material, surface, geometry and temperature dependent. Here we will only focus on the material dependence, which is easy to treat on a level of some percent precision but which will turn out difficult to tackle on a high level of precision since different uncontrolled factors are involved.

In its original form, the Casimir force per unit surface [@Cas48]

$$F_{c}\left( a\right) =-\frac{\pi ^{2}}{240}\frac{\hbar c}{L^{4}}
\label{Fc}$$

was calculated between ideal metals. It depends only on the fundamental constants and the distance between the plates $L$. The force between real materials differs significantly from (\[Fc\]) for mirror separations smaller than 1 $\mu$m.

For mirrors of arbitrary material, which can be described by reflection coefficients, the force per unit area can be written as [@Lam00]:

$$\begin{aligned}
F&=& 2\sum_{\mu}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{0}^{\infty }\frac{\mathrm{d}\zeta }{2\pi } \hbar\kappa
\frac{r_{\mu}\left[ i\zeta ,\mathbf{k} \right]^2 e^{-2\kappa
L}}{1-r_{\mu}
\left[ i\zeta ,\mathbf{k} \right]^2 e^{-2\kappa L}}\nonumber \\
&&\kappa=\sqrt{\mathbf{k}^{2}+ \frac{\zeta ^{2}}{c^2}} \label{Force}\end{aligned}$$

where $r_{\mu}=(r_s,r_p)$ denotes the reflection amplitude for a given polarization $\mu=s,\;p$

$$\begin{aligned}
r_{s } &=&-\frac{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}-c\kappa } {\sqrt{\mathbf{k}^{2}+
\varepsilon \left( i\zeta \right) \frac{\zeta ^{2}}{c^2}}+c\kappa }
\nonumber \\
r_{p} &=&\frac{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}-c\kappa \varepsilon \left( i\zeta
\right) }{\sqrt{\mathbf{k}^{2}+ \varepsilon \left( i\zeta
\right)\frac{\zeta ^{2}}{c^2}}+c\kappa \varepsilon \left( i\zeta
\right) 