Atomistic modelling and characterizaion of light sources based on small-amplitude short-period periodically bent crystals

Trajectories of 10 GeV electrons and positrons have been simulated in $L=200$ microns thick oriented diamond(110), silicon(110) and germanium(110) crystals. The simulations have been performed for the linear crystals as well as for the SASP periodically bent crystals. In the latter case the harmonic profile $a\cos 2\pi{z/{\lambda_{\rm u}}}$ of the periodic bending of the (110) planes has been considered with the $z$ coordinate measured along the non-deformed ($a=0$) plane. The direction of the $z$-axis was chosen well away from crystallographic axes to avoid the axial channeling regime. The $y$-axis was chosen along the $\langle 110\rangle$ axial direction. The bending amplitude $a$ was considered within the range $0-0.5$ Å, the values of the bending period ${\lambda_{\rm u}}$ were $200,400$ and 600 microns. We note that these values of $a$ and ${\lambda_{\rm u}}$ are close to those used in the channeling experiments with the SASP crystals [38, 39, 40].

In the course of simulations the positions atoms in either linear or periodically bent crystalline structure was generated with account for their random displacement from the nodes due to thermal vibrations at temperature $T=300$ K. For each atom the displacement from the node was generated using the normal distribution with the root-mean-square amplitude of the thermal vibrations equal to 0.04, 0.075 and 0.085 Å [46]. More details on the simulation procedure as well on the formalism behind it can be found in Ref. [43] and in the review paper [35].

For each trajectory simulated the initial conditions at the crystal entrance (i.e. $x$ and $y$ coordinates and the corresponding velocities) were generated assuming the Gaussian distributions of the beam particles in the transverse coordinates and velocities. The root mean square (rms) beam sizes $\sigma_{x,y}$ and angular divergences $\sigma_{\phi_{x,y}}$ used in the simulations are presented in Table 1 along with other parameters that correspond to the ${\varepsilon}=10$ GeV (the relativistic Lorentz factor is $\gamma=1.96\times 10^{4}$) electron and positron beams. The data on the beam size, normalized rms emittance $\gamma\epsilon_{x,y}=\gamma\sigma_{\phi_{x,y}}\sigma_{x,y}$, and peak current $I_{\rm peak}$ are taken from Table 4.7 in Technical Design Report for the FACET-II beam at the SLAC facility, Ref. [4]. The divergencies $\sigma_{\phi_{x,y}}$ indicated in the table were calculated from the $\sigma_{x,y}$ and $\gamma\epsilon_{x,y}$ values.

As mentioned, in the simulations the $y$-axis was chosen perpendicular to the $(110)$ planar direction. This choice ensures that the biggest fraction $\xi$ of a Gaussian beam with angular divergence $\sigma_{\phi_{y}}=10$ $\mu$rad enters the crystal having the incident angle with respect the $(110)$ plane much less that Lindhard’s critical angle $\Theta_{\rm L}$, i.e. such particles can be accepted in the channeling mode at the entrance. This fraction can be estimated as $\xi=(2\pi\sigma_{\phi_{y}}^{2})^{-1/2}\int_{-\Theta_{\rm L}}^{\Theta_{\rm L}}\exp\left(-\phi^{2}/2\sigma_{\phi_{y}}^{2}\right){\rm d}\phi$. Using the vaues $U_{0}\approx 22,23,39$ eV for the interplanar potential depth in (110) planar channels in the diamond, silicon and germanium one finds, respectively, $\Theta_{\rm L}=(2U_{0}/{\varepsilon})^{1/2}\approx 66,68$ and 88 $\mu$rad that exceeds by large factors the divergence $\sigma_{\phi_{y}}$, hence $\xi\approx 1$ in all cases. The beams divergence $\phi_{x}=30$ $\mu$rad along the $x$ direction is smaller than the natural emission angle $\gamma^{-1}\approx 50$ $\mu$rad. As a result, one can expect that big fraction of radiation emitted by the channeling particles will be collected within the cone $\theta_{0}\leq\theta_{\gamma}$ centered along the incident beam.

Due to the randomization of the conditions at the crystal entrance as well as of the positions of the lattice atoms that are subject to thermal fluctuations, each trajectory simulated corresponds to a unique crystalline environment. Statistical independence of the trajectories allows one to quantify the channeling process in terms of several functional dependencies which appear as a result of statistical analysis of the trajectories.

