Abstract

The two-dimensional (2-D) finite-difference time-domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions and extinction and absorption efficiencies, are derived by use of an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than ~0.1%. The errors in the scattering phase functions are typically smaller than ~4%. With the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. C. Mundy, J. A. Roux, A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64, 1593–1597 (1974).
    [CrossRef]
  2. P. Chylek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67, 561–563 (1977).
    [CrossRef]
  3. C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
    [CrossRef]
  4. M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
    [CrossRef]
  5. A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).
  6. W. Sun, “Light scattering by nonspherical particles: numerical simulation and applications,” Ph.D. dissertation (Dalhousie University, Halifax, Nova Scotia, Canada, 2000).
  7. Q. Fu, W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40, 1354–1361 (2001).
    [CrossRef]
  8. I. W. Sudiarta, P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. A 18, 1275–1278 (2001).
    [CrossRef]
  9. I. W. Sudiarta, P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quat. Spectrosc. Radiat. Transfer 70, 709–714 (2001).
    [CrossRef]
  10. Y. Ping, B. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H. Huang, B. A. Baum, Y. Hu, D. M. Winker, S. Tsay, S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41, 2740–2759 (2002).
    [CrossRef]
  11. G. Videen, W. Sun, “Yet another look at light scattering from particles in absorbing media,” Appl. Opt. 42, 6724–6727 (2003).
    [CrossRef] [PubMed]
  12. W. Sun, N. G. Loeb, B. Lin, “Light scattering by an infinite circular cylinder immersed in an absorbing medium,” Appl. Opt. (to be published).
  13. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  14. 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]
  15. 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]
  16. W. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
    [CrossRef]
  17. W. Sun, N. G. Loeb, Q. Fu, “Finite-difference time domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41, 5728–5743 (2002).
    [CrossRef] [PubMed]
  18. Z. S. Sacks, D. M. Kingsland, R. Lee, J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  19. S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]
  20. A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 2000).
  21. W. Sun, N. G. Loeb, G. Videen, Q. Fu, “Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm,” Appl. Opt. 43, 1957–1964 (2004).
    [CrossRef] [PubMed]
  22. D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
    [CrossRef]
  23. 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 (1981).
    [CrossRef]
  24. A. Taflove, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995)
  25. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]

2004 (1)

2003 (1)

2002 (2)

2001 (3)

1999 (2)

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

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

1996 (3)

S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

1995 (1)

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

1994 (1)

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

1981 (1)

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 (1981).
[CrossRef]

1980 (1)

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

1979 (1)

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

1977 (1)

1975 (1)

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]

1974 (1)

1966 (1)

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

Baum, B. A.

Berenger, J. P.

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

Bohren, C. F.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[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]

Chen, Z.

Chylek, P.

Fisher, R.

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

Fu, Q.

Gao, B.

Gartz, M.

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Hagness, S.

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 2000).

Hu, Y.

Huang, H.

Kingsland, D. M.

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

Kreibig, U.

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

Lebedev, A. N.

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

Lee, J. F.

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

Lee, R.

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

Lin, B.

W. Sun, N. G. Loeb, B. Lin, “Light scattering by an infinite circular cylinder immersed in an absorbing medium,” Appl. Opt. (to be published).

Liou, K. N.

Loeb, N. G.

Merewether, D. E.

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

Mishchenko, M. I.

Mundy, W. C.

Mur, G.

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 (1981).
[CrossRef]

Park, S. K.

Ping, Y.

Platnick, S. E.

Quinten, M.

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

Rostalski, J.

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

Roux, J. A.

Sacks, Z. S.

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

Smith, A. M.

Smith, F. W.

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

Stenzel, O.

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

Sudiarta, I. W.

I. W. Sudiarta, P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. A 18, 1275–1278 (2001).
[CrossRef]

I. W. Sudiarta, P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quat. Spectrosc. Radiat. Transfer 70, 709–714 (2001).
[CrossRef]

Sun, W.

Taflove, A.

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]

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 2000).

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

Tsay, S.

Videen, G.

Winker, D. M.

Wiscombe, W. J.

Yang, P.

Yee, K. S.

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

Appl. Opt. (6)

Eur. Phys. J. (1)

A. N. Lebedev, M. Gartz, U. Kreibig, O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. 6, 365–373 (1999).

IEEE Trans. Antennas Propag. (3)

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

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

S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

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 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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]

IEEE Trans. Nucl. Sci. (1)

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

J. Colloid Interface Sci. (1)

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

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

I. W. Sudiarta, P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quat. Spectrosc. Radiat. Transfer 70, 709–714 (2001).
[CrossRef]

Part. Part. Syst. Charact. (1)

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

Other (4)

W. Sun, “Light scattering by nonspherical particles: numerical simulation and applications,” Ph.D. dissertation (Dalhousie University, Halifax, Nova Scotia, Canada, 2000).

W. Sun, N. G. Loeb, B. Lin, “Light scattering by an infinite circular cylinder immersed in an absorbing medium,” Appl. Opt. (to be published).

