the short-distance range $d<
100\; nm$. It is known that for these separations and at room temperature the thermal correction is small [@Mil04]. Physically it means that the zero-point quantum fluctuations of the electromagnetic field give the main contribution to the interaction, while thermally excited fields can be neglected. In this case instead of summation over the discrete set of Matsubara frequencies, one can integrate over a continuous imaginary frequency [@Lif56]. The resulting interaction energy corresponding to zero temperature can be presented in the following form: $$\label{Lif}
    E_{pp}^{flat}(d)=\frac{\hbar}{2\pi}\sum\limits_{\mu}\int
    \limits_0^{\infty}d\zeta\int\frac{d^2q}{(2\pi)^2}
    \ln\left(1-R_{\mu}e^{-2dk_0} \right).$$ Here the index $\mu=s,p$ is running two possible polarization states of the electromagnetic field, and $R_{\mu}$ is the product of the Fresnel reflection coefficients for plates 1 and 2: $R_{\mu}=r_{1\mu}r_{2\mu}$. The integration variables have the physical meaning of the imaginary frequency $\zeta$, and the wave vector $q=|{\rm{ \bf q}}|$ along the plates.

Here it will be assumed that both interacting surfaces are the same: $r_{1\mu}=r_{2\mu}=r_{\mu}$. For what follows it will be convenient to use index [*2*]{} for quantities related to gold. If no additional layer on the gold surface exists, then the Fresnel coefficients can be written as $$\label{r_def}
    r_s=\frac{k_0-k_2}{k_0+k_2},\ \ \
    r_p=\frac{\varepsilon_2 k_0-k_2}{\varepsilon_2 k_0+k_2},$$ where $k_0$ and $k_2$ 