Abstract

In this paper, a new finite element method (FEM) is proposed to analyse time domain wave propagation in photonic devices. Dissimilar to conventional FEM, efficient ”inter-element” matrices are accurately formed through smoothing the field derivatives across element boundaries. In this sense, the new approach is termed ”smoothed FEM” (SFETD). For time domain analysis, the propagation is made via the time domain beam propagation method (TD-BPM). Relying on first order elements, our suggested SFETD-BPM enjoys accuracy levels comparable to second-order conventional FEM; thanks to the element smoothing. The proposed method numerical performance is tested through applicating on analysis of a single mode slab waveguide, optical grating structure, and photonic crystal cavity. It is clearly demonstrated that our method is not only accurate but also more computationally efficient (far few run time, and memory requirements) than the conventional FEM approach. The SFETD-BPM is also extended to deal with the very challenging problem of dispersive materials. The material dispersion is smartly utilized to enhance the quality factor of photonic crystal cavity.

© 2015 Optical Society of America

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References

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  1. A. taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 1995)
  2. M. F. O. Hameed, S. S. A. Obayya, and H. A. El-Mikati, “Passive polarization converters based on photonic crystal fiber with L-shaped core region,” J. Lightwave Technol. 30(3), 283–289 (2012).
    [Crossref]
  3. F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
    [Crossref]
  4. A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
    [Crossref]
  5. D. Pinto and S. S. A. Obayya, “Improved complex-envelope alternating-direction-implicit finite-difference-time-domain method for photonic-bandgap cavities,” J. Lightwave Technol. 25(1), 440–447 (2007).
    [Crossref]
  6. M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
    [Crossref]
  7. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18(1), 102–110 (2000).
    [Crossref]
  8. V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Finite element analysis of photonic crystal cavities:time and frequency domain,” J. Lightwave Technol. 23(3), 1514–1521 (2005).
    [Crossref]
  9. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits,” J. Lightwave Technol. 22(2), 684–691 (2004).
    [Crossref]
  10. V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Frequency-dependent envelope finite element time domain analysis of dispersion materials,” Microwave and Opt. Technol. Lett. 44(1), 13–16 (2004).
    [Crossref]
  11. A. Niiyama, M. Koshiba, and Y. Tsuji, “An efficient scalar finite element formulation for nonlinear optical channel waveguides,” J. Lightwave Technol. 13(9), 1919–1925 (1995).
    [Crossref]
  12. G. R. Liu, “A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods,” International Journal of Computational Methods 5(2), 199–236 (2008).
    [Crossref]
  13. G. R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method (CRC Press, 2010).
  14. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge, 2002)
    [Crossref]
  15. G. R. Liu and N. T. Trung, Smoothed Finite Element Methods (CRC Press, 2010)
    [Crossref]
  16. P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
    [Crossref]
  17. G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
    [Crossref]
  18. J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158(2), 485–505 (2003).
    [Crossref]
  19. M. Movahhedi and A. Abdipour, “Alternating direction implicit formulation for the finite element time domain method,” IEEE Trans. Microwave Theory Technology 55(6), 1322–1331 (2007).
    [Crossref]
  20. J. F. Lee, “WETD-A finite element time-domain approach for solving Maxwell’s equations,” IEEE Microwave and Guided wave Letters 4(1), 11–13 (1994).
    [Crossref]
  21. S. S. A. Obayya, “Efficient finite-element-based time-domain beam propagation analysis of Optical integrated circuits,” IEEE J. Quantum Electron. 40(5), 591–595 (2004).
    [Crossref]
  22. V. F. R.- Esquerre and H. E. H.- Figueroa, “Novel time-domain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett. 13(4), 311–313 (2001).
    [Crossref]
  23. G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics - A Meshfree Method (World Scientific: Singapore, 2003).
  24. H. A. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. and Stat. 13(2), 631–644 (1992).
  25. J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Yong-Hee Lee, “Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition,” J. Opt. Soc. Am. B 15(8), 2316–2324 (1998).
    [Crossref]
  26. I. Ahmed and E. Li, “Time domain simulation of dispersive materials from microwave to optical frequencies,” Proceedings of IEEE 7th International Conference on Emerging Technologies (ICET) (IEEE, 2011) 1–5.
  27. S.J. Orfanidis, “Electromagnetic Waves and Antenna,” http://www.ece.rutgers.edu/orfanidi/ewa .

