Action of a relativistically intense subpicosecond laser pulse on the near critical density (NCD) plasma can give rise to the formation of ion channel inside the plasma and effective acceleration of background electrons. Thus, one can produce high current (electron charges from tens nCl to several mkCl), high energy (from several MeV to several hundreds of MeV) electron bunches, demanded in different practical applications. Synchrotron (betatron) radiation of these electrons can serve as an important tool both for practical applications and also for diagnostic of the process in laser plasma, which is important for better understanding of these processes and for optimization of experimental conditions. For the last goals, an approximate model is proposed for calculating the spatial and energy characteristics of a bunch of DLA (direct laser accelerated) electrons in the ion channel formed in the NCD plasma and the characteristics describing the spectrum of their synchrotron radiation. For the considered example of a powerful laser pulse action on NCD plasma, the predictions of the proposed model are in good agreement with the results of particles in cell simulations and with the experimental measurements of the synchrotron radiation specter. It is shown that with the assumption of the rotation of the initial plane of motion of DLA electrons, the experimental data on the measurements of synchrotron radiation specters can be explained on the basis of the concept of betatron radiation of electrons accelerated in NCD plasma by DLA mechanism.
I. INTRODUCTION
It has been shown in recent works that electrons accelerated in near critical density (NCD) plasma by intense laser fields can serve as an effective mean for the production of highenergy photon and particle fluxes.^{1–9} Electron bunches with simultaneously relatively high energies $Wb\u223c102$ MeV and high currents (charges $Qb\u223c102\u2009nCl\u2009\xf71\u2009mkCl$), required for the effective generation of xray and gamma radiation, can be generated by the action of powerful laser pulses on NCD plasma.^{1–5,10,11}
Numerical particles in cell (PIC) simulations are widely used for calculations of spatial and energy characteristics of laseraccelerated electrons^{10–14} as well as synchrotron radiation specters^{8,15} of those electrons. However, analytic and semianalytic estimates of the spatial and energy characteristics of accelerated electrons and their radiation spectrum and dependencies of those values on the main parameters of electron bunches and the medium where they are accelerated are necessary for better understanding of the physical processes involved and prediction of the results of future experiments.^{9,15,16} Theoretical analyses are necessary for better understanding and separating of different mechanisms of electron acceleration and their radiation. Particularly, besides DLA (direct laser accelerated), the mechanism of stochastic acceleration (stochastic heating) can play a role in high power laser pulse interactions with plasmas. Stochastic heating can be realized in the case of laser pulse action on the plasma with moderate density (of a few percent of the critical density) when the amplitude of the Ramanbackscattered wave is high enough to serve as the second counterpropagating pulse to trigger the electron stochastic motion; see Ref. 17. Another mechanism can be realized in the case of powerful (Pw) laser pulse action on longscale length preplasma or homogeneous NCD plasma in the regime leading to the formation of multiple filaments. In this case, electrons can jump across the filaments during the acceleration and their motion becomes stochastic.^{18}
The model proposed in the present paper permits one to estimate the contribution of DLA process into the energy spectrum of accelerated electrons and into the spectrum of their radiation. Particularly, it is shown that the results of the recent experiment^{19} on the registration of synchrotron radiation spectrum can be explained on the base of assumption of betatron radiation of DLA accelerated electrons as the main contribution to the radiation spectrum.
Preliminary analytical estimates can also be used as a starting point for the choice of parameters of more precise PIC simulations of electron acceleration and simulations of their specters of synchrotron radiation, with the aim to optimize the obtained parameters of electron bunches and radiation specters. Using the proposed model, one can estimate beforehand the parameters of plasma and laser pulses necessary for obtaining electron bunches and flashes of synchrotron radiation with required characteristics. Further, those parameters can be refined in PIC simulations.
Below it is shown that basic parameters, which determine the horizontal position of the maximum and the slope of the spectral curve (which represents the dependence of the number of quanta $N0.1%BW$ emitted in thin energy interval $10\u22123E$ on the energy E of quanta), are:

the characteristic energy of electrons in accelerated bunch or the temperature of hot electrons T_{h},

characteristic amplitudes of their betatron oscillations $r\beta $,

characteristic value of background electron concentration n_{0} and the fraction of electrons remaining in the created ion channel f, which together with T_{h} determine the characteristic value of the frequency $\omega \beta $ of electron betatron oscillations.