In the course of passing through an oriented crystal a charged projectile can experience different types of motion including the channeling and overbarrier motion. The transition from the channeling to the overbarrier motions, - the dechanneling process, as well as the opposite transition, - the rechanneling process, occur due to uncorrelated collisions of the projectile with the crystal atoms. To quantify these processes the following two dependencies, termed channeling fractions, can be considered [43, 47]. One of these is defined as $f_{\rm ch0}(z)=N_{\rm ch0}(z)/N_{\rm acc}$ where $N_{\rm acc}$ is the number of accepted particles, i.e. those which experience the channeling motion at the crystal entrance, $z=0$. The numerator, $N_{\rm ch0}(z)\leq N_{\rm acc}$, stands for the number of accepted particles that channel up to the distance $z$ where they dechannel. To quantify the efficiency of the rechanneling process one can construct another fraction, $f_{\rm ch}(z)=N_{\rm ch}(z)/N_{\rm acc}$ with $N_{\rm ch}(z)$ being the total number of particles which move in the channeling mode at distance $z$.

The fraction $f_{\rm ch0}(z)$ is a decreasing function of the penetration distance due to gradual dechanneling of the accepted particles. At sufficiently large distances this fraction decreases following the exponential decay law, $\propto\exp(-z/{L_{\rm d}})$ [48], where ${L_{\rm d}}$ stands for the dechanneling length. In contrast, the fraction $f_{\rm ch}(z)$ is a non-monotonous function: in the vicinity of the entrance point it increases due to the rechanneling of the non-accepted particles and at larger distances, as the beam becomes more divergent because of the multiple scattering and thus dechanneling rate exceeds the rechanneling one, it decreases. In the region $z\sim{L_{\rm d}}$ and above this fraction can be be approximately written in terms of the error function $f_{\rm ch}(z)={\rm erf}\left(A/\sqrt{z}\right)$ where $A$ is the fitting parameter [35].

Figure 1 shows the dependencies $f_{\rm ch0}(z)$ (solid lines with symbols) and $f_{\rm ch}(z)$ (solid lines without symbols) obtained from the simulated trajectories of electrons (upper row) and positrons (lower row) in linear diamond, silicon and germanium crystals. A striking difference in the behaviour of the two fractions as functions of the penetration distance $z$ is mostly pronounced for electrons (note different scale in the vertical axes in the graphs for electrons and positrons). Away from the entrance point the fraction $f_{\rm ch0}(z)$ follows approximately the exponential decay law, see red dashed lines without symbols. The values of the dechanneling lengths ${L_{\rm d}}$ used to construct the fits are indicated in the caption. Red dashed lines with squares show the fits for $f_{\rm ch}(z)$ with the fitting parameters $A$ as indicated in the caption. It is seen that this fraction decreases much slower than $f_{\rm ch0}(z)$, which does not account for the rechanneling. At large distances $f_{\rm ch}(z)\propto z^{-1/2}$. The crystal thickness $L=200$ $\mu$m used in the simulations is order of magnitude smaller than the positron dechanneling lengths ${L_{\rm d}}$ estimated from the dependencies $f_{\rm ch0}(z)$. As a result, both fractions do not differ much even in the case of a higher-$Z$ germanium crystal. On this scale of $z$ the fraction $f_{\rm ch}(z)$ does not exhibit its asymptotic behaviour, therefore, no fitting curves are presented.

Spectral distribution (per particle) of the radiant energy $E$ emitted within the cone $\theta\leq\theta_{0}\ll 1$ along the incident beam is evaluated numerically by means of the formula

Here, ${\omega}$ is the radiation frequency, ${\Omega}$ is the solid angle corresponding to the emission angles $\theta$ and $\phi$. The sum is carried out over all simulated trajectories of the total number $N$ (in the current simulations $N\approx 4\times 10^{3}$). For each trajectory the spectral-angular distribution ${\rm d}^{3}E_{n}/{\rm d}{\omega}\,{\rm d}{\Omega}$ of the radiation is computed by numerical integration as described in detail in [43].

To compute the spectral distribution of radiation emitted the quasi-classical formalism [49] is used. It accounts for the quantum recoil that becomes important when the photon energies are no longer negligible compared to the energies of projectiles, $\hbar{\omega}\lesssim{\varepsilon}$. In Refs. [34, 35], it was demonstrated that photons of such energies are emitted by multi-GeV electrons and positrons passing through SASP periodically bent crystals.

