Abstract

The three-dimensional (3-D) finite-difference time-domain (FDTD) technique has been extended to simulate light scattering and absorption by nonspherical particles embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. When computing the single-scattering properties of a particle in an absorbing dielectric medium, we derive the single-scattering properties including scattering phase functions, extinction, and absorption efficiencies using a volume integration of the internal field. A Mie solution for light scattering and absorption by spherical particles in an absorbing medium is used to examine the accuracy of the 3-D UPML FDTD code. It is found that the errors in the extinction and absorption efficiencies from the 3-D UPML FDTD are less than ∼2%. The errors in the scattering phase functions are typically less than ∼5%. The errors in the asymmetry factors are less than ∼0.1%. For light scattering by particles in free space, the accuracy of the 3-D UPML FDTD scheme is similar to a previous model [Appl. Opt. 38, 3141 (1999)].

© 2002 Optical Society of America

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    [CrossRef]
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    [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. D 6, 365–373 (1999).
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    [CrossRef]
  7. 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]
  8. I. W. Sudiarta, P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70, 709–714 (2001).
    [CrossRef]
  9. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
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    [CrossRef]
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    [CrossRef]
  12. 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]
  13. 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]
  14. 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]
  15. P. Yang, K. N. Liou, M. I. Mishchenko, B. C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: applications to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, New York, 2000).
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    [CrossRef]
  25. 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]
  26. Z. Liao, H. L. Wong, B. Yang, Y. Yuan, “A transmitting boundary for transient wave analyses,” Sci. Sin. 27, 1063–1076 (1984).
  27. R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation,” Math. Comput. 47, 437–459 (1986).
  28. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
    [CrossRef]
  29. D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guid. Wave Lett. 6, 97–99 (1996).
    [CrossRef]
  30. 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]
  31. K. Umashanker, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
    [CrossRef]
  32. A. Taflove, Computational Electrodynamics: the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).
  33. 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]
  34. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
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    [CrossRef] [PubMed]
  36. G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
    [CrossRef]

2001 (3)

2000 (2)

1999 (2)

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. D 6, 365–373 (1999).

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]

1996 (6)

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]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guid. Wave Lett. 6, 97–99 (1996).
[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]

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (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]

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]

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 (4)

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

R. Holland, “Finite-difference time domain (FDTD) analysis of magnetic diffusion,” IEEE Trans. Electromagn. Compat. 36, 32–39 (1994).
[CrossRef]

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

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

1988 (1)

1986 (1)

R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation,” Math. Comput. 47, 437–459 (1986).

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 (1)

K. Umashanker, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[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 (2)

A. Bayliss, E. Turkel, “Radiation boundary conditions for wavelike equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

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)

1971 (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1971).

1966 (1)

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

Bayliss, A.

A. Bayliss, E. Turkel, “Radiation boundary conditions for wavelike equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

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]

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

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.

Engquist, B.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1971).

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.

Gandhi, O. P.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

Gao, B. C.

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. D 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]

Goedecke, G. H.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).

Higdon, R. L.

R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation,” Math. Comput. 47, 437–459 (1986).

Holland, R.

R. Holland, “Finite-difference time domain (FDTD) analysis of magnetic diffusion,” IEEE Trans. Electromagn. Compat. 36, 32–39 (1994).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, New York, 2000).

Huffman, D. R.

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

Joseph, R. M.

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

Katz, D. S.

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

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. D 6, 365–373 (1999).

Lazzi, G.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

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. D 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]

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.

Majda, A.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1971).

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]

O’Brien, S. G.

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]

Reuter, C. E.

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[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. D 6, 365–373 (1999).

Sudiarta, I. W.

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

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]

Sullivan, D. M.

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guid. Wave Lett. 6, 97–99 (1996).
[CrossRef]

Sun, W.

Taflove, A.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

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

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. C. Hagness, Computational Electrodynamics: the Finite-Difference Time Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).

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

Taflove, T.

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

Thiele, E. T.

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, New York, 2000).

Turkel, E.

A. Bayliss, E. Turkel, “Radiation boundary conditions for wavelike equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

Umashanker, K.

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

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).

Yang, P.

Yee, K. S.

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

Yuan, Y.

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

Appl. Opt. (5)

Commun. Pure Appl. Math. (1)

A. Bayliss, E. Turkel, “Radiation boundary conditions for wavelike equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

Eur. Phys. J. D (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. D 6, 365–373 (1999).

