## 2 Theoretical Model  
### The Equation of Motion of Electron Layer  
In the one-dimensional space approximation, and in the limit of zero electron layer width, it appears to be possible to integrate overcharge distribution in one-dimensional Lienard-Wiechert potentials, to express the Eigen-fields of the layer using its mechanical variables and, as a result, to write an equation for the dynamics including only the external field, velocity and layer coordinates. The radiation friction force (self-action) can also be expressed using the layer velocity. It is convenient to write the equation of layer motion using the dimensionless parameters \(P=\dfrac{u_{{}_{y}}}{\sqrt{1-u_{{}_{y}}^{{}^{2}}-\dot{X}^{{}^{2}}}}-a_{{}_{y}} ^{{}^{(ext)}}(\theta)\), \(\Gamma=\dfrac{1-\dot{X}}{\sqrt{1-u_{{}_{y}}^{{}^{2}}-\dot{X}^{{}^{2}}}}\) and being the constants of motion for a single electron in an electromagnetic wave down, where \(a_{{}_{y}}^{{}^{(ext)}}=\left|e\right|A_{{}_{y}}^{{}^{(ext)}}(\theta)/\,m_{e}c ^{{}^{2}}\) is a dimensionless vector potential representing the incident wave; \(\theta=\omega t-k\pi\), \(u_{{}_{y}}=v_{{}_{y}}/\,c\) is the dimensionless layer velocity component along the polarization direction (y); and, \(\dot{X}=v_{{}_{x}}/\,c\) is the dimensionless layer velocity component along the wave vector (x). The equation of motion of the thin electron layer using these variables has the following form:\[ \begin{split}\frac{dP}{d\theta}=-\varepsilon_{0}\,\frac{P+a_{y}^{\ (ext)}( \theta)}{\Gamma},\frac{d\Gamma}{d\theta}=-2\varepsilon_{0}\,\frac{(P+a_{y}^{ \ (ext)}(\theta))^{2}}{\Gamma^{2}+(P+a_{y}^{\ (ext)}(\theta))^{2}+1}\\ \frac{dX}{d\theta}=\frac{1+(P+a_{y}^{\ (ext)}(\theta))^{2}-\Gamma^{2} }{2\Gamma^{2}},\frac{d\tau}{d\theta}=\frac{1+(P+a_{y}^{\ (ext)}(\theta))^{2}+\Gamma^{2}}{2\Gamma^{2}}\end{split}  \tag{1}\]  
where \(\tau=\omega t\), \(X=\omega x\) / \(c\) are the dimensionless time and coordinate of the layer, and the dimensionless parameter \(\varepsilon_{0}=\pi n_{e}l_{f}\) / \(n_{cr}\lambda_{ L}\) defines both the width \(l_{\mathrm{f}}\) and the electron layer number density \(n_{\mathrm{c}}\). The equations of motion (1) correspond to the equations of motion of an extended electron layer [6] if one makes a transition \(l_{\mathrm{f}}\)\(\rightarrow\)0 in the latter equations and considers the movement of the central region of the layer. Let the electron layer be still at the initial instant: \(P(0)=0\), \(\Gamma(0)=1\), \(a_{y}^{\ (ext)}(0)=0\), \(X(0)=0\), \(\tau(0)=0\). Then, let the electron layer be acted upon by a pulse of limited duration \(a_{y}^{\ (ext)}(\theta)=a_{0}\sin(\theta)\), \(\theta\in[0;2\pi N]\), which is switched off after \(N\) cycles. Let us define the finite electron energy at \(\theta=2\pi N\), _i.e._, at the time moment of pulse switching-off. This energy (Lorentz factor) can be expressed through the variables \(P\), \(\Gamma\) in the following way:  
\[ \begin{split}\gamma&=\frac{1}{\sqrt{1-u_{y}^{\ 2}-\dot{X}^{2}}}=\frac{1+(P+a_{y}^{\ (ext)}(\theta))^{2}+\Gamma^{2}}{2\Gamma}\\ \gamma_{x}&=\frac{1}{\sqrt{1-\dot{X}^{2}}}=\frac{1+( P+a_{y}^{\ (ext)}(\theta))^{2}+\Gamma^{2}}{2\Gamma\sqrt{1+(P+a_{y}^{\ (ext)}(\theta))^{2}}}\end{split}  \tag{2}\]  
In a case for \(\varepsilon_{0}=0\), when \(P=0\), \(\Gamma=1\), these equations let us obtain the energy of a single electron in a linearly polarized wave [9]. At the instant of pulse switching-off \(\theta=2\pi N\), the initial single electron energy remains the same and no energy exchange between the electron and pulse occurs. Let us find the solution of the system of Equation (1) and consider the case close to the experiment parameters of the laser pulse a\({}_{0}=10\), \(N=10\). The dependence of the layer energy at the instant of pulse switching-off is shown in Figure 1.  
Figure 1: The dependence of the layer energy on its width at the instant of pulse switching-off. When \(\varepsilon_{0}=0\), the layer energy can be determined from Equation (2).