2015 (1)

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

2014 (1)

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

2013 (1)

A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
[Crossref]

2012 (1)

2008 (1)

G. R. Liu, “A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods,” International Journal of Computational Methods 5(2), 199–236 (2008).
[Crossref]

2007 (2)

M. Movahhedi and A. Abdipour, “Alternating direction implicit formulation for the finite element time domain method,” IEEE Trans. Microwave Theory Technology 55(6), 1322–1331 (2007).
[Crossref]

D. Pinto and S. S. A. Obayya, “Improved complex-envelope alternating-direction-implicit finite-difference-time-domain method for photonic-bandgap cavities,” J. Lightwave Technol. 25(1), 440–447 (2007).
[Crossref]

2005 (1)

2004 (3)

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Frequency-dependent envelope finite element time domain analysis of dispersion materials,” Microwave and Opt. Technol. Lett. 44(1), 13–16 (2004).
[Crossref]

S. S. A. Obayya, “Efficient finite-element-based time-domain beam propagation analysis of Optical integrated circuits,” IEEE J. Quantum Electron. 40(5), 591–595 (2004).
[Crossref]

T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits,” J. Lightwave Technol. 22(2), 684–691 (2004).
[Crossref]

2003 (1)

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158(2), 485–505 (2003).
[Crossref]

2001 (1)

V. F. R.- Esquerre and H. E. H.- Figueroa, “Novel time-domain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett. 13(4), 311–313 (2001).
[Crossref]

2000 (1)

1998 (1)

1997 (1)

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

1995 (2)

A. Niiyama, M. Koshiba, and Y. Tsuji, “An efficient scalar finite element formulation for nonlinear optical channel waveguides,” J. Lightwave Technol. 13(9), 1919–1925 (1995).
[Crossref]

P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
[Crossref]

1994 (1)

J. F. Lee, “WETD-A finite element time-domain approach for solving Maxwell’s equations,” IEEE Microwave and Guided wave Letters 4(1), 11–13 (1994).
[Crossref]

1992 (1)

H. A. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. and Stat. 13(2), 631–644 (1992).

Abdelaziz, W. S.

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

Abdipour, A.

M. Movahhedi and A. Abdipour, “Alternating direction implicit formulation for the finite element time domain method,” IEEE Trans. Microwave Theory Technology 55(6), 1322–1331 (2007).
[Crossref]

Ahmed, I.

I. Ahmed and E. Li, “Time domain simulation of dispersive materials from microwave to optical frequencies,” Proceedings of IEEE 7th International Conference on Emerging Technologies (ICET) (IEEE, 2011) 1–5.

Choa, F.-S.

P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
[Crossref]

Decoster, D.

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

El-Azab, J.

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

El-Mikati, H. A.

Esquerre, V. F. R.-

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Finite element analysis of photonic crystal cavities:time and frequency domain,” J. Lightwave Technol. 23(3), 1514–1521 (2005).
[Crossref]

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Frequency-dependent envelope finite element time domain analysis of dispersion materials,” Microwave and Opt. Technol. Lett. 44(1), 13–16 (2004).
[Crossref]

V. F. R.- Esquerre and H. E. H.- Figueroa, “Novel time-domain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett. 13(4), 311–313 (2001).
[Crossref]

Figueroa, H. E. H.-

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Finite element analysis of photonic crystal cavities:time and frequency domain,” J. Lightwave Technol. 23(3), 1514–1521 (2005).
[Crossref]

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Frequency-dependent envelope finite element time domain analysis of dispersion materials,” Microwave and Opt. Technol. Lett. 44(1), 13–16 (2004).
[Crossref]

V. F. R.- Esquerre and H. E. H.- Figueroa, “Novel time-domain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett. 13(4), 311–313 (2001).
[Crossref]

Fornberg, B.

