# A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Model of Plasma Media

#### 2.2. Governing Equations

**D**and

**E**in the frequency domain is expressed as

**D**($\mathsf{\omega}$) = $\mathsf{\epsilon}\left(\mathsf{\omega}\right)E\left(\mathsf{\omega}\right)$. The relationship in the time domain of $D\left(x,t\right)$ and $E\left(x,t\right)$ could be written as

#### 2.3. The Extended LBM

#### 2.4. Chapman–Enskog Expansion

^{2}× Equation (23)), we have:

## 3. Results

#### 3.1. Electromagnetic Pulse in Non-Dispersive Media

#### 3.2. Effects of Plasma Frequency on EM Waves

#### 3.3. Effects of Layer Thickness on EM Waves in 1D Plasma PhCs

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Haslinger, M.J.; Sivun, D.; Pohl, H.; Munkhbat, B.; Muhlberger, M.; Klar, T.A.; Scharber, M.C.; Hrelescu, C. Plasmon assistend direction and polarization sensitive organic thin film detector. Nanomaterials
**2020**, 10, 1866. [Google Scholar] [CrossRef] - Kyaw, C.; Yahiaoui, R.; Chase, Z.A.; Tran, V.; Baydin, A.; Tay, F.; Kono, J.; Manjappa, M.; Singh, R.; Abeysinghe, D.C.; et al. Guided-mode resonances in flexible 2D terahertz photonic crystals. Optica
**2020**, 7, 537–541. [Google Scholar] [CrossRef] - Shi, J.; Li, Z.; Sang, D.K.; Xiang, Y.; Li, J.; Zhang, S.; Zhang, H. THZ photonics in two dimensional materials and metamaterials: Properties, devices and prospects. J. Mater. Chem. C
**2018**, 6, 1291–1306. [Google Scholar] [CrossRef] - Xiao, M.; Zhang, Z.Q.; Chan, C.T. Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems. Phys. Rev. X
**2014**, 4, 021017. [Google Scholar] [CrossRef][Green Version] - Gao, W.; Hu, X.; Li, C.; Yang, J.; Chai, Z.; Xie, J.; Gong, Q. Fano-resonance in one-dimensional topological photonic crystal heterostructure. Opt. Express
**2018**, 26, 8634–8644. [Google Scholar] [CrossRef] [PubMed] - Ahmed, A.M.; Mehaney, A. Ultra-high sensitive 1D porous silicon photonic crystal sensor based on the coupling of Tamm/Fano resonances in the mid-infrared region. Sci. Rep.
**2019**, 9, 6973. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ming Qing, Y.; Feng Ma, H.; Wei Wu, L.; Jun Cui, T. Manipulating the light-matter interaction in a topological photonic crystal heterostructure. Opt. Express
**2020**, 28, 34904–34915. [Google Scholar] [CrossRef] - Dong, J.-W.; Chang, M.-L.; Huang, X.-Q.; Hang, Z.H.; Zhong, Z.-C.; Chen, W.-J.; Huang, Z.-Y.; Chan, C.T. Conical Dispersion and Effective Zero Refractive Index in Photonic Quasicrystals. Phys. Rev. Lett.
**2015**, 114, 163901. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xu, L.; Wang, H.-X.; Xu, Y.-D.; Chen, H.-Y.; Jiang, J.-H. Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals. Opt. Express
**2016**, 24, 18059–18071. [Google Scholar] [CrossRef] - Son, J.H. Terahertz electromagnetic interactions with biological matter and their aaplications. J. Appl. Phys.
**2009**, 105, 102033. [Google Scholar] [CrossRef] - Yang, Z.; Zhang, M.; Li, D.; Chen, L.; Fu, A.; Liang, Y.; Wang, H. Study on an artificial phenomenon observed in terahertz biological imaging. Biomed. Opt. Express
**2021**, 12, 3133–3141. [Google Scholar] [CrossRef] - Erez, E.; Leviatan, Y. Current model analysis of electromagnetic scattering from objects containing a variety of length scales. J. Opt. Soc. Am. A
**1994**, 11, 1500–1504. [Google Scholar] [CrossRef] - Shao, Y.; Yang, J.; Huang, M. A review of computational electromagnetic methods for graphene modeling. Int. J. Antennas Propag.
**2016**, 1, 7478621. [Google Scholar] [CrossRef][Green Version] - Sumithra, P.; Thiripurasundari, D. A review of computational electromagnetics methods. Adv. Electromagn.
**2017**, 6, 42–55. [Google Scholar] [CrossRef] - Yee, K.S. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag.
**1966**, 39, 302–307. [Google Scholar] - Chen, P.; Wang, C.; Ho, J. A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium. Comput. Math. Appl.
**2013**, 65, 961–973. [Google Scholar] [CrossRef] - Ganhi, O.P.; Gao, B.Q.; Chen, J.Y. A frequency dependent finite difference time domain formulation for general dispersive media. IEEE Trans. Microw. Theory Tech.
**1993**, 41, 658–665. [Google Scholar] [CrossRef] - Nickisck, L.J.; Franke, P.M. Finite difference time domain solution of Maxwell’s equations for the dispersive ionosphere. IEEE Antennas Propag. Mag.
**1992**, 34, 33–39. [Google Scholar] [CrossRef] - Kashiwa, T.; Yoshida, N.; Fukai, I. A treatment by the finite difference time domain method of the dispersive characteristics associated with orientation polarization. IEICE Trans.
