## One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media |

Optics Express, Vol. 21, Issue 18, pp. 20565-20576 (2013)

http://dx.doi.org/10.1364/OE.21.020565

Acrobat PDF (1940 KB)

### Abstract

The one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is reformulated for simulating general electrically dispersive media. It models material dispersive properties with equivalent polarization currents. These currents are then solved with the auxiliary differential equation (ADE) and then incorporated into the one-step leapfrog ADI-FDTD method. The final equations are presented in the form similar to that of the conventional FDTD method but with second-order perturbation. The adapted method is then applied to characterize (a) electromagnetic wave propagation in a rectangular waveguide loaded with a magnetized plasma slab, (b) transmission coefficient of a plane wave normally incident on a monolayer graphene sheet biased by a magnetostatic field, and (c) surface plasmon polaritons (SPPs) propagation along a monolayer graphene sheet biased by an electrostatic field. The numerical results verify the stability, accuracy and computational efficiency of the proposed one-step leapfrog ADI-FDTD algorithm in comparison with analytical results and the results obtained with the other methods.

© 2013 Optical Society of America

## 1. Introduction

10. S. Zhao, “High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media,” Opt. Lett. **36**(16), 3245–3247 (2011). [CrossRef] [PubMed]

11. 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]

*Z*-transform technique [12

12. D. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE Trans. Microw. Theory Tech. **43**(3), 676–682 (1995). [CrossRef]

13. R. J. Luebbers and F. Hunsberger, “FDTD *N*th-order dispersive media,” IEEE Trans. Antenn. Propag. **40**(11), 1297–1301 (1992). [CrossRef]

14. M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microw. Wirel. Compon. Lett. **16**(3), 119–121 (2006). [CrossRef]

16. L. Han, D. Zhou, K. Li, X. Li, and W. P. Huang, “A rational-fraction dispersion model for efficient simulation of dispersive material in FDTD method,” IEEE J. Light. Tech. **30**(13), 2216–2225 (2012). [CrossRef]

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

18. 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]

19. L. J. Xu and N. C. Yuan, “FDTD formulations for scattering from 3-D anisotropic magnetized plasma objects,” IEEE Antennas Wirel. Propag. Lett. **5**(1), 335–338 (2006). [CrossRef]

20. D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag. **44**(6), 792–797 (1996). [CrossRef]

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

22. S. Liu and S. Liu, “Runge-Kutta exponential time differencing FDTD method for anisotropic magnetized plasma,” IEEE Antennas Wirel. Propag. Lett. **7**, 306–309 (2008). [CrossRef]

23. D. Y. Heh and E. L. Tan, “FDTD modeling for dispersive media using matrix exponential method,” IEEE Microw. Wirel. Compon. Lett. **19**(2), 53–55 (2009). [CrossRef]

24. S. Huang and F. Li, “FDTD implementation for magnetoplasma medium using exponential time differencing,” IEEE Microw. Wirel. Compon. Lett. **15**(3), 183–185 (2005). [CrossRef]

*etc*. Unfortunately, they cannot be easily used to model the general dispersive media.

25. 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]

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

27. 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]

28. X. H. Wang, W. Y. Yin, Y. Q. Yu, Z. Chen, J. Wang, and Y. Guo, “A convolutional perfect matched layer (CPML) for one-step leapfrog ADI-FDTD method and its applications to EMC problems,” IEEE Trans. Electromagn. Compat. **54**(5), 1066–1076 (2012). [CrossRef]

29. T. H. Gan and E. T. Tan, “Mur absorbing boundary condition for 2-D leapfrog ADI-FDTD method,” in *proceedings of IEEE Conference on Asia-Pacific Antennas and Propagation*, 3–4 (2012). [CrossRef]

30. Y. F. Mao, B. Chen, J. L. Xia, J. Chen, and J. Z. Tang, “Application of the leapfrog ADI-FDTD method to periodic structures,” IEEE Antennas Wirel. Propag. Lett. **12**, 599–602 (2013). [CrossRef]

31. 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]

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

18. 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]

## 2. The proposed leapfrog ADI-FDTD method for general dispersive media

*i.e.*Therefore, the

*x*-component of

18. 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]

*x*-component of the polarization current density, andSimilarly, we also have Therefore, Eq. (1a) can be expressed asConverting Eq. (5) into the time domain by using

*μ*is the permeability in free space.

