## 2 Theoretical Model
### The Influence of a Second Plasma Layer on the Parameters of the Accelerated Electrons  
However, there is the possibility of conserving the maximum layer energy locally in time by settling at some point a second foil, which would cut off the laser field and would conserve the maximum layer energy [10,11]. Let us determine from Equation (1) this position and the laboratory time interval during which this energy can be gained. Solution \(\gamma(X)\) of the system has the following form, shown in Figure 2.  
In Figure 2, the spatial stretching-out of the electron energy peak is quite reasonable, thereby explaining why there is no necessity for the exact determination of the plasma screen position at high intensities and layer energies. The dimensionless coordinate \(X=\omega x\,/\,c=2\pi x\,/\,\lambda_{{}_{L}}\). Hence, \(X\approx 5\cdot 10^{4}\) corresponds to \(x\,/\,\lambda_{{}_{L}}=\)7960 or \(x=6.4\) mm. It is sufficient to determine the mirror position to approximately mm accuracy. The layer takeoff angle (3) at such energy is only 0.1\({}^{\circ}\). As a result, the plasma screen enables us to fix the local temporal maximum layer energy.  
We would like to point out that the target optimization with respect to \(\varepsilon_{0}\) for the maximum electron energy gain in the layer in its turn will not be optimal for the whole maximum layer energy, _i.e._, \(\gamma\varepsilon_{0}\). If one increases \(\varepsilon_{0}\), then this value increases as well, reaching its maximum at \(\varepsilon_{0}\approx 0.1\) before dropping; the energy of a single electron corresponding to the maximum layer energy being then 558 MeV instead of 1 GeV. Hence, the optimal target with respect to the layer energy is two times thicker than that being optimal with respect to \(\gamma\), and the electron energy is two times lower. In addition, we would like to point out that the maximum layer energy with respect to \(\varepsilon_{0}\) is quite smooth, but \(\gamma\) depends on \(\varepsilon_{0}\) more.  
For circular polarization, the equations for the \(z\)-components of laser field and electron momentum should be added to the system (1). The solution of the new, but similar, system is shown in Figure 3. Here, one can see the dynamics of electron acceleration of the same foil by laser pulses of the same intensity but different polarization. One can see that the layer (initially at rest) energy is oscillating for both polarizations. For the linear polarization, the number of oscillations is equal to the number of semi-periods of laser pulse and is connected to double frequency of laser ponderomotive pressure. However, for the circular polarization, the number of oscillation is equal to the number of pulse periods. In the case of the circular polarization, maximal energy is higher and the electron layer can propagate at  
Figure 2: The dependence of the layer energy on its longitudinal coordinates at \(\mathrm{a}_{0}=19\), linear polarization, pulse duration = five periods and \(\varepsilon_{0}=0.04\).  
a longer distance compared to the linear one. The spatial domain of maximal energy for the circular polarization is wider, thus the requirements for the positioning of the second plasma mirror are lower.  
The selection of a target material (atomic number) has influence on initial electron density, _i.e._, the parameter \(\varepsilon_{0}\). Now the manufacturing of foils of sub-nanometer size (up to 0.5 nm) is possible only for carbon (Grafen) and plastic targets. We therefore consider, in our numerical modeling, fully ionized carbon targets.