ISSN 1063-780X, Plasma Physics Reports, 2015, Vol. 41, No. 4, pp. 343–349. © Pleiades Publishing, Ltd., 2015. Original Russian Text © V.F. Kovalev, V.Yu. Bychenkov, 2015, published in Fizika Plazmy, 2015, Vol. 41, No. 4, pp. 374–380.
PLASMA DYNAMICS
Dynamics of Ponderomotive Ion Acceleration in a Laser-Plasma Channel V. F. Kovaleva, c and V. Yu. Bychenkovb, c a Keldysh b
Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119991 Russia c Center for Fundamental and Applied Research, All-Russia Research Institute of Automatics, Sushchevskaya ul. 22, Moscow, 127055 Russia e-mail:
[email protected],
[email protected] Received August 28, 2014
Abstract—Analytical solution to the Cauchy problem for the kinetic equation describing the radial acceleration of ions under the action of the ponderomotive force of a laser beam undergoing guided propagation in transparent plasma is constructed. Spatial and temporal dependences of the ion distribution function and the integral ion characteristics, such as the density, average velocity, and energy spectrum, are obtained for an axisymmetric laser-plasma channel. The formation of a density peak near the channel boundary and the effect of ion flow breaking for a quasi-stationary laser beam are described analytically. DOI: 10.1134/S1063780X15040017
1. INTRODUCTION Among various schemes used for generation of high-energy ions under the action of a high-power laser pulse, the scheme in which ions undergo radial acceleration from the laser-plasma channel formed by a laser beam propagating in underdense transparent plasma is discussed extensively [1–5]. The use of laser-accelerated ions produced in underdense plasma is of interest from the standpoint of obtaining neutrons and various radionuclides, in particular, those employed in medicine [1, 6–8]. To a large extent, the dynamics of radial acceleration of ions by laser radiation in underdense plasma (produced, e.g, from a gas jet [1], evaporated foil [9], or nanoporous carbon [10]) is determined by the ponderomotive action of the laser pulse on the plasma electrons that are ejected from the plasma channel by the ponderomotive force. As a result, there appears a strong radial charge-separation electric field, which accelerates ions. A characteristic example is the problem of ion acceleration analyzed in [2, 5], where it was shown that analytical description of particle acceleration by a laser pulse from the plasma channel formed as a result of self-focusing is a rather complicated problem even when using approximate methods. That is why the existing theoretical models [2, 3, 5] characterize radial acceleration of particles from the laserplasma channel only qualitatively. Accordingly, the spatiotemporal distribution of laser-accelerated particles is analyzed mainly by using kinetic numerical modeling. In so doing, as a rule, the particle-in-cell
(PIC) method is applied [11–13]. The problem can be somewhat simplified by using a one-dimensional numerical electrostatic model of ponderomotive acceleration describing the dynamics of plasma expansion in a given field [2, 5], wherein only slow dynamics of plasma electrons is taken into consideration, which corresponds to the averaging over fast electron oscillations in the laser field. It is difficult to predict how the spatiotemporal and spectral characteristics of ions accelerated from the laser channel depend on the laser and plasma parameters, because the main results obtained for the electrostatic ponderomotive model are based on PIC simulations. Numerical simulations of ion acceleration under the action of the electric field induced by the ponderomotive force revealed a large jump of the plasma density at the laser channel boundary—a well-pronounced dense cylindrical wall of the channel [2, 5]. The appearance of such a jump is accompanied by strong local heating of electrons, the subsequent breaking of the channel wall, and the inversion of the charge-separation field [5]. Numerical simulations [2, 5] revealed that electrons adiabatically respond to the ponderomotive action of the laser pulse; as a result the electric field in the channel is determined with a high degree of accuracy by the balance between the electrostatic force returning electrons back to the channel and the ponderomotive force pushing them out. Therefore, it may be expected that, as long as such balance is maintained, the acceleration dynamics of ions can be studied analytically by using the kinetic equation for ions