The absolute value of the maximum of spectral curve is determined by the total number of accelerated electrons which, in terms, is determined by the characteristic radius of the laser pulse waist r_{L}, the effective time of acceleration $teff$, the fraction of electrons remaining in the created ion channel f, and the fraction α_{h} of those electrons, which are effectively accelerated.
In Ref. 11, the scaling for the dependence of the temperature T_{h} of DLA electrons on laser pulse intensity was proposed. In Ref. 20, using the theory developed in Ref. 21, it was shown that the similar scaling holds for electron maximum energy. Also in Ref. 20, the scaling of the dependencies of the temperature and the maximum energy of DLA electrons on the laser pulse duration was proposed on the base of the results of PIC simulations. In the present paper, using the same theoretical approach, we refine the expression for the frequency of betatron oscillations, taking into account the abovementioned coefficients f and α_{h}. In addition to that, we estimate the characteristic value of the amplitude $r\beta $ of betatron oscillations. We show that $r\beta $ is determined by the characteristic value of the radial position of an electron $r\sigma $ at the moment when it starts to accelerate.
In the case $f\u226a1$, the shape of the spectral curve is determined by T_{h}, $r\sigma $, and n_{0}. This makes it possible to estimate $r\sigma $ from betatron radiation spectrum data, assuming that the characteristic temperature T_{h} of DLA electrons is derived independently of the abovementioned scaling and that n_{0} can also be estimated independently. Note that the radiation of electrons undergoing betatron oscillations can also be used for the diagnostic of the emittance of accelerated electron bunches.^{22,23} In this paper, we show how the synchrotron radiation spectrum can be used to withdraw information about $r\sigma $ and $r\beta $ and about the mechanism of acceleration itself.
In this paper, in Sec. II, we present the general equations describing the betatron radiation of electrons in ion channel and expressions for the estimation of the main parameters of radiation spectrum.
II. BASIC EQUATIONS FOR SYNCHROTRON RADIATION SPECTRUM AND ESTIMATES OF ITS MAIN PARAMETERS
It should be noted that the expression (2) implies an incoherent summation of the contributions of different parts of the trajectory of an individual electron and different electrons to the resulting spectrum. The respective conditions are discussed in Appendix A.
Results on PIC simulations^{8,19} of the trajectories of electrons selfinjected and accelerated in laser plasma at the laser and plasma parameters close to considered here^{6} show that up to the moment of exit from the ion channel, the characteristic oscillation amplitude of most accelerated electrons is not higher than several μm, and the characteristic path length of accelerated electrons before leaving the acceleration zone is about $Lch\u2243LL=c\tau L$ for optimal parameters of plasma fitted to laser parameters; see Appendix B and expression (B23); τ_{L} is the FWHM laser pulse duration. Below, we assume that the parameter $\alpha eff=teff/\tau L\u2243Lch/LL\u22431$.
A. Radiation in perpendicular plane
B. Radiation in plane of motion
C. Critical frequency and the number of emitted quanta in 0.1% of bandwidth
D. Total emitted energy and number of quanta
III. COMPARISON OF ESTIMATES, EXPERIMENTAL RESULTS, AND PIC SIMULATIONS
A. Measurements and calculations of the number of quanta per energy interval and per solid angle
Figure 1 shows the dependence of number of quanta $N\gamma =S\gamma /E$ per unit energy interval and unit solid angle, as a function of the quantum energy $E=\u210f\omega $, obtained in the experiment and in PIC simulations, described in Refs. 19 and 29, as well as calculated by the model (18) and (13) or (14), for angle θ = 0. In the experiment, laser and plasma parameters were similar to those described earlier in Ref. 6. Here, the intense relativistic laser pulse with wavelength $\lambda 0=1\mu $ m, duration $\tau L=700$ fs, and peak intensity $I0=2.5\u20091019$ W/cm^{2} and FWHM spot diameters $Dx=15\mu $ m and $Dy=11\mu $ m, acts on NCD plasma with thickness $l=325\mu $ m, created from trinitrocellulose ( $C12H16O8$) target. The density profile of the plasma had smooth forward (faced to the laser pulse) front and a plateau with electron density $nmx/nc=0.65$.
