Abstract

A three-dimensional fourth-order finite-difference time-domain (FDTD) program with a symplectic integrator scheme has been developed to solve the problem of light scattering by small particles. The symplectic scheme is nondissipative and requires no more storage than the conventional second-order FDTD scheme. The total-field and scattered-field technique is generalized to provide the incident wave source conditions in the symplectic FDTD (SFDTD) scheme. The perfectly matched layer absorbing boundary condition is employed to truncate the computational domain. Numerical examples demonstrate that the fourth-order SFDTD scheme substantially improves the precision of the near-field calculation. The major shortcoming of the fourth-order SFDTD scheme is that it requires more computer CPU time than a conventional second-order FDTD scheme if the same grid size is used. Thus, to make the SFDTD method efficient for practical applications, one needs to parallelize the corresponding computational code.

© 2005 Optical Society of America

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References

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  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).
  2. K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EC-24, 397–405 (1982).
    [CrossRef]
  3. A. Taflove, S. C. Hagness, Computational Electromagnetics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).
  4. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  5. 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]
  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]
  7. 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]
  8. P. Yang, K. N. Liou, “Finite-difference time-domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 173–221.
    [CrossRef]
  9. W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
    [CrossRef]
  10. S. C. Hill, G. Videen, W. Sun, Q. Fu, “Scattering and internal fields of a microsphere that contains a saturable absorber: finite-difference time-domain simulations,” Appl. Opt. 40, 5487–5494 (2001).
    [CrossRef]
  11. K. L. Shlager, J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag. 51, 642–653 (2003).
    [CrossRef]
  12. J. Fang, “Time domain finite difference computation for Maxwell’s equations,” Ph.D. dissertation (Department of Electrical Engineering, University of California at Berkeley, Berkeley, Calif., 1989).
  13. T. Deveze, L. Beaulieu, W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” in Proceedings of the 1992 International IEEE Antennas and Propagation Society Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 346–349.
    [CrossRef]
  14. C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).
  15. J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
    [CrossRef]
  16. M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulation,” Phys. Lett. A 146, 319–323 (1990).
    [CrossRef]
  17. H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
    [CrossRef]
  18. M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. Lett. 32, 400–407 (1991).
    [CrossRef]
  19. T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
    [CrossRef]
  20. I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
    [CrossRef]
  21. T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
    [CrossRef]
  22. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EC-23, 377–382 (1981).
    [CrossRef]
  23. Ref. 3, pp. 175–233.
  24. P.-W. Zhai, Y.-K. Lee, G. W. Kattawar, P. Yang, “Implementing the near- to far-field transformation in the finite-difference time-domain method,” Appl. Opt. 43, 3738–3746 (2004).
    [CrossRef] [PubMed]
  25. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  26. T. Hirono, W. W. Lui, S. Seki, “Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space,” in Vol. 3 of Microwave Symposium Digest, 1999 IEEE MTT-S International (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1293–1296.
    [CrossRef]

2004 (1)

2003 (1)

K. L. Shlager, J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag. 51, 642–653 (2003).
[CrossRef]

2001 (3)

I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
[CrossRef]

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[CrossRef]

S. C. Hill, G. Videen, W. Sun, Q. Fu, “Scattering and internal fields of a microsphere that contains a saturable absorber: finite-difference time-domain simulations,” Appl. Opt. 40, 5487–5494 (2001).
[CrossRef]

2000 (1)

1999 (1)

1997 (2)

J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
[CrossRef]

T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
[CrossRef]

1996 (1)

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]

C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).

1994 (1)

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

1991 (1)

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. Lett. 32, 400–407 (1991).
[CrossRef]

1990 (2)

M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulation,” Phys. Lett. A 146, 319–323 (1990).
[CrossRef]

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

1982 (1)

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

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EC-23, 377–382 (1981).
[CrossRef]

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).

Beaulieu, L.

T. Deveze, L. Beaulieu, W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” in Proceedings of the 1992 International IEEE Antennas and Propagation Society Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 346–349.
[CrossRef]

Berenger, J.-P.

