are defined as the normal components of the wave vector in vacuum and in $Au$, respectively. These components are $$\label{k_def}
    k_0=\sqrt{\zeta^2/c^2+q^2},\ \ \
    k_2=\sqrt{\varepsilon_2\zeta^2/c^2+q^2}.$$ In Eqs. (\[Lif\])-(\[k\_def\]) the dielectric function of gold $\varepsilon_2$ has to be understood as the function at imaginary frequencies: $\varepsilon_2 = \varepsilon_2(i\zeta)$. This function cannot be directly measured, but it can be expressed with the Kramers-Kronig relation via the observable dielectric function $\varepsilon_2(\omega)$ at real frequencies $\omega$ $$\label{K-K}
    \varepsilon_2(i\zeta)=1+\frac{2}{\pi}\int\limits_0^{\infty}d\omega
    \frac{\omega\varepsilon_2^{\prime\prime}(\omega)}{\omega^2+\zeta^2}.$$ Note that only the imaginary part of the dielectric function $\varepsilon_2^{\prime\prime}(\omega)$ contributes to $\varepsilon_2(i\zeta)$. The fact that the dispersive force depends on $\varepsilon_2(i\zeta)$ makes the material dependence of the force sometimes confusing. For example, thin metallic film transparent for visible light gives very significant contribution to the force [@Sve00a]. On the other hand, a hydrogen-switchable mirror that changes from reflection to transmission in visible light does not give a measurable effect [@Ian04].

For practical evaluation of the interaction energy it is convenient to change variables in (\[Lif\]). Namely, instead of $q$ one can introduce $x=2dk_0$. With this variable the integral will run from $\xi=\zeta/\omega_c$ to $\infty$, where $$\label{omc_def}
    \omega_c=\frac{c}{2d}$$ is the characteristic imaginary frequency (it is not a characteristic frequency in real domain). Because of exponent 