IEEE Microwave Guid. Wave Lett. (3)

C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, T. Taflove, “Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations,” IEEE Microwave Guid. Wave Lett. 4, 344–346 (1994).
[CrossRef]

D. M. Sullivan, “A simplified PML for use with the FDTD method,” IEEE Microwave Guid. Wave Lett. 6, 97–99 (1996).
[CrossRef]

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations 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]

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[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. (3)

K. Umashanker, 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 (1981).
[CrossRef]

R. Holland, “Finite-difference time domain (FDTD) analysis of magnetic diffusion,” IEEE Trans. Electromagn. Compat. 36, 32–39 (1994).
[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. (2)

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

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 (3)

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

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

Math. Comput. (2)

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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]

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).

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M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles (Academic, New York, 2000).

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

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

A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).

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Figures (15)

Fig. 1
Fig. 1

Geometries for the one-dimensional auxiliary FDTD grid and the closed rectangular interface of the total field and the scattered field.

Fig. 2
Fig. 2

Incident and scattering geometries for the scattered far-field formulation following Yang and Liou.13

Fig. 3
Fig. 3

Comparison of phase functions for a spherical particle embedded in an absorbing medium with a size parameter of 2πa0 = 6, where a is the radius of the spherical particle and λ0 is the incident wavelength in free space. These results were computed with Mie theory and the FDTD method for a host refractive index of 1.2 + 0.05i and a particle refractive index of 1.3 + 0.1i. For the FDTD calculations, we used a FDTD cell size of Δs = λ0/30 and the UPML parameters are (a), κmax = 1 and R(0) = 10-8 and (b), κmax = 15 and R(0)] = 10-5. Also shown are the absolute and the relative errors of the FDTD results.

Fig. 4
Fig. 4

Extinction and absorption efficiencies and asymmetry factors for spherical particles embedded in an absorbing medium as functions of size parameter 2πa0, where a is the radius of the spherical particle and λ0 is the incident wavelength in free space. These results were computed by the FDTD method and Mie theory for a host refractive index of 1.2 + 0.05i and a particle refractive index of 1.3 + 0.1i. Also shown are the absolute and the relative errors of the FDTD results. For the FDTD calculation, we used a FDTD cell size of Δs = λ0/30.

Fig. 5
Fig. 5

Scattering phase functions for spherical particles embedded in an absorbing medium with size parameters of 2, 4, and 6. These results were computed by the FDTD method and Mie theory for a host refractive index of 1.2 + 0.05i and a particle refractive index of 1.3 + 0.1i. Also shown are the absolute and the relative errors of the FDTD results. For the FDTD calculation, we used a FDTD cell size of Δs = λ0/30.

Fig. 6
Fig. 6

Same as Fig. 5, but for size parameters of 8, 10, and 12.

Fig. 7
Fig. 7

Same as Fig. 4, but for a host refractive index of 1.2 + 0.1i and a particle refractive index of 1.1 + 0.05i.

Fig. 8
Fig. 8

Same as Fig. 5, but for a host refractive index of 1.2 + 0.1i and a particle refractive index of 1.1 + 0.05i.

Fig. 9
Fig. 9

Same as Fig. 6, but for a host refractive index of 1.2 + 0.1i and a particle refractive index of 1.1 + 0.05i.

Fig. 10
Fig. 10

Extinction (scattering) efficiencies and asymmetry factors for spherical air bubbles embedded in an absorbing medium as functions of size parameter 2πa0, where a is the radius of the spherical bubble and λ0 is the incident wavelength in free space. These results were computed by the FDTD method and the Mie theory for a host refractive index of 1.2 + 0.05i. Also shown are the absolute and the relative errors of the FDTD results. For the FDTD calculation, we used a FDTD cell size of Δs = λ0/30.

Fig. 11
Fig. 11

Same as Fig. 5, but for spherical air bubbles embedded in a host medium with a refractive index of 1.2 + 0.05i.

Fig. 12
Fig. 12

Same as Fig. 6, but for spherical air bubbles embedded in a host medium with a refractive index of 1.2 + 0.05i.

Fig. 13
Fig. 13

Scattering phase functions for a spherical particle in free space with a size parameter of 20. These results were computed by the FDTD method and Mie theory for a particle refractive index of 1.0925 + 0.248i. Also shown are the absolute and the relative errors of the FDTD results. For the FDTD calculation, we used a FDTD cell size of Δs = λ0/20.

Fig. 14
Fig. 14

Geometries of a spheroid with half-axes a, b, and c in a coordinate system. The light is incident on the spheroid in the direction of a zenith angle of 45° and an azimuth angle of 45°.

Fig. 15
Fig. 15

Azimuthally averaged elements of the scattering phase matrix for the spheroid and incidence geometries illustrated in Fig. 14. The spheroid has half-axes a = 3b = 6c and a size parameter of 2πa0 of 30. The refractive index of the spheroid is 1.407. The refractive indices of the host medium are 1.34 and 1.34 + 0.05i. For the FDTD calculation, we used a FDTD cell size of Δs = λ0/30.