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

Broschat, S. L.

C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).

Chen, Z.

Deveze, T.

T. Deveze, L. Beaulieu, W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” in Proceedings of the 1992 International IEEE Antennas and Propagation Society Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 346–349.
[CrossRef]

Fang, J.

J. Fang, “Time domain finite difference computation for Maxwell’s equations,” Ph.D. dissertation (Department of Electrical Engineering, University of California at Berkeley, Berkeley, Calif., 1989).

Fu, Q.

Gaitonde, D.

J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
[CrossRef]

Hagness, S. C.

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

Hill, S. C.

Hirono, T.

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[CrossRef]

T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
[CrossRef]

T. Hirono, W. W. Lui, S. Seki, “Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space,” in Vol. 3 of Microwave Symposium Digest, 1999 IEEE MTT-S International (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1293–1296.
[CrossRef]

Kattawar, G. W.

Kunz, K. S.

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

Lee, Y.-K.

Liou, K. N.

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

P. Yang, K. N. Liou, “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]

P. Yang, K. N. Liou, “Finite-difference time-domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 173–221.
[CrossRef]

Luebbers, R. J.

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

Lui, W.

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[CrossRef]

Lui, W. W.

T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
[CrossRef]

T. Hirono, W. W. Lui, S. Seki, “Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space,” in Vol. 3 of Microwave Symposium Digest, 1999 IEEE MTT-S International (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1293–1296.
[CrossRef]

Manry, C. W.

C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EC-23, 377–382 (1981).
[CrossRef]

Saitoh, I.

I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
[CrossRef]

Schneider, J. B.

K. L. Shlager, J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag. 51, 642–653 (2003).
[CrossRef]

C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).

Seki, S.

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[CrossRef]

T. Hirono, W. W. Lui, S. Seki, “Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space,” in Vol. 3 of Microwave Symposium Digest, 1999 IEEE MTT-S International (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1293–1296.
[CrossRef]

Shang, J. S.

J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
[CrossRef]

Shlager, K. L.

K. L. Shlager, J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag. 51, 642–653 (2003).
[CrossRef]

Sun, W.

Suzuki, M.

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. Lett. 32, 400–407 (1991).
[CrossRef]

M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulation,” Phys. Lett. A 146, 319–323 (1990).
[CrossRef]

Suzuki, Y.

I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
[CrossRef]

Tabbara, W.

T. Deveze, L. Beaulieu, W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” in Proceedings of the 1992 International IEEE Antennas and Propagation Society Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 346–349.
[CrossRef]

Taflove, A.

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

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

Takahashi, N.

I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
[CrossRef]

Umashankar, K. R.

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

Videen, G.

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).

Yokoyama, K.

T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
[CrossRef]

Yoshida, H.

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

Yoshikuni, Y.

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[CrossRef]

Young, J. L.

J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
[CrossRef]

Zhai, P.-W.

Appl. Comput. Electromagn. Soc. J. (1)

C. W. Manry, S. L. Broschat, J. B. Schneider, “Higher-order FDTD methods for large problems,” Appl. Comput. Electromagn. Soc. J. 10, 17–29 (1995).

Appl. Opt. (4)

IEEE Microwave Guid. Wave Lett. (1)

T. Hirono, W. W. Lui, K. Yokoyama, “Time-domain simulation of electromagnetic field using a symplectic integrator,” IEEE Microwave Guid. Wave Lett. 7, 279–281 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

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).

K. L. Shlager, J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag. 51, 642–653 (2003).
[CrossRef]

J. L. Young, D. Gaitonde, J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach,” IEEE Trans. Antennas Propag. 45, 1573–1580 (1997).
[CrossRef]

IEEE Trans. Electromagn. Compat. (2)

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

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EC-23, 377–382 (1981).
[CrossRef]

IEEE Trans. Magn. (1)

I. Saitoh, Y. Suzuki, N. Takahashi, “The symplectic finite difference time domain method,” IEEE Trans. Magn. 37, 3251–3254 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

T. Hirono, W. Lui, S. Seki, Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microwave Theory Tech. 49, 1640–1648 (2001).
[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. Math. Phys. Lett. (1)

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. Lett. 32, 400–407 (1991).
[CrossRef]

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

Phys. Lett. A (2)

M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulation,” Phys. Lett. A 146, 319–323 (1990).
[CrossRef]

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

Other (7)

J. Fang, “Time domain finite difference computation for Maxwell’s equations,” Ph.D. dissertation (Department of Electrical Engineering, University of California at Berkeley, Berkeley, Calif., 1989).