To create such a plasma, a foam target irradiated by an ns pulse preceding the relativistic main pulse was used. The intensity of this ns laser pulse was about $5\u20091013$ W/cm^{2} to initiate a supersonic ionization wave propagating at about $2\u2009107$ cm/s in the target. The main pulse creates an ion channel in the NCD plasma and accelerates electrons trapped from the plasma in the quasistatic electric and magnetic fields inside this ion channel by the DLA acceleration mechanism.^{4–6}
The number of quanta $N\gamma $ is determined by the following main parameters:

the total number N_{h} of superponderomotive (hot) electrons, estimated by Eq. (7); $N\gamma \u223cNh$;

the frequency $\omega \beta $ of betatron oscillations of electrons, determined by their background concentration n_{0} and by their fraction f in ion channel through Eq. (1);

the amplitude of betatron oscillations $r\beta $;

gamma factor of electrons γ, where the characteristic value $\gamma h=Th/(mc2)$ is determined by the temperature of superponderomotive electrons.
All last three parameters determine the main parameters of the radiation spectrum: $a\beta $ and $\omega eff$ or ω_{c}; see Eqs. (5) and (23). From Eqs. (18) and (23), it is clear that the curves $S\gamma (\omega )$ and $N\gamma (\omega )$ for $\omega >\omega c$ are sharply decreasing functions of quantum energy $E=\u210f\omega $ and that the rate of decrease in those curves with E or the slope of those curves is higher for:

lower values of background electron concentration n_{0} and higher values of the fraction f of electrons remaining in the ion channel [which determines the factor κ_{h}, see Eq. (1)];

lower temperatures T_{h} of hot electrons and lower rates of increase of T_{h} with time t, i.e., lower rates of acceleration, which determines lower effective temperatures $Th,ef=\kappa tTh,mx,\u2009\kappa t<1$; see below;

lower amplitudes of electron betatron oscillations $r\beta $ determined by the characteristic value $r\sigma $; see expressions (6) and (B18).
It is also clear that averaging on amplitudes $r\beta $ of betatron oscillations as well as on energies W of electrons in expression (13) decrease the slope of the curves $S\gamma (\omega )$ and $N\gamma (\omega )$.
The above peculiarities are clearly illustrated in Fig. 1 by calculations of $N\gamma (E)$ for different background electron concentrations in the ion channel $n0/nc$, different temperatures of hot electrons T_{h,} and different betatron oscillations radius, determined by the parameter $r\sigma $ in the distribution (6) and by the formula (B18). The parameters $\alpha eff=1$, f = 0.3, $\alpha h=0.25,\u2009\alpha rad=0.6$ were used in calculations.
To approximately take into account the smooth variation of the background electron concentration n_{0} in the experiment^{19} from 0 till $nmx=0.65nc$, we calculate $N\gamma $ for $n0=0.32nc\u2248nmx/2$ and for $n0=0.16nc\u2248nmx/4$. The electron temperature dynamic was calculated by Eq. (10) with $Th,mx$ given by Eq. (B17) and the coefficient of selffocusing $Ksf=3$. This value of $Ksf$ was chosen taking into account the results of relevant PIC simulations.^{20} In the case of calculations using Eq. (14) expression (15) for $Th*$ was used. In accordance with Eq. (B17), we get $Th,mx\u224813.4$ MeV for $n0=0.32nc$ and $Th,mx\u224819$ MeV for $n0=0.16nc$. $N\gamma \u223cn0$ in accordance with Eqs. (7) and (13), but due to higher $Th,mx$, the values of $N\gamma $ for $n0=0.16nc$ are not significantly lower than for $n0=0.32nc$.
The obtained value of $Th,mx$ is close to the value of $Th\u224816$ MeV registered in the experiment.^{19} The initial laser spot waist was chosen as $r0=6.5\mu $ m, in accordance with expression (8) [the value of r_{0} enters expression (7) for N_{h}]. The values of other parameters were: $f=\alpha h=0.3,\u2009teff=\tau L=210\tau 0$; other parameters are shown in the legend of the figure.
