## 3 Results and Discussion
### Scattering of a Counter-Propagating Laser Pulse from the Relativistic Electron Mirror  
The generated stable thin relativistic electron layer can be used as a source of coherent hard radiation produced by scattering of a counter-propagating laser pulse by the electron layer (see examples in [13, 14]). Simple estimations indicate that in the rest frame of reference of a moving layer, the frequency of an incident pulse gets increased by the factor \([\left(1+\dot{X}\right)/\left(1-\dot{X}\right)]^{1/2}\approx 2\gamma_{x}\) times. The scattering in the rest frame of reference occurs without frequency change. The following recalculation of the frequency of scattered radiation in the original laboratory frame of reference gives again the factor \(2\gamma_{x}\), so that the reflected pulse has a quantum energy \(h\omega=4\gamma_{x}^{\ 2}h\omega_{x}\), with \(\omega_{x}\) being the frequency of the incident counter-propagating laser pulse. Note that a frequency shift is appearing not only due to  
Figure 4: The spatial distributions of all electrons at time \(t=0\)fs (black color) and fast electrons (\(>\)10 MeV) at \(t=35\) fs (before the second target is reached, red color) and at \(t=57\) fs (blue color) when the electron layer from the first target has reached the second one. [PERSON]'s effect of radiation reflected off a moving overdense electron layer, but also due to change of electron density inside a layer [14]. In our case, the number of electrons at laser front does not change after propagation out of the second target, and the effect connected with change of electron density is absent during the process of counter-propagating laser pulse reflection. In order to estimate the reflection coefficient of the thin relativistic electron layer we would like to make use of the well-known Fresnel reflection coefficient of the thin relativistic layer of a still plasma \(R^{\prime}={\varepsilon_{0}}^{{}^{\prime}2}/({\varepsilon_{0}}^{{}^{\prime}2 }+1)\)[15]. The surface electron number density \(n_{e}^{{}^{\prime}}l_{f}^{{}^{\prime}}\) contained in \({\varepsilon_{0}}^{{}^{\prime}}\)(see definition of \({\varepsilon_{0}}\) in Equation (2)) does not depend on the type of a reference system, which is why \(n_{e}^{{}^{\prime}}l_{f}^{{}^{\prime}}=n_{e}l_{f}\). Except for \(n_{e}^{{}^{\prime}}l_{f}^{{}^{\prime}}\), the denominator of \({\varepsilon_{0}}^{{}^{\prime}}\) contains only the frequency of the incident pulse \({\omega_{L}}^{{}^{\prime}}\), which in the rest frame of reference of the electron layer is equal to \({\omega_{L}}^{{}^{\prime}}={\omega_{L}}\sqrt{\left(1+\dot{X}\right)/\left(1- \dot{X}\right)}=\left(1+\dot{X}\right){\gamma_{x}}^{{}^{2}}\)[9]. As a result, in the rest frame of reference of the electron layer, the reflection coefficient expressed using the variables in the laboratory frame of reference is equal to \(R^{\prime}={\varepsilon_{0}}^{{}^{2}}/({\varepsilon_{0}}^{{}^{2}}+\left(1+ \dot{X}\right)^{2}{\gamma_{x}^{2}})\approx{\varepsilon_{0}}^{{}^{2}}/({ \varepsilon_{0}}^{{}^{2}}+4{\gamma_{x}^{2}})\). In the rest frame of reference of the layer, the reflection occurs without frequency change, thus explaining why the number of reflected quanta \(N_{h}\) (hard quanta in the laboratory frame of reference) can be expressed through the number of incident laser quanta \(N_{s}\) as following \(N_{{}_{h}}=R^{\prime}N_{{}_{s}}\). Since the absolute numbers of quanta are the relativistic invariants, then \(R^{\prime}\) is also the reflection coefficient with respect to the number of quanta (but not with respect to the pulses energies) in laboratory frame of reference. We would like to emphasize that the reflection coefficient \(R^{\prime}\) implies the non-relativistic (\(<\)10\({}^{18}\) W/cm\({}^{2}\)) intensity \(I_{{}_{s}}^{{}^{\prime}}\) of the incident radiation in the rest frame of reference of the layer.