T. Deveze, L. Beaulieu, W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” in Proceedings of the 1992 International IEEE Antennas and Propagation Society Symposium (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 346–349.
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time-domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 173–221.
[CrossRef]

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

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

T. Hirono, W. W. Lui, S. Seki, “Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space,” in Vol. 3 of Microwave Symposium Digest, 1999 IEEE MTT-S International (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 1293–1296.
[CrossRef]

Ref. 3, pp. 175–233.

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Figures (5)

Fig. 1
Fig. 1

(a) Six-sided TF–SF interface surface for the three-dimensional symplectic FDTD space lattice. (b) A detailed top view of the Yee structure in region b, where the arrows represent Ex, the circles represent Hz, and the solid line represents the TF–SF interface. At the left of the solid line is the SF region and at the right of that line is the TF region.

Fig. 2
Fig. 2

Snapshots of the Ex field distribution in a one-dimensional grid in a time domain calculated by Mie theory and FDTD and SFDTD methods. cΔtz = 0.5 and free space are assumed. (a) n = 250 and (b) n = 1000.

Fig. 3
Fig. 3

TF–SF grid zoning for a pulsed plane wave propagating in free space. The three snapshots calculated by the SFDTD scheme show the Ex field distribution at the xz plane at three time steps.

Fig. 4
Fig. 4

Ex field distribution in the frequency domain along the z axis inside the spherical particle calculated by the Mie theory and the FDTD and SFDTD methods. The refractive index is m = 1.0925 + i0.248, the size parameter is X = 10, and we used Δ = λ/20. (a) Amplitude of the Ex field. (b) Percentage errors of the Ex fields calculated by the FDTD and SFDTD methods relative to the Mie theory. (c) Phase of the Ex field. (d) Phase difference of the Ex field calculated by the FDTD and SFDTD methods relative to Mie theory.

Fig. 5
Fig. 5

Same as Fig. 4 except the refractive index is m = 1.5710 + i0.1756, and we used Δ = λ/25.

Tables (1)

Tables Icon

Table 1 Coefficients of the Symplectic Integrator Propagators cp = cm+1−p(0 < p < m + 1), dp = dmp(0 < p < m), dm = 0

Equations (29)

