Abstract

The unconditional stable finite-difference time-domain (FDTD) method based on field expansion with weighted Laguerre polynomials (WLPs) is applied to model electromagnetic wave propagation in gyrotropic materials. The conventional Yee cell is modified to have the tightly coupled current density components located at the same spatial position. The perfectly matched layer (PML) is formulated in a stretched-coordinate (SC) system with the complex-frequency-shifted (CFS) factor to achieve good absorption performance. Numerical examples are shown to validate the accuracy and efficiency of the proposed method.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media

Xiang-Hua Wang, Wen-Yan Yin, and Zhi Zhang (David) Chen
Opt. Express 21(18) 20565-20576 (2013)

Unconditionally stable FDTD algorithm for 3-D electromagnetic simulation of nonlinear media

Mohammad Moradi, Seyyed-Mohammad Pourangha, Vahid Nayyeri, Mohammad Soleimani, and Omar M. Ramahi
Opt. Express 27(10) 15018-15031 (2019)

Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method

Ding Yu Heh and Eng Leong Tan
Biomed. Opt. Express 2(5) 1169-1183 (2011)

References

  • View by:
  • |
  • |
  • |

  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  2. R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
    [Crossref]
  3. M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photonics Technol. Lett. 21(12), 817–819 (2009).
    [Crossref]
  4. S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antenn. Propag. 45(3), 392–400 (1997).
    [Crossref]
  5. W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antenn. Propag. 45(3), 401–410 (1997).
    [Crossref]
  6. F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
    [Crossref]
  7. A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
    [Crossref]
  8. Y. Yu and J. Simpson, “An JE collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. 58(2), 469–478 (2010).
    [Crossref]
  9. C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18(20), 21427–21448 (2010).
    [Crossref] [PubMed]
  10. I. R. Capoğlu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express 21(1), 87–101 (2013).
    [Crossref] [PubMed]
  11. S. Zhao, “High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media,” Opt. Lett. 36(16), 3245–3247 (2011).
    [Crossref] [PubMed]
  12. S. Buil, J. Laverdant, B. Berini, P. Maso, J. P. Hermier, and X. Quélin, “FDTD simulations of localization and enhancements on fractal plasmonics nanostructures,” Opt. Express 20(11), 11968–11975 (2012).
    [Crossref] [PubMed]
  13. F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
    [Crossref]
  14. E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antenn. Propag. 56(1), 170–177 (2008).
    [Crossref]
  15. Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
    [Crossref]
  16. G. Singh, K. Ravi, Q. Wang, and S. T. Ho, “Complex-envelope alternating-direction-implicit FDTD method for simulating active photonic devices with semiconductor/solid-state media,” Opt. Lett. 37(12), 2361–2363 (2012).
    [Crossref] [PubMed]
  17. G. Singh, E. L. Tan, and Z. N. Chen, “Split-step finite-difference time-domain method with perfectly matched layers for efficient analysis of two-dimensional photonic crystals with anisotropic media,” Opt. Lett. 37(3), 326–328 (2012).
    [Crossref] [PubMed]
  18. S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
    [Crossref]
  19. X. H. Wang, W. Y. Yin, and Z. Z. Chen, “One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media,” Opt. Express 21(18), 20565–20576 (2013).
    [Crossref] [PubMed]
  20. M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
    [Crossref]
  21. S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
    [Crossref]
  22. T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. 60(12), 5801–5808 (2012).
    [Crossref]
  23. W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
    [Crossref]
  24. W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
    [Crossref]
  25. W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
    [Crossref]
  26. Y. Fang, Radiowave Xi’an University of Technology, 5 Golden flower south road, Xi’an, Shan Xi 710048, and J. F. Liu, X. L. Xi, Y. R. Pu are preparing a manuscript to be called “A J-E collocated WLP-FDTD model of wave propagation in isotropic cold plasma.”
  27. J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).
  28. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
    [Crossref]
  29. S. D. Gedney, “Scaled CFS-PML: it is more robust, more accurate, more efficient, and simple to implement. Why aren’t you using it?” in Antennas and Propagation Society International Symposium, (IEEE 2005), 364–367.
    [Crossref]
  30. D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. 60(4), 901–914 (2012).
    [Crossref]
  31. H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
    [Crossref]
  32. J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
    [Crossref]
  33. X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
    [Crossref]
  34. J. F. Liu, Y. Fang, and X. L. Xi, “A new effective SC-PML Implementation for WLP-FDTD Method in Plasma medium,” in Proceedings of International Symposium on Antennas and Propagation (IEEE, 2014), pp.149–150.
    [Crossref]
  35. J. F. Liu, Y. Fang, Z. B. Zhu, and X. L. Xi, “WLP-FDTD implementation of CFS-PML for plasma media,” in 31st International Review of Progress in Applied Computational Electromagnetics (ACES), (2015).

