ,
P d ( Θ ) = 1 2 π χ ( 1 + cos Θ ) 2 × 0 π / 6 [ χ cos α sin ( χ cos α sin Θ ) χ cos α sin Θ ] 2 d α ,
e ^ p r = e ^ p i - 2 ( e ^ p i · n ^ p ) n ^ p ,             p = 1 , 2 , 3 , ,
e ^ p t = 1 m p { e ^ p i - ( e ^ p i · n ^ p ) n ^ p - [ m p 2 - 1 + ( e ^ p i · n ^ p ) 2 ] 1 / 2 n ^ p } ,             p = 1 , 2 , 3 , ,
e ^ p i = { e ^ i , p = 1 e ^ 1 t , p = 2 e ^ p - 1 r , p > 2 ,
H = m e ^ × E             inside the scatterer ,
H = e ^ × E             outside the scatterer ,
E i ( r 1 ) = A e ^ z exp ( i k e ^ i · r 1 ) ,
E r ( r 1 ) = R 1 A e ^ z exp ( i k e ^ i · r 1 ) ,
E t ( r 2 ) = T 1 T 2 A e ^ z exp { i k [ e ^ i · r 1 + m e ^ 1 t · ( r 2 - r 1 ) ] } ,
E total ( r ) = { E i ( r ) + E r ( r ) + E t ( r ) , r ( illuminated surface ) , E t ( r ) , r ( shadowed surface ) .
σ r = - σ i / ( n ^ 1 · e ^ i ) .
σ p t = σ i [ m 2 - 1 + ( n ^ 1 · e ^ i ) 2 ] 1 / 2 [ m ( n ^ 1 · e ^ i ) ( n ^ p + 1 · e ^ p + 1 i ) ] - 1 ,             P 1.
J ( r ) = n ^ × H ( r ) ,
M ( r ) = - n ^ × E ( r ) .
F TM ( s ^ ) = k 4 [ J z ( r ) - s x M y ( r ) + s y M x ( r ) ] × exp ( - i k s ^ · r ) d r = k 4 [ n x H y ( r ) - n y H x ( r ) - ( n ^ · s ^ ) E z ( r ) ] exp ( - i k s ^ · r ) d r ,
F TE ( s ^ ) = k 4 [ - M z ( r ) - s x J y ( r ) + s y J x ( r ) ] × exp ( - i k s ^ · r ) d r = k 4 [ n x E y ( r ) - n y E x ( r ) + ( n ^ · s ^ ) H z ( r ) ] exp ( - i k s ^ · r ) d r ,
σ TM s = 2 π k 0 2 π F TM ( Θ ) 2 d Θ ,
σ TE s = 2 π k 0 2 π F TE ( Θ ) 2 d Θ .
k i = k 0 e ^ i ,             k r = k 0 e ^ r ,             k t = k t e ^ t + i k α e ^ α ,
E i ( r , t ) = A i exp [ i ( k 0 r · e ^ i - ω t ) ] ,
E r ( r , t ) = A r exp [ i ( k 0 r · e ^ r - ω t ) ] ,
E t ( r , t ) = A t exp { i [ ( k t e ^ t + i k α e ^ α ) · r - ω t ] } ,
N r = k t / k 0 ,             N i * = k α / k 0 .
e ^ i · r s = e ^ r · r s = N r ( e ^ t · r s ) + i N i * ( e ^ α · r s ) .
e ^ i · r s = e ^ r · r s = N r ( e ^ t · r s ) ,
e ^ α · r s = 0.
sin Θ i = sin Θ r ,             sin Θ t = sin Θ i / N r ,
2 E t ( r , t ) - ( m r + i m i ) 2 c 2 2 E t ( r , t ) t 2 = 0 ,
N r 2 - N i * 2 = m r 2 - m i 2 ,             N r N i * cos Θ t = m r m i .
N r = { m r 2 - m i 2 + sin 2 Θ i + [ ( m r 2 - m i 2 - sin 2 Θ i ) 2 + 4 m r 2 m i 2 ] 1 / 2 2 } 1 / 2 ,
N i = m r m i / N r .
E t ( r , t ) = A t exp ( - k 0 N i l ) exp [ i ( k 0 N r e ^ t · r - ω t ) ] ,
E t ( r , t ) = E t ( r ) exp [ i ( N r k 0 e ^ t · r - ω t ) ] .
H t ( r , t ) = H t ( r ) exp [ i ( N r k 0 e ^ t · r - ω t ) ] .
N r e ^ t × H t ( r ) + E t ( r ) = i k 0 × H t ( r ) ,
N r e ^ t × E t ( r ) - H t ( r ) = i k 0 × E t ( r ) ,
N r e ^ t · E t ( r ) = i k 0 · E t ( r ) ,
N r e ^ t · H t ( r ) = i k 0 · H t ( r ) ,
e ^ t × H t ( r ) + N r E t ( r ) 0 ,
N r e ^ t × E t ( r ) - H t ( r ) 0 ,
e ^ t · E t ( r ) 0 ,
e ^ t · H t ( r ) 0 ,
R l = N r cos θ i - cos θ t N r cos θ i + cos θ t ,             T l = 2 cos θ i N r cos θ i + cos θ t ,
R r = cos θ i - N r cos θ t cos θ i + N r cos θ t ,             T r = 2 cos θ i cos θ i + N r cos θ t ,

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