The energy $\hbar{\omega}_{1}$ of the first harmonic of radiation due to the SASP modulations of the particle’s trajectory depends rather weakly on the bending amplitude [34] and can be estimated from the following relation [49]:

where ${\omega}_{1}^{\prime}\approx 2\pi\gamma^{2}c/{\lambda_{\rm u}}$ is the peak position of the radiation if the quantum recoil is neglected. Table 2 presents the values of $\hbar{\omega}_{1}$ and $\hbar{\omega}_{1}^{\prime}$ calculated for ${\varepsilon}=10$ GeV projectiles.

For the sake of reference let us estimate the photon energies $\hbar{\omega}_{\rm ch}$ that correspond to the peaks of the channeling radiation in linear crystals. This can be done applying the harmonic approximation for the interplanar potential (this model is adequate for positively charged projectiles):

where $U_{0}$ and $d$ stand for the interplanar potential well depth and the interplanar spacing. For the (110) planes in diamond, silicon and germanium crystals these parameters are as follows²²2The indicated values of $U_{0}$ correspond to the Moliére continuous potential calculated at the crystal temperature $T=300$ K, Ref. [35]: $d=1.26,1.92,2.00$ Å, $U_{0}=22,23,39$ eV. Using these values in Eq. (2) one finds that for all aforementioned crystals the peaks of ChR emitted by a 10 GeV positron is located at $\approx 100$ MeV i.e. order of magnitude lower than the photon energies due to the SASP bending.

To establish the ranges of bending amplitudes and periods that provide the largest values of the radiated energy, we have opted to study first radiation for comparatively thin, $L=12$ $\mu$m, crystals. Figures 2-4 present the spectral distribution of radiation emitted within the cone $\theta_{0}=1/\gamma\approx 50$ $\mu$rad by electrons (left column) and positrons (right column) incident along the (110) crystallographic planes in diamond, silicon and germanium crystals. In each figure three rows of graphs refer to different values of the bending period ${\lambda_{\rm u}}$ as indicated. In each graph the curves refer to the amplitudes $a$ as specified in the common legend located in the middle row graphs.

The spectra formed in the linear crystals, $a=0$, (black solid-line curves) are dominated by the peaks of ChR. The peak intensity greatly exceeds the intensity of the incoherent bremsstrahlung radiation (its values are indicated in the caption) emitted by the projectiles in the corresponding amorphous media. In the case of positron channeling, the peaks exhibit some internal structure due to the emission in the first and the second harmonics of ChR. This feature is more pronounced for the germanium target where the two maxima are clearly seen whereas in the spectra for diamond and silicon the second harmonics reveals itself as a step-like structure on the right shoulder. The positions of the first-harmonic peaks correlate with the estimates presented above, see Eq. (3). For the electrons, the ChR peaks are broadened and less intensive as a result of strong anharmonicity of the channeling oscillations.

The radiation spectra produced in the SASP bent crystals ($a>0$) display two novel features. These are (i) lowering and, in the electron case, red-shifting of the ChR peaks [33], and (ii) emergence of additional peaks that are due to the short-period modulations of the projectile trajectories [32, 34].

Qualitative explanation of the changes in the ChR peaks is as follows.