2015 (3)

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

2014 (1)

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

2013 (5)

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
[Crossref]

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

I. R. Capoğlu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express 21(1), 87–101 (2013).
[Crossref] [PubMed]

X. H. Wang, W. Y. Yin, and Z. Z. Chen, “One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media,” Opt. Express 21(18), 20565–20576 (2013).
[Crossref] [PubMed]

2012 (6)

G. Singh, K. Ravi, Q. Wang, and S. T. Ho, “Complex-envelope alternating-direction-implicit FDTD method for simulating active photonic devices with semiconductor/solid-state media,” Opt. Lett. 37(12), 2361–2363 (2012).
[Crossref] [PubMed]

G. Singh, E. L. Tan, and Z. N. Chen, “Split-step finite-difference time-domain method with perfectly matched layers for efficient analysis of two-dimensional photonic crystals with anisotropic media,” Opt. Lett. 37(3), 326–328 (2012).
[Crossref] [PubMed]

S. Buil, J. Laverdant, B. Berini, P. Maso, J. P. Hermier, and X. Quélin, “FDTD simulations of localization and enhancements on fractal plasmonics nanostructures,” Opt. Express 20(11), 11968–11975 (2012).
[Crossref] [PubMed]

D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. 60(4), 901–914 (2012).
[Crossref]

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. 60(12), 5801–5808 (2012).
[Crossref]

2011 (1)

2010 (2)

Y. Yu and J. Simpson, “An JE collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. 58(2), 469–478 (2010).
[Crossref]

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18(20), 21427–21448 (2010).
[Crossref] [PubMed]

2009 (2)

M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photonics Technol. Lett. 21(12), 817–819 (2009).
[Crossref]

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

2008 (1)

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antenn. Propag. 56(1), 170–177 (2008).
[Crossref]

2006 (1)

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

2003 (1)

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

2000 (1)

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
[Crossref]

1997 (2)

S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antenn. Propag. 45(3), 392–400 (1997).
[Crossref]

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antenn. Propag. 45(3), 401–410 (1997).
[Crossref]

1994 (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

1992 (1)

F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
[Crossref]

1991 (1)

R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
[Crossref]

Agrawal, G. P.

Al-Jabr, A.

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photonics Technol. Lett. 21(12), 817–819 (2009).
[Crossref]

Alsunaidi, M.

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photonics Technol. Lett. 21(12), 817–819 (2009).
[Crossref]

Alvarez, J.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

Angulo, L. D.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

Antonsen, T. M.

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

Backman, V.

Baek, H. J.

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

Berini, B.

Botton, M.

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

Bretones, A. R.

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

Buil, S.

Caloz, C.

D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. 60(4), 901–914 (2012).
[Crossref]

Capoglu, I. R.

Chen, H.

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

Chen, W. J.

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
[Crossref]

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

Chen, Z.

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
[Crossref]

Chen, Z. N.

Chen, Z. Z.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Chung, Y.

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

Cooke, S. J.

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

Cummer, S. A.

S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antenn. Propag. 45(3), 392–400 (1997).
[Crossref]

Dissanayake, C. M.

Du, Y. X.

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

Fang, Y.

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

J. F. Liu, Y. Fang, and X. L. Xi, “A new effective SC-PML Implementation for WLP-FDTD Method in Plasma medium,” in Proceedings of International Symposium on Antennas and Propagation (IEEE, 2014), pp.149–150.
[Crossref]

Gan, T. H.

T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. 60(12), 5801–5808 (2012).
[Crossref]

Garcia, S. G.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

Hermier, J. P.

Ho, S. T.

Hunsberger, F.

F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
[Crossref]

R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
[Crossref]

Khee, T.

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

Kunz, K.

F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
[Crossref]

Kunz, K. S.

R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
[Crossref]

Laverdant, J.

Levush, B.

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

Li, J. L.

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

Lin, H.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

Liu, J. F.

