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.

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 high-energy photon and particle fluxes.1–9 Electron bunches with simultaneously relatively high energies Wb102 MeV and high currents (charges Qb102nCl÷1mkCl), required for the effective generation of x-ray 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 laser-accelerated electrons10–14 as well as synchrotron radiation specters8,15 of those electrons. However, analytic and semi-analytic 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 Raman-backscattered 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 long-scale 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 experiment19 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 103E on the energy E of quanta), are:

  • the characteristic energy of electrons in accelerated bunch or the temperature of hot electrons Th,

  • characteristic amplitudes of their betatron oscillations rβ,

  • characteristic value of background electron concentration n0 and the fraction of electrons remaining in the created ion channel f, which together with Th determine the characteristic value of the frequency ωβ 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 rL, the effective time of acceleration teff, the fraction of electrons remaining in the created ion channel f, and the fraction αh of those electrons, which are effectively accelerated.

In Ref. 11, the scaling for the dependence of the temperature Th 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 above-mentioned coefficients f and αh. In addition to that, we estimate the characteristic value of the amplitude rβ of betatron oscillations. We show that rβ is determined by the characteristic value of the radial position of an electron rσ at the moment when it starts to accelerate.

In the case f1, the shape of the spectral curve is determined by Th, rσ, and n0. This makes it possible to estimate rσ from betatron radiation spectrum data, assuming that the characteristic temperature Th of DLA electrons is derived independently of the above-mentioned scaling and that n0 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 rσ and rβ 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.

In Sec. III, we make a comparison of predictions of the proposed theoretical model with the experimental results and the results of numerical simulations of synchrotron radiation of electrons accelerated in laser plasma, with PIC simulations of electron dynamic.8,19

Section IV summarizes the results.  Appendix contains estimates of the main characteristics of spatial and energy spectrum of DLA electrons as well as the number of those electrons and the length of ion channel.

To determine the radiation spectrum of a bunch of accelerated electrons with a total number of particles Ne, we assume that the frequency ωβ of betatron oscillations of an electron and its gamma factor γ are slowly changing at the betatron period Tβ=2π/ωβ,
(1)
is the betatron frequency of electrons in the ion channel,15 where ωp=4πn02/m, n0 is the electron concentration in the background plasma, γ=We/(mc2) is the electron gamma-factor; for derivation of Eq. (1), see  Appendix.
The betatron oscillation period Tβ estimated by Eq. (1) is Tβ=2γ(n0/nc)1κh1τ0[2÷30]τ0 for considered below plasma with n0/nc=0.1÷0.65 and electron energies We=[0.525] MeV, where τ0=2π/ω0 is the laser period, nc=mω02/(4πe2), ω0 is the laser frequency, while the characteristic time teff of electron acceleration is teff102τ0 for the considered below parameters, i.e., the condition of the slow change of ωβ and γ during the betatron period Tβ is fulfilled. In this case, one can show that the radiation spectrum Sγ=d2Eωb/dEdΘ of an electron bunch can be written approximately as
(2)
where d2EωidEdΘ is the radiation spectrum of the ith electron over the one betatron period Tβ [in Eq. (2), the sum over different betatron periods is replaced by the integral on t]. In the considered case of adiabatic variation of laser and plasma parameters over the time interval Tβ, one can consider the frequency ωβ and amplitude rβ of the transverse betatron oscillations x(t)=rβcos(ωβt) of an electron along the 0x axis in a stationary electrostatic field inside a homogeneous ion channel as constant at each of betatron periods. In this case, the spectrum d2EωidEdΘ of radiation of an electron moving with the velocity close to the speed of light along the 0z axis and making transverse betatron oscillation along the 0x axis can be written as24 
(3)
(4)
where index “i” is omitted for brevity while writing the values in the r.h. side of Eqs. (3) and (4); Jk is the Bessel function of the first kind of the order k, φ is the angle between the plane x0z of electron's motion and the plane that contains the vectors n and ez,ez||0z, θ is the angle between n and ez (θ, φ, r determines spherical coordinates); below, it is assumed that θ1; θ̂=θγ; αf=e2/(c);
(5)
where k=ω/c,kβ=ωβ/c,aβ is the basic parameter of the betatron radiation, and the parameters ωeff, αz, αx, and w define the characteristic frequencies of the peaks of the synchrotron radiation spectrum of an individual electron.

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.

Below, we assume that the sum ∑i over the electrons in Eq. (2) can be replaced by a double integral: over the electron distribution fe(rin) over distances from electron initial positions rin to ion channel axis:
(6)
and over the distribution fW(W) over electron energies W. While calculating fW, one should take into account that “ponderomotive” electrons escape from the ion channel without betatron oscillations and do not contribute to synchrotron radiation spectrum; hence, fW(W)=fh(W), where fh(W) is given by Eq. (B13). Using Eq. (B13), we assume the same value of the gamma factor γ=W/(mc2) for all electrons in the same bunch cross section, which allows us to use a simple product of the distribution functions fe(rin) and fW(W). We also assume that we can change the order of integration with respect to time, electron energies, and transverse positions.
The study of the dynamic of individual electrons as well as the dynamic of the average electron characteristics like rσ and Th and the dynamic of the number of accelerated electrons will be the subject of our further activity. Presently, we assume that the most rapidly changing in time quantities are the effective number of “hot” accelerated electrons Nh and their temperature Th. Nh is changing while accelerating laser pulse propagates through NCD plasma, and it can be estimated as
(7)
where αh<1 is the characteristic fraction of the accelerated electrons from the total number of electrons remaining inside the ion channel and Lch is the ion channel length, teffLch/c is some effective time of electron DLA process; NΣ is the total amount of electrons in an ion channel volume,25 
(8)
is the initial effective laser spot radius, where Dx and Dy are the initial laser FWHM spot diameters in the parallel and perpendicular to the polarization direction, respectively.