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158(2), 485–505 (2003).
[Crossref]

Fujisawa, T.

Hameed, M. F. O.

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
[Crossref]

M. F. O. Hameed, S. S. A. Obayya, and H. A. El-Mikati, “Passive polarization converters based on photonic crystal fiber with L-shaped core region,” J. Lightwave Technol. 30(3), 283–289 (2012).
[Crossref]

Harari, J.

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

Heikal, A. M.

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
[Crossref]

Hikari, M.

Hussain, F. F. K.

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

Hwang, J. K.

Hyun, S. B.

Jin, G. H.

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

Koshiba, M.

Lee, J.

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158(2), 485–505 (2003).
[Crossref]

Lee, J. F.

J. F. Lee, “WETD-A finite element time-domain approach for solving Maxwell’s equations,” IEEE Microwave and Guided wave Letters 4(1), 11–13 (1994).
[Crossref]

Lee, Yong-Hee

LeVeque, R. J.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge, 2002)
[Crossref]

Li, E.

I. Ahmed and E. Li, “Time domain simulation of dispersive materials from microwave to optical frequencies,” Proceedings of IEEE 7th International Conference on Emerging Technologies (ICET) (IEEE, 2011) 1–5.

Liu, G. R.

G. R. Liu, “A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods,” International Journal of Computational Methods 5(2), 199–236 (2008).
[Crossref]

G. R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method (CRC Press, 2010).

G. R. Liu and N. T. Trung, Smoothed Finite Element Methods (CRC Press, 2010)
[Crossref]

G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics - A Meshfree Method (World Scientific: Singapore, 2003).

Liu, M. B.

G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics - A Meshfree Method (World Scientific: Singapore, 2003).

Liu, P.-L.

P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
[Crossref]

Movahhedi, M.

M. Movahhedi and A. Abdipour, “Alternating direction implicit formulation for the finite element time domain method,” IEEE Trans. Microwave Theory Technology 55(6), 1322–1331 (2007).
[Crossref]

Muhammad, M.

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

Niiyama, A.

A. Niiyama, M. Koshiba, and Y. Tsuji, “An efficient scalar finite element formulation for nonlinear optical channel waveguides,” J. Lightwave Technol. 13(9), 1919–1925 (1995).
[Crossref]

Obayya, S. S. A.

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
[Crossref]

M. F. O. Hameed, S. S. A. Obayya, and H. A. El-Mikati, “Passive polarization converters based on photonic crystal fiber with L-shaped core region,” J. Lightwave Technol. 30(3), 283–289 (2012).
[Crossref]

D. Pinto and S. S. A. Obayya, “Improved complex-envelope alternating-direction-implicit finite-difference-time-domain method for photonic-bandgap cavities,” J. Lightwave Technol. 25(1), 440–447 (2007).
[Crossref]

S. S. A. Obayya, “Efficient finite-element-based time-domain beam propagation analysis of Optical integrated circuits,” IEEE J. Quantum Electron. 40(5), 591–595 (2004).
[Crossref]

Pinto, D.

Ryu, H. Y.

taflove, A.

A. taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 1995)

Trung, N. T.

G. R. Liu and N. T. Trung, Smoothed Finite Element Methods (CRC Press, 2010)
[Crossref]

Tsuji, Y.

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18(1), 102–110 (2000).
[Crossref]

A. Niiyama, M. Koshiba, and Y. Tsuji, “An efficient scalar finite element formulation for nonlinear optical channel waveguides,” J. Lightwave Technol. 13(9), 1919–1925 (1995).
[Crossref]

Van der Vorst, H. A.