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

J. F. Liu, Y. Fang, and X. L. Xi, “A new effective SC-PML Implementation for WLP-FDTD Method in Plasma medium,” in Proceedings of International Symposium on Antennas and Propagation (IEEE, 2014), pp.149–150.
[Crossref]

J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).

Luebbers, R.

F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
[Crossref]

Luebbers, R. J.

R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
[Crossref]

Martin, R. G.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

Martín, R. G.

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

Maso, P.

Ooi, B. S.

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

Pantoja, M. F.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

Peng, H. L.

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

Premaratne, M.

Pu, Y. R.

J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).

Quélin, X.

Rappaport, C. M.

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antenn. Propag. 45(3), 401–410 (1997).
[Crossref]

Ravi, K.

Rubio, R. G.

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

Rukhlenko, I. D.

Salazar-Palma, M.

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

Sarkar, T. K.

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

Shao, W.

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
[Crossref]

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

Simpson, J.

Y. Yu and J. Simpson, “An JE collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. 58(2), 469–478 (2010).
[Crossref]

Singh, G.

Sounas, D. L.

D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. 60(4), 901–914 (2012).
[Crossref]

Taflove, A.

Tan, E. L.

G. Singh, E. L. Tan, and Z. N. Chen, “Split-step finite-difference time-domain method with perfectly matched layers for efficient analysis of two-dimensional photonic crystals with anisotropic media,” Opt. Lett. 37(3), 326–328 (2012).
[Crossref] [PubMed]

T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. 60(12), 5801–5808 (2012).
[Crossref]

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antenn. Propag. 56(1), 170–177 (2008).
[Crossref]

Wang, B. Z.

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
[Crossref]

Wang, Q.

Wang, X. H.

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

X. H. Wang, W. Y. Yin, and Z. Z. Chen, “One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media,” Opt. Express 21(18), 20565–20576 (2013).
[Crossref] [PubMed]

Weedon, W. H.

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antenn. Propag. 45(3), 401–410 (1997).
[Crossref]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Xi, X. L.

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

J. F. Liu, Y. Fang, and X. L. Xi, “A new effective SC-PML Implementation for WLP-FDTD Method in Plasma medium,” in Proceedings of International Symposium on Antennas and Propagation (IEEE, 2014), pp.149–150.
[Crossref]

J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).

Yin, W. Y.

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

X. H. Wang, W. Y. Yin, and Z. Z. Chen, “One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media,” Opt. Express 21(18), 20565–20576 (2013).
[Crossref] [PubMed]

Yu, Y.

Y. Yu and J. Simpson, “An JE collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. 58(2), 469–478 (2010).
[Crossref]

Zhai, M. L.

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

Zhang, J.

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
[Crossref]

Zhang, J. S.

J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).

Zhao, S.

Zheng, F.

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
[Crossref]

Zhu, Z. B.

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (6)

W. J. Chen, W. Shao, J. L. Li, and B. Z. Wang, “Numerical dispersion analysis and key parameter selection in Laguerre-FDTD method,” IEEE Microw. Wirel. Compon. Lett. 23(12), 629–631 (2013).
[Crossref]

W. J. Chen, W. Shao, and B. Z. Wang, “ADE-Laguerre-FDTD Method for Wave Propagation in General Dispersive Materials,” IEEE Microw. Wirel. Compon. Lett. 23(5), 228–230 (2013).
[Crossref]

W. J. Chen, W. Shao, H. Chen, and B. Z. Wang, “Nearly PML for ADE-WLP-FDTD Modeling in Two-Dimensional Dispersive Media,” IEEE Microw. Wirel. Compon. Lett. 24(2), 75–77 (2014).
[Crossref]

S. G. Garcia, R. G. Rubio, A. R. Bretones, and R. G. Martín, “On the dispersion relation of ADI-FDTD,” IEEE Microw. Wirel. Compon. Lett. 16(6), 354–356 (2006).
[Crossref]

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “FDTD modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Wirel. Compon. Lett. 22(12), 612–614 (2012).
[Crossref]

J. F. Liu, Y. Fang, X. L. Xi, Z. B. Zhu, and Y. X. Du, “A New Effective SC-PML Implementation for WLP-FDTD Method,” IEEE Microw. Wirel. Compon. Lett. 25(8), 499–501 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photonics Technol. Lett. 21(12), 817–819 (2009).
[Crossref]

IEEE Trans. Antenn. Propag. (8)