**1990**, 73, 1326–1328. [Google Scholar] - Kashiwa, T.; Fukai, I. A treatment by the FDTD method of the dispersive characteristics associated with electronic polarization. Microw. Opt. Technol. Lett.
**1990**, 3, 203–205. [Google Scholar] [CrossRef] - Feliziani, M.; Cruciani, S.; Santis, V.D.; Maradei, F. FD
^{2}TD Analysis of electromagnetic field propagation in multipole Debye media with and without convolution. Prog. Electromagn. Res.**2012**, 42, 181–205. [Google Scholar] [CrossRef][Green Version] - Luebbers, R.; Kunz, K.S. A frequency dependent finite difference time domain formulation for dispersive materials. IEEE Trans. Electromagn. Compat.
**1990**, 32, 222–227. [Google Scholar] [CrossRef] - Tang, M.; Zhan, H.; Ma, H.; Lu, S. Upscaling of dynamic capillary pressure of two-phase flow in sandstone. Water Resour. Res.
**2019**, 55, 426–443. [Google Scholar] [CrossRef][Green Version] - Tang, M.; Zhan, H.; Lu, S.; Ma, H.; Tan, H. Pore-scale CO
_{2}displacement simulation based on the three fluid phase lattice Boltzmann method. Energy Fuels**2019**, 33, 10039–10055. [Google Scholar] [CrossRef] - Tang, M.; Lu, S.; Zhan, H.; Guo, W.; Ma, H. The effect of a microscale fracture on dynamic capillary pressure of two phase flow in porous media. Adv. Water Resour.
**2018**, 113, 272–284. [Google Scholar] [CrossRef] - Chopard, B.; Luthi, P.; Wagen, J. Lattice Boltzmann method for wave propagation in urban microcells. IEEE Proc. Microwa. Antennas Propag.
**1997**, 144, 251–255. [Google Scholar] [CrossRef] - Succi, S. Lattice Boltzmann schemes for quantum applications. Comput. Phys. Commun.
**2002**, 146, 317–323. [Google Scholar] [CrossRef] - Succi, S.; Benzi, R. Lattice Boltzmann equation for quantum mechanics. Phys. D
**1993**, 69, 327–332. [Google Scholar] [CrossRef] - Zhang, J.; Yan, G. A lattice Boltzmann model for the nonlinear Schrodinger equation. J. Phys. A Math. Theor.
**2007**, 40, 10393–10405. [Google Scholar] [CrossRef] - Sajjadi, H.; Atashafrooz, M.; Delouei, A.A.; Wang, Y. The effect of indoor heating system location on particle deposition and convection heat transfer: DMRT-LBM. Comput. Math. Appl.
**2021**, 86, 90–105. [Google Scholar] [CrossRef] - Jalali, A.; Delouei, A.A.; Khorashadizadeh, M.; Golmohammadi, A.M.; Karimnejad, S. Mesoscopic simulation of forced convective heat transfer of Carreau-Yasuda fluid flow over an inclined square: Temperature-dependent viscosity. J. Appl. Comput. Mech.
**2020**, 6, 307–319. [Google Scholar] - Karimnejad, S.; Delouei, A.A.; Nazari, M.; Shahmardan, M.M.; Rashidi, M.M.; Wongwsies, S. Immersed boundary-thermal lattice Boltzmann method for moving simulation of non-isothermal elliptical particle. J. Therm. Anal. Calorim.
**2019**, 138, 4003–4017. [Google Scholar] [CrossRef] - Chen, Y.; Wang, X.; Zhu, H. A general single-node second-order boundary condition for the lattice Boltzmann method. Phys. Fluids
**2021**, 33, 043317. [Google Scholar] [CrossRef] - Hanasoge, S.M.; Succi, S.; Orszag, S.A. Lattice Boltzmann method for electromagnetic wave propagation. A Lett. J. Explor. Front. Phys.
**2011**, 96, 14002. [Google Scholar] [CrossRef][Green Version] - Lin, Z.; Fang, H.; Xu, J.; Zi, J.; Zhang, X. Lattice Boltzmann model for photonic band gap materials. Phys. Rev. E
**2003**, 67, 025701. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mendoza, M.; Munoz, J.D. Three dimensional Lattice-Boltzmann model for electrodynamics. Phys. Rev. E
**2010**, 82, 56708. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cole, K.S.; Cole, R.H. Dispersion and absorption in dielectrics. J. Chem. Phys.
**1941**, 9, 341–351. [Google Scholar] [CrossRef][Green Version] - Dhuri, D.B.; Hanasoge, S. Numerical analysis of the lattice Boltzmann method for simulation of linear acoustic waves. Phys. Rev. E
**2017**, 95, 043306. [Google Scholar] [CrossRef][Green Version] - Liu, Q.H. The PSTD algorithm: A time domain method requiring only two cells per wavelength. Microw. Opt. Technol. Lett.
**1997**, 15, 158–165. [Google Scholar] [CrossRef]

**Figure 1.**Distribution of electric pulse crossing a dielectric interface. The shadow zone is the dielectric media, with dielectric constant ${\epsilon}_{r}=2.0$ and the other one corresponds to the media with ${\epsilon}_{r}=1.0$. The curves are the intensity of the electric field at t = 0 (dashed line), and t = 300 (solid line).

