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In the present paper, the possibility of generation of thin dense relativistic electron layers is shown using the analytical and numerical modeling of laser pulse interaction with ultra-thin layers. It was shown that the maximum electron energy can be gained by optimal tuning between the target width, intensity and laser pulse duration. The optimal parameters were obtained from a self-consistent system of Maxwell equations and the equation of motion of electron layer. For thin relativistic electron layers, the gaining of maximum electron energies requires a second additional overdense plasma layer, thus cutting the laser radiation off the plasma screen at the instant of gaining the maximum energy (DREAM-schema).

Generation of short-wavelength (less than the laser wavelength) dense relativistic electron bunches is of high interest for study and diagnostics of superfast physical processes. Such bunches make possible the direct electron microscopy, as well as the generation of short (atto-second range) pulses of soft X-ray and further X-ray microscopy. Short, harder X-ray pulses can be also generated during the scattering of additional laser pulses by the electron bunches. Generation of thin electron layers is possible using different methods, such as in a laser gas target, whereby the oscillations of nonlinear electron density occur leading to the generation of a sequence of thin electron layers [

In the present work, the possibility of generation of thin dense relativistic electron layers is shown using the analytical and numerical modeling of the interaction of laser pulses with ultra-thin solid foils. The first ultrathin foil is semitransparent for laser radiation, but its electrons are removed by laser pulse and propagate together with them. The electric field of target ion core is much weaker than that of the laser one, thus movement of an electron bunch separated from the target can be described by the used self-consistent system of Maxwell equations and the electron equation of motion. The maximum electron energy can be gained by optimal tuning between the target width, intensity and laser pulse duration. The optimal parameters can be determined from our self-consistent system of Maxwell equations and the equation of motion of electron layer. For thin relativistic electron layers, the gaining of maximum electron energies requires an additional overdense plasma layer, which cuts the laser radiation off the plasma screen at the instant of gaining the maximum energy (Double Relativistic Electron Accelerating Mirror (DREAM) schema). The scattering of counter-propagating probe laser pulse by the generated relativistic electrons mirror makes possible the generation of the hard coherent electromagnetic radiation with quant energy of 1 keV and efficiency of 0.1% with respect to the energy of initial laser pulse generating the 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 Liénard–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 _{y}^{(ext)} = |_{y}^{(ext)}(_{e}c^{2 }is a dimensionless vector potential representing the incident wave; _{y}_{y}_{0} = _{e}l_{f}_{cr}λ_{L}_{f} and the electron layer number density _{e}. The equations of motion (1) correspond to the equations of motion of an extended electron layer [_{f}→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: _{y}^{(ext)}(0) = 0, _{y}^{(}^{ext}^{)}(_{0} sin(_{0} = 0, when _{0} = 10,

The dependence of the layer energy on its width at the instant of pulse switching-off.

When ε_{0} = 0, the layer energy can be determined from Equation (2). As one can clearly see, during an increase of ε_{0}, the layer starts to speed up and its energy begins to increase in comparison with the energy of a single electron. When ε_{0} = 0.02, the target electrons speed up until the energy, which is significantly higher than the energy of a single electron, is obtained in the wave. During the pulse action period, the maximum energy oscillates and the Lorentz factor gains the magnitude 1000. At a fixed value of a_{0}, when the target width increases further, the layer stops to speed up and its energy begins to oscillate in time with the same amplitudes. Hence, the equations for the layer motion demonstrate that there is an optimum width for acceleration, and that the instantaneous layer energy during the laser pulse action period can be significantly higher than that when the pulse is switched off. This is why, when solving the acceleration optimization problem, it is reasonable to terminate the pulse action at the instant of the maximum gained energy. As will be shown, this can be done either by settling an additional plasma layer cutting the laser field off the electrons. It is noteworthy that the following formula for the angle of electron layer propagation (with respect to the axis x) out of the laser field is valid