Results on PIC simulations8,19 of the trajectories of electrons self-injected and accelerated in laser plasma at the laser and plasma parameters close to considered here6 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 LchLL=cτL for optimal parameters of plasma fitted to laser parameters; see  Appendix B and expression (B23); τL is the FWHM laser pulse duration. Below, we assume that the parameter αeff=teff/τLLch/LL1.

For estimates, it is convenient to rewrite expression (7) for the maximum number of accelerated hot electrons Nh,mxNh(teff) in the form Nh,mx=(π2/3)fαh(r0/λ0)2(n0/nc)LL/λe,λe=e2/mc2 is the classical electron length. The charge of these electrons can be written as
(9)
An increase of the energy of DLA electrons gives rise to an increase in the temperature Th of their distribution on energies. We assume that this increase can be approximated as
(10)
where pT<1 is some numerical constant, and Th,mx is given by Eq. (B17).
Thus, one can have from Eqs. (2) and (3) the following expression for the dimensionless spectrum Sγ of the emission of different electrons along the ion channel depth, in a unit solid angle for the unit interval of quanta energy, during the time tteff of their motion in ion channel:
(11)
where the designation like A denotes averaging over electron distribution functions
(12)
with fW = fh (B13), with the effective temperature Th=Th(t); tradteffτL is some effective time of electron betatron radiation; NΣ is given by Eq. (7), Fω is dependent on betatron oscillation amplitude rβ(rin) (B18) and on W=mc2γ through the respective dependence of the parameters aβ and ωeff (5); rβrin for most of electrons, see  Appendix B; Wmx is the maximum energy of accelerated electrons determined by expression (B12), and rchrL is the ion channel radius.
One can suppose that with the accuracy sufficient under the scope of the present simplified model one can simplify the above double integral in time in Eq. (11) in the form
(13)
where Nh*=αhNΣ/2 is the “average” number of electrons participating in the process of betatron radiation. Below results of calculations have shown that one can further simplify expression (13) omitting time dependence:
(14)
where the designation A* denotes averaging over electron distribution functions fe(rin) and fW*(W), with fW* determined by Eq. (B13) with fixed electron temperature
(15)
where kt is some numerical coefficient and the second equality is written in accordance with Eq. (10). In the below calculations, we had used the value κt=0.81 that corresponds to the choice26 of the parameter pT=0.3.
In most cases in expression (3) for Fω entering (13), the sum over m has a large peak at m[w] due to the function Rm (where [x] denotes the integer part of x). Taking this into account and assuming the wiggler regime of synchrotron radiation,8 when aβ1,w1 and the emission spectrum of a single electron consists of many peaks and forms a quasi-continuous broad band,24,27 one can write the following equations:
(16)
where w is given by Eq. (4) and Fm is given by Eq. (3).
The most simple case is when the radius vector n lies in the plane perpendicular to the plane of electron betatron oscillations, with φ=π/2. In this case, from Eqs. (3) and (16), one has αx=0, Cx,m=2l+1=(1)l[JlJl+1], Cz,m=2l=(1)lJl, Fk=Cx,k2+Cz,k2μ2,
(17)
where l=[w/2] and w is given by Eq. (3).
Further simplification of the last expression taking in mind that aβ1,w1,JlJl+1Jl and l1,l>αz leads to a well-known equality8,24
(18)
where
(19)
is the critical frequency of radiation. Note that the definition of ωc (19) is coincident with that in Ref. 24 but is two times higher than in Ref. 6 and in Ref. 8.
Expression (18) gives one an estimate for the characteristic angular width of the radiation spectrum as
(20)
For φ=0 (radiation in plane of motion), one has from Eqs. (3) and (16):
(21)
For
(22)
in the sums over k in Eq. (21), there are terms JlkJ2k,Jlk+1J2k,JlkJ2k±1 with lk+1<αz and 2k+1<αx, which considerably exceed another terms. These other terms are of the same order, as terms in expression (17). The last ones are infinitivally small for ωωc and θ>1/γ, while the terms JlkJ2k,Jlk+1J2k,JlkJ2k±1 can be essential even for θ>1/γ provided condition (22) is fulfilled.
From Eqs. (1), (5), and (19), one can write the critical photon energy of betatron radiation as
(23)
where the amplitude of betatron oscillations rβ is given by expression (B18). Alternatively, for the estimation of ωc, one can find the maximum of radiation specter integrated over all directions, using properties of a universal spectral curve S(ω/ωc), where
(24)
where K5/3(t) is a modified Bessel function of the second kind. The expression (24) is obtained after integrating the emission spectrum of a single electron (18) over angles θ.6,24,28 One can also use the maximum of the spectrum of radiation in the forward direction, using the function S̃(ω/ωc)=Fω(θ=0,ω/ωc), obtained from Eq. (18):
(25)
The curves S(x) and S̃(x) have maxima at
(26)
where ωSmax and ωS̃max are the radiation frequencies corresponding to the maximum of the universal spectral curve (24) and forward specter curve (25), respectively.
Using Eq. (13), one can estimate the number of emitted quanta N0.1%BW=E0.1%BW/ω and Ñ0.1%BW=Ẽ0.1%BW/ω, where E0.1%BW is the energy emitted in the frequency range Δω=103ω during the considered laser–plasma acceleration of electrons. Here, the values without tilde refer to the emission into all directions, while values with tilde refer to the emission into a cone in the forward direction. In the case of DLA of ultra-relativistic electrons, the radiation is mainly concentrated in a cone of approximately θeff3050° around the propagation direction of electrons 0x.4,5,8,13 Taking this into account, one can write
(27)
where Sγ(ω,θ=0) is calculated using Eqs. (13) and (18) with θ = 0 and αθ=1cos(θeff); αθ0.2 for θeffπ/5. The uncertainty in the multiplier αθ does not create problems in the considered case, since, in accordance with expressions (13) and (7), it enters the final expression (28) for N0.1%BW only as a product with another three coefficients αh, f and αeffteff/τL.
For radiation in all angles, one can write
(28)
where the notation Sγ(FωS(ω/ωc)) means that in the expression (13), the function Fω dependent on θ is replaced by the universal spectral curve (24), which is the result of integration over θ.
One can also estimate the total energy Ẽγ,1,Eγ,1 and the total number Ñγ,10.1ωc,Nγ,10.1ωc of quanta with frequencies ω0.1ωc emitted by an electron with energy mc2γ:
(29)
where the values with tilde give estimates on the base of calculations with the value of spectral curve Fω given by Eq. (18), while the values without tilde give estimates on the base of universal spectral curve (24),
Taking into account the distributions of electrons on amplitudes of their betatron oscillations and on their energies and accounting approximation (10) for time dependence of the temperature Th of hot electrons, one gets from Eq. (29) the following estimates for the total emitted energy of betatron radiation ẼγΣ,EγΣ, as a fraction of the initial laser energy WL and the total number of quanta emitted by a bunch of electrons accelerated in the ion channel:
(30)
(31)
(32)
(33)
where
(34)
αh is defined above Eq. (1), αeff=teff/τL,αrad=trad/τL, ƛ=/(mc) is the Compton wavelength, λ0 is the laser wavelength, γh=Th,mx/(mc2), and constants