Tables (2)

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Table 1 Single-Scattering Properties of a Spherical Particlea

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Table 2 Single-Scattering Properties Computed by the FDTD Method and by Mie Theorya

Equations (74)

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Hxn+1/2i, j+1/2, k+1/2=Hxn-1/2i, j+1/2, k+1/2+Δtμi, j+1/2, k+1/2Δs×Eyni, j+1/2, k+1-Eyni, j+1/2, k+Ezni, j, k+1/2-Ezni, j+1, k+1/2,
Exn+1i+1/2, j, k=exp-εii+1/2, j, kεri+1/2, j, k ωΔtExni+1/2, j, k+exp-εii+1/2, j, kεri+1/2, j, k ωΔt/2Δtεri+1/2, j, kΔsHyn+1/2i+1/2, j, k-1/2-Hyn+1/2i+1/2, j, k+1/2+Hzn+1/2i+1/2, j+1/2, k-Hzn+1/2i+1/2, j-1/2, k,
Hxn+1/2i, ja-1/2, k=Hxn+1/2i, ja-1/2, k1a+Δtμi, ja-1/2, kΔs×Ez,incni, ja, k.
Hxn+1/2i, jb+1/2, k=Hxn+1/2i, jb+1/2, k1a-Δtμi, jb+1/2, kΔs×Ez,incni, jb, k.
Hxn+1/2i, j, ka-1/2=Hxn+1/2i, j, ka-1/21a-Δtμi, j, ka-1/2Δs×Ey,incni, j, ka.
Hxn+1/2i, j, kb+1/2=Hxn+1/2i, j, kb+1/21a+Δtμi, j, kb+1/2Δs×Ey,incni, j, kb.
Exn+1i, ja, k=Exn+1i, ja, k1b-exp-εii, ja, kεri, ja, k ωΔt/2×Δtεri, ja, kΔs×Hz,incn+1/2i, ja-1/2, k.
Exn+1i, jb, k=Exn+1i, jb, k1b+exp-εii, jb, kεri, jb, k ωΔt/2×Δtεri, jb, kΔs×Hz,incn+1/2i, jb+1/2, k.
Exn+1i, j, ka=Exn+1i, j, ka1b+exp-εii, j, kaεri, j, ka ωΔt/2×Δtεri, j, kaΔs×Hy,incn+1/2i, j, ka-1/2.
Exn+1i, j, kb=Exn+1i, j, kb1b-exp-εii, j, kbεri, j, kb ωΔt/2×Δtεri, j, kbΔs×Hy,incn+1/2i, j, kb+1/2.
Eincnm=2=exp-t30 Δt-52.
Hincn+1/2m+1/2=Hincn-1/2m+1/2+Δtμm+1/2Δsvpθ=0, ϕ=0vpθ, ϕ ×Eincnm-Eincnm+1,
Eincn+1m=exp-εimεrm ωΔtEincnm+exp-εimεrm ωΔt/2×ΔtεrmΔsvpθ=0, ϕ=0vpθ, ϕ×Hincn+1/2m-1/2-Hincn+1/2m+1/2,
×Hx, y, z=iωε+σs¯¯Ex, y, z,
×Ex, y, z=-iωμs¯¯Hx, y, z,
s¯¯=sx-1000sx000sxsy000sy-1000sysz000sz000sz-1 =syszsx-1000sxszsy-1000sxsysz-1,
sx=κx+σxiωε0, sy=κy+σyiωε0, sz=κz+σziωε0.
κxx=1+x/dmκx,max-1,
σxx=x/dmσx,max,
Rθ=exp-2 cos θε0c0d σxdx=exp-2σx,maxd cos θε0cm+1.
σx,max=-m+1lnR02ε0cd.
s=E×H*,
×H=-iωεr+iεiE,
×E=iωμH,
·s=·E×H*=H*·×E-E·×H* =iωμH·H*-εrE·E*-ωεiE·E*.
wa=-12Resn·sξd2ξ =-12Rev ·sξd3ξ =ω2v εtiξEξ·E*ξd3ξ,
Es=E-Ei,
Hs=H-Hi,
ws=12Resn·Es×Hs*d2ξ =12Resn·E-Ei×H*-Hi*d2ξ.
we=ws+wa =12Resn·E-Ei×H*-Hi*d2ξ-12Resn·E×H*d2ξ =12Resn·Ei×Hi*-Ei×H*-E×Hi*d2ξ =12Rev ·Ei×Hi*-Ei×H*-E×Hi*d3ξ.