Let us determine the optimal width of the electron layer for maximum energy gain. Target width should be thicker than 0.1 nm (one-atomic layer) and _{e}_{cr}_{0} ≈ 0.04. In order for such a target to gain the energy of 1 GeV, it is sufficient to have a pulse with a_{0} = 19 and duration 5 laser periods. At laser wavelength 0.8 μm, the corresponding intensity becomes 8 × 10^{20} W/cm^{2}. In order to tune the electron energy peak to the pulse end, one needs to alter the pulse duration and slightly changes ε_{0}. For example, for the pulse intensity 2 × 10^{21} W/cm^{2} (_{0} = 30) and duration of exactly 4 periods the target with ε_{0} = 0.042 gains a maximum energy of 2.06 GeV, and the energy becomes 1.8 GeV at the end of the 4th period, _{0}, for example, to 0.05, then the takeoff energy drops to only 306 MeV. Hence, the electron energy peak related to the end of the laser pulse exhibits a resonance character and its optimization requires the exact tuning of all parameters. It is significant that the high takeoff electron energies can be gained through the abrupt switching-off of the laser pulse. If one considers the major (during 2 and 4 periods) switching-off of the laser pulse, then the takeoff energies are one order lower than the maximum one. Real laser pulses, unless special technical tricks are employed, have the time growth and drop-off of the order of several periods and these times are not determined with high accuracy, _{e}_{f}_{e}_{f}

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 [

The dependence of the layer energy on its longitudinal coordinates at a_{0} = 19, linear polarization, pulse duration = five periods and ε_{0} = 0.04.

In _{L}^{4 }corresponds to _{L}

We would like to point out that the target optimization with respect to ε_{0} for the maximum electron energy gain in the layer in its turn will not be optimal for the whole maximum layer energy, _{0 }If one increases ε0, then this value increases as well, reaching its maximum at _{0}≈ 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 γ, and the electron energy is two times lower. In addition, we would like to point out that the maximum layer energy with respect to ε_{0} is quite smooth, but γ depends on ε_{0} more.

For circular polarization, the equations for the

The dependence of the layer’s energy on its longitudinal coordinate. The red color denotes the linear polarization case at _{0} = 23 and _{0} = 0.02; the blue color denotes the circular polarization case at _{0} = 0.02. Pulse duration is four laser periods for both cases.

The selection of a target material (atomic number) has influence on initial electron density, _{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.

Let us consider the propagation of the electron bunch through the plasma layer using numerical modeling by the modified code [^{2}° W/cm^{2}, duration 15 fs and diameter of 7 μm, interacts with two sequentially located С^{+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 μm, thus representing the screen cutting off the laser pulse; the distance between targets was 11 μm. The numerical step was 1 nm; 40 particles were allocated in the cell.

In ^{23} cm^{−3}, ^{2} / 2, [

The spatial distributions of all electrons at time

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 [_{x }_{s }being the frequency of the incident counter-propagating laser pulse. Note that a frequency shift is appearing not only due to Doppler’s effect of radiation reflected off a moving overdense electron layer, but also due to change of electron density inside a layer [

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 _{0}΄^{2} / (_{0}΄^{2} + 1) [_{e}_{f}_{0}´ (see definition of _{0 }in Equation (2)) does not depend on the type of a reference system, which is why _{e}_{f}_{e}l_{f}_{e}_{f}_{0}´ ontains only the frequency of the incident pulse _{L}_{h}_{s}_{h}_{S}^{18} W/cm^{2}) intensity _{s}_{s}^{18} W/cm^{2} times (a more detailed and complex formula for _{0}΄, _{L}^{21} cm^{−3}, and this inequality was valid for the whole spectrum of electron energy. If, for some reason (e.g. durable movement of the electron layer), the electron number density will be low _{T} = 6.6 × 10^{−25} cm^{2} is Thomson scattering cross-section. The reflection coefficient of non-coherent scattering is obviously equal to _{T}_{T}n_{e}l_{f}_{s}