_{0}*n to n*+ 1/2,

*n*+ 1/2 to

*n*+ 1, where

*p*

_{1}and

*p*

_{2}are the time index within single time step, and they are within the ranges of [0, 1/2] and [1/2, 1], respectively. With the algebraic manipulations used in [25

25. 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]

*n*with

*n*-1, we obtainAdding Eq. (9) and Eq. (10) on the both corresponding sides, we obtain the iterative equation for the electric field in the leapfrog ADI-FDTD method as

*p*

_{1}=

*p*

_{2}= 1/2 as Here the Taylor series approximation

**e**and

**h**such that whereEquations (17a) and (17b) show that the leapfrog ADI-FDTD can be regarded as the conventional FDTD method, with the second-order perturbation terms

### 2.1 Lossy media

### 2.2 Magnetized plasma

*z*-axis direction, we then have [18

**61**(3), 1321–1326 (2013). [CrossRef]

*ω*is the plasma frequency,

_{p}*ω*is the cyclotron frequency, and

_{b}*ν*is the electron collision rate. By substituting Eqs. (21a)-(21d) into Eq. (4), we obtainAs discussed in [18

_{c}**61**(3), 1321–1326 (2013). [CrossRef]

*i.e.*J x n = J x x n + J x y n . Both

*n*-1, we obtainand then, one iterative equation for

*J*is derived as

### 2.3 Graphene biased by a magnetostatic field

**B**in the

_{o}*z*-axis direction, and it is placed in the

*x*-

*y*plane. Then, its volume conductivity

35. 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]

*v*is the Fermi velocity,

_{F}*k*is the Boltzmann constant,

_{B}*T*is the operating temperature, and

## 3. Numerical results and discussion

### 3.1 The PEC Waveguide Loaded with Magnetized Plasma

*f*

_{0}= 10 GHz,

_{10}-mode was excited and its cutoff frequency is 3.75GHz. The plasma frequency

*ω*= 2π × 10 Grad/s,

_{p}*ν*= 10 GHz, and

*ω*= 10 Grad/s. An observation point is set at the waveguide center, and it is ten cells away from the magnetized plasma slab. Figure 2 shows the recorded

_{b}*E*-component at the observation point verses time with different CFLNs. It is seen that for CFLN = 1, the recorded

_{z}*E*–component agrees well with that of the conventional FDTD. As the CFLN increases from 1 to 3, the errors are increased slightly, but they are in an acceptable range. To check the stability of our developed algorithm, we run the simulation to one million time steps with the time step CFLN = 2 and 3, respectively; the recorded field was found to be always stable.

_{z}### 3.2 Magnetized graphene sheet

**B**

_{0}, it produces gyrotropy [35

35. 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]

*z*-direction in the

*x*-

*y*plane as in Fig. 3(a). We chose the maximum frequency

*f*=

_{max}*c*

_{0}/

*λ*= 2 THz, the FDTD cells ∆x = ∆y = ∆z = ∆ =

_{min}*λ*/20,

_{min}*ν*= 20THz,

*T*= 300K,

*μ*= 0.1 eV, and

_{c}28. X. H. Wang, W. Y. Yin, Y. Q. Yu, Z. Chen, J. Wang, and Y. Guo, “A convolutional perfect matched layer (CPML) for one-step leapfrog ADI-FDTD method and its applications to EMC problems,” IEEE Trans. Electromagn. Compat. **54**(5), 1066–1076 (2012). [CrossRef]

*x*-direction and described bywhere

*x*-

*y*plane and 4 cells away from the graphene sheet. The analytical transmission coefficient of the infinite graphene sheet can be calculated as [35

35. 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]

*FFT*represents the fast Fourier transformation.

_{0}= 1T and CFLN = 1, 2, and 3, respectively. The analytical result is also plotted for comparison. It is seen that, when CFLN = 1, the transmission coefficient |T| and the analytical solution agrees very well up to 2 THz. When CFLN = 2, the value of |T| still agrees well with the analytical solution at low frequencies. When CFLN = 3, the errors become much larger at high frequencies. This means that the leapfrog ADI-FDTD is quite suitable for computing the transmission coefficient at low frequencies.

*E*

_{y}– component at time

*t*= 2.5

*t*

_{0}for CFLN = 1 and 3, respectively. The field is recorded at 10 cells away from the CPML in the

*y*-

*z*plane. It is seen that the field distribution computed with CFLN = 3 is slightly different from that of CFLN = 1. On the other hand, it is noted that, as the incident EM wave is polarized in the

*x*-axis direction, the

*E*– component remains zero before it interacts with the graphene sheet. Therefore, Fig. 4 shows the gyrotropic property of the magnetized graphene sheet.

