## 3 Results and Discussion
### Scattering of a Counter-Propagating Laser Pulse from the Relativistic Electron Mirror
In the opposite case, as it was shown in [15], the decrease of the reflection coefficient occurs by \(I_{{}_{s}}^{{}^{\prime}}/10^{18}\) W/cm\({}^{2}\) times (a more detailed and complex formula for \(R^{\prime}({\varepsilon_{0}  
\[B\Bigg{[}\frac{photons}{mm^{2}\ mrad^{2}\ s\ 0.1\%\ width}\Bigg{]}{=}\frac{10^{-6}}{S[mm^{2}]}\frac{3 {\gamma_{x}}^{2}}{2\pi}\Phi=\frac{10^{-6}}{S[mm^{2}]}\frac{3{\gamma_{x}}^{2}}{2 \pi}fN_{s}R \tag{4}\]  
In the paper [13], instead of the average luminosity (4) one considers the peak luminosity \(B_{\max}\) during the scattering time \(\tau_{s}\ /\ 4{\gamma_{x}}^{2}\) (during the scattering the number of pulse periods remains constant) of a single laser pulse of duration \(\tau_{s}\). This luminosity differs in the definition of the flow of hard quanta as \(\Phi_{\max}=4{\gamma_{x}}^{2}R\ N_{s}\ /\ \tau_{s}\) and is \(4{\gamma_{x}}^{2}(f\tau_{s})^{-1}\) times higher than the average one (4). Further, in our numerical modeling, using Equation (3), we will make estimations of the average and peak luminosities of a source of hard quanta. The presented estimates of the reflection coefficient of the relativistic electron layer do not take into account some important physical effects occurring during scattering, such as the smearing of the electrical charge and slowing down of electrons by the counter-propagating pulse. For more accurate calculations of the energy of a reflected quantum and the reflection coefficient, the one-dimensional PIC simulations of reflection of a \(10^{18}\) W/cm\({}^{2}\) intense and 16 fs long laser pulse off the thin electron layer have been carried out; the layer was generated out of 0.6 nm thick C\({}^{+6}\) target irradiated by a \(5\ \times\ 10^{19}\) W/cm\({}^{2}\) intense and 16 fs long laser pulse. In Figure 5a, the pulse fields and electron layer number density (red) before the interaction are shown. The main pulse (blue) propagates from left to right, the counter (black)--in the opposite direction. **Figure 5.** The fields of the main (blue) and counter-propagating (black) pulses, the electrical number density of the thin layer (red) (**a**) before the scattering from the counter-propagating pulse (20 fs); (**b**) at the instant of pulses overlap (35 fs) and (**c**) after the formation of the reflected hard pulse (50 fs). During the movement of the electron layer in a superimposed field of two counter-propagating laser pulses of different amplitudes, the intense smearing of electron density occurs, which is clearly seen in Figure 5b. In the same figure, the small part of the counter-propagating pulse reflected off the electron front and propagating onto the front from left to right can be seen. In Figure 5b, the field of reflected pulse is shown the next time in a magnified scale, _i.e._, when the reflected pulse has overtaken the relativistic electrons. For determination of the reflection coefficient \(R\), frequency \(\Omega\) (in the initial frequency \(\omega_{s}\) units) of scattered radiation and determination of their dependences on the width of the initial target and initial pulse intensity, similar calculations have been made for intensities of \(5\,\times\,10^{18}\), \(10^{20}\), \(5\,\times\,10^{20}\) W/cm\({}^{2}\) and target widths of 0.4, 1, 5, 10 nm. The counter-propagating pulse had the same intensity, \(10^{18}\) W/cm\({}^{2}\). The results of the calculations are presented in the Figure 6 as the functions \(R(I,l_{f})\), \(\Omega(I,l_{f})\). These figures show that for generation of hard quanta ( \(\Omega\) > 100 ) the thin laser targets < 1 nm and high intensity laser targets > \(10^{20}\) W/cm\({}^{2}\) are optimal. However, the reflection coefficient of the test pulse is small and amounts to only a small percentage. The reflection coefficient \(R\) enables us to determine the conversion coefficient \(\chi\) of laser energy \(\varepsilon_{L}\) of the main pulse to the energy of hard radiation \(\chi=\Omega R\varepsilon_{s}\) / \(\varepsilon_{L}\), where \(\varepsilon_{s}\) is the energy of reflected pulse.