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in a given electrostatic field. This work is aimed at performing such analysis. 2. KINETIC MODEL WITH A GIVEN ACCELERATING ELECTRIC FIELD Let us analyze the propagation of an axisymmetric laser beam in transparent plasma. Due to the radial inhomogeneity of the laser electric field, plasma electrons are expelled from the region of strong electric field. This results in the inhomogeneity of the electron density and the appearance of a charge-separation electric field, which, in turn, leads to the redistribution of the ion density and radial ion acceleration up to high energies. The dynamics of this process can be described in terms of the kinetic equations for the electron and ions distribution functions. The influence of the laser electric field in the kinetic equation for electrons is taken into account by introducing an additional electrostatic force corresponding to the action of the ponderomotive force averaged over the period of the laser field [14, 15]. However, the results of numerical simulations [5] show that, as long as there are no substantial temperature gradients in plasma and the radial profile of the ion flow does not experience breaking, the electric field in the beam propagation region is determined by the balance between the electrostatic force −e E , returning electrons back to the channel, and the ponderomotive force
F p = −mc ∇ 1 + a /2 (where a is the dimensionless vector potential of laser radiation), pushing them out, i.e., e E = F p . This condition is quite natural, because, due to the small mass of electrons, they tend to establish such an equilibrium state while rapidly oscillating at the laser and plasma frequencies. Taking this into account, the mathematical model describing ion acceleration can be significantly simplified by leaving only the kinetic equation for ions in which the electric field is assumed to be given. Taking into consideration the symmetry of the laser beam propagating along the z axis, we will analyze the ion kinetic equation for the ion distribution function integrated over the longitudinal and axial velocity components in cylindrical coordinates {r, ϕ, z} . The distribution function is thus assumed to be dependent only on the time t , the radial coordinate r , and the radial component of the ion velocity v r . As a result, we arrive at the Cauchy problem for the kinetic equation for the ion distribution function f i (τ, x,v ) integrated over v ϕ and v z (for the sake of definiteness, the thermal spread over v ϕ and v z is neglected), 2
2
∂ τ f + v ∂ x f + p∂ v f = 0, f = xf , p = p(τ, x), f τ=0 = f 0( x, v ).
(1)
Here, τ = ω Li t is the dimensionless time (where ω Li is the Langmuir frequency of ions with the mass
M i = AM and charge e i = Ze , with A , Z , and M being the mass and charge numbers of ions and the proton mass, respectively), x = r / L is the dimensionless coordinate (with L being the characteristic scale of the laser beam localization over the radius), v = v r /(ω Li L) is the dimensionless ion velocity, p = (ZeE /(M i Lω2Li )) is the dimensionless electric field, and f i = (n 0 /(ω Li L)) f is the dimensionless ion distribution function (with n 0 being the unperturbed ion density). The function f i in Eq. (1) is normalized to the characteristic ion velocity ω Li L , rather than to the commonly used thermal velocity, because, in the given formulation of the problem, the thermal spread in ion velocities is insignificant and, accordingly, the ion thermal velocity is much smaller than the quantity ω Li L , used for normalization. Taking into account the relation between the electric field in plasma and the gradient of the laser intensity, the expression governing the electric field p takes the form 1/ 2
2 ⎡ a 2(τ, x )⎤ p = − 2c 2 ∂ x ⎢1 + . 2 ⎥⎦ ω Le L ⎣
(2)
Here, ω Le is the Langmuir frequency of electrons with the mass m and charge e e = −e ; the quantity a 2(τ, x) = A(τ)a 2( x) characterizes the dimensionless laser intensity, a 2( x) = a02I 0( x, a02 ), with a0 =
0 . 85 × 10 −9 λ [μ m] I 00 [W/cm 2 ], where λ is the laser wavelength in μm and I 00 is the peak laser pulse intensity in W/cm2; and the function A(τ) describes the laser pulse shape. The function a 2( x ) characterizes the radial distribution of the laser intensity. In numerical simulations performed in [5], it was assumed that a 2( x) = a02 exp(− x 2 ), so that I 0 = exp(− x 2 ) was independent of a02 and was equal to unity on the beam axis, I 0(0, a02 ) = 1. Equations of form (1) are usually investigated by using the method of characteristics. The analysis is substantially simplified when the electric field is independent of time, i.e., p = p( x), which corresponds to the quasi-stationary regime in which the duration of the laser pulse exceeds the characteristic ion acceleration time. In this case, the characteristics of Eq. (1) can be integrated and the solution to Eq. (1), f = F ( j1, j2 ) , is expressed in terms of the invariants j1 and j2 , 2
j1 = v − Φ, 2
j2 = τ −
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∫ Ψ(Φ)
dΦ , 2( j1 + Φ)
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ν 1.0
ν ν3
(a)
345
(b)
0.8 2
5
10
15
0.6 0.4
ν2 1
0.2 0.5
ν1 2
4
6
8
10
x
12
xl
xi
xu
x
Fig. 1. (a) Solutions to Eq. (11) on the phase plane ( x, ν) for different times τ = 0 . 5 , 1, 2, 5, 10, and 15 and (b) boundaries of the existence domain of the three-flow ion motion, x l < x < x u , for τ > τ * .
where the potential Φ and Ψ(Φ) are coupled by the following implicit relationships:
p( x) = ∂ x Φ = Ψ(Φ).