Figure 1 shows the dependence of number of quanta Nγ=Sγ/E per unit energy interval and unit solid angle, as a function of the quantum energy E=ω, 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 λ0=1μ m, duration τL=700 fs, and peak intensity I0=2.51019 W/cm2 and FWHM spot diameters Dx=15μ m and Dy=11μ m, acts on NCD plasma with thickness l=325μ 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.

FIG. 1.

Number of quanta per unit energy interval and unit solid angle, as a function of quantum energy ω, for the angle of incidence θ = 0. Other parameters are indicated in the text. Experimental results and PIC calculations of Ref. 19 are shown together with calculations by the model, described here, on the basis of Eq. (13) or Eq. (14); for different parameters, see the legend. Short dash-dotted, dotted, and marked thin gray curves are calculated with fixed electron energies W (without integration over W); short and long dash-dotted and marked thin gray curves are calculated with fixed radial coordinates rin of electrons (without integration over rin).

FIG. 1.

Number of quanta per unit energy interval and unit solid angle, as a function of quantum energy ω, for the angle of incidence θ = 0. Other parameters are indicated in the text. Experimental results and PIC calculations of Ref. 19 are shown together with calculations by the model, described here, on the basis of Eq. (13) or Eq. (14); for different parameters, see the legend. Short dash-dotted, dotted, and marked thin gray curves are calculated with fixed electron energies W (without integration over W); short and long dash-dotted and marked thin gray curves are calculated with fixed radial coordinates rin of electrons (without integration over rin).

Close modal

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 51013 W/cm2 to initiate a supersonic ionization wave propagating at about 2107 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 quasi-static electric and magnetic fields inside this ion channel by the DLA acceleration mechanism.4–6 

The number of quanta Nγ is determined by the following main parameters:

  • the total number Nh of super-ponderomotive (hot) electrons, estimated by Eq. (7); NγNh;

  • the frequency ωβ of betatron oscillations of electrons, determined by their background concentration n0 and by their fraction f in ion channel through Eq. (1);

  • the amplitude of betatron oscillations rβ;

  • gamma factor of electrons γ, where the characteristic value γh=Th/(mc2) is determined by the temperature of superponderomotive electrons.

All last three parameters determine the main parameters of the radiation spectrum: aβ and ωeff or ωc; see Eqs. (5) and (23). From Eqs. (18) and (23), it is clear that the curves Sγ(ω) and Nγ(ω) for ω>ωc are sharply decreasing functions of quantum energy E=ω and that the rate of decrease in those curves with E or the slope of those curves is higher for:

  1. lower values of background electron concentration n0 and higher values of the fraction f of electrons remaining in the ion channel [which determines the factor κh, see Eq. (1)];

  2. lower temperatures Th of hot electrons and lower rates of increase of Th with time t, i.e., lower rates of acceleration, which determines lower effective temperatures Th,ef=κtTh,mx,κt<1; see below;

  3. lower amplitudes of electron betatron oscillations rβ determined by the characteristic value rσ; see expressions (6) and (B18).