The results obtained by expression (13) with the dynamically changed temperature T_{h} (10) are practically the same as the results calculated by the simplified equation (14) with fixed $Th*$ given by Eq. (15); see the example represented by an orange solid curve, calculated by Eq. (13), and orange rectangular markers, calculated by Eq. (14).
In accordance with the peculiarities stated above, the lower values of T_{h}, $r\sigma $, and $n0/nc$ increase the slope of the curve $N\gamma (E)$, as well as the exclusion of the averaging over electron energies (dotted gray curve calculated for the fixed energy of electrons $W=Th,mx$) or exclusion of the averaging over both electron energies and amplitudes of betatron oscillations (short dashdotted gray curve in Fig. 1 calculated for fixed $W=Th,mx$ and $rin=r\sigma $). It is also clear that the lowerenergy photons are generated by lowerenergy electrons (with $W\u2264Th,mx$), while highenergy photons are generated by highenergy electrons (with $W>Th,mx$), compare the short dashdotted gray curve calculated with $W=Th,mx$ and the marked gray curve calculated with $W=1.4Th,mx$. The averaging on amplitudes of betatron oscillations of electrons is much less essential than the averaging on electron energies, compare the long dashdotted gray curve calculated for fixed $rin=r\sigma $, but with averaging on electron energies, with other gray curves calculated for fixed electron energies W.
The results of calculations are in good agreement with the experimental results and are in reasonable agreement with the results of PIC simulations. The slope of the PIC simulation curve $N\gamma (E)$ is somewhat higher than one in calculations by our model, at least for $r\sigma >1\mu $ m. This can be connected with the approximation of the constant electron density and respective approximation (7) of the linear increase in time of the number of hot electrons. In addition to that, the model (7) assumes conservation of the number of captured and accelerated electrons. In fact, some electrons continue to accelerate even at times $t\u2273teff$, while others leave the acceleration zone before and after $t=teff$; see respective Fig. 4 in Ref. 8 and Fig. 7 in Ref. 19. This also leads to a slowly increasing value of $N0.1%BW$ for $t>teff$, as shown in PIC simulations in Refs. 8 and 19: according to these simulations, $N0.1%BW$ is a slowly increasing function of t for $t>teff$ with saturation reached at approximately $t=600\tau 0$.
Figure 2 shows the same experimental data, as for Fig. 1, but for detectors placed at $\theta =\xb110\xb0$ relatively to the direction of laser propagation and in the plane perpendicular to the laser polarization plane, i.e., at $\phi =\pi /2$; see Ref. 19 for details. Calculations by expressions (14) and (18) for $\phi =\pi /2$ and $\theta =0,10\xb0$ are shown by lines; see the legend.
From Fig. 2 and its comparison with Fig. 1, it is clear that registered in the experiment decrease in radiation due to nonzero angle $\theta =10\xb0$ is only about a factor of 0.25. Such large values of $N\gamma $ for $\theta =10\xb0$ cannot be explained by the radiation of electrons moving in the plane of laser pulse polarization. For those electrons, the directions to the detectors under the condition of the experiment^{19} are characterized by $\phi =\pi /2$. In this case, expressions (14) and (18) give values of $N\gamma $ similar to the experimental ones only for $\theta \u22721\xb0$; see the dashed and dashdotted curves at Fig. 2. Note that from Eq. (20), one have a similar critical angle $\theta \u22a5\u2243mc2/Th,mx\u22482\xb0$ for $Th,mx=13.4$ MeV calculated above. On the other hand, critical angle (22) for radiation in the polarization plane is, taking into account (1) and (5), $\theta \u22432k\beta r\beta \u2243kpr\sigma 2\kappa hmc2/Th,mx\u224850\xb0$ for chosen f = 0.2, $r\sigma =1\mu $ m, $Th,mx=13.4$ MeV, $n0/nc=0.32$, i.e., for radiation of electrons in the plane of their motion, the angle $\theta =10\xb0\u226a\theta $ is practically the same as $\theta =0\xb0$.
Taking into account the above information, one can suppose that $\u224325%$ of electrons, which originally move in the plane of laser polarization, change their plane of motion to that close to the perpendicular plane. The rotation of the plane of motion can be connected, e.g., with the action of stochastic electromagnetic fields in the ion channel.