**Figure 2.**Comparison of dispersion relations in the FDTD and proposed LBM method in this study with the exact dispersion relation in non-dispersive media.

**Figure 4.**Schematic computational domain and incident Gaussian-modulated sinusoidal pulse through a film of the dispersive medium.

**Figure 5.**Simulation of a wave propagating in air and striking a plasma medium. The plasma has the properties of silver: ${\omega}_{p}=2000$ THz, ${\vartheta}_{c}=50$ THz. The propagating wave has a center frequency of 500 THz. (

**a**,

**b**) show electric field in the computational domain at different times (

**a**) = 0 fs, (

**b**) = 11 fs, and (

**c**) = 21 fs. (Left side with film thickness of 1000 cells, right side with film thickness of 2000 cells).

**Figure 6.**Simulation of a wave propagating in air and striking a plasma medium. The plasma has the properties of silver: ${\omega}_{p}=2000$ THz, ${\vartheta}_{c}=50$ THz. The propagating wave has a center frequency of 4000 THz. (

**a**,

**b**) show electric field in the computational domain at different times (

**a**) = 0 fs, (

**b**) = 17 fs, and (

**c**) = 23 fs. (Left side with film thickness of 1000 cells, right side with film thickness of 2000 cells).

**Figure 8.**Schematic computational domain and incident Gaussian-modulated sinusoidal pulse through 1D plasma PhCs.

**Figure 9.**Simulation of a wave propagating in air and striking 1D plasma PhCs with plasma layer thickness $d=200\mathrm{nm}$. The plasma has the properties of silver: ${\omega}_{p}=2000$ THz, ${\vartheta}_{c}=50$ THz. The propagating wave has a center frequency of 4000 THz. (

**a**,

**b**) show electric field in the computational domain at different times (

**a**) = 0 fs, (

**b**) = 15 fs, and (

**c**) = 23 fs. (Left side with film thickness of 1000 cells, right side with film thickness of 2000 cells).

**Figure 10.**Simulation of a wave propagating in air and striking 1D plasma PhCs with plasma layer thickness $d=20\mathrm{nm}$. The plasma has the properties of silver: ${\omega}_{p}=2000$ THz, ${\vartheta}_{c}=50$ THz. The propagating wave has a center frequency of 4000 THz. (

**a**,

**b**) show electric field in the computational domain at different times (

**a**) = 0 fs, (

**b**) = 15 fs, and (

**c**) = 23 fs. (Left side with film thickness of 1000 cells, right side with film thickness of 2000 cells).

**Figure 11.**Simulation of a wave propagating in air and striking 1D plasma PhCs with plasma layer thickness $d=2\mathrm{nm}$. The plasma has the properties of silver: ${\omega}_{p}=2000$ THz, ${\vartheta}_{c}=50$ THz. The propagating wave has a center frequency of 4000 THz. (

**a**,

**b**) show electric field in the computational domain at different times (

**a**) = 0 fs, (

**b**) = 15 fs, and (

**c**) = 23 fs. (Left side with film thickness of 1000 cells, right side with film thickness of 2000 cells).

Denomination | LBM Context | Physical Context |
---|---|---|

Space step | $\Delta {x}^{LB}=1$ | $\Delta {x}^{py}=\Delta {x}^{LB}\frac{DL}{L}$ |

Time step | $\Delta {t}^{LB}=1$ | $\Delta {t}^{py}=\Delta {t}^{LB}\frac{\Delta {x}^{py}}{\Delta {x}^{LB}}\frac{{c}^{LB}}{{c}^{py}}$ |

Light speed | ${c}^{LB}=1$ | ${c}^{py}=c$ |

Electric field density | ${E}^{LB}=1$ | ${E}^{py}={E}^{LB}{\theta}^{V}$ |

Frequency | ${f}^{LB}=1$ | ${f}^{py}={f}^{LB}\frac{\Delta {t}^{LB}}{\Delta {t}^{py}}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ma, H.; Wu, B.; Wang, Y.; Ren, H.; Jiang, W.; Tang, M.; Guo, W. A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium. *Electronics* **2022**, *11*, 882.
https://doi.org/10.3390/electronics11060882

**AMA Style**

Ma H, Wu B, Wang Y, Ren H, Jiang W, Tang M, Guo W. A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium. *Electronics*. 2022; 11(6):882.
https://doi.org/10.3390/electronics11060882

**Chicago/Turabian Style**

Ma, Huifang, Bin Wu, Ying Wang, Hao Ren, Wanshun Jiang, Mingming Tang, and Wenyue Guo. 2022. "A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium" *Electronics* 11, no. 6: 882.
https://doi.org/10.3390/electronics11060882