(4)
To better understand the ion acceleration process, the case where integrals (3) are calculated explicitly is of the highest interest. Accordingly, the spatial distribution of the electric field p should be specified in such a way that it corresponds to a physically meaningful profile of the laser intensity. Let us consider a simple example of calculating the invariants and finding the solution to kinetic equation (1) for the case of a radially localized distribution of the electric field of the form
⎛ ⎞ ρ = 22c 2 ⎜ 1 + − 1⎟ . (5) ⎜ ⎟ 2 ω Le L ⎝ ⎠ Such a dependence of the electric field on the x coordinate corresponds to a linear increase in the electric field with x inside the plasma cylinder at x Ⰶ 1 and its exponential decay at x Ⰷ 1. Similar to expression (2), the electric field in formula (5) can be represented in the form of a spatial gradient of the electric field potential induced by the ponderomotive force, p = ρ tanh x cosh
−2
x,
2
a02
ρ (6) p = − ∂ x cosh −2 x. 2 Comparison of expressions (2) and (6) shows that the intensity of the laser field creating electric field (5) has the form 2 ⎡⎛ ⎛ ⎤ ⎞ ⎞ a02 −2 ⎢ a ( x) = 2 ⎜1 + ⎜ 1 + − 1⎟ cosh x ⎟ − 1⎥ . (7) ⎟ ⎟ 2 ⎢⎜⎝ ⎜⎝ ⎥ ⎠ ⎠ ⎣ ⎦ Similar to the exponential dependence of the laser intensity on x used in numerical simulations [5], I 0 = exp(− x 2 ) , dependence (7) is also characterized
by a monotonic decrease in the intensity I 0( x, a02 ) at infinity and has a maximum on the beam axis,
I 0(0, a02 ) = 1. Substituting formula (5) into expressions (3), we obtain the following solution to kinetic equation (1):
f = F ( x, ν),
2
ν2 = v , ρ
sinh x = sinh x cosh τ − ν cosh x sinh τ , ν 2 + cosh −2 x ν cosh x = ν cosh x cosh τ − ν 2 + cosh −2 x sinh x sinh τ.
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For a negligibly low ion temperature (the limit of cold ions) and a homogeneous initial ion density, we have F τ=0 = n0δ(v ) and solution (8) takes the form
f = n0 x δ ( ρν ) . x
(9)
Multiplying Eq. (9) by 1 and v and integrating with respect to v , we obtain the following expressions for the integral characteristics of ions having distribution function (9), i.e., for their density and average velocity:
2
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τ = τ ρ ν 2 + cosh −2 x,
nav = n0
∑ xx { ∂ k
ν av
n = 0 nav
∑ k
k
ν
ν x =const
}
−1
x k ρ ∂ ν ν k ν x =const x
{
ν=ν k
}
, (10)
−1
ν=ν k
.
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Here, x k = x( x, ν k ) and summation is performed over all roots ν k of the equation ν = 0 , which, being written in variables x and ν , takes the following simple form: ν ν + cosh −2 x 2 −2 − tanh x tanh(τ ν + cosh x) = 0,
calculated characteristic of ions, which leads to a discontinuity of this characteristic. Using Eq. (11) in formulas (10) greatly simplifies the expressions for the ion density and average ion velocity,
2
(11)
τ = τ ρ.
Figure 1a shows the curves ν = 0 on the phase plane {x, ν} for different values of the time τ . It is seen that, at long times τ > τ *, the dependence ν( x) is not single valued in the entire interval of the variable x and resembles, e.g., the profile of the velocity distribution in the case of Coulomb plasma explosion [18], in which case the peripheral particles are taken over by the inner particles that experience the action of a stronger accelerating field. The upper, x u , and lower, x l , boundaries of the region x l < x < x u in which the solution is multivalued are determined from the condition that the derivative of the inverse function is zero, ∂ ν x = 0 . This yields the following two equations for the coordinates of the boundaries: − ω tanh x tanh ω = 0, τν tanh x tanh ω = 0, ω − τν
(12)
2 −2 ω = τ ν + cosh x.
