} \label{rThick}\end{aligned}$$

The force between dielectric materials had first been derived by Lifshitz [@Lif56; @LP9]. The material properties enter these formulas via the dielectric function $\varepsilon \left( i\zeta
\right) $ at angular imaginary frequencies $\omega=i\zeta $, which is related to the physical quantity $\varepsilon ^{\prime \prime
}\left( \omega \right)= \mathrm{Im}\left( \varepsilon \left( \omega
\right)\right) $ with the help of the dispersion relation

$$\varepsilon \left( i\zeta \right) -1=\frac{2}{\pi
}\int\limits_{0}^{\infty } d\omega\frac{\omega \varepsilon ^{\prime
\prime }\left( \omega \right) }{\omega ^{2}+\zeta ^{2}}.
\label{K-K}$$

For metals $\varepsilon ^{\prime \prime }\left( \omega
\right)$ is large at low frequencies, thus the main contribution to the integral in Eq. (\[K-K\]) comes from the low frequencies even if $\zeta $ corresponds to the visible frequency range. For this reason the low-frequency behavior of $\varepsilon(\omega)$ is of primary importance.

The Casimir force is often calculated using the optical data taken from [@HB1], which provides real and imaginary parts of the dielectric function within some frequency range, typically between 0.1 and $10^4$ eV for the most commonly used metals, Au, Cu and Al, corresponding to a frequency interval $[1.519\cdot 10^{14},1.519
\cdot10^{19}]$ rad/s (1 eV=$1.519 \cdot10^{15}$ rad/s [^1]). When the two plates are separated by a distance $L$, one may introduce 