by their polarization $p$. Time invariance of the problem, as well as transverse spatial translation invariance (along $x$ and $y$) ensure that the frequency, the transverse wavevector and the polarization are conserved quantities throughout the scattering process on the cavity. The scattering couples only the free vacuum modes with opposite signs for the component $k_{z}$ of the wavevector along the longitudinal $z$ axis of the cavity. We write $r_{\bf k}^{p}[\omega]$ the reflection amplitude of the mirror $i=1,2$ as seen from the inner side of the cavity. These amplitudes obey general physical properties of causality, unitarity and high frequency transparency.

The spectral density of the vacuum intracavity fields is changed with respect to that of free-fields outside the cavity. The ratio of energy inside the cavity to energy outside the cavity is fixed, for a given mode, by the following function $$\begin{aligned}
g_{\bf k}^{p}[\omega] = \frac{1-|\rho_{\bf k}^{p}[\omega]|^{2}}{|1-\rho_{\bf k}^{p}[\omega]|^{2}} \ \ , \ \  \rho_{\bf k}^{p}[\omega] = r_{\bf k}^{p}[\omega]_{1}r_{\bf k}^{p}[\omega]_{2}e^{2ik_{z}L}.\end{aligned}$$ This statement constitues a theorem which has been demonstrated for lossless as well as lossy mirrors [@GenetPRA03; @Barnett96]. It does not depend on the state of the fields and is therefore valid for vacuum fluctuations as well as for thermal fluctuations, assuming 