The time τ * corresponding to the onset of multiflow ion motion can be found from the condition that the following derivatives vanish simultaneously: ∂ ν x = ∂ νν x = 0 . These conditions are expressed in the form of three equations combining Eq. (12) and the equation
cosh ω* tanh ω* 2 2 + ω* 1 + 2 cosh x*ν *(1 − ω* tanh ω* ) = 0, 2
(
)
(13)
ω* = τ * ν *2 + cosh −2 x*. Numerical solution of Eqs. (12) and (13) yields the following “universal” (i.e., valid at any value of ρ) values: τ * ≈ 2.78989, x* ≈ 2.01412 , and ν * ≈ 0.55715. At τ > τ *, three values of the variable ν (ν = ν1 , ν = ν 2 , and ν = ν 3) correspond to each value of x i from the interval x l < x < x u . At the lower and upper boundaries of the multiflow region, x = x l and x = x u , two of three roots merge into one. The presence of the multiflow region becomes clearly evident when calculating the average characteristics of accelerated ions, namely, their average density and velocity. Their calculation includes integration (summation in case (9)) of the ion velocity distribution function. Crossing the boundary of the multiflow region is accompanied by an abrupt change in the number of particles contributing to the
nav = n0
⎛
⎞ ⎟ ∂ ν ν ν=ν k k⎠
sinh x ∑ 1x Arcsinh ⎜⎝ cosh ω k
ν av =
n0 nav
νk
⎛
−1
⎞ ⎟ ∂ ν ν ν=ν k k⎠
sinh x ∑ x Arcsinh ⎜⎝ cosh ω k
∂ ν ν ν= = 0
,
−1
,
(14)
cosh x 2 sinh x + cosh ω 2 2 2 × cosh ω − tanh x sinh ω + ω tanh ω .
(
2
))
(
In addition to expressions (14) determining the spatial distribution of the ion density and average ion velocity, expressions for the energy spectrum of accelerated ions, N ⑀ , are also of interest. The spectrum is defined in such a way that integration of N ⑀ with respect to all possible values of the ion energy, 0 < ⑀ < ∞ , yields the total number of plasma ions, ∞
N ⑀ = π d xx 2 ⎡⎣ f ( x, 2ε) + f ( x, − 2ε)⎤⎦ . ε
∫
(15)
0
Substituting expression (9) into formula (15), we obtain the spectrum of the laser-accelerated ions,
N ε = 2π 2 ε=ν , 2
n0 2ε
⎛
⎞ ⎟ ∂ x ν x = xk k⎠
sinh x ∑ Arcsinh ⎜⎝ cosh ω k
−1
,
∂ x ν ν= = 0
τ (16) 2 2 cosh x sinh x + cosh ω 2 2 2 2 × tanh x − cosh ω tanh ω (1 + ν − 2 tanh x) . ω In contrast to expressions (14), summation in Eq. (16) is performed over all roots x k of the equation ν = 0 . The maximum energy of ions, ⑀ m = ν 2m /2, at any time moment is found from the set of two equations,
(
)
m − ω m tanh x m tanh ω m = 0, τν ω m = τ ν 2m + cosh −2 x m,
(17)
ω m tanh 2 x m − cosh ω m tanh ω m ⎛⎜⎝1 + ν m − 2 tanh x m ⎞⎟⎠ = 0 . 2
2
2
Typical distributions of the ion density and average ion velocity, as well as the ion energy spectrum, are shown in Fig. 2. It can be seen that, at τ ⲏ 5, the ion energy reaches its peak value of ⑀ m 1/2 at the right boundary of the spectrum. As τ increases further, this value remains practically uncharged. The transition to the multiflow regime is accompanied by the formation of singularities in the ion density distribution and jumps in the distribution of the average ion velocity. Such PLASMA PHYSICS REPORTS
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10
nav 102
(a)
nav 101
347
nav 104
(b)
101
102
100
100
(c)
0
10−1 1 av 100
10−2
10−1
1
2 2
3
10−2 4 x av 100
4 x
3
1
2
3
4 x
1
2
1
2
3
4 x av 100
1
2
10−1
10−1
10−1
10−2
10−2
10−2
10−3
10−3
10−3
N⑀ 105
N⑀ 105
N⑀ 105
104
104
104
103
103
103
102
102
102
0.05 0.10 0.15 0.20 0.25 ⑀
0.1
0.2
0.3
0.4 ⑀
3 3
4
6 x
5 4
5
6 x
0.1 0.2 0.3 0.4 0.5 ⑀
Fig. 2. Spatial distributions of the ion density and average ion velocity along the x coordinate and the ion energy spectrum for three moments of time: τ = (a) 2 , (b) 2 . 788 , and (c) 5 . The time τ = 2 . 788 is close to τ * , i.e., to the time corresponding to the onset of multiflow ion motion.
