## 3 Results and Discussion  
### 2D Numerical Modeling of the Electron and Plasma Layers Interaction  
Let us consider the propagation of the electron bunch through the plasma layer using numerical modeling by the modified code [12]. In the calculations, the laser pulse of intensity \(10^{2\circ}\) W/cm\({}^{2}\), duration 15 fs and diameter of 7 \(\mu\)m, interacts with two sequentially located C\({}^{+6}\) targets having an ion number density of \(10^{23}\) cm\({}^{-3}\). The width of the first target was 5 nm, representing an electron layer source, the second being 1 \(\mu\)m, thus representing the screen cutting off the laser pulse; the distance between targets was 11 \(\mu\)m. The numerical step was 1 nm; 40 particles were allocated in the cell.  
In Figure 4 the black color illustrates the initial location of targets, and the red color is the spatial distribution of fast electrons (\(>\)10 MeV) at the time \(t=35\) fs (before the second target is reached). One can see that the relativistic electron layer has been generated. The plane cross section at \(y=9\)\(\mu\)m shows that the electron bunch number density is 0.06 of the initial one \(10^{23}\) cm\({}^{-3}\), _i.e._, overcritical. The cut-off energy of the calculated electron distribution function exceeds the energy of a single electron in the field of plane EM. wave (\(\gamma=1+a^{2}\) / 2, [9]). For \(a=8.5\) we have \(\gamma=38\), corresponding to the energy of \(\sim\)20 MeV. In the simulations, one can easily see the cut-off at 28 MeV, which can be explained by the  
Figure 3: The dependence of the layer’s energy on its longitudinal coordinate. The red color denotes the linear polarization case at \(a_{0}=23\) and \(\varepsilon_{0}=0.02\); the blue color denotes the circular polarization case at \(a_{0}=23/\sqrt{2}\) and \(\varepsilon_{0}=0.02\). Pulse duration is four laser periods for both cases.  
difference of the field amplitude from that in a vacuum during laser pulse penetration through the target. Such self-consisted field behavior is taken into account by the system (1). The calculated field shows that the amplitude was increased by 1.2 times. Estimation of the energy from the formula (2) demonstrates increase of electron energy up to 30 MeV, which correlates with the calculated value \(-\)28 MeV. When the laser pulse reaches the second target, the characteristic energy of new, generated in this target fast electrons, (with \(\gamma=\sqrt{1+a^{2}}\approx\) 8.6 ) becomes \(\sim\)4 MeV, correspondingly, the \"tail\" of the electron distribution function from the first target should not significantly change when passing through the second target. In Figure 4, the blue color illustrates the electron number density (with electrons energy \(>\) 15 MeV) at the time \(t=57\) fs when the electron layer from the first target reaches the second one. One can clearly see that the electron bunch has passed through the second target without any loss of energy and number of electrons. The laser pulse appears to be cut off by the dense plasma with 0.1 \(\mu\)m width. Thus, the plasma layer enables us to effectively separate the thin relativistic electron layer from the laser pulse without any loss of energy or number of electrons. It is noteworthy that the thickness of the electron bunch after propagation for the distance of 11 microns is about 60 nm in accordance with our calculations and estimations [8]. It is larger compared to the initial thickness, but is, in any case, still less than the laser wavelength.