_{y}### 3.3 SPPs propagating along the graphene sheet

36. B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. **100**(13), 131111 (2012). [CrossRef]

37. 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]

*ν*= 1 THz,

*T*= 300 K,

*μ*= 0.15 eV,and

_{c}31. 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]

*f*

_{0}= 6 THz. Therefore, the grid resolution is

*λ*

_{0}/∆ =

*c*

_{0}/(

*f*

_{0}∆) = 1000, and for such a fine structure the large CFLN is needed so as to reduce the simulation time.

*E*– and

_{z}*E*–components at time

_{y}*t*= 2.887 ps. In Figs. 6(a) and 6(b), the numerical results were obtained with the leapfrog ADI-FDTD with CFLN = 10, and in Figs. 6(c) and 6(d), they were computed using the conventional ADE-FDTD [17

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

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Taflove and S. C. Hagness, |

2. | C. Lundgren, R. Lopez, J. Redwing, and K. Melde, “FDTD modeling of solar energy absorption in silicon branched nanowires,” Opt. Express |

3. | E. H. Khoo, I. Ahmed, R. S. M. Goh, K. H. Lee, T. G. G. Hung, and E. P. Li, “Efficient analysis of mode profiles in elliptical microcavity using dynamic-thermal electron-quantum medium FDTD method,” Opt. Express |

4. | D. T. Nguyen and R. A. Norwood, “Label-free, single-object sensing with a microring resonator: FDTD simulation,” Opt. Express |

5. | I. R. Çapoğlu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express |

6. | 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 |

7. | C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express |

8. | 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. |

9. | 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. |

10. | S. Zhao, “High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media,” Opt. Lett. |

11. | 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. |

12. | D. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE Trans. Microw. Theory Tech. |

13. | R. J. Luebbers and F. Hunsberger, “FDTD |

14. | M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microw. Wirel. Compon. Lett. |

15. | P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. |

16. | L. Han, D. Zhou, K. Li, X. Li, and W. P. Huang, “A rational-fraction dispersion model for efficient simulation of dispersive material in FDTD method,” IEEE J. Light. Tech. |

17. | M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photon. Technol. Lett. |

18. | 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. |

19. | L. J. Xu and N. C. Yuan, “FDTD formulations for scattering from 3-D anisotropic magnetized plasma objects,” IEEE Antennas Wirel. Propag. Lett. |

20. | D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag. |

21. | Y. Yu and J. Simpson, “An EJ collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag. |

22. | S. Liu and S. Liu, “Runge-Kutta exponential time differencing FDTD method for anisotropic magnetized plasma,” IEEE Antennas Wirel. Propag. Lett. |

23. | D. Y. Heh and E. L. Tan, “FDTD modeling for dispersive media using matrix exponential method,” IEEE Microw. Wirel. Compon. Lett. |

24. | S. Huang and F. Li, “FDTD implementation for magnetoplasma medium using exponential time differencing,” IEEE Microw. Wirel. Compon. Lett. |

25. | S. J. Cooke, M. Botton, T. M. Antonsen, and B. Levush, “A leapfrog formulation of the 3D ADI-FDTD algorithm,” Int. J. Numer. Model. |

26. | F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-different time-domain method,” IEEE Trans. Microw. Theory Tech. |

27. | E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antenn. Propag. |

28. | X. H. Wang, W. Y. Yin, Y. Q. Yu, Z. Chen, J. Wang, and Y. Guo, “A convolutional perfect matched layer (CPML) for one-step leapfrog ADI-FDTD method and its applications to EMC problems,” IEEE Trans. Electromagn. Compat. |

29. | T. H. Gan and E. T. Tan, “Mur absorbing boundary condition for 2-D leapfrog ADI-FDTD method,” in |

30. | Y. F. Mao, B. Chen, J. L. Xia, J. Chen, and J. Z. Tang, “Application of the leapfrog ADI-FDTD method to periodic structures,” IEEE Antennas Wirel. Propag. Lett. |

31. | T. H. Gan and E. L. Tan, “Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method,” IEEE Trans. Antenn. Propag. |

32. | T. H. Gan and E. L. Tan, “Unconditionally stable leapfrog ADI-FDTD method for lossy media,” Progress In Electromagnetics Research M |

33. | X. H. Wang, W. Y. Yin, and Z. Chen, “One-step leapfrog ADI-FDTD method for anisotropic magnetized plasma,” in |

34. | J. Y. Gao and H. X. Zheng, “One-Step leapfrog ADI-FDTD method for lossy media and its stability analysis,” Progress In Electromagnetics Research Letters |

35. | D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech. |

36. | B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. |

37. | 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. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(160.1190) Materials : Anisotropic optical materials