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

t ( H E ) = W ( H E ) ,
W = U + V ,
U = [ ( 0 ) - c R ( 0 ) ( 0 ) ] ,
V = [ ( 0 ) ( 0 ) c R / ɛ r - τ · I 3 ] ,
R = ( 0 - z y y 0 - x - y x 0 ) .
( H E ) ( Δ t ) = exp ( Δ t W ) ( H E ) .
exp ( Δ t W ) = II p = 1 m exp ( d p Δ t V ) exp ( c p Δ t U ) + O [ ( Δ t ) n + 1 ] ,
exp ( Δ t U ) = I 6 + Δ t U ,
exp ( Δ t V ) = n = 0 ( Δ t V ) n / n ! = ( I 3 ( 0 ) 1 - exp ( - τ Δ t ) τ c R / ɛ r exp ( - τ Δ t ) · I 3 ) .
( f x ) i 27 ( f i + 1 / 2 - f i - 1 / 2 ) - f i + 3 / 2 + f i - 3 / 2 24 Δ x .
H y n + p / 5 ( i , j + 1 2 , k ) = H y n + ( p - 1 ) / 5 ( i , j + 1 2 , k ) - c Δ t c p Δ z { 9 8 [ E x n + ( p - 1 ) / 5 ( i , j + 1 2 , k + 1 2 ) - E x n + ( p - 1 ) / 5 ( i , j + 1 2 , k - 1 2 ) ] - 1 24 [ E x n + ( p - 1 ) / 5 ( i , j + 1 2 , k + 3 2 ) - E x n + ( p - 1 ) / 5 ( i , j + 1 2 , k - 3 2 ) ] } + c Δ t c p Δ x { 9 8 [ E z n + ( p - 1 ) / 5 ( i + 1 2 , j + 1 2 , k ) - E z n + ( p - 1 ) / 5 ( i - 1 2 , j + 1 2 , k ) ] - 1 24 [ E z n + ( p - 1 ) / 5 ( i + 3 2 , j + 1 2 , k ) - E z n + ( p - 1 ) / 5 ( i - 3 2 , j + 1 2 , k ) ] } ,
E x n + p / 5 ( i , j + 1 2 , k + 1 2 ) = exp ( - τ d p Δ t ) E x n + ( p - 1 ) / 5 ( i , j + 1 2 , k + 1 2 ) + 1 - exp ( - τ d p Δ t ) τ ɛ r Δ y c { 9 8 [ H z n + p / 5 ( i , j + 1 , k + 1 2 ) - H z n + p / 5 ( i , j , k + 1 2 ) ] - 1 24 [ H z n + p / 5 ( i , j + 2 , k + 1 2 ) - H z n + p / 5 ( i , j - 1 , k + 1 2 ) ] } + 1 - exp ( - τ d p Δ t ) τ ɛ r Δ z c { 9 8 [ H y n + p / 5 ( i , j + 1 2 , k + 1 ) - H y n + p / 5 ( i , j + 1 2 , k ) ] - 1 24 [ H y n + p / 5 ( i , j + 1 2 , k + 2 ) - H y n + p / 5 ( i , j + 1 2 , k - 1 ) ] } ,
E x n + p / 5 ( i , j 0 + 1 / 2 , k + 1 / 2 ) = E x n + p / 5 ( i , j 0 + 1 / 2 , k + 1 / 2 ) + c Δ t d p 24 Δ y H i , z n + p / 5 ( i , j 0 - 1 , k + 1 / 2 ) ,
E x n + p / 5 ( i , j 0 - 1 / 2 , k + 1 / 2 ) = E x n + p / 5 ( i , j 0 - 1 / 2 , k + 1 / 2 ) - c Δ t d p Δ y × [ 9 8 H i , z n + p / 5 ( i , j 0 - 1 , k + 1 / 2 ) - 1 24 H i , z n + p / 5 ( i , j 0 - 2 , k + 1 / 2 ) ] ,
E x n + p / 5 ( i , j 0 - 3 / 2 , k + 1 / 2 ) = E x n + p / 5 ( i , j 0 - 3 / 2 , k + 1 / 2 ) + c Δ t d p 24 Δ y H i , z n + p / 5 ( i , j 0 , k + 1 / 2 ) ,
E x n + p / 5 ( k s ) = g [ n ( p ) Δ t ] ,
t ( H E ) = W ( H E ) ,
W = ( - σ - R R - σ ) ,
σ 11 = σ 66 = 4 π σ y c ,
σ 22 = σ 33 = 4 π σ z c ,
σ 44 = σ 55 = 4 π σ x c ,
σ i j = 0 , i j ,
R 15 = R 16 = - R 61 = - R 62 = y ,
R 31 = R 32 = - R 23 = - R 24 = z ,
R 53 = R 54 = - R 45 = - R 46 = x ,
R i j = 0
W = [ - σ - R ( 0 ) ( 0 ) ] + [ ( 0 ) ( 0 ) R - σ ]
H x y n + p / 5 ( i + 1 / 2 , j , k ) = exp ( - σ y c Δ t c p ) H x y n + ( p - 1 ) / 5 ( i + 2 , j , k ) - 1 - exp ( - σ y c Δ t c p ) σ y E z n + ( p - 1 ) / 5 y ( i + 1 / 2 , j , k ) ,
E x n ( k s + 1 / 2 ) = exp [ - ( n / n decay - n 0 ) 2 ] ,

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