In a linear crystal, due to thermal vibrations, the nuclei of atoms of a crystal plane are distributed within a layer of the width to $2u_{\rm T}$, where $u_{\rm T}$ is the rms amplitude of the vibrations. Using the Thomas-Fermi model one estimates the atomic radius as $a_{\rm TF}=0.8853Z^{-1/3}a_{0}$ where $Z$ is the nucleus charge and $a_{0}$ is the Bohr radius. Hence, in the linear crystal the density of atomic nuclei and electrons is concentrated primarily within the layer $\Delta_{0}\approx 2\left(u_{\rm T}^{2}+a_{\rm TF}^{2}\right)^{1/2}$. In this region a channeling particle experiences frequent collisions with the atoms, which result in the dechanneling event. Hence, stable channeling motion, which leads to the intensive ChR, occurs if the particle’s trajectory (or most of it) is outside this regions. Positrons channel between two adjacent plains, therefore, this condition sets an upper bound on the amplitude of channeling oscillations, $a_{\rm ch}\lesssim(d-\Delta_{0})/2$. In the case of electron channeling, which occurs in the vicinity of a plane, the amplitude’s value acquires the lower bound equal to $\Delta_{0}/2$. When the plane is subjected to the SASP bending with amplitude $a$, the layer of the increased density becomes wider, $\Delta_{a}\approx\left(4a^{2}+\Delta_{0}^{2}\right)^{1/2}>\Delta_{0}$. As a result, the number of channeling particles decreases and so does the intensity of ChR for the projectiles of both types. However, the mechanisms of the additional modification of the ChR peak are different for positively and for negatively charged projectiles. The SASP bending lowers the upper bound of $a_{\rm ch}$ for positrons making it equal to $(d-\Delta_{a})/2$ but for electrons the effect is the opposite: more favourable are channeling trajectories with the amplitudes larger than in the linear channel, $a_{\rm ch}\gtrsim\Delta_{a}/2$. Due to nearly harmonic character of the positron channeling oscillations the decrease in $a_{\rm ch}$ leads to lowering the intensity of the ChR peak but does not affect its position. Strong anharmonicity of the electron channeling oscillations makes their frequency ${\Omega}_{\rm ch}$ dependent on the amplitude. In particular, ${\Omega}_{\rm ch}$ is a monotonously decreasing function of $a_{\rm ch}$ for the (110) planar channels in diamond, silicon and germanium crystals [28]. Taking into account that the photon energy is proportional to $2\gamma^{2}{\Omega}_{\rm ch}$ one concludes that the increase in $a$ leads to stronger suppression of the ChR in the region of higher photon energies. As a result, the ChR peaks become less intensive and red-shifted.

The second feature in the emission spectra presented in Fig. 2-4 is the peaks due to the SASP periodic bending. These peaks are located in the domain of higher photon energies and their positions agree well with the estimated values of $\hbar{\omega}_{1}$ indicated in Table 2. As it is seen from the figures, for both types projectiles and for all three crystals considered the peak intensity at a fixed value of the amplitude $a$ is inversely proportional to the bending period ${\lambda_{\rm u}}$. For a given ${\lambda_{\rm u}}$ the peak intensity increases from zero at $a=0$ up to the maximum value at some amplitude $a_{0}$ and then decreases with further increase in $a$ [33]. For the case studies considered here the $a_{0}$ values were found to be equal to 0.3 Å for diamond and silicon, and to 0.5 Å for germanium.

In what follows we focus on the characteristics of radiation (spectral distribution, brilliance, power and number of photons) emitted in linear crystals and in the periodically bent crystals with amplitude $a=0.3$ Å and period ${\lambda_{\rm u}}=600$ nm.

Figures 5-7 compare the spectral distributions of radiation formed in linear (upper rows) and periodically bent (lower rows) oriented diamond, silicon and germanium crystals of different thickness $L$ ranging from 6 $\mu$m up to 96 $\mu$m. In each figure the left and right columns refer to the electron and positron channeling, respectively. All spectra correspond to the emission cone $\theta_{0}=50\ \mbox{$\mu$rad}\approx 1/\gamma$ along the incident beam.

The following features of the spectra are to be noted.

In the linear crystals the (main) peak of ChR emitted by positrons is higher and narrower that the corresponding peak for electrons in a crystal of the same type and thickness. In the shorter crystals both of these features are due to different character (harmonic or anharmonic) of the channeling oscillations experienced by positively and negatively charged projectiles. As the crystal thickness increases, the intensity of ChR by electrons is additionally suppressed due to much higher dechanneling rate. The particles that leave the channeling mode of motion produce much less intensive radiation as compared to those that continue the channeling motion. The rechanneling process brings part of the particles back to the channeling mode mode, nevertheless, the fraction of channeling electrons decreases with the penetration distance much faster than that of the positrons, see Fig. 1. This effect is more pronounced for the high-$Z$ crystal, germanium. As a result, for all crystals the emission spectra of positrons scale linearly with thickness in the whole interval of the $L$ values. For electrons, this is true only in the case of the low-$Z$ target, diamond. For silicon and, especially, germanium, the linear dependence is seen over shorter intervals of $L$.