It is also clear that averaging on amplitudes rβ of betatron oscillations as well as on energies W of electrons in expression (13) decrease the slope of the curves Sγ(ω) and Nγ(ω).

The above peculiarities are clearly illustrated in Fig. 1 by calculations of Nγ(E) for different background electron concentrations in the ion channel n0/nc, different temperatures of hot electrons Th, and different betatron oscillations radius, determined by the parameter rσ in the distribution (6) and by the formula (B18). The parameters αeff=1, f = 0.3, αh=0.25,αrad=0.6 were used in calculations.

To approximately take into account the smooth variation of the background electron concentration n0 in the experiment19 from 0 till nmx=0.65nc, we calculate Nγ for n0=0.32ncnmx/2 and for n0=0.16ncnmx/4. The electron temperature dynamic was calculated by Eq. (10) with Th,mx given by Eq. (B17) and the coefficient of self-focusing 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,mx13.4 MeV for n0=0.32nc and Th,mx19 MeV for n0=0.16nc. Nγn0 in accordance with Eqs. (7) and (13), but due to higher Th,mx, the values of Nγ 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 Th16 MeV registered in the experiment.19 The initial laser spot waist was chosen as r0=6.5μ m, in accordance with expression (8) [the value of r0 enters expression (7) for Nh]. The values of other parameters were: f=αh=0.3,teff=τL=210τ0; other parameters are shown in the legend of the figure.

The results obtained by expression (13) with the dynamically changed temperature Th (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 Th, rσ, and n0/nc increase the slope of the curve Nγ(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 dash-dotted gray curve in Fig. 1 calculated for fixed W=Th,mx and rin=rσ). It is also clear that the lower-energy photons are generated by lower-energy electrons (with WTh,mx), while high-energy photons are generated by high-energy electrons (with W>Th,mx), compare the short dash-dotted 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 dash-dotted gray curve calculated for fixed rin=rσ, 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 Nγ(E) is somewhat higher than one in calculations by our model, at least for rσ>1μ 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 tteff, 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=600τ0.

Figure 2 shows the same experimental data, as for Fig. 1, but for detectors placed at θ=±10° relatively to the direction of laser propagation and in the plane perpendicular to the laser polarization plane, i.e., at φ=π/2; see Ref. 19 for details. Calculations by expressions (14) and (18) for φ=π/2 and θ=0,10° are shown by lines; see the legend.

FIG. 2.

Number of quanta per unit energy interval and unit solid angle, as a function of quantum energy ω, for the angle of incidence θ=±10°,φ=π/2 (experiment19) and angles of incidence θ=0°,10°,φ=π/2 [calculations by formula (14)]. Th=κtTh,mx=10.9 MeV (κt=0.81), rσ=1μ m, n0/nc=0.32. Other parameters are the same as for Fig. 1. Label “Ross” means Ross filters used in the experiment; see Ref. 19.

FIG. 2.

Number of quanta per unit energy interval and unit solid angle, as a function of quantum energy ω, for the angle of incidence θ=±10°,φ=π/2 (experiment19) and angles of incidence θ=0°,10°,φ=π/2 [calculations by formula (14)]. Th=κtTh,mx=10.9 MeV (κt=0.81), rσ=1μ m, n0/nc=0.32. Other parameters are the same as for Fig. 1. Label “Ross” means Ross filters used in the experiment; see Ref. 19.

Close modal

From Fig. 2 and its comparison with Fig. 1, it is clear that registered in the experiment decrease in radiation due to nonzero angle θ=10° is only about a factor of 0.25. Such large values of Nγ for θ=10° 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 experiment19 are characterized by φ=π/2. In this case, expressions (14) and (18) give values of Nγ similar to the experimental ones only for θ1°; see the dashed and dash-dotted curves at Fig. 2. Note that from Eq. (20), one have a similar critical angle θmc2/Th,mx2° 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), θ||2kβrβkprσ2κhmc2/Th,mx50° for chosen f = 0.2, rσ=1μ m, Th,mx=13.4 MeV, n0/nc=0.32, i.e., for radiation of electrons in the plane of their motion, the angle θ=10°θ|| is practically the same as θ=0°.

Taking into account the above information, one can suppose that 25% 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=3048nC were obtained. In comparison, expression (9) gives Qh,mx40nC and QW>10=exp(10MeV/Th,mx)19nC for parameters n0/nc=0.32 and r0=6.5μ m relevant to this experiment (with the same f=0.3,αh=0.25 as used for calculations depicted at Figs. 1 and 2) and for Th,mx=13.4MeV calculated above for these parameters.

The number of emitted quanta Ñ0.1%BW and N0.1%BW calculated by formulas (27) and (28), respectively, with Sγ calculated by Eq. (14), is shown in Fig. 3 by lines. Results of PIC simulations of the work8 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 Sγ by the formula (14), we take Th=κtTh,mx with the same κt=0.81.

FIG. 3.