Note also that in the experiment,^{19} the charge of accelerated electrons with energies higher than 10 MeV, $QW>10$ was measured. The values $QW>10=30\u201348\u2009nC$ were obtained. In comparison, expression (9) gives $Qh,mx\u224340\u2009nC$ and $QW>10=exp(\u221210\u2009MeV/Th,mx)\u224319\u2009nC$ for parameters $n0/nc=0.32$ and $r0=6.5\mu $ m relevant to this experiment (with the same $f=0.3,\alpha h=0.25$ as used for calculations depicted at Figs. 1 and 2) and for $Th,mx=13.4\u2009MeV$ calculated above for these parameters.
B. Calculation of the number of emitted quanta $N0.1%BW$
The number of emitted quanta $N\u03030.1%BW$ and $N0.1%BW$ calculated by formulas (27) and (28), respectively, with $S\gamma $ calculated by Eq. (14), is shown in Fig. 3 by lines. Results of PIC simulations of the work^{8} are shown by dotted markers. Formulation of the task and all parameters are the same as for the case of calculations discussed in Sec. III A and depicted in Figs. 1 and 2, with the exception that in Ref. 8 and hence in calculations by formulas (27) and (28), the background plasma, unlike the above considered case of the experiment,^{19} was assumed to have a uniform density with $n0/nc=0.65$. The respective $Th,mx$ calculated by Eq. (B17) was 9.4 MeV. Similarly as in the case considered in Sec. III A, while calculating $S\gamma $ by the formula (14), we take $Th=\kappa tTh,mx$ with the same $\kappa t=0.81$.
The results of calculations of $N0.1%BW$ by the model (14), (18), and (27) for betatron radiation in the forward direction are in reasonable agreement with the results obtained from the PIC simulations^{8} for $r\sigma =1\mu $ m. According to the peculiarities discussed in Sec. III A, smaller values of $r\sigma $ ensure larger slopes of the curve $N0.1%BW(E)$. Good agreement with PIC simulations is obtained with $r\sigma =0.5\mu $ m. This value does not contradict expression (B20), which gives $r\sigma \u22720.9\mu $ m for considered parameters.
The value $r\sigma $ determines not only the slope of the curve $N0.1%BW(E)$, but also the position of its maximum, closely related to the critical frequency of betatron radiation through expressions (26). Another parameter of the model, which determines the slope and position of the maximum of the curve $N0.1%BW(E)$, is the effective temperature $Th,mx$. However, this parameter can be determined independently, using expression (B17). The value $Th,mx=9.4$ MeV calculated for the case depicted at Fig. 3 is close to the value $Th,mx\u224810$ MeV obtained from the results of PIC simulations in Ref. 8.
An increase of 30% of the value of $Th,mx$ leads to a noticeable displacement of the maximum of the curve $N0.1%BW(E)$, compared to the dashdotted curve with one with crest markers in Fig. 3. This follows from the respective increase in the critical energy $\u210f\omega c$; see Eq. (23). An increase of $r\sigma $ produces similar effect, compared to the solid (violet) curve with other ones.
In addition to the effective temperature T_{h} of hot electrons and the effective radius of their initial transverse distribution $r\sigma $, which can be estimated as some fraction of the laser pulse waist radius, with account for the process of selffocusing (see the Appendix), other free parameters of the model are the fraction f of electrons remaining in the ion channel and the part α_{h} of those electrons that are effectively accelerated. The values f and α_{h} determine mainly the vertical position of the curve $N0.1%BW(E)$, though they (mainly f) influence also on the betatron frequency $\omega \beta $ (1) and hence on the slope of the curve, as stated in Sec. III A.
The model (14), (18), and (27) described above gives proper estimation of the critical frequency of betatron radiation as long as for $r\sigma \u22431\mu $ m, it gives the maximum of the curve $N0.1%BW(E)$ approximately at the same quant energy E, as PIC simulations, as shown in Fig. 3, where $\omega S\u0303max\u22481.7$ keV. From this value and expressions (26), one has $\u210f\omega c\u22484.3$ keV. On the other hand, one can get this value of ω_{c} from Eq. (23) with $r\beta =2\mu $ m, $\gamma =Th,mx/mc2$ [with $Th,mx\u22489.4$ MeV, in accordance with Eq. (B17)] or with $r\beta =1\mu $ m, $\gamma =1.4Th,mx/mc2$, and with same other parameters, as those used in the calculations depicted by curves in Fig. 3. Note that the gray marked curve calculated with fixed $rin=r\sigma $ and fixed $W=1.4Th,mx$ (i.e., without averaging on the distributions of electrons) reveals a much more sharp peak than curves calculated using averaging with distribution functions $fe(rin)$ and $fW(W)$.
