describing a optical response of the material inside the mirrors. Taken together, relations (\[FEscatt\]) and (\[Fresnel\]) reproduce the Lifshitz expression [@Lifshitz56]. They are known to tend to the original Casimir expression in the limit $\varepsilon \rightarrow \infty$ which produces perfectly reflecting mirrors [@Schwinger78].

We may emphasize at this point that relations (\[FEscatt\]) are more general than Lifshitz expression which, incidentally, were not written originally in terms of reflection amplitudes [@Katz77]. They are valid for example for non-local optical responses of the mirrors provided the reflection amplitudes are substituted by their possibly more complicated expressions. The only limitation, discussed below, is associated with the assumption of specular scattering.

Finite conductivity corrections {#sec:4}
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We now review the corrections to the Casimir expression coming from the finite conductivity of the bulk material. Here, these corrections are deduced from relations (\[FEscatt\]), assuming Fresnel laws (\[Fresnel\]) for a local optical response of the bulk material. This function may be given by a simple description of the conduction electrons in terms of a plasma model $$\begin{aligned}
\varepsilon[i\xi]=1+\frac{\omega_{\rm P}^{2}}{\xi^2},\end{aligned}$$ characterized by a plasma frequency $\omega_{\rm P}$ and wavelength $\lambda_{\rm P}\equiv 2\pi c/\omega_{\rm P}$. It may be given by a more realistic representation based upon tabulated optical 