(240.0310) Optics at surfaces : Thin films

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 8, 2013

Revised Manuscript: August 15, 2013

Manuscript Accepted: August 15, 2013

Published: August 26, 2013

**Citation**

Xiang-Hua Wang, Wen-Yan Yin, and Zhi Zhang (David) Chen, "One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media," Opt. Express **21**, 20565-20576 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20565

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### References

- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- C. Lundgren, R. Lopez, J. Redwing, and K. Melde, “FDTD modeling of solar energy absorption in silicon branched nanowires,” Opt. Express21(9), 329–400 (2013). [PubMed]
- E. H. Khoo, I. Ahmed, R. S. M. Goh, K. H. Lee, T. G. G. Hung, and E. P. Li, “Efficient analysis of mode profiles in elliptical microcavity using dynamic-thermal electron-quantum medium FDTD method,” Opt. Express21(5), 5910–5923 (2013). [CrossRef] [PubMed]
- D. T. Nguyen and R. A. Norwood, “Label-free, single-object sensing with a microring resonator: FDTD simulation,” Opt. Express21(1), 49–59 (2013). [CrossRef] [PubMed]
- I. R. Çapoğlu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express21(1), 87–101 (2013). [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. Express20(11), 11968–11975 (2012). [CrossRef] [PubMed]
- C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express18(20), 21427–21448 (2010). [CrossRef] [PubMed]
- 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. Zhao, “High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media,” Opt. Lett.36(16), 3245–3247 (2011). [CrossRef] [PubMed]
- 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]
- D. Sullivan, “Nonlinear FDTD formulations using Z transforms,” IEEE Trans. Microw. Theory Tech.43(3), 676–682 (1995). [CrossRef]
- R. J. Luebbers and F. Hunsberger, “FDTD Nth-order dispersive media,” IEEE Trans. Antenn. Propag.40(11), 1297–1301 (1992). [CrossRef]
- M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microw. Wirel. Compon. Lett.16(3), 119–121 (2006). [CrossRef]
- P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys.125(16), 164705 (2006). [CrossRef] [PubMed]
- L. Han, D. Zhou, K. Li, X. Li, and W. P. Huang, “A rational-fraction dispersion model for efficient simulation of dispersive material in FDTD method,” IEEE J. Light. Tech.30(13), 2216–2225 (2012). [CrossRef]
- M. Alsunaidi and A. Al-Jabr, “A general ADE-FDTD algorithm for the simulation of dispersive structures,” IEEE Photon. Technol. Lett.21(12), 817–819 (2009). [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]
- L. J. Xu and N. C. Yuan, “FDTD formulations for scattering from 3-D anisotropic magnetized plasma objects,” IEEE Antennas Wirel. Propag. Lett.5(1), 335–338 (2006). [CrossRef]
- D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag.44(6), 792–797 (1996). [CrossRef]
- Y. Yu and J. Simpson, “An EJ collocated 3-D FDTD model of electromagnetic wave propagation in magnetized cold plasma,” IEEE Trans. Antenn. Propag.58(2), 469–478 (2010). [CrossRef]
- S. Liu and S. Liu, “Runge-Kutta exponential time differencing FDTD method for anisotropic magnetized plasma,” IEEE Antennas Wirel. Propag. Lett.7, 306–309 (2008). [CrossRef]
- D. Y. Heh and E. L. Tan, “FDTD modeling for dispersive media using matrix exponential method,” IEEE Microw. Wirel. Compon. Lett.19(2), 53–55 (2009). [CrossRef]
- S. Huang and F. Li, “FDTD implementation for magnetoplasma medium using exponential time differencing,” IEEE Microw. Wirel. Compon. Lett.15(3), 183–185 (2005). [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]
- F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three dimensional unconditionally stable finite-different time-domain method,” IEEE Trans. Microw. Theory Tech.48(9), 1550–1558 (2000). [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]
- X. H. Wang, W. Y. Yin, Y. Q. Yu, Z. Chen, J. Wang, and Y. Guo, “A convolutional perfect matched layer (CPML) for one-step leapfrog ADI-FDTD method and its applications to EMC problems,” IEEE Trans. Electromagn. Compat.54(5), 1066–1076 (2012). [CrossRef]
- T. H. Gan and E. T. Tan, “Mur absorbing boundary condition for 2-D leapfrog ADI-FDTD method,” in proceedings of IEEE Conference on Asia-Pacific Antennas and Propagation, 3–4 (2012). [CrossRef]
- Y. F. Mao, B. Chen, J. L. Xia, J. Chen, and J. Z. Tang, “Application of the leapfrog ADI-FDTD method to periodic structures,” IEEE Antennas Wirel. Propag. Lett.12, 599–602 (2013). [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]
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