The values of $E\u0303\gamma \Sigma ,\u2009E\gamma \Sigma ,\u2009N\u0303\gamma \Sigma 0.1\omega c,\u2009N\gamma \Sigma 0.1\omega c$, estimated by formulas (30)–(33), respectively, for parameters corresponding to Fig. 3, with $r\sigma =1\mu $ m, are $E\u0303\gamma \Sigma \u22482.9\u200910\u22125,\u2009E\gamma \Sigma \u22482.5\u200910\u22125,\u2009N\u0303\gamma \Sigma 0.1\omega c\u22484.5\u20091012,N\gamma \Sigma 0.1\omega c\u22482.5\u20091012$. The value $N\gamma \Sigma 0.1\omega c$ is close to the number of photons with energies E > 1 keV, $N\gamma \Sigma 1\u2009keV\u22437\u20091011$, obtained in Ref. 8 in PIC calculations.
C. Simplified expressions for estimations under experimental parameters
In addition to γ_{h,} another important parameter, which should be estimated, is $r\sigma $, which determines through Eqs. (6) and (B18) the characteristic amplitude of betatron oscillations $r\beta $. The latter determines (together with the values of f, $\alpha h,\u2009n0/nc$, and γ) the value of the critical frequency ω_{c} (23), which, in terms, determines the shapes of spectral curves in Eqs. (35)–(38). One can use Eq. (B20) to estimate $r\sigma $ in terms of laser and plasma parameters. Particularly, from expression (B20), one has $r\sigma \u22480.9\mu $ m under parameters of calculations corresponding to Fig. 3 and $r\sigma \u22481.2\mu $ m under parameters of Figs. 1 and 2. Both values do not contradict to results of calculations depicted in Figs. 1–3.
One should discuss free parameters of the presented model. The value $\alpha eff=1$ can be fixed in accordance with the above discussion. The product $f\alpha h$ of parameters f and α_{h} can be fixed using expression (9) (if one can estimate or measure the charge of accelerated electrons). This product is also connected with the parameter $\u03f0\u2323e$ or $\u03f0e$ (34). The last one is also connected with the parameter $\alpha rad$, which determines the effective temperature $Th*$ of radiating electrons, calculated by Eq. (15), and the number of emitted quanta, as long as $\u03f0e\u223c\alpha rad$.
Thus, one can have three independent parameters of the model: $\u03f0\u2323e,\alpha rad,f$ or $\u03f0\u2323e,\alpha rad,\alpha h$. From physical reasons, and taking into account the results of PIC calculations^{8} for powerful laser pulses, parameters f and $\alpha h$ are restricted by requirements $f\u226a1$ and $\alpha h<1$. Taking into account that the variation of f and $\alpha h$ within the requirement $f\alpha h=const$ has a small effect on the betatron radiation spectrum (see Fig. 4), one has actually only two essential parameters of the present model of the betatron spectrum: $\u03f0\u2323e$ and $\alpha rad$, where $\u03f0\u2323e$ is the main one; see the results of parameter variation in Fig. 4.
Another parameter $Ksf$ concerns the model of electron heating due to DLA; see Eq. (B17). The value $Ksf\u22433$ was chosen taking into account the results of PIC simulations for laser pulses with intensities $IL\u223c1019\u20131020$ W/cm^{2} and powers $P\u22720.1\u20131$ PW. The question about the dependence of $Ksf$ on laser and plasma parameters needs further studies.
IV. DISCUSSION AND CONCLUSIONS
The proposed model permits one to estimate the main characteristics of radiation spectra for electrons accelerating by the DLA mechanism in the ion channel produced by powerful laser pulse in the NCD plasma: the slope of the curve $N\gamma (E)$ (number of quanta per unit energy interval of quanta per solid angle), the slope of the curve $N\u03030.1%BW(E)$ (the number of quanta emitted in 0.1% of bandwidth), and the position of its maximum, the number of quanta with energies higher than $0.1\u210f\omega c$ emitted in the forward direction $N\u0303\gamma \Sigma 0.1\omega c$ and in all directions $N\gamma \Sigma 0.1\omega c$, the critical frequency of the betatron radiation ω_{c}.