we=wa+ws =ω2vεtiξ+εhiξReEiξ·E*ξd3ξ-ω2vεtrξ-εhrξImEiξ·E*ξd3ξ-ω2v εhiξEiξ·Ei*ξd3ξ.
f=2πa2η2 I01+η-1eη,
s2s3s4s1=Fα,yFα,xFβ,yFβ,xβ·xβ·y-β·yβ·x,
Fα,xFβ,x=ikh34πv1-εξεh×α·Eξβ·Eξexp-ikhr·ξd3ξ.
Fα,yFβ,y=ikh34πv1-εξεh×α·Eξβ·Eξexp-ikhr·ξd3ξ.
x2a2+y2b2+z2c2=1,
Bxx, y, z=μszsxHxx, y, z,
Byx, y, z=μsxsyHyx, y, z,
Bzx, y, z=μsyszHzx, y, z.
Eyx, y, zz-Ezx, y, zyEzx, y, zx-Exx, y, zzExx, y, zy-Eyx, y, zx=iωsy000sz000sxBxx, y, zByx, y, zBzx, y, z.
iωκx+σxε0Bxx, y, z=iωκz+σzε0μHxx, y, z,
iωκy+σyε0Byx, y, z=iωκx+σxε0μHyx, y, z,
iωκz+σzε0Bzx, y, z=iωκy+σyε0μHzx, y, z,
Eyx, y, z, tz-Ezx, y, z, tyEzx, y, z, tx-Exx, y, z, tzExx, y, z, ty-Eyx, y, z, tx=tκy000κz000κxBxx, y, z, tByx, y, z, tBzx, y, z, t+1ε0σy000σz000σxBxx, y, z, tByx, y, z, tBzx, y, z, t,
κxBxx, y, z, tt+σxε0 Bxx, y, z, t=μκzHxx, y, z, tt+μ σzε0 Hxx, y, z, t,
κyByx, y, z, tt+σyε0 Byx, y, z, t=μκxHyx, y, z, tt+μ σxε0 Hyx, y, z, t,
κzBzx, y, z, tt+σzε0 Bzx, y, z, t=μκyHzx, y, z, tt+μ σyε0 Hzx, y, z, t.
Bxn+1/2i, j+1/2, k+1/2=2ε0κy-σyΔt2ε0κy+σyΔtBxn-1/2×i, j+1/2, k+1/2+2ε0Δt/Δs2ε0κy+σyΔt×Eyni, j+1/2, k+1-Eyni, j+1/2, k+Ezni, j, k+1/2-Ezni, j+1, k+1/2,
Hxn+1/2i, j+1/2, k+1/2=2ε0κz-σzΔt2ε0κz+σzΔtHxn-1/2×i, j+1/2, k+1/2+1/μ2ε0κz+σzΔt×2ε0κx+σxΔtBxn+1/2i, j+1/2, k+1/2-2ε0κx-σxΔtBxn-1/2i, j+1/2, k+1/2.
Pxx, y, z=syszsxExx, y, z,
Pyx, y, z=sxszsyEyx, y, z,
Pzx, y, z=sxsyszEzx, y, z,
Qxx, y, z=1syPxx, y, z,
Qyx, y, z=1szPyx, y, z,
Qzx, y, z=1sxPzx, y, z,
Pxn+1i+1/2, j, k=2ε-σΔt2ε+σΔtPxni+1/2, j, k+2Δt/Δs2ε+σΔt×Hyn+1/2i+1/2, j, k-1/2-Hyn+1/2i+1/2, j, k+1/2+Hzn+1/2i+1/2, j+1/2, k-Hzn+1/2i+1/2, j-1/2, k.
Qxn+1i+1/2, j, k=2ε0κy-σyΔt2ε0κy+σyΔt×Qxni+1/2, j, k+2ε02ε0κy+σyΔt×Pxn+1i+1/2, j, k-Pxni+1/2, j, k.
Exn+1i+1/2, j, k=2ε0κz-σzΔt2ε0κz+σzΔt×Exni+1/2, j, k+12ε0κz+σzΔt×2ε0κx+σxΔt×Qxn+1i+1/2, j, k-2ε0κx-σxΔt×Qxni+1/2, j, k.
·D=0,
·H=0,
×E=iωμH,
×H=-iωD,
D=εhE+P=εE,
·E=-1εh ·P,
×H=-iωεhE+P.
2+kh2E=-1εhkh2P+·P,
2+kh2E=-1εhkh2II+·P.
ER=E0R+v GR, ξ×kh2II+ξξ·P/εhd3ξ,
GR, ξ=expikh|R-ξ|4π|R-ξ|.
EsR|khR=kh2 expikhR4πRvεξεh-1×Eξ-rr·Eξ×exp-ikhr·ξd3ξ.
EsR=αEs,αR+βEs,βR.
Es,αREs,βR=expikhR-ikhRs2s3s4s1E0,αE0,β,
E0,αE0,β=β·xˆ-β·yˆβ·yˆβ·xˆE0,yE0,x.
Es,αREs,βR=kh2 expikhR4πR×vεξεh-1α·Eξβ·Eξ×exp-ikhr·ξd3ξ.

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