# 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

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

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**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