Number of quanta emitted in the frequency interval 0.1% of the quantum frequency ω, as a function of ω. The dot markers are for PIC calculations of Ref. 8 for t=230τ0, curves are calculated using Eqs. (14) and (27) with Fω given by Eq. (18), with αθ=0.2 (solid, dashed, dash-dotted curves, and the curve with crest markers) and using Eqs. (14) and (28) with the universal spectral curve (24) (dotted curve). Electron concentration n0/nc=0.65, temperature Th,mx=9.4 MeV, values of rσ are indicated on the legend; other parameters are the same as in Fig. 1. A thin gray curve with markers is calculated by the same way as a similar curve as in Fig. 1.

FIG. 3.

Number of quanta emitted in the frequency interval 0.1% of the quantum frequency ω, as a function of ω. The dot markers are for PIC calculations of Ref. 8 for t=230τ0, curves are calculated using Eqs. (14) and (27) with Fω given by Eq. (18), with αθ=0.2 (solid, dashed, dash-dotted curves, and the curve with crest markers) and using Eqs. (14) and (28) with the universal spectral curve (24) (dotted curve). Electron concentration n0/nc=0.65, temperature Th,mx=9.4 MeV, values of rσ are indicated on the legend; other parameters are the same as in Fig. 1. A thin gray curve with markers is calculated by the same way as a similar curve as in Fig. 1.

Close modal

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 simulations8 for rσ=1μ m. According to the peculiarities discussed in Sec. III A, smaller values of rσ ensure larger slopes of the curve N0.1%BW(E). Good agreement with PIC simulations is obtained with rσ=0.5μ m. This value does not contradict expression (B20), which gives rσ0.9μ m for considered parameters.

The value rσ 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,mx10 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 dash-dotted curve with one with crest markers in Fig. 3. This follows from the respective increase in the critical energy ωc; see Eq. (23). An increase of rσ produces similar effect, compared to the solid (violet) curve with other ones.

In addition to the effective temperature Th of hot electrons and the effective radius of their initial transverse distribution rσ, which can be estimated as some fraction of the laser pulse waist radius, with account for the process of self-focusing (see the  Appendix), other free parameters of the model are the fraction f of electrons remaining in the ion channel and the part αh of those electrons that are effectively accelerated. The values f and αh determine mainly the vertical position of the curve N0.1%BW(E), though they (mainly f) influence also on the betatron frequency ωβ (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 rσ1μ 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 ωS̃max1.7 keV. From this value and expressions (26), one has ωc4.3 keV. On the other hand, one can get this value of ωc from Eq. (23) with rβ=2μ m, γ=Th,mx/mc2 [with Th,mx9.4 MeV, in accordance with Eq. (B17)] or with rβ=1μ m, γ=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=rσ 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 ẼγΣ,EγΣ,ÑγΣ0.1ωc,NγΣ0.1ωc, estimated by formulas (30)–(33), respectively, for parameters corresponding to Fig. 3, with rσ=1μ m, are ẼγΣ2.9105,EγΣ2.5105,ÑγΣ0.1ωc4.51012,NγΣ0.1ωc2.51012. The value NγΣ0.1ωc is close to the number of photons with energies E > 1 keV, NγΣ1keV71011, obtained in Ref. 8 in PIC calculations.

For estimation of the spectrum data under experimental conditions, it is helpful to decrease the number of free parameters of the model. For this goal, one can rewrite the expressions derived above in the form:
(35)
(36)
where ϰe1 is some constant equal to the relation of the parameter ϰe (34) to its value ϰe4.0102 at parameters chosen above in calculations: f=0.3,αh=0.25,αeff=1,αrad=0.6 (ϰe=1 for calculations under parameters considered here); the designation A* denotes averaging over electron distribution functions fe(rin) and fW*(W) with the fixed effective temperature Th* (15)
(37)
(38)
Maximum values of terms in Eqs. (35)–(38) are of the order of 1, and these terms determine the shape of spectral curves. All other terms in Eqs. (35)–(38) are determined by laser and plasma parameters and by γh, which, in accordance with Eq. (B17), can also be estimated in terms of laser and plasma parameters as
(39)

In addition to γh, another important parameter, which should be estimated, is rσ, which determines through Eqs. (6) and (B18) the characteristic amplitude of betatron oscillations rβ. The latter determines (together with the values of f, αh,n0/nc, and γ) the value of the critical frequency ωc (23), which, in terms, determines the shapes of spectral curves in Eqs. (35)–(38). One can use Eq. (B20) to estimate rσ in terms of laser and plasma parameters. Particularly, from expression (B20), one has rσ0.9μ m under parameters of calculations corresponding to Fig. 3 and rσ1.2μ m under parameters of Figs. 1 and 2. Both values do not contradict to results of calculations depicted in Figs. 1–3.

For the estimation of the critical frequency ωc of betatron radiation, which determines the shapes of spectral curves, one can rewrite expression (23) in the form:
(40)
where γ1.4γh was substituted into Eq. (23) in accordance with the above considerations. Substituting scaling (B20) for rσ and scaling (39) for γh into Eq. (40), one get the following scaling for ωc:
(41)
which show a strong dependence of ωc on the laser pulse intensity and the relatively weak dependence on plasma density.
Expressions (30)–(33) can also be rewritten in terms of laser and plasma parameters and free parameter of the model ϰe as
(42)
(43)
(44)
(45)
where the value pT=0.3 was used for the calculation of numerical coefficients.