S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antenn. Propag. 45(3), 392–400 (1997).
[Crossref]

W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antenn. Propag. 45(3), 401–410 (1997).
[Crossref]

F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma,” IEEE Trans. Antenn. Propag. 40(12), 1489–1495 (1992).
[Crossref]

A. Al-Jabr, M. Alsunaidi, T. Khee, and B. S. Ooi, “A simple FDTD algorithm for simulating EM-wave propagation in general dispersive anisotropic material,” IEEE Trans. Antenn. Propag. 61(3), 1321–1326 (2013).
[Crossref]

Y. Yu and J. Simpson, “An JE collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. 58(2), 469–478 (2010).
[Crossref]

R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antenn. Propag. 39(1), 29–34 (1991).
[Crossref]

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antenn. Propag. 56(1), 170–177 (2008).
[Crossref]

T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. 60(12), 5801–5808 (2012).
[Crossref]

IEEE Trans. Microw. Theory Tech. (3)

Y. Chung, T. K. Sarkar, H. J. Baek, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 51(3), 697–704 (2003).
[Crossref]

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48(9), 1550–1558 (2000).
[Crossref]

D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. 60(4), 901–914 (2012).
[Crossref]

IEEE Trans. Terahertz Sci. Technol. (1)

M. L. Zhai, H. L. Peng, X. H. Wang, Z. Chen, and W. Y. Yin, “The conformal HIE-FDTD method for simulating tunable graphene-based couplers for THz applications,” IEEE Trans. Terahertz Sci. Technol. 5(3), 368–376 (2015).
[Crossref]

IEICE Electron. Express (1)

X. L. Xi, Y. Fang, J. F. Liu, and Z. B. Zhu, “An effective CFS-PML implementation for 2-D WLP-FDTD method,” IEICE Electron. Express 12(7), 20150191 (2015).
[Crossref]

Int. J. Numer. Model. (1)

S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. 22(2), 187–200 (2009).
[Crossref]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Other (6)

J. F. Liu, Y. Fang, and X. L. Xi, “A new effective SC-PML Implementation for WLP-FDTD Method in Plasma medium,” in Proceedings of International Symposium on Antennas and Propagation (IEEE, 2014), pp.149–150.
[Crossref]

J. F. Liu, Y. Fang, Z. B. Zhu, and X. L. Xi, “WLP-FDTD implementation of CFS-PML for plasma media,” in 31st International Review of Progress in Applied Computational Electromagnetics (ACES), (2015).

S. D. Gedney, “Scaled CFS-PML: it is more robust, more accurate, more efficient, and simple to implement. Why aren’t you using it?” in Antennas and Propagation Society International Symposium, (IEEE 2005), 364–367.
[Crossref]

Y. Fang, Radiowave Xi’an University of Technology, 5 Golden flower south road, Xi’an, Shan Xi 710048, and J. F. Liu, X. L. Xi, Y. R. Pu are preparing a manuscript to be called “A J-E collocated WLP-FDTD model of wave propagation in isotropic cold plasma.”

J. S. Zhang, X. L. Xi, J. F. Liu, and Y. R. Pu, “An unconditionally stable WLP-FDTD model of wave propagation in magnetized plasma,” IEEE Trans. Plasma Sci. (to be published).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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 (5)

Fig. 1
Fig. 1 Spatial positioning of the field components.
Fig. 2
Fig. 2 Simulation model of the plane wave incidence on graphene sheet.
Fig. 3
Fig. 3 A plane wave incident normally on an infinite graphene sheet. (a) The transmission amplitude versus μ c and B 0 at 30 GHz. (b) The transmission amplitude as a function of frequency when μ c = 0.37 eV , and B 0 = 0.5 T .
Fig. 4
Fig. 4 Simulation model of a plane wave incident on magnetoplasma cylinders.
Fig. 5
Fig. 5 Snapshots of the magnetic field H z at time instances of (a) 15 ps, (b) 17.5 ps, (c) 20 ps, (d) 22.5 ps, and (e) 25 ps.

