Drude model in FDTD solutions
The Plasma model is used to create a material with the the following relative permittivity.
\[\varepsilon_{total}(f)=\varepsilon-\frac{\omega_p^2}{2\pi \cdot f(i\nu_c+2\pi \cdot f)}\]- \(\varepsilon\) : permittivity
- \(\omega_p\) : plasma resonance in units of rad/s
- \(\nu_c\) : plasma collision in units of rad/s
Drude model
The dielectric function: \(\varepsilon(\omega)=\varepsilon_1(\omega)+i\varepsilon_2(\omega)\)
where:
\[\varepsilon_1 = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2+\gamma^2}\] \[\varepsilon_2 = \frac{\omega_p^2 \gamma}{\omega(\omega^2+\gamma^2)}\]In where:
\(\varepsilon_\infty\) is the high frequency dielectric permittivity, and \(\omega_p\) is the plasma frequency, and \(\gamma\) is the plasma damping.
Deformation of Drude model equation in FDTD
\[\begin{aligned} \varepsilon_{total}(f) & =\varepsilon-\frac{\omega_p^2}{2\pi \cdot f(i\nu_c+2\pi \cdot f)}\\ & = \varepsilon-\frac{\omega_p^2}{2\pi \cdot f}\cdot \frac{1}{i\nu_c+2\pi \cdot f}\\ & = \varepsilon-\frac{\omega_p^2}{2\pi \cdot f}\cdot \frac{i\nu_c-2\pi \cdot f}{-\nu_c^2-4\pi^2\cdot f^2}\\ & = \varepsilon-\frac{\omega_p^2}{2\pi \cdot f}\cdot \frac{-2\pi \cdot f}{-\nu_c^2-4\pi^2\cdot f^2}-\frac{\omega_p^2}{2\pi \cdot f}\cdot \frac{i\nu_c}{-\nu_c^2-4\pi^2\cdot f^2}\\ & = \varepsilon - \frac{\omega_p^2}{\nu_c^2+4\pi^2\cdot f^2}+i\frac{\omega_p^2 \nu_c}{2\pi\cdot f(\nu_c^2+4\pi^2\cdot f^2)} \end{aligned}\]Drude model and Drude model in FDTD
I.e.:
\[\varepsilon_{total}(f)=\varepsilon - \frac{\omega_p^2}{4\pi^2\cdot f^2+\nu_c^2}+i\frac{\omega_p^2 \nu_c}{2\pi\cdot f(4\pi^2\cdot f^2+\nu_c^2)} \tag{1}\] \[\varepsilon(\omega)=\varepsilon_\infty - \frac{\omega_p^2}{\omega^2+\gamma^2}+i\frac{\omega_p^2 \gamma}{\omega(\omega^2+\gamma^2)} \tag{2}\]compared (1) and (2), we will get that:
- Permittivity \(\varepsilon\) in FDTD equals to the \(\varepsilon_\infty\) in Drude model.
- Plasma frequency \(\omega_p\) (unit: rad/s) in FDTD equals to the \(\omega_p / 2\pi\) in Drude model.
- Plasma collision \(\nu_c\) (unit: rad/s) in FDTD equals to the \(\gamma\) in Drude model.
- Frequency \(f\) (unit: rad/s) in FDTD equals to the \(\omega/2\pi\) in Drude model.