Another feature seen in the positron spectra in both the linear and periodically bent crystals is the emergence of the peaks corresponding to higher harmonics of ChR. The peaks associated with the second harmonic are well pronounced in the spectra starting with $L=24$ $\mu$m, in some cases (linear diamond, linear and periodically bent germanium) the emission in the third harmonic is also seen. To provide an explanation one recalls that channeling oscillations of positrons are nearly harmonic. This results in the emission of the undulator-type radiation. The width $\Delta{\omega}_{n}$ of the peak of the $n$the harmonic of the undulator radiation as well as its natural emission cone $\Delta\theta_{n}$ decrease with the number $N_{\rm u}$ of undulator periods: $\Delta{\omega}_{n}\propto N_{\rm u}^{-1}$ and $\Delta\theta_{n}\propto N_{\rm u}^{-1/2}$ (see, e.g., [50]). As a result, the larger $N_{\rm u}$ is the more separated the harmonics become, especially if the radiation is collected within a (comparatively) narrow cone along the undulator axis. Applying these arguments to the positron spectra one concludes that as $L$ increases so does the number of periods of channeling oscillations. Hence, the emission in higher harmonics becomes more pronounced.³³3The smaller the emission cone $\theta_{0}$ is the larger number of the higher harmonic peaks are resolved in the ChR spectrum, see Section B.

The spectra presented in the lower graphs in Figs. 5-7 demonstrate the notable increase in the intensity in the photon energy range 0.8…1.4 GeV due to the SASP periodic bending. The position and the height of the SASP peaks are virtually not sensitive to the projectile’s charge but strongly depend on the crystal type and thickness.

To further characterize the emission produced in linear and periodically bent crystals we consider the largest value of the target thickness, $L=96$ $\mu$m.

One of the quantities used to compare light sources of different nature is called brilliance, $B_{{\omega}}$. It is defined in terms of the number of photons $\Delta N_{{\omega}}=\left({\rm d}E(\theta\leq\theta_{0})/{\rm d}{\omega}\right)(\Delta{\omega}/{\omega})$ emitted by the particles of the beam within the interval ${\omega}\pm\Delta{\omega}/2$ in the solid angle $\Delta{\Omega}\approx 2\pi\theta_{0}^{2}/2$ (the emission cone is assumed to be small, $\theta_{0}\ll 1$) per unit time interval, unit source area, unit solid angle and per a bandwidth (BW) $\Delta{\omega}/{\omega}$ [51]. To calculate brilliance of a photon source of a finite size it is necessary to know the beam current $I$, beam sizes $\sigma_{x,y}$ and angular divergence $\phi_{x,y}$ in the transverse directions as well as the divergence angle $\phi$ and the ’size’ $\sigma$ of the photon beam.

Explicit expression for brilliance written in terms of the spectral distribution reads

Here ${\cal E}_{x,y}$ stand for the total emittance of the photon source in the transverse $x,y$ directions

where $\phi=\sqrt{\Delta{\Omega}/2\pi}=\theta_{0}/\sqrt{2}$ and $\sigma=\lambda/4\pi\phi$ is the ’apparent’ source size taken in the diffraction limit [52] ($\lambda$ is the radiation wavelength).

Commonly, brilliance is measured in $\left[\hbox{photons/s/mrad}^{2}\hbox{/mm}^{2}/0.1\,\%\,\hbox{BW}\right]$. To achieve this in Eq. (4) the current is measured in amperes, $\sigma,\,\sigma_{x,y}$ in millimeters, and $\phi,\,\sigma_{\phi_{x,y}}$ in milliradians.

If one uses the peak value of the current, $I_{\rm max}$, then the corresponding quantity is called peak brilliance, $B_{\rm peak}$.

Figures 8-10 show the peak brilliance of radiation emitted by the electron (left columns) and positron (right columns) beams channeling along the (110) planar direction in $L=96$ $\mu$m thick linear (upper rows) and SASP bent (lower rows) diamond, silicon and germanium crystals. In each graph the curves correspond to different emission cones $\theta_{0}$ that measured in units of the natural cone $1/\gamma\approx 50\ \mbox{$\mu$rad}$ as indicated in the common legend in each figure. The dependencies $B_{\rm peak}({\omega})$ presented have been obtained from Eq. (4) with the total emittances ${\cal E}_{x,y}$ (5) calculated using the values of $\sigma_{x,y}$ and $\sigma_{\phi_{x,y}}$ indicated in Table 1. The values $I_{\rm peak}=64$ and 5.8 kA were used for the electron and positron beams, therefore, the dependencies presented in the figures refer to the brilliance of radiation emitted by a single bunch of the beams.