The main parameters that determine the specter of betatron radiation of electrons accelerated in ion channels created in NCD plasma by powerful laser pulses are temperature T_{h} of hot (superponderomotive) electrons, amplitude $r\beta $ of their betatron oscillations determined by the characteristic initial distance $r\sigma $ of electrons from the ion channel axis, in accordance with Eqs. (B19) and (B20), background concentration n_{0} of electrons in plasma and the fraction f of electrons remaining in the ion channel, which determine the frequency of betatron oscillations $\omega \beta $ (1).
It is demonstrated that the slopes of the curves $N\gamma (E)$ and $N\u03030.1%BW(E)$ for $E\u226b\u210f\omega c$ are higher for lower background electron concentrations n_{0} and higher fractions f of electrons remaining in the ion channel; lower temperatures T_{h} of hot electrons and lower rates of their acceleration, i.e., lower effective temperatures $Th,ef=\kappa tTh,mx,\u2009\kappa t<1$; lower amplitudes of electron betatron oscillations $r\beta $ determined by the characteristic distance $r\sigma $ from electron initial positions to the ion channel axes.
The value of the maximum temperature of hot electrons $Th,mx$ can be estimated by Eq. (B17). Estimation of $r\sigma $ is a more complex question; for crude estimation, one can take $r\sigma $ as some part of the laser pulse waist radius, with the account for selffocusing; see Eq. (B19). Alternatively, $r\sigma $ can be estimated from the above as the value corresponding to the maximum energy acquired by electrons; see Eq. (B20).
As long as the position of the maximum of the curve $N0.1%BW(E)$ and the slopes of curves $N\gamma (E)$ and $N0.1%BW(E)$ are sensitive to $Th,mx$ and $r\sigma $, one can use spectral data on betatron radiation of electrons in NCD plasma for the diagnostic of those values as well as the value of the critical frequency of betatron radiation ω_{c}; see Eqs. (23) and (26). The information on free parameters of the model: the fraction of electrons f remaining in the ion channel and the fraction α_{h} of these electrons effectively accelerated can be withdrawn from (i) the total charge of the accelerated hot electrons $Qh,mx$ (9) or from (ii) absolute values of spectral curves: $N\u03030.1%BW(E)$ and $N\gamma (E),\u2009N\u03030.1%BW(E)$ or (iii) from the values of emitted energies $E\u0303\gamma \Sigma $ (30), $E\gamma \Sigma $ (31) or numbers of quanta with energies higher than the fixed one ( $0.1\u210f\omega c$ in the represented case), $N\u0303\gamma \Sigma 0.1\omega c$ (32), $N\gamma \Sigma 0.1\omega c$ (33).
All quantities (ii) and (iii) are proportional to the main free parameter of the model $\u03f0\u2323e$ or $\u03f0e$ (34). The results obtained had demonstrated that for a fixed product $f\alpha h$, spectral curves are not sensitive to the particular choice of f or $\alpha h$. Another free parameter of the model $\alpha rad=trad/\tau L$ stipulated by the effective time of electron betatron radiation $trad$ can be determined (together with $\u03f0\u2323e$ or $\u03f0e$) if information on both the charge of accelerated hot electrons and absolute values of emitted quanta or their energies or spectral curves is available.
The results of calculations by the proposed model are in reasonable agreement with both PIC simulations and experimental results for angle θ = 0. For nonzero angles, θ calculations conducted under the assumption of the radiation of electrons moving in the plane of laser polarization, which is perpendicular to the plane of detectors (i.e., for $\phi =\pi /2$), give much lower values of radiation specter curve $N\gamma (E)$, than measured in the experiment. This means that some electrons change their plane of motion from the initial laser polarization plane to another one, including the perpendicular plane. This change is quite possible, taking into account the results of PIC simulations available in the literature (e.g., Refs. 8, 13, and 19), which show that a powerful laser pulse created in the plasma ion channel is not the perfect one, with the structure, which always reveals the presence of some kind of filaments. This means that some stochastic fields are always available in ion channels and can change the original plane of electron motion. Taking into account this notation, the results obtained make one possible to draw a conclusion, that the concept of betatron radiation of electrons accelerated in NCD plasma by DLA mechanism can explain the respective experimental data on measurements of radiation specters of accelerated electrons.