Equations (26)

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

× H = ε 0 E t + J × E = μ 0 H t d J d t + v J = ε 0 ω p 2 E + ω b × J .
d J x d t + v J x = ε 0 ω p 2 E x ω b J y d J y d t + v J y = ε 0 ω p 2 E y + ω b J x d J z d t + v J z = ε 0 ω p 2 E z
ε 0 ( 0.5 s E q + s k = 0 , q > 0 q 1 E k ) + J q = × H q μ 0 ( 0.5 s H q + s k = 0 , q > 0 q 1 H k ) = × E q ( 0.5 s + v ) J q + s k = 0 , q > 0 q 1 J k = ε 0 ω p 2 E q + ω b × J q .
× H q = ( A B ) H q , × E q = ( B A ) E q
A = [ / y / z / x ] , B = [ / z / x / y ] .
E q + 2 ε 0 s J ¯ q = 2 k = 0 , q > 0 q 1 E k + 2 ε 0 s ( A B ) H q H q = 2 k = 0 , q > 0 q 1 H k + 2 μ 0 s ( B A ) E q .
( 0.5 s + v ) J q = ε 0 ω p 2 E ¯ q + ω b × J q s k = 0 , q > 0 q 1 J k .
J q = 0.5 ε 0 s ( P E ¯ q + Q k = 0 , q > 0 q 1 J k )
P = [ p 2 p 1 p 2 p 1 p 2 p 2 p 2 ] , Q = [ p 3 p 1 p 3 p 1 p 3 p 3 p 3 ] .
p 1 = ω b 0.5 s + v , p 2 = 4 ω p 2 ( s 2 + 2 s v ) ( 1 + p 1 2 ) , p 3 = 4 ε 0 ( s + 2 v ) ( 1 + p 1 2 ) .
E q + a ¯ E P a ¯ J E q = 2 k = 0 , q > 0 q 1 E k + 2 ε 0 s ( A B ) H q a ¯ E Q k = 0 , q > 0 q 1 J k .
( I + a ¯ E P a ¯ J + 4 ε 0 μ 0 s 2 ( A B ) 2 ) E q = 2 k = 0 , q > 0 q 1 E k a ¯ E Q k = 0 , q > 0 q 1 J k 4 ε 0 s ( A B ) k = 0 , q > 0 q 1 H k
˜ = x ^ 1 s x x + y ^ 1 s y x + z ^ 1 s z x .
s ζ = k ζ + σ ζ / ( α ζ + j ω ε 0 ) .
˜ × H = H ˜ A H ˜ B , ˜ × E = E ˜ B E ˜ A
s A H ˜ A = A H , s B H ˜ B = B H , s A E ˜ A = A E , s B E ˜ B = B E
( ) a = [ ( ) y ( ) z ( ) x ] , ( ) b = [ ( ) z ( ) x ( ) y ]
H ˜ A q = c A A H q + c 1 A q 1 ( H A ) , H ˜ B q = c B B H q + c 1 B q 1 ( H B )
q 1 ( H A ) = A k = 0 , q > 0 q 1 H k k a k = 0 , q > 0 q 1 H ˜ A q 1 ( H B ) = B k = 0 , q > 0 q 1 H k k b k = 0 , q > 0 q 1 H ˜ B .
E q + a ¯ E P a ¯ J E q = 2 k = 0 , q > 0 q 1 E k a ¯ E Q k = 0 , q > 0 q 1 J k + 2 ε 0 s [ ( c A A c B B ) H q + q 1 ( H A ) q 1 ( H B ) ] .
H q = 2 k = 0 , q > 0 q 1 H k + 2 μ 0 s [ ( c B B c A A ) E q + q 1 ( E B ) q 1 ( E A ) ] .
[ I + a ¯ E P a ¯ J + 4 ε 0 μ 0 s 2 ( c A A c B B ) 2 ] E q = 2 k = 0 , q > 0 q 1 E k 4 ε 0 s ( c A A c B B ) k = 0 , q > 0 q 1 H k a ¯ E Q k = 0 , q > 0 q 1 J k + 4 ε 0 μ 0 s 2 ( c A A c B B ) [ q 1 ( E B ) q 1 ( E A ) ] + 2 ε 0 s [ q 1 ( H A ) q 1 ( H B ) ] .
q ( H A ) = ( I k c 1 a ) q 1 ( H A ) + ( I k c a ) A H q .
E x ( t ) = exp [ ( t t 0 ) 2 / τ 2 ]
σ 0 = 2 e 2 τ π 2 k B T ln ( 2 cos h μ c 2 k B T ) , σ = σ 0 ε 0 Δ z , and ω c = e B 0 ν F 2
ω p = 2 π × 50 × 10 9 rad/s, ω b = 3 × 10 11 rad/s, v = 2 × 10 10 Hz .

Metrics