One should discuss free parameters of the presented model. The value αeff=1 can be fixed in accordance with the above discussion. The product fαh of parameters f and αh 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 ϰe or ϰe (34). The last one is also connected with the parameter αrad, which determines the effective temperature Th* of radiating electrons, calculated by Eq. (15), and the number of emitted quanta, as long as ϰeαrad.

Thus, one can have three independent parameters of the model: ϰe,αrad,f or ϰe,αrad,αh. From physical reasons, and taking into account the results of PIC calculations8 for powerful laser pulses, parameters f and αh are restricted by requirements f1 and αh<1. Taking into account that the variation of f and αh within the requirement fαh=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: ϰe and αrad, where ϰe is the main one; see the results of parameter variation in Fig. 4.

FIG. 4.

The same calculations as in 3 for rσ=1μ m, Th=Th* but for different parameters f, αh and αrad; see the legend. The parameters f and αh were varied so as to fix the product fαh=0.075. Figure (b) shows the same curves for changed αrad, as figure (a), but with values N0.1%BW normalized to take into account that N0.1%BWαrad; the respective values of Th*, calculated by Eq. (15), are also shown in the legend.

FIG. 4.

The same calculations as in 3 for rσ=1μ m, Th=Th* but for different parameters f, αh and αrad; see the legend. The parameters f and αh were varied so as to fix the product fαh=0.075. Figure (b) shows the same curves for changed αrad, as figure (a), but with values N0.1%BW normalized to take into account that N0.1%BWαrad; the respective values of Th*, calculated by Eq. (15), are also shown in the legend.

Close modal

Another parameter Ksf concerns the model of electron heating due to DLA; see Eq. (B17). The value Ksf3 was chosen taking into account the results of PIC simulations for laser pulses with intensities IL10191020 W/cm2 and powers P0.11 PW. The question about the dependence of Ksf on laser and plasma parameters needs further studies.

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 Nγ(E) (number of quanta per unit energy interval of quanta per solid angle), the slope of the curve Ñ0.1%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 0.1ωc emitted in the forward direction ÑγΣ0.1ωc and in all directions NγΣ0.1ωc, the critical frequency of the betatron radiation ωc.

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 Th of hot (superponderomotive) electrons, amplitude rβ of their betatron oscillations determined by the characteristic initial distance rσ of electrons from the ion channel axis, in accordance with Eqs. (B19) and (B20), background concentration n0 of electrons in plasma and the fraction f of electrons remaining in the ion channel, which determine the frequency of betatron oscillations ωβ (1).

It is demonstrated that the slopes of the curves Nγ(E) and Ñ0.1%BW(E) for Eωc are higher for lower background electron concentrations n0 and higher fractions f of electrons remaining in the ion channel; lower temperatures Th of hot electrons and lower rates of their acceleration, i.e., lower effective temperatures Th,ef=κtTh,mx,κt<1; lower amplitudes of electron betatron oscillations rβ determined by the characteristic distance rσ 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 rσ is a more complex question; for crude estimation, one can take rσ as some part of the laser pulse waist radius, with the account for self-focusing; see Eq. (B19). Alternatively, rσ 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 Nγ(E) and N0.1%BW(E) are sensitive to Th,mx and rσ, 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 ωc; 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 αh 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: Ñ0.1%BW(E) and Nγ(E),Ñ0.1%BW(E) or (iii) from the values of emitted energies ẼγΣ (30), EγΣ (31) or numbers of quanta with energies higher than the fixed one (0.1ωc in the represented case), ÑγΣ0.1ωc (32), NγΣ0.1ωc (33).

All quantities (ii) and (iii) are proportional to the main free parameter of the model ϰe or ϰe (34). The results obtained had demonstrated that for a fixed product fαh, spectral curves are not sensitive to the particular choice of f or αh. Another free parameter of the model αrad=trad/τL stipulated by the effective time of electron betatron radiation trad can be determined (together with ϰe or ϰe) 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 φ=π/2), give much lower values of radiation specter curve Nγ(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.

This work was carried out with partial support from the State Atomic Energy Corporation Rosatom (agreement dated June 22, 2021 No. 17706413348210001390/226/3462-D).

The author is grateful to Professor N.E. Andreev for his fruitful discussions.

The author has no conflicts to disclose.

M. E. Veysman: Investigation (lead).

The data that support the findings of this study are available within the article.

Let us consider the radiation of a group of Ng electrons moving along similar orbits near the central orbit characterized by coordinates and velocities (r,v). Taking into account that difference in the motion of different electrons can only impede the conditions of coherence of their radiation, we consider idealized situation when electron trajectories differ only by constant vectors Rq and by time displacement: rq(t)=Rq+r0(ttq), vq(t)=v0(ttq) for the qth electron. In this case from the expression for the Linard–Wichert potentials for these electrons, it follows that the energy radiated into the solid angle dΘ in the interval of quantum energies dE=dω in the direction of the unit vector n is28,30
(A1)
where αf=e2/(c),E=ω and
(A2)
is the coherence factor,30  Rq and tq are the space and time displacement of the qth electron trajectory from the central one. If the length of the electron bunch Lb, which emits radiation, is much shorter than the emitted wavelength λ, when the module of the argument of the exponent in Eq. (A2) is much less than 1 and C=Ng2 (coherent summation). In the opposite case, Lbλ, the summation is coherent for the frequency ωb=2πc/λb only if ctqnRq=λbNq, where NqN. Unlike the case of crystals, in plasmas, electrons are distributed randomly, ctqnRqλbNq for arbitrary q and C(ω)=Ng (non-coherent summation) for this case of a relatively long electron bunch Lbλ, which we assume in this paper, taking into account that Lb is of the order of μm, while λ is of the order of nm (see also Ref. 31).

