# Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model

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## Abstract

**:**

## 1. Introduction

## 2. Formulation

- Update ${H}_{x}$, ${H}_{y}$ and ${H}_{z}$;
- Store the current values of ${E}_{x}$, ${E}_{y}$ and ${E}_{z}$;
- Update ${E}_{x}$, ${E}_{y}$ and ${E}_{z}$;
- Update ${Q}_{pxx}$, ${Q}_{pxy}$ and ${Q}_{pxz}$;
- Update ${Q}_{pyx}$, ${Q}_{pyy}$ and ${Q}_{pyz}$;
- Update ${Q}_{pzx}$, ${Q}_{pzy}$ and ${Q}_{pzz}$.

## 3. Applications and Numerical Results

#### 3.1. Propagation in Magnetized Plasma

#### 3.2. Terahertz Wave Propagation through a Nematic Liquid Crystal Cell

#### 3.3. Propagation in Ferrites

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The magnitude and phase of the transmission/reflection coefficients for RCP/LCP waves propagating through magnetized plasma calculated by the proposed FDTD method and the Berreman $4\times 4$ matrix method.

**Figure 2.**(

**a**) The ordinary and extraordinary relative permittivities of 5CB in the low THz spectrum. (

**b**) Transmittance calculated by the proposed FDTD method and the Berreman method.

Model | Expression | Parameters of the CCPR Model: $\mathit{\sigma}$, (${\mathit{c}}_{\mathit{p}},{\mathit{a}}_{\mathit{p}}$) |
---|---|---|

Drude | $\frac{{\omega}_{p}^{2}}{j\omega (j\omega +\nu )}$ | ${\epsilon}_{0}{\omega}_{p}^{2}/\nu $, ($-0.5{\omega}_{p}^{2}/\nu ,-\nu $) |

Debye | $\frac{\Delta \epsilon}{1+j\omega \tau}$ | 0, $(0.5\Delta \epsilon /\tau ,-1/\tau $) |

Lorentz | $\frac{\Delta \epsilon {\omega}_{p}^{2}}{{\omega}_{p}^{2}+j\omega \delta +{\left(j\omega \right)}^{2}}$ | 0, $\left(\right)$ |

Critical Points | ${A}_{p}{\Omega}_{p}\left({\displaystyle \frac{{e}^{j{\varphi}_{p}}}{{\Omega}_{p}+\omega -j{\Gamma}_{p}}}\right)$ $\left(\right)$ | 0, $\left(\right)$ |

Sellmeier | $\frac{B{\lambda}^{2}}{{\lambda}^{2}-C}},\lambda ={\displaystyle \frac{\omega}{2\pi c}$ | 0, $\left(\right)$ |

Modified Lorentz | $\frac{{a}_{0}+j{a}_{1}\omega}{{b}_{0}+j\omega {b}_{1}+{\left(j\omega \right)}^{2}}$ | 0, $({c}_{p}={\displaystyle \frac{{a}_{0}+{a}_{p}\phantom{\rule{0.166667em}{0ex}}{a}_{1}}{{a}_{p}-{a}_{p}^{*}}}$, ${a}_{p}={\displaystyle \frac{1}{2}}\left(\right)open="("\; close=")">-{b}_{1}+j\sqrt{4{b}_{0}-{b}_{1}^{2}}$ or 0, $({c}_{p1}={\displaystyle \frac{{a}_{0}+{a}_{p1}\phantom{\rule{0.166667em}{0ex}}{a}_{1}}{{a}_{p1}-{a}_{p2}}},{a}_{p1}=0.5\left(\right)open="("\; close=")">-{b}_{1}+\sqrt{{b}_{1}^{2}-4{b}_{0}}$, ${c}_{p2}={a}_{1}-{c}_{p1}$, ${a}_{p2}=0.5\left(\right)open="("\; close=")">-{b}_{1}+\sqrt{{b}_{1}^{2}-4{b}_{0}}$ |

Tensor Element | Parameters: $\mathit{\sigma}$, (${\mathit{c}}_{\mathit{p}},{\mathit{a}}_{\mathit{p}}$) |
---|---|

${\epsilon}_{xx}$ | ${\epsilon}_{0}{\displaystyle \frac{{\omega}_{p}^{2}v}{{v}^{2}+{\omega}_{b}^{2}}}$, $\left(\right)$ |

${\epsilon}_{xy}$ | $-{\epsilon}_{0}{\displaystyle \frac{{\omega}_{p}^{2}{\omega}_{b}}{{v}^{2}+{\omega}_{b}^{2}}}$, $\left(\right)$ |

${\epsilon}_{zz}$ | ${\epsilon}_{0}{\omega}_{p}^{2}/v$, ($-0.5{\omega}_{p}^{2}/{v}^{2},-v$) |

Tensor’s Element | Parameters: $\mathit{\sigma}$, (${\mathit{c}}_{\mathit{p}},{\mathit{a}}_{\mathit{p}}$) |
---|---|

${\mu}_{xx},{\mu}_{yy}$ | $0,(-0.5j{\omega}_{M},j{\omega}_{0})$ |

${\mu}_{xy}$ | $0,(0.5{\omega}_{M},j{\omega}_{0})$ |

${\mu}_{yx}$ | $0,(-0.5{\omega}_{M},j{\omega}_{0})$ |

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**MDPI and ACS Style**

Prokopidis, K.P.; Zografopoulos, D.C.
Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model. *Appl. Sci.* **2021**, *11*, 3844.
https://doi.org/10.3390/app11093844

**AMA Style**

Prokopidis KP, Zografopoulos DC.
Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model. *Applied Sciences*. 2021; 11(9):3844.
https://doi.org/10.3390/app11093844

**Chicago/Turabian Style**

Prokopidis, Konstantinos P., and Dimitrios C. Zografopoulos.
2021. "Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model" *Applied Sciences* 11, no. 9: 3844.
https://doi.org/10.3390/app11093844