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

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometries of the incident direction and the closed rectangular interface of the total field and the scattered field.

Fig. 2
Fig. 2

Geometries of the incident direction, the unit vector in the scattering direction, and the unit vector perpendicular to the scattering direction.

Fig. 3
Fig. 3

Extinction efficiency (Qe), absorption efficiency (Qa), and asymmetry factor (g) for circular cylinders embedded in an absorbing medium as functions of size parameter (2πa0) calculated by the 2-D UPML FDTD method and the exact solution for a host refractive index of 1.2 + 0.01i and a particle refractive index of 1.4 + 0.05i. Also shown are the relative errors of the FDTD results.

Fig. 4
Fig. 4

Comparison of the scattering phase functions of a circular cylinder embedded in an absorbing medium with a size parameter of 30 calculated by the 2-D UPML FDTD method and the exact solution for a host refractive index of 1.2 + 0.01i and a particle refractive index of 1.4 + 0.05i. Also shown are the relative errors of the FDTD results.

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

Geometries of incidence on an infinite elliptic column.

Fig. 7
Fig. 7

Scattering phase matrix elements of an elliptic column with a size parameter of 2πa0 = 60 immersed in a host medium, where a is the longer half-axis of the ellipse. The shorter half-axis of the ellipse b = a/2. As shown in Fig. 6, the light is incident on the column at an angle of 45° from the x axis. The host medium is assumed to have a refractive index of 1.2. The refractive index of the column is assumed to be 1.4 + 0.05i.

Fig. 8
Fig. 8

Same as Fig. 7, but the refractive index of the host medium is assumed to be 1.2 + 0.01i.

Equations (28)