The spectral distributions that have been used to calculate the brilliance are shown in Figs. 15-17 in Section B. It is seen that in contrast to ${\rm d}E(\theta\leq\theta_{0})/{\rm d}(\hbar{\omega})$, which is an increasing function of the emission cone for all values of ${\omega}$, the peak brilliance can exhibit a non-monotonous behaviour due to the presence of the terms proportional to $\theta_{0}$ in the total emittance. The case studies presented by the figures demonstrate that by means of the FACET-II electron beam it is possible to achieve peak value $B_{\rm peak}\approx 10^{24}$ photons/s/mrad²mm²/0.1 % BW for the channeling radiation in the linear diamond crystal in the photons energy range $\hbar{\omega}\approx 100-300$ MeV and the value $B_{\rm peak}\approx 3\times 10^{23}$ photons/s/mrad²mm²/0.1 % BW in the SASP bent diamond crystal for much higher photon energies, $\hbar{\omega}\approx 1-1.3$ GeV. The peak brilliance achievable by means of the positron beam are noticeably smaller. This is despite the fact that the values of ${\rm d}E(\theta\leq\theta_{0})/{\rm d}(\hbar{\omega})$ for positrons are either higher (in the vicinity of the channeling peak) or approximately equal to (for the peak due to the SASP bending) those for electrons, see Figs. 15-17. This is a direct consequence of the difference in the parameters of the two beams, namely, the factor $I_{\rm peak}/{\cal E}_{x}{\cal E}_{y}$, which enters the right-hand side of (4), is two orders of magnitude larger in the case of the electron beam.

Figure 11 compares the peak brilliance of radiation emitted by the FACET-II electron beam incident along the (110) directions at linear (left graph) and periodically bent (right graph) diamond, silicon and germanium crystals. The highest values of $B_{\rm peak}$ are achievable for the diamond target. In the linear crystal, the values of $B_{\rm peak}$ achievable in the photon energy range 100-400 MeV are compared to those predicted in Ref. [12] for a high-brilliant light source based on a two-stage laser-wakefield accelerator driven by a single multi-petawatt laser pulse. However, in the photon energy range about 1 GeV the SASE CLS peak brilliance is orders of magnitude higher than the quoted valus (see Figure 2 in the cited paper).

To characterize the radiation emitted by a beam of particles it is instructive to calculate, in addition to spectral distribution and brilliance, two other commonly used characteristics - the number of photons per unit time ${\cal N}_{{\omega}}$ and the power $P_{{\omega}}$ of radiation emitted within the cone $\theta_{0}$ and the bandwidth $\Delta{\omega}/{\omega}$. Both of these quantities are expressed in terms of spectral distribution ${\rm d}E(\theta\leq\theta_{0})/{\rm d}(\hbar{\omega})$ and beam current $I$. The corresponding relations are as follows:

Using the peak current $I_{\rm peak}$ on the right-hand side of (6) one determines the instantaneous value of ${\cal N}_{{\omega}}$. Calculated for the FACET-II electron beam this quantity as a function of the photon energy is shown in Fig. 12 two panels of which refer to the linear (left) and periodically bent (right) (110) channels in $L=96$ $\mu$m thick diamond, silicon and germanium crystals. The curves presented correspond to the bandwidth $\Delta{\omega}/{\omega}=0.01$ and to the natural emission cone $1/\gamma$.

It is instructive to compare the values ${\cal N}_{{\omega}}$ obtained here with the number of photons per second predicted in the Gamma Factory (GF) proposal for CERN [5, 6], where a light source based on the resonant absorption of laser photons by ultra-relativistic ions has been elaborated. Within the GF proposal it is expected to achieve the average number of photons $\langle{\cal N}_{{\omega}}\rangle$ on the level of $10^{17}$ photons/s in the photon energy range 1-400 MeV. This value refers to the average beam current. To estimate the corresponding instantaneous value one multiplies $\langle{\cal N}_{{\omega}}\rangle$ by a factor $100/0.64\approx 150$, which is the ratio of the bunch spacing ($\approx 100$ ns) to the bunch length ($\approx 0.64$ ns) [1, 2]. The dashed line drawn in both graphs in Fig. 12 indicates the estimated instantaneous value of $1.5\times 10^{19}$ photons/s in the photon energy range specified above. To be noted is that the value of the FACET-II peak current correspond to a much shorter bunch (several picoseconds [4].

The peak power of radiation from the electron beam is presented in Fig. 13. It is seen that by means of the SASP bending it is possible to achieve the same values of the power as in the linear crystal but for the photons of much higher energy.