H. A. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. and Stat. 13(2), 631–644 (1992).

Vilcot, J. P.

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

Zhao, Q.

P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
[Crossref]

IEEE J. Quantum Electron. (3)

S. S. A. Obayya, “Efficient finite-element-based time-domain beam propagation analysis of Optical integrated circuits,” IEEE J. Quantum Electron. 40(5), 591–595 (2004).
[Crossref]

F. F. K. Hussain, A. M. Heikal, M. F. O. Hameed, J. El-Azab, W. S. Abdelaziz, and S. S. A. Obayya, “Dispersion characteristics of asymmetric channel plasmon polariton waveguides,” IEEE J. Quantum Electron. 50(6), 474–482 (2014).
[Crossref]

A. M. Heikal, M. F. O. Hameed, and S. S. A. Obayya, “Coupling characteristic of a novel hybrid long-range plasmonic waveguide including bends,” IEEE J. Quantum Electron. 49(8), 621–627 (2013).
[Crossref]

IEEE Microwave and Guided wave Letters (1)

J. F. Lee, “WETD-A finite element time-domain approach for solving Maxwell’s equations,” IEEE Microwave and Guided wave Letters 4(1), 11–13 (1994).
[Crossref]

IEEE Photon. Technol. Lett. (3)

V. F. R.- Esquerre and H. E. H.- Figueroa, “Novel time-domain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett. 13(4), 311–313 (2001).
[Crossref]

P.-L. Liu, Q. Zhao, and F.-S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7(8), 890–892 (1995).
[Crossref]

G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “An improved time domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett. 9(3), 117–122 (1997).
[Crossref]

IEEE Trans. Microwave Theory Technology (1)

M. Movahhedi and A. Abdipour, “Alternating direction implicit formulation for the finite element time domain method,” IEEE Trans. Microwave Theory Technology 55(6), 1322–1331 (2007).
[Crossref]

IET Optoelectronics (1)

M. Muhammad, S. S. A. Obayya, A. M. Heikal, and M. F. O. Hameed, “Porous core photonic crystal fibre with metal-coated central hole for terahertz applications,” IET Optoelectronics 9, 37–42 (2015).
[Crossref]

International Journal of Computational Methods (1)

G. R. Liu, “A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods,” International Journal of Computational Methods 5(2), 199–236 (2008).
[Crossref]

J. Comput. Appl. Math. (1)

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158(2), 485–505 (2003).
[Crossref]

J. Lightwave Technol. (6)

J. Opt. Soc. Am. B (1)

Microwave and Opt. Technol. Lett. (1)

V. F. R.- Esquerre, M. Koshiba, and H. E. H.- Figueroa, “Frequency-dependent envelope finite element time domain analysis of dispersion materials,” Microwave and Opt. Technol. Lett. 44(1), 13–16 (2004).
[Crossref]

SIAM J. Sci. and Stat. (1)

H. A. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. and Stat. 13(2), 631–644 (1992).

Other (7)

A. taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 1995)

G. R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method (CRC Press, 2010).

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge, 2002)
[Crossref]

G. R. Liu and N. T. Trung, Smoothed Finite Element Methods (CRC Press, 2010)
[Crossref]

I. Ahmed and E. Li, “Time domain simulation of dispersive materials from microwave to optical frequencies,” Proceedings of IEEE 7th International Conference on Emerging Technologies (ICET) (IEEE, 2011) 1–5.

S.J. Orfanidis, “Electromagnetic Waves and Antenna,” http://www.ece.rutgers.edu/orfanidi/ewa .

G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics - A Meshfree Method (World Scientific: Singapore, 2003).

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Figures (9)

Fig. 1
Fig. 1 The computational domain (Ω) divided into triangular elements (solid black lines). On top of the element mesh, smoothing domains is constructed (dashed red lines) to calculate the smoothed stiffness matrix. The filled circles (•) represent the field nodes and the empty circles (○) represent the centroid of the element.