Finally, we can note that the proposed simple model permits one to estimate the dependencies of the main parameters of DLA electrons and their radiation spectrum on the parameters of laser pulse and plasma. Its comparison with experimental data on synchrotron radiation specters of accelerated electrons can also be used for preliminary diagnostic of the process inside NCD plasma.
ACKNOWLEDGMENTS
This work was carried out with partial support from the State Atomic Energy Corporation Rosatom (agreement dated June 22, 2021 No. 17706413348210001390/226/3462D).
The author is grateful to Professor N.E. Andreev for his fruitful discussions.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
M. E. Veysman: Investigation (lead).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.
APPENDIX A: COHERENT AND NONCOHERENT SUMMATION OF CONTRIBUTIONS OF ELECTRONS TO RADIATION SPECTRUM
In addition to the problem of summation of contributions of different electrons to radiation specter, there is a question of coherent or noncoherent contribution to the specter from different parts of the trajectory of the one and the same electron. The expression (2) implies an incoherent summation of the contributions of different parts of the electron trajectory to the resulting spectrum. It can be shown that the basic condition for such a summation can be written as the inequality $\omega \beta \u22121\Delta \omega \beta T\beta >(8n)\u22121$, where $\Delta AT\beta $ denotes the change in the magnitude of A over the betatron period $T\beta $, and n is the number of the harmonic under consideration in the sum (3). This inequality is valid for the conditions considered in the text.
APPENDIX B: ESTIMATION OF ENERGY DISTRIBUTION OF DLA ELECTRONS AND THEIR DISTRIBUTION ON AMPLITUDES OF BETATRON OSCILLATIONS
1. Conditions for ion channel formation
For the case of NCD plasmas considered here, the ions cannot be always treated as immobile ones. However, for the considered powerful (sub or over Petawatt) laser pulses and NCD plasmas, the inequality (B1) can be treated as satisfied with a large margin.
It should be noted that for the formation of highly directional fluxes of accelerated electrons, one has to ensure that a single ion channel will be formed inside the plasma, i.e., filamentation instability^{37–39} will not develop. The respective conditions for that are described in Ref. 18. Suppression of the filamentation instability can be achieved, for example, by making a transverse profile of the laser pulse to be the elliptical one.^{40} Throughout the text, it is assumed that filamentation does not take place.
2. Distributions of DLA electrons on energies and betatron oscillation amplitudes
In order to estimate the spatial and energetic characteristics of a bunch of electrons, trapped and accelerated inside an ion channel, we will briefly analyze the dynamics of electrons, following consideration of the works.^{11,15} Consider a simplified twodimensional geometry, when the electromagnetic field of a plane laser wave $ExL=EL\u2009cos(\varphi ),\u2009ByL=ELc/vph\u2009cos(\varphi )$ interacts with the electrons moving at a speed $vz\u2248c$ along the 0z axis in a flat z–x channel. Here, E_{L}, $\varphi =kL(z\u2212vpht),\u2009vph=\omega 0/kL$ are, respectively, the amplitude, the phase, and the phase velocity of the laser wave, ω_{0} is the laser frequency, and k_{L} is the laser wave vector. The electron concentration profile along the 0x axis can be approximated by the step function $ne=n0[f+(1\u2212f)\theta (x\u2212rch)]$, where n_{0} is an unperturbed electron concentration, and $f\u223c10\u22121$ is the fraction of electrons remaining in the channel.
From Eq. (B10), the condition for the resonant excitation of betatron oscillations of an electron in the ion channel by a laser field follows as $\omega \beta =\omega D$. Acceleration of electrons under the combined action of laser and quasistationary fields, described by the above equations, can be treated as the socalled regime of direct laser acceleration (DLA) of electrons in an ion channel by laser fields.^{11}
In addition to the characteristics of the energy spectrum of DLA electrons, for calculations of radiation specter, one should know their spatial–angular characteristics.