In the paper [_{max}during the scattering time _{s} / 4_{x}^{2 }(during the scattering the number of pulse periods remains constant) of a single laser pulse of duration τ_{s}. This luminosity differs in the definition of the flow of hard quanta as Φ_{max}= 4_{x}^{2}_{S}_{S }_{x}^{2}(_{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 × 10^{19} W/cm^{2} intense and 16 fs long laser pulse. In

The fields of the main (blue) and counter-propagating (black) pulses, the electrical number density of the thin layer (red) (

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 _{s}^{18}, 10^{20}, 5 × 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 _{f}_{f}^{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 _{L}_{S}_{L}_{s}

The dependence of the frequency Ω (^{19} W/cm^{2}.

The dependence of the energy conversion coefficient of the main laser pulse to the hard quanta on the laser intensity at 0.6 nm target width.

One can clearly see that the optimal conversion can be reached at comparatively low energies of hard quanta 50 eV (Ω ≈ 30), which is why, in our case, the generation of hard quanta of high energies (higher than 1 keV) occurs at the non-optimal coefficient of energy conversion to that of the scattered radiation. In our calculations, at an intensity of 5 × 10^{19} W/cm^{2} and width of Carbon target of 0.6 nm, the conversion coefficient is χ ≈ 0.1%. Such a conversion coefficient exceeds that of laser X-ray line conversion in the same spectrum of quanta energies. Let us estimate, using Equation (3), the source luminosity corresponding to the parameters of _{s}/^{18} W/cm^{2}, 0.8 μm) amounts to 5.3 × 10^{20} photons/mm^{2}. In accordance with _{x}^{2}≈ 40. At the scattering pulse repetition rate ^{15} photons/(sec mm^{2} mrad^{2}). This is significantly higher than that of conventional X-ray tubes (10^{8}) and laser-electron generators on a base of accelerators (10^{12}), but lower than the average luminosity of contemporary synchrotrons (10^{21}) in the approximate energy range of hard quanta. In ^{29} photons/(sec mm^{2} mrad^{2}), which is 8 orders higher than that of synchrotron luminosity. One can increase average luminosity by increasing the pulse repetition rate to 10 kHz, thereby leading to a doubling of the average luminosity of such a scheme.

Concluding the above results, the relativistic intense laser pulse enables us to form a relativistic dense and thin electron layer when interacting with the ultrathin target. Optimum foil thickness for layer production is much smaller than targets for fast ion generation and the optimal parameter _{0} = _{e}l_{f}_{cr}λ _{0} ≈ 0.01 can be optimal. At the instant of termination of a laser pulse the electron energy in a layer does not come back to its initial value in an optimum range of thickness. Therefore, the plasma screening for an optimum target ceases to be necessary. At the same time, final magnitude of energy of a layer can be less than maximum possible magnitude reached during the pulse action. Then, by means of the plasma screening (DREAM scheme), it is possible to fix the maximum value. Because energy of a layer’s maximum is considerably extended in the spatial area, there is therefore no necessity for high accuracy for the screen positioning. Maximum energy in the case of the circular polarization is higher, and electron layer can propagate for a longer distance compared to the linear one. The spatial domain of maximum energy for the circular polarization is wider, thus the requirements for the location of secondary plasma mirror are lower. Reflection of a counter-propagating laser pulse off the relativistic mirror, produced by the main pulse in DREAM scheme, enables the production of KeV coherent radiation with an efficiency of about 0.1% in respect to the energy of the main laser pulse. Peak brightness of a source of the hard quanta gained in DREAM scheme exceeds that of known sources of hard radiation, and average brightness exceeds that of X-ray tubes and laser-electron generators, but it is less than that of a synchrotron X-ray source.

A. A. Andreev acknowledges the provided computation resources of JSC at project HBUIS.

The authors declare no conflict of interest.

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