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Fig. 2
Fig. 2 Schematic diagram elucidates the increase in matrix bandwidth of the proposed method over FEM-T3 and FEM-T6.

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Fig. 3
Fig. 3 Two adjacent elements describe the mesh distortion immunity.

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Fig. 4
Fig. 4 (a) Schematic diagram of a single mode slab waveguide. (b) Numerical dispersion calculated using the proposed method, FETD-BPM-T6 and FETD-BPM-T3 [7].

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Fig. 5
Fig. 5 (a) Schematic diagram of the optical grating structure. (b) The wavelength of maximum reflectivity computed using SFETD-BPM-T3, FETD-BPM-T6 and FETD-BPM-T3. (c) Optical gratting Reflection characteristics for TE mode. (d) Optical gratting Reflection characteristics for TM mode.

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Fig. 6
Fig. 6 (a) Schematic diagram of 5 × 5 PC cavity. (b) Time variation of the envelop of the electric field monitored at the center of the cavity. (c) The spectral distribution of the resonant mode energy for the square cavity computed using the proposed method. (d) The electric field profile of the resonant mode inside the square cavity computed using the proposed method.

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Fig. 7
Fig. 7 Schematic diagram of the computational domain used to calculate the reflection and the transmission of plasma material.

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Fig. 8
Fig. 8 (a) Reflection and (b) transmission of a plane wave incidents normally from air to 20 mm slab of plasma material.

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Fig. 9
Fig. 9 (a) Schematic diagram of 7 × 7 PC cavity modified by replacing 4 rods in the second ring by rods of diameter di = 0.1a surrounded by silver so that the total diameter becomes d = 0.2a. (b) Time variation of the envelop of the electric field monitored at the center of the cavity for the conventional 7 × 7 PC cavity, 7 × 7 PC cavity with 4 coated rods in the second ring and 7 × 7 PC cavity with 4 coated rods in the first ring.

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Tables (3)

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Table 1 The PML parameters sy and sz definition in each region.

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Table 2 Comparison between SFETD-BPM-T3, FETD-BPM-T3, and FETD-BPM-T6 in computing the reflection characteristics of TE-mode and TM-mode.

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Table 3 Comparison between the SFETD-BPM-T3, FETD-BPM-T6, and FETD-BPM-T3 in calculating the resonance frequency (F = a/λ) of the PC cavity.

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Equations (32)