singularities related to the appearance of multiflow motion were discussed in many publications devoted to studying the dynamics of noninteracting particles [16], the dynamics of dissipation-free gas in the expanding Universe [17], and the Coulomb plasma explosion (see, e.g., [18, 19]). Note that the distance between the peaks of the average ion velocity increases rapidly with time, while the peaks themselves are relatively narrow, which can be understood from the anal ysis of the curve ν = 0 on phase plane {x, ν} . For regions where the derivative ∂ ν x is small, which corresponds to large values of the average ion density, the interval of variation of the x coordinate is typically small (this is especially pronounced for the peak lying further away from the beam axis). The widths of these regions decrease with time. As a result, to find the peaks of the ion density in numerical simulations is a rather difficult task [19] (it is often hard to reveal the second (external) peak in simulations, due to which a singly peaked density distribution similar to that obtained in [1] is usually observed). Therefore, predictions of the theoretical model (i.e., the solution to Eqs. (12)) could be useful for determining the spatial coordinates of these peaks. PLASMA PHYSICS REPORTS
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3. DISCUSSION AND CONCLUSIONS It follows from the above analysis that maximum energy ⑀ m of ions accelerated in a given quasi-stationary plasma field does not exceed 1/2 . This value is reached over the time τ , close to the time τ * corresponding to the onset of multiflow ion motion. This means that, in order to achieve the maximum ion energy, the laser pulse duration τ L should be equal to or longer than τ * . In the dimensional variables, this limitation is expressed as
τ t L ⭓ t* = 2 A M L * , 3 3 Z m c pmax
(18)
where pmax is the peak value of the normalized electric field amplitude p0( x) determined from the relationship p = (c 2 / ω2Le L2 ) p0 . For electric field (5) under consideration, the peak value of p0( x) ,
pmax =
(
)
2ρ ω Le L = 4 1 + a02 /2 − 1 , 2 3 3 c 3 3 2
2
(19)
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is achieved at x max = Arcsinh(1/ 2) . When transforming from the case of a0 Ⰶ 1 to the case of a0 Ⰷ 1, the quadratic dependence pmax = (1/3 3)a02 on a0 is replaced with the linear dependence pmax = (2/3)3 / 2 a0 . Similarly, in the dimensional variables, the peak value of the ion energy as a function of the laser and plasma parameters takes the form
(20) Ᏹ m = 3 3 ⑀ m Zpmax mc 2, 2 which yields the upper estimate for the energy of accel erated ions Ᏹ max = 3 3 pmax Zmc 2 /4 at ⑀ m = 1/2. When the ion acceleration time is shorter than that required to achieve the maximum energy ⑀ m = 1/2, the upper cutoff of the ion energy spectrum can be found from Eqs. (17). Note that expression (18) is obviously universal; i.e., although the numerical value of τ * depends on the details of the spatial structure of electric field (2), the dependence of pmax on the laser intensity a02 in Eq. (19) is apparently more important for determining the time corresponding to the onset of multiflow ion motion. In this case, reasonable estimates for t* for close spatial distributions of the laser intensity can be obtained from Eq. (18). As an example, let us analyze the case of the exponential distribution of the laser intensity a 2 = a02 exp(− x 2 ). In the limit of low laser intensity, a02 Ⰶ 1, the peak value of the electric field gs found from Eq. (2) is pmax ≈ a02 / 8e ≈ 0 . 21a02 , which nearly coincides with the value 2 2 pmax ≈ a0 /3 3 ≈ 0 . 19a0 found from Eq. (19). In the opposite limiting case of high laser intensity, a02 Ⰷ 1, gs we have pmax ≈ a0 / 2e ≈ 0 . 43a0 , which also correlates with the value pmax ≈ (2/3)3 / 2 a0 ≈ 0 . 54a0 gs found from Eq. (19). Substitution of pmax ≈ a02 / 8e into Eq. (18) yields an estimate for the time corresponding to the onset of multiflow ion motion at a low laser intensity, t*gs ≈ 3 . 78 AM / Zm(L/ a0c), which is comparable with the value of t b given by relationship (A.5) in [5] (taking into consideration the normalization of the laser intensity used in [5]), which was obtained in the low-intensity limit a 2 Ⰶ 1. In the case of a high intensity (i.e., for a0 exceeding unity), for which simulation was performed in [5], general expressions (18) and (19) should be used to find t . In particular, for the * plasma and laser beam parameters used in [5], we find that the breaking time is t* ≈ 665TL , which agrees with the simulation result t L = 660TL < t* ; i.e., there is no ion breaking before the end of the laser pulse. The esti-