In addition to the problem of summation of contributions of different electrons to radiation specter, there is a question of coherent or non-coherent 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 ωβ1Δωβ|Tβ>(8n)1, where ΔA|Tβ denotes the change in the magnitude of A over the betatron period Tβ, 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.

1. Conditions for ion channel formation
Almost complete displacement of electrons from the created ion channel (that corresponds to f1) is ensured provided the laser radiation power PL significantly exceeds the power of relativistic self-focusing:32,33
(B1)
where the coefficient αP30 for the case of immobile ions.34,35

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.

In addition to that, we assume that the laser pulse FWHM duration τL or the laser pulse length LL=cτL is large in comparison with the plasma wavelength λp=2πc/ωp and the laser pulse transverse size rL,
(B2)
where kp=2π/λp. Under conditions (B1) and (B2), which are satisfied for the conditions of the experiment and PIC calculations considered in the paper, the ponderomotive force of the laser field forms an ion channel in the plasma that is almost completely void of electrons.10,36

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 instability37–39 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 two-dimensional geometry, when the electromagnetic field of a plane laser wave ExL=ELcos(ϕ),ByL=ELc/vphcos(ϕ) interacts with the electrons moving at a speed vzc along the 0z axis in a flat zx channel. Here, EL, ϕ=kL(zvpht),vph=ω0/kL are, respectively, the amplitude, the phase, and the phase velocity of the laser wave, ω0 is the laser frequency, and kL is the laser wave vector. The electron concentration profile along the 0x axis can be approximated by the step function ne=n0[f+(1f)θ(|x|rch)], where n0 is an unperturbed electron concentration, and f101 is the fraction of electrons remaining in the channel.

In the channel, the charge separation leads to a static electric field Exs, while the current due to motion of electrons along the 0z axis causes a static magnetic field Bys:
(B3)
Expressions (B3) are derived under the assumption of a stationary channel during laser pulse propagation through a plasma, where αh101 is the relative part of hot (effectively accelerated) electrons among the electrons remaining in the channel, vz, and below vx are the respective components of the electron velocity. Here and below are the dimensionless coordinates tω0t,xω0x/c,vv/c,E,BeE,eB/(mω0c),pγv/c are used; nc=mω02/(4πe2).
Equations of motion of electrons inside the ion channel under the action of electromagnetic fields of the laser pulse and stationary electric and magnetic fields (B3) have the form11,15
(B4)
(B5)
(B6)
where γ=1+pz2+px2,η=1vz/vph. From Eqs. (B3), (B4), and (B6), one has
(B7)
Taking into account the above equations, the term in the r.h. side of Eq. (B7) can be written as
One can disregard this term due to the smallness of vphvz1 and x2dx2/dt. In this case, (B7) leads to integral of motion
(B8)
Substituting expressions (B3) for kE and kB into Eq. (B8) leads to the integral of motion for the considered planar geometry. In true cylindrical geometry, one has additional factors 1/2 for kE and kB and one has finally
(B9)
For an electron moving near the channel axis with vzc, one can approximate the phase of the wave as ϕ=kL(zvpht)ωηt. Then, under the condition of a slow change of electron energy γ over a period of its betatron oscillations, after differentiation of Eq. (B5), it is reduced to the form
(B10)
where ωD=ηω0 is the Doppler-shifted frequency of laser wave and ωβ is given by Eq. (1). For αh=1 Eq. (1) for ωβ is coincident with the ones derived in Refs. 11 and 15.

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 ωβ=ωD. Acceleration of electrons under the combined action of laser and quasistationary fields, described by the above equations, can be treated as the so-called regime of direct laser acceleration (DLA) of electrons in an ion channel by laser fields.11 

