#### 3.1. 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}° 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

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 μm 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 (γ = 1 +

α^{2} / 2, [

9]). For

а = 8.5 we have γ = 38, corresponding to the energy of ~20 MeV. In the simulations, one can easily see the cut-off at 28 MeV, which can be explained by the 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

) becomes ~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 μm 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.

**Figure 4.**
The spatial distributions of all electrons at time t = 0fs (black color) and fast electrons (>10 MeV) at t = 35 fs (before the second target is reached, red color) and at t = 57 fs (blue color) when the electron layer from the first target has reached the second one.

**Figure 4.**
The spatial distributions of all electrons at time t = 0fs (black color) and fast electrons (>10 MeV) at t = 35 fs (before the second target is reached, red color) and at t = 57 fs (blue color) when the electron layer from the first target has reached the second one.

#### 3.2. Scattering of a Counter-Propagating Laser Pulse from the Relativistic Electron Mirror

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 [

13,

14]). Simple estimations indicate that in the rest frame of reference of a moving layer, the frequency of an incident pulse gets increased by the factor

times. The scattering in the rest frame of reference occurs without frequency change. The following recalculation of the frequency of scattered radiation in the original laboratory frame of reference gives again the factor 2

γ_{x }so that the reflected pulse has a quantum energy

, with

ω_{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 [

14]. In our case, the number of electrons at laser front does not change after propagation out of the second target, and the effect connected with change of electron density is absent during the process of counter-propagating laser pulse reflection.

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

R΄ =

ε_{0}΄

^{2} / (

ε_{0}΄

^{2} + 1) [

15]. The surface electron number density

n_{e}΄

l_{f}΄ contained in

ε_{0}´ (see definition of

ε_{0 }in Equation (2)) does not depend on the type of a reference system, which is why

n_{e}΄

l_{f}΄ =

n_{e}l_{f}. Except for

n_{e}΄

l_{f}΄, the denominator of

ε_{0}´ ontains only the frequency of the incident pulse

ω_{L}΄, which in the rest frame of reference of the electron layer is equal to

[

9]. As a result, in the rest frame of reference of the electron layer, the reflection coefficient expressed using the variables in the laboratory frame of reference is equal to

. In the rest frame of reference of the layer, the reflection occurs without frequency change, thus explaining why the number of reflected quanta

N_{h} (hard quanta in the laboratory frame of reference) can be expressed through the number of incident laser quanta

N_{s} as following

N_{h} =

R´

N_{S}. Since the absolute numbers of quanta are the relativistic invariants, then

R´ is also the reflection coefficient with respect to the number of quanta (but not with respect to the pulses energies) in laboratory frame of reference. We would like to emphasize that the reflection coefficient

R´ implies the non-relativistic (<10

^{18} W/cm

^{2}) intensity

I_{s}´ of the incident radiation in the rest frame of reference of the layer. In the opposite case, as it was shown in [

15], the decrease of the reflection coefficient occurs by

I_{s}´ / 10

^{18} W/cm

^{2} times (a more detailed and complex formula for

R΄(

ε_{0}΄,

I_{L}΄)is given in this paper). The employment of the coefficient from the equation for the reflection coefficient

R also implies the coherent character of the scattering process (this also follows from the fact that

,

i.e., the squared number of electrons in the layer) and the assumption of thinness of the layer (the width is less than the wavelength in the rest frame of reference). In order to validate these approximations, one needs to satisfy the inequality

for the electron number density of the thin layer at the instant of reflection. In the numerical modeling, results of which are presented in

Figure 5, the electron number density in the layer at

t = 57 fs was estimated to 6 × 10

^{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

, then the coherent scattering will switch to the non-coherent Thomson scattering by single electrons. In this case, it is obvious that

, where

S is the scattering spot area, and σ

_{T} = 6.6 × 10

^{−25} cm

^{2} is Thomson scattering cross-section. The reflection coefficient of non-coherent scattering is obviously equal to

R_{T} =

σ_{T}n_{e}l_{f} and its absolute magnitudes is substantially lower than

R. The main important characteristic of the hard radiation source is its luminosity

B (number of photons radiated from unit area per unit solid angle and per unit time). When the laser pulse is repeatedly irradiated with the frequency

f (for the considered laser tens of Hz) the average flow (number of quanta per unit time) of the hard radiation will be Φ =

fRN_{s}. The average luminosity

B of forward in the layer movement direction radiation is related to the flow Ф through the following relation:

In the paper [

13], instead of the average luminosity (4) one considers the peak luminosity

B_{max}during the scattering time

τ_{s} / 4

γ_{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

γ_{x}^{2}RN_{S} /

τ_{S }and is 4

γ_{x}^{2}(

fτ_{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

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).

**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 Ω (in the initial frequency

ω_{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 × 10

^{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

Figure 6 as the functions

R(

I,

l_{f}), Ω(

I,

l_{f}). These figures show that for generation of hard quanta (Ω >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 χ of laser energy

ε_{L}of the main pulse to the energy of hard radiation

χ= Ω

Rε_{S} /

ε_{L}, where

ε_{s}is the energy of reflected pulse. The contrary behavior of dependences of the reflection coefficient and hard quanta energy on the laser intensity and target width shown in

Figure 6 indicates the existence of an optimum with respect to the width and intensity at which the conversion coefficient of laser radiation energy to the energy of hard radiation reaches its maximum. In

Figure 7, the dependence of the energy conversion coefficient of the main laser pulse to the hard quanta on the laser intensity is shown.

**Figure 6.**
The dependence of the frequency Ω (**a**) of scattered hard radiation and the reflection coefficient R on the intensity of the main laser pulse. The target width is 0.6 nm. The dependence of the frequency Ω (**b**) of scattered hard radiation R and the reflection coefficient on the width of initial target at laser intensity of 5 × 10^{19} W/cm^{2}.

**Figure 6.**
The dependence of the frequency Ω (**a**) of scattered hard radiation and the reflection coefficient R on the intensity of the main laser pulse. The target width is 0.6 nm. The dependence of the frequency Ω (**b**) of scattered hard radiation R and the reflection coefficient on the width of initial target at laser intensity of 5 × 10^{19} W/cm^{2}.

**Figure 7.**
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.

**Figure 7.**
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

Figure 5. The number of photons per unit area

N_{s}/

S for a given laser pulse (16 fs, 10

^{18} W/cm

^{2}, 0.8 μm) amounts to 5.3 × 10

^{20} photons/mm

^{2}. In accordance with

Figure 6a, the reflection coefficient is

R≈ 0.05 and the value Ω = 4

r_{x}^{2}≈ 40. At the scattering pulse repetition rate

f = 10 Hz, the average luminosity is about 1.3 × 10

^{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

Figure 5, the magnitude of the source peak luminosity reaches 3 × 10

^{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.