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

H x n + 1 / 2 ( i ,     j + 1 / 2 ) = H x n - 1 / 2 ( i ,     j + 1 / 2 ) - Δ t μ 0 Δ s [ E z n ( i ,     j + 1 ) - E z n ( i , j ) ] ,
H y n + 1 / 2 ( i + 1 / 2 ,     j ) = H y n - 1 / 2 ( i + 1 / 2 ,     j ) + Δ t μ 0 Δ s [ E z n ( i + 1 ,     j ) - E z n ( i ,     j ) ] ,
E z n + 1 ( i ,     j ) = exp [ - ɛ i ( i ,     j ) ɛ r ( i ,     j ) ω Δ t ] E z n ( i ,     j ) + exp [ - ɛ i ( i ,     j ) ɛ r ( i ,     j ) ω Δ t / 2 ] Δ t ɛ r ( i ,     j ) Δ s × [ H y n + 1 / 2 ( i + 1 / 2 ,     j ) - H y n + 1 / 2 ( i - 1 / 2 ,     j ) - H x n + 1 / 2 × ( i ,     j + 1 / 2 ) + H x n + 1 / 2 ( i ,     j - 1 / 2 ) ] ,
H x n + 1 / 2 ( i ,     j a - 1 / 2 ) = [ H x n + 1 / 2 ( i ,     j a - 1 / 2 ) ] ( 1 a ) + Δ t μ 0 Δ s E z , inc n ( i ,     j a ) ,
E z n + 1 ( i ,     j a ) = [ E z n + 1 ( i ,     j a ) ] ( 1 c ) - exp [ - ɛ i ( i ,     j a ) ɛ r ( i ,     j a ) ω Δ t / 2 ] × Δ t ɛ r ( i ,     j a ) Δ s H x , inc n + 1 / 2 ( i ,     j a - 1 / 2 ) .
H x n + 1 / 2 ( i ,     j b + 1 / 2 ) = [ H x n + 1 / 2 ( i ,     j b + 1 / 2 ) ] ( 1 a ) - Δ t μ 0 Δ s E z , inc n ( i ,     j b ) ,
E z n + 1 ( i ,     j b ) = [ E z n + 1 ( i ,     j b ) ] ( 1 c ) + exp [ - ɛ i ( i ,     j b ) ɛ r ( i ,     j b ) ω Δ t / 2 ] × Δ t ɛ r ( i ,     j b ) Δ s H x , inc n + 1 / 2 ( i ,     j b + 1 / 2 ) .
H y n + 1 / 2 ( i a - 1 / 2 ,     j ) = [ H y n + 1 / 2 ( i a - 1 / 2 ,     j ) ] ( 1 b ) - Δ t μ 0 Δ s E z , inc n ( i a ,     j ) ,
E z n + 1 ( i a ,     j ) = [ E z n + 1 ( i a ,     j ) ] ( 1 c ) + exp [ - ɛ i ( i a ,     j ) ɛ r ( i a ,     j ) ω Δ t / 2 ] × Δ t ɛ r ( i a ,     j ) Δ s H y , i n c n + 1 / 2 ( i a - 1 / 2 ,     j ) .
H y n + 1 / 2 ( i b + 1 / 2 ,     j ) = [ H y n + 1 / 2 ( i b + 1 / 2 ,     j ) ] ( 1 b ) + Δ t μ 0 Δ s E z , inc n ( i b ,     j ) ,
E z n + 1 ( i b ,     j ) = [ E z n + 1 ( i b ,     j ) ] ( 1 c ) - exp [ - ɛ i ( i b , j ) ɛ r ( i b ,     j ) ω Δ t / 2 ] × Δ t ɛ r ( i b ,     j ) Δ s H y , inc n + 1 / 2 ( i b + 1 / 2 ,     j ) .
B x n + 1 / 2 ( i ,     j + 1 / 2 ) = ( 2 ɛ 0 κ y - σ y Δ t 2 ɛ 0 κ y + σ y Δ t ) B x n - 1 / 2 ( i ,     j + 1 / 2 ) - ( 2 ɛ 0 Δ t / Δ s 2 ɛ 0 κ y + σ y Δ t ) × [ E z n ( i ,     j + 1 ) - E z n ( i ,     j ) ] ,
H x n + 1 / 2 ( i ,     j + 1 / 2 ) = H x n - 1 / 2 ( i ,     j + 1 / 2 ) + ( 1 2 μ 0 ɛ 0 ) × [ ( 2 ɛ 0 κ x + σ x Δ t ) B x n + 1 / 2 ( i ,     j + 1 / 2 ) - ( 2 ɛ 0 κ x - σ x Δ t ) B x n - 1 / 2 ( i ,     j + 1 / 2 ) ] ,
B y n + 1 / 2 ( i + 1 / 2 ,     j ) = B y n - 1 / 2 ( i + 1 / 2 ,     j ) + ( Δ t Δ s ) × [ E z n ( i + 1 ,     j ) - E z n ( i ,     j ) ] ,
H y n + 1 / 2 ( i + 1 / 2 ,     j ) = ( 2 ɛ 0 κ x - σ x Δ t 2 ɛ 0 κ x + σ x Δ t ) H y n - 1 / 2 ( i + 1 / 2 ,     j ) + ( 1 / μ 0 2 ɛ 0 κ x + σ x Δ t ) × [ ( 2 ɛ 0 κ y + σ y Δ t ) B y n + 1 / 2 ( i + 1 / 2 ,     j ) - ( 2 ɛ 0 κ y - σ y Δ t ) B y n - 1 / 2 ( i + 1 / 2 ,     j ) ] ,
P z n + 1 ( i ,     j ) = ( 2 ɛ - σ Δ t 2 ɛ + σ Δ t ) P z n ( i ,     j ) + ( 2 Δ t / Δ s 2 ɛ + σ Δ t ) × [ H y n + 1 / 2 ( i + 1 / 2 ,     j ) - H y n + 1 / 2 ( i - 1 / 2 , j ) - H x n + 1 / 2 ( i , j + 1 / 2 ) + H x n + 1 / 2 ( i ,     j - 1 / 2 ) ] ,
Q z n + 1 ( i ,     j ) = ( 2 ɛ 0 κ x - σ x Δ t 2 ɛ 0 κ x + σ x Δ t ) Q z n ( i ,     j ) + ( 2 ɛ 0 2 ɛ 0 κ x + σ x Δ t ) × [ P z n + 1 ( i ,     j ) - P z n ( i ,     j ) ] ,
E z n + 1 ( i ,     j ) = ( 2 ɛ 0 κ y - σ y Δ t 2 ɛ 0 κ y + σ y Δ t ) E z n ( i ,     j ) + ( 2 ɛ 0 2 ɛ 0 κ y + σ y Δ t ) [ Q z n + 1 ( i ,     j ) - Q z n ( i ,     j ) ] ,
κ x ( x ) = 1 + ( x / d ) m ( κ x , max - 1 ) ,
σ x ( x ) = ( x / d ) m σ x , max ,
σ x , max = - ( m + 1 ) ln [ R ( 0 ) ] 2 ɛ 0 c d .
w e = ω 2 s [ ɛ t i ( ξ ) + ɛ h i ( ξ ) ] Re [ E i ( ξ ) · E * ( ξ ) ] d 2 ξ - ω 2 s [ ɛ t r ( ξ ) - ɛ h r ( ξ ) ] Im [ E i ( ξ ) · E * ( ξ ) ] d 2 ξ - ω 2 s ɛ h i ( ξ ) [ E i ( ξ ) · E i * ( ξ ) ] d 2 ξ ,
w a = ω 2 ɛ t i ( ξ ) E ( ξ ) · E * ( ξ ) d 2 ξ .
E S ( R ) = s G ( R ,     ξ ) ( k h 2 II + ξ ξ ) · ( P / ɛ h ) d 2 ξ ,
G ( R ,     ξ ) = i 3 / 2 8 π k h exp ( i k h R - ξ ) R - ξ .
s 1 = i 3 / 2 k h 2 8 π k h S [ 1 - ɛ t ( ξ ) / ɛ h ] E z exp ( - i k h r · ξ ) d 2 ξ ,
s 2 = i 3 / 2 k h 2 8 π k h S [ 1 - ɛ t ( ξ ) / ɛ h ] a · ( E x , E y ) × exp ( - i k h r · ξ ) d 2 ξ
x 2 a 2 + y 2 b 2 = 1 ,

Metrics