Maximum electron energy γmax in DLA is determined by the period of time during which the resonance condition ωβωD is fulfilled together with the requirement that the electron velocity is anti-parallel to the electric field of laser radiation. The acceleration process stops when dephasing between betatron oscillations of an electron and the force experienced by it from the electromagnetic field approaches the value π/2. In Ref. 21, an estimate for γmax is done provided χ=I0cvph1|1c/vph|ω2/(2ωp2)1, which means that vphc:
(B11)
where κ0.35 is the numerical coefficient, and the dimensionless quantity AL=aLωpωvphcI03/2 is proportional to the laser wave amplitude.
From Eq. (B11) for the considered case of NCD plasma and relativistically intense laser pulses, when ALaLωp/ω0>1, one get an estimate for the maximum energy of DLA electrons as follows:
(B12)
where a0=aL(t=0) is the initial dimensionless laser field strength; accelerated electrons are assumed to start with the zero velocity near the ion channel axis, for which the integral of motion is I01. The multiplier κ0 is introduced in Eq. (B12) to approximately take into account the change of characteristics of electrons and laser fields in the ion channel due to laser field self-focusing and heating of electrons.
In the DLA process, different electrons start acceleration at different times and with different initial conditions. Therefore, the spectrum of accelerated electrons is characterized by some distribution falling with energy, with cut off approximately at electron energies We=Wmax. Studding of the results of PIC simulations4,5,11 shows that the energy distributions of electrons obtained in PIC simulations can be approximated by a 2-temperature Maxwell-like function f̃W(W,t). Taking into account that electrons escaping from the ion channel can acquire an energy of the order of ponderomotive energy and electrons inside the channel can be accelerated by DLA mechanism, one can write the expression for the distribution function of accelerated electrons f̃W as
(B13)
where Tp=Tp(t) and Th=Th(t) are, respectively, some temperatures characterizing the spectrum of ponderomotive electrons, acquiring ponderomotive oscillation energy in the laser field after leaving the region of a strong laser field near the ion channel axis, and hot (“superponderomotive”) electrons, acquiring energy in the DLA process.
The effective temperature Tp can be estimated as
(B14)
Here and below, we use the coefficient Ksf=aL¯/a0 of laser field increase due to self-focusing, where aL¯ is the average magnitude of the dimensionless laser field strength in the ion channel. PIC calculations had shown that for the considered parameters of the problem, we can set Ksf3. This value of Ksf is used in the text.
In Ref. 11, it is shown that ThILaL (where IL is the laser radiation intensity), which corresponds to the scaling for the maximum energy Wmax (B12). Therefore, one can assume that the dependence of Th on the parameters of plasma and laser pulse is described, like Wmax, by a relation of the form (B12). Calculations under parameters corresponding to publications5,10,13,14 (see Ref. 20) have shown the proportionality ThWmax,
(B15)
In Ref. 20, it was shown that the coefficient κ0 in the formula (B12) can be approximated as
(B16)
and hence, taking into account (B12), (B16), and (B15), the effective temperature of high-energy electrons can be approximated as
(B17)

In addition to the characteristics of the energy spectrum of DLA electrons, for calculations of radiation specter, one should know their spatial–angular characteristics.

In order to estimate the characteristic amplitude of electron betatron oscillations rβ, one can use the expression for the integral of motion I0 (B9). Assuming that electrons start to accelerate with zero velocity and taking into account that for the ultra-relativistic electrons γpx/(mc) (paraxial approximation21), one has from Eq. (B9):
(B18)
where κh is given in Eq. (1); rin is the distance from the initial position of the electron to the ion channel axis. One can assume that electrons most effectively accelerated to high energies in the strong fields near the ion channel axis when rin constituent some small part of the ion channel radius rch. The radius rch is determined by the balance between the transverse ponderomotive force and the force arising from the charge separation field. One can suppose that at the beginning of the laser pulse entrance into the plasma rchr0, while at a later time of laser pulse propagation in the ion channel rchr0/Ksf, r0 is the initial value of the laser pulse waist radius and Ksf is the above-mentioned coefficient of self-focusing. From these considerations, one can have
(B19)
where αin1 is the numerical coefficient and rσ is some characteristic value of rin. We assume that this value determines the distribution of electrons over the values of their distances rin from their initial positions to the ion channel axis given by Eq. (6).
Alternatively, one can estimate from the above rσ using the transverse distance of an electron from the ion channel axis, which corresponds to the maximum energy acquired by electrons:41,
(B20)
where ϵcr is constant, ϵcr0.2; see Ref. 41. This choice of rσ is stipulated by the results of PIC simulations, which show that most of the hot electrons oscillates with amplitudes lower than rWmx; see Figs. 3(c) and 3(d) of the work.41 
From Eq. (B18), it is also clear that for
(B21)
the values of rin do not influence on rβ. In the opposite case, the value of rβ is determined mainly by rin. The main conclusion from the above consideration is that rβ does not exceed rin for rinr*, which is confirmed by the PIC calculation data on the characteristic betatron oscillations for high-energy electrons; see, e.g., Ref. 19 (Fig. 7) and Ref. 8 (Fig. 4).
For estimation of the phase velocity vph in Eq. (B18), one can use linear theory and take into account the relativistic gamma factor of electrons remaining in the channel. The influence of the walls of the ion channel on the first mode of laser field propagating inside this channel can be taken into account through the perpendicular wave vector r,1=u1/rch, where u12.4 is the first root of the zero-order Bessel function42 and rchr0/Ksf is the ion channel radius; see above. Thus, one can write the following estimate under assumption vph11:
(B22)
where k0=ω0/c is the vacuum wave vector.
For the most effective acceleration of a bunch of electrons in the ion channel, its length Lch should be approximately equal to the length of full absorption of laser pulse energy. Equating the laser pulse energy to the full energy of electrons in the ion channel determined as the first energy moment of the distribution function (B13), one can have
(B23)
where αL0.10.7 is the part of laser energy converted to ponderomotive and superponderomotive electrons (other part of laser energy is passed due to leakage of laser energy from the ion channel and is converted to the energy of plasma fields). Note that in accordance with Eq. (B23) and expression (B17) for Th, Lcha0 for the case when most of laser energy is converted to electron energy (αL0.5). Such a dependence of Lch on the dimensionless laser amplitude a0 is similar to that followed from the expression for the optimal target thickness for the case a01 given in Ref. 43.
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