mate of the peak energy acquired by ions by the end of the laser pulse t = t L yields Ᏹ m = 3 × 10 −4 Mc 2 , which is comparable with the energy of 2 . 2 × 10 −4 Mc 2 found by numerical simulation. The model of ion dynamics in the accelerating field determined by the ponderomotive force of the laser beam allows one to find the spatiotemporal and spectral characteristics of accelerated ions. The applicability of this model may be limited by the violation of the balance between the force created by the charge-separation field and the ponderomotive force of the laser beam due to, e.g., an increase in the electron thermal pressure, as was observed in numerical simulations [2, 5] at the outer boundary of the beam. A consistent approach with allowance for this effect implies the study of electron kinetics and requires performing additional analysis based on the complete set of kinetic equations for plasma particles in a self-consistent field. Thus, in this work, we have analyzed the radial acceleration of ions from a plasma channel under the action of a propagating laser beam. The ions experience acceleration as a result of the ponderomotive laser action on plasma electrons, due to which electrons are pushed out of the region occupied by the electric field that is highly nonuniform in the radial direction. As an example, we have considered a physically meaningful radial profile of the laser intensity, which makes it possible to perform a relatively simple quantitative analysis. We have analyzed the spatiotemporal behavior of the ion distribution function as a function of the plasma and laser beam parameters and plotted the spatial distributions of the density and average velocity of plasma ions for different moments of time. The energy distributions of accelerated ions are found, and the existence of the upper cutoff of their energy spectrum is revealed. It is also demonstrated, due to the expansion of ions, a region with a low ion density forms on the beam axis and peaks of the ion density appear at the beam periphery. The condition for the appearance of the region of multiflow ion motion is investigated, the spatial boundaries of this region are determined, and the time corresponding to the onset of such a regime is found. It is demonstrated that, after the breaking of the ion flow, two characteristic peaks of the ion density there appear, the distance between which gradually increases, while their widths decrease. The obtained results are analyzed for the case of a quasi-stationary beam, although the proposed approach is also applicable to a nonstationary laser pulse of finite duration. The analysis performed in this work provide theoretical explanation of the results of numerical simulations [2, 5] and can be used as a basis for further analytical studies of the behavior of plasma particles (ions and electrons) both in the stage of laser PLASMA PHYSICS REPORTS
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pulse action and after the end of the laser pulse. We plan to perform such a study in the future. The proposed simple analytical model can also be used to estimate the parameters of accelerated ions in experiments. For instance, under the experimental conditions of [2], where a relativistic laser pulse (a = 2 . 2 ) interacted with gaseous (helium) plasma, we find from Eq. (5) that τ τ=ω t ≈ 1. Such a value of the Li L characteristic time of ion acceleration yields the following estimate for the maximum velocity of accelerated ions: v max ≈ 2 . 5 × 10 8 cm/s, which is close to experimentally obtained value v max ≈ 3 . 8 × 10 8 cm/s. ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation, project no. 14-12-00194. REFERENCES 1. G. S. Sarkisov, V. Yu. Bychenkov, and V. T. Tikhonchuk, JETP Lett. 69, 20 (1999). 2. G. S. Sarkisov, V. Yu. Bychenkov, V. N. Novikov, V. T. Tikhonchuk, A. Maksimchuk, S.-Y. Chen, R. Wagner, G. Mourou, and D. Umstadter, Phys. Rev. E 59, 7042 (1999). 3. V. K. Tripathi, T. Taguchi, and C. S. Liu, Phys. Plasmas 12, 043106 (2005). 4. A. Lifschitz, F. Sylla, S. Kahaly, A. Flacco, M. Veltcheva, G. Sanchez-Arriaga, E. Lefebvre, and V. Malka, New J. Phys. 16, 033031 (2014). 5. A. Macchi, F. Ceccherini, F. Cornolti, S. Kar, and M. Borghesi, Plasma Phys. Controlled Fusion 51, 024005 (2009).
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Translated by I. Shumai