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∇ [ D ] ∇ T ϕ + s ω o 2 q c 2 ϕ − 2 j s ω o q c 2 ∂ ϕ ∂ t − s q c 2 ∂ 2 ϕ ∂ t 2 = 0 ,
[ D ] = [ p s y 2 s 0 0 p s z 2 s ] ,
∇ = [ ∂ ∂ y ∂ ∂ z ] ,
ϕ = E x ,         p = 1 ,     q = n 2 ,                 for     TE   modes
ϕ = H x ,         p = 1 / n 2 ,     q = 1 ,                 for     TM   modes ,
s = { 1 − j 3 c 2 ω o n d ( ρ d ) 2 ln ( 1 R ) , in     PML   region 1 , in     non-PML   region ,
− ∬ Ω { B } [ D ] { B } T { ϕ } d Ω + ω o 2 c 2 ∬ Ω sq { N } { N } T { ϕ } d Ω − 2 j ω o c 2 ∬ Ω sq { N } { N } T ∂ { ϕ } ∂ t d Ω − 1 c 2 ∬ Ω sq { N } { N } T ∂ 2 { ϕ } ∂ t 2 d Ω = 0 ,
{ B } = { ∇ N } .
[ M ] = ∬ Ω sq { N } { N } T d Ω = ∑ N e ∬ Ω e sq { N } { N } T d Ω e ,
[ K ] = − ∬ Ω sq { B } [ D ] { B } T d Ω = − ∑ N e ∬ Ω e sq { B } [ D ] { B } T d Ω e ,
− 1 c 2 [ M ] d 2 { ϕ } d t 2 − 2 j ω o c 2 [ M ] d { ϕ } d t + ( [ K ] + ω o 2 c 2 [ M ] )     { ϕ } = { 0 } ,
[ K ¯ ] = − ∬ Ω sq { B } [ D ] { B } T d Ω = − ∑ N s ∬ Ω k s { B } [ D ] { B } T d Ω k s ,
{ B } = { ∇ N } ≈ ∬ Ω k s W ^ { ∇ N } d Ω k s .
{ B } ≈ − ∬ Ω k s ∇ W ^ { N } d Ω k s + ∫ Γ k s W ^ { N } u → d Γ k s ,
{ B } = 1 A k s ∑ i n s ∫ Γ k , i s { N ^ } d Γ k , i s [ u y , i u z , i ] ,
[ K ¯ ] = − ∑ N s { B }     [ ( A k , 1 s [ D ] 1 ) + ( A k , 2 s [ D ] 2 ) ]   { B } T ,
− 2 j ω o c 2 [ M ^ ] d { ϕ } d t + ( [ K ¯ ] + ω o 2 c 2 [ M ] )     { ϕ } = 0 ,
[ M ^ ] = [ M ] = c 2 4 ω o 2     ( [ K ¯ ] − ω o 2 c 2 [ M ] ) .
[ R ] i { ϕ } i + 1 = [ Q ] i { ϕ } i ,
[ R ] i = − 2 j ω o c 2 [ M ^ ] i + 0.5 Δ t   ( [ K ¯ ] i + ω o 2 c 2 [ M ] i ) ,
[ Q ] i = − 2 j ω o c 2 [ M ^ ] i − 0.5 Δ t   ( [ K ¯ ] i + ω o 2 c 2 [ M ] i ) ,
ϕ ( y , z , t = 0 ) = ϕ o ( y ) exp [ − ( z − z o W o ) 2 ] ,
ϕ ( x = x o , y = y o , t ) = e − ( t − t o T o ) 2 ,
ε r ( ω ) = ε ∞ + ω p 2 j ω ( j ω + ν c ) ,
∇ D ∇ T E x + s ω o 2 c 2 ( ε ∞ − ω p 2 ω o 2 ) E x − 2 j s ω o ε ∞ c 2 ∂ E x ∂ t − s ε ∞ c 2 ∂ 2 E x ∂ t 2 = − ν c ω p 2 c 2 ψ ,
{ ψ } i + 1 = e − ( ν c + j ω o ) Δ t { ψ } i + 1 − e − j ω o Δ t 2 j ω o [ { E x } i + 1 + e − ν c Δ t { E x } i ] .
− 2 j ω o c 2 [ M ^ d ] d { E x } d t + ( [ K ] + ω o 2 c 2 [ G ] )       { E x } = − ω p 2 ν c c 2 [ W ] { Ψ } ,
[ G ] = ∑ N e ∬ Ω e s ( ε ∞ − ω p 2 ω o 2 )     g { N } { N } T d Ω e ,
[ M ^ d ] = [ M ] − c 2 4 ω o 2     ( [ K ] − ω o 2 c 2 [ G ] ) ,
[ W ] = ∑ N e ∬ Ω e { N } sg { N } T d Ω e ,     and     g = { 1 , in   plasma   material , 0 , in   non-plasma   material .
{ B } = 1 A k s ( ∑ i = 1 2 ∫ Γ k , i s { N ^ } d Γ k , i s [ ε r 1 u y , i u z , i ] + ∑ i = 3 4 ∫ Γ k , i s { N ^ } d Γ k , i s [ ε r 2 u y , i u z , i ] ) ,
[ K ¯ ] = − ∑ N s { B }     [ A k s 0 0 A k , 1 s ε r 1 + A k , 2 s ε r 2 ]     { B } T ,

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