{{projectinfo|V&V|Modeling Dispersive Materials Using EM.Tempo|ART DISP title.png|In this article, several periodic structures involving different types of dispersive materials are simulated using EM.Tempo and EM.Picasso, and the results are validated by the published data.|
*[[EM.Tempo]]
*Dispersive Material
*Debye Pole
:<math> \varepsilon (\omega) = \varepsilon_\infty + \sum_{p=1}^N \dfrac{\Delta \varepsilon_p}{1 + j\omega \tau_p}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_\infty </math>
where <math>\varepsilon_{\infty}</math> is the value of the permittivity at infinite frequency, <math>\tau_p</math> is the relaxation time corresponding to the p''th'' pole having the unit of seconds, and <math>\varepsilon_{sp}</math> is the value of the static permittivity (at DC) corresponding to the p''th'' pole. <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole. Water has a Debye pole with parameters τ<sub>p</sub> = 9.4×10<sup>-12</sup> s, ε<sub>sps</sub> = 81 and ε<sub>∞</sub> = 1.8. In this example, we consider a laterally infinite slab of water with a finite thickness of 6mm. A periodic unit cell with lateral periods of 3mm along both X and Y directions are assumed.
Figure 1 shows the geometry setup for the periodic unit cell of the water slab in [[EM.Tempo]]. The top and bottom domain walls are assumed to be convolutional perfectly matched layers (PML). The periodic structure is excited using a normally incident plane wave source.
== Drude Plasma Slab ==
Next, [[EM.Tempo]]'s periodic boundary condition will be used to simulate a plasma slab with infinite extents in the X and Y directions, but a finite Z-thickness of 1.5 cm. The Drude model often provides a good abstraction for an unmagnetized isotorpic non-magnetized plasma. The complex permittivity of a Drude material with N poles is given by:
:<math> \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{{\omega_p}^2}{\omega^2 - j\omega \nu_p} </math>
where <math>\omega_p</math> and <math>\nu_p</math> are the angular plasma frequency and angular collision frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s. For an unmagnetized isotropic non-magnetized plasma, <math>\varepsilon_{\infty} =</math> 1. A Drude pole with ω<sub>p</sub> = 1.803×10<sup>11</sup> rad/s, and ν<sub>p</sub> = 2×10<sup>10</sup> radcollisions/s is used as the dispersive model for this project.
Figure 3 shows the geometry setup for the periodic unit cell of the Drude plasma slab in [[EM.Tempo]]. A box of dimensions 10mm × 10mm × 15mm is considered, with lateral periods of 10mm along both X and Y directions. The top and bottom domain walls are assumed to be convolutional perfectly matched layers (PML). The periodic structure is excited using a normally incident plane wave source.
</table>
In this project, the Debye pole has parameters τ<sub>p</sub> = 5.27×10<sup>-10</sup>s, ε<sub>sps</sub> = 5.2 and ε<sub>∞</sub> = 3.7. The table below summarizes the simulation parameters:
{| class="wikitable"
:<math> \varepsilon(\omega) = \varepsilon_{\infty} - G_1 \dfrac{(\varepsilon_{s1} - \varepsilon_{\infty}){\omega_1}^2}{\omega^2 - 2j\omega \delta_1 - {\omega_1}^2} - G_2 \dfrac{(\varepsilon_{s2} - \varepsilon_{\infty}){\omega_2}^2}{\omega^2 - 2j\omega \delta_2 - {\omega_2}^2} </math>
where <math>\omega _p</math> and <math>\delta_p</math> are the angular resonant frequency and angular damping frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s. Similar to a Debye material, <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole. The coefficients G<sub>1</sub> and G<sub>2</sub> are the weights used for the two pole terms. In order to model the half-space, a periodic unit cell of dimensions 2mm × ; 2mm × ; 50mm is considered as shown in Figure 14. The lateral periods are 2mm in both X and Y directions. The values of the parameters in the above expression are given in the table below:
{| class="wikitable"
<hr>
[[Image:Top_icon.png|48px30px]] '''[[#Introduction | Back to the Top of the Page]]'''
[[Image:Back_icon.png|40px30px]] '''[[EM.Cube#EM.Cube Verification Articles & Validation Notes | Check out more Articles & Notes]]''' [[Image:Back_icon.png|30px]] '''[[EM.Cube | Back to V&V Articles GatewayEM.Cube Main Page]]'''