Time t is normalized to the inverse electron plasma frequency ${\omega }_{pe}^{-1}$, length is normalized to $l_0=c{\omega }_{pe}^{-1}$, velocity and momentum are normalized respectively to the velocity of light c and to $m_{e}c$. The general form of the Vlasov equation is written for the present problem first in a 4D phase-space for the electron distribution function $F_e\left(x,p_{xe},p_{ye},p_{ze},t\right)$ and the ion distribution function $F_i\left(x,p_{xi},p_{yi},p_{zi},t\right)$ (one spatial dimension) as follows:

$\frac{\partial F_{e,i}}{\partial t}+M_{e,i}\frac{p_{xe,i}}{{\gamma }_{e,i}}\frac{\partial F_{e,i}}{\partial x}\mp \left(E+\frac{p_{e,i}\times B}{{\gamma }_{e,i}}\right)\cdot \frac{\partial F_{e,i}}{\partial p_{e,i}}=0$ (1)

With

${\gamma }_{e,i}={\left(1+M_{e,i}^2\left(p_{xe,i}^2+p_{ye,i}^2+p_{ze,i}^2\right)\right)}^{1/2}$ (2)

(the upper sign in Equation (1) is for electrons and the lower sign for ions, and subscripts e or i denote electrons or ions respectively). In our normalized units $M_e=1$ and $M_i=\frac{m_e}{m_i}$.

We write the Hamiltonian of a particle in the electromagnetic field of the wave:

$H_{e,i}=\frac{1}{M_{e,i}}\left({\gamma }_{e,i}-1\right)\mp \phi \left(x,t\right)$. (3)

where $\phi \left(x,t\right)$ is the scalar potential. Equation (1) can be reduced to a two-dimensional phase-space Vlasov equation as follows. The canonical momentum $P_{ce,i}$ is connected to the particle momentum $p_{e,i}$ by the relation $P_{ce,i}=p_{e,i}\mp a$. $a=eA/m_{e}c$ is the normalized vector potential. From Equation (3), we can write:

$H_{e,i}=\frac{1}{M_{e,i}}\left({\left(1+M_{e,i}^2{\left(P_{ce,i}\pm a\right)}^2\right)}^{1/2}-1\right)\mp \phi \left(x,t\right)$. (4)

Choosing the Coulomb gauge ( $diva=0$), we have for our one dimensional problem $\frac{\partial a_x}{\partial x}=0$, hence $a_x=0$. The vector potential $a=a_{\perp }\left(x,t\right)$, and we also have the following relation along the longitudinal direction:

$\frac{\text{d}{P}_{cxe,i}}{\text{d}t}=-\frac{\partial H_{e,i}}{\partial x}$ (5)

And since there is no transverse dependence

$\frac{\text{d}{P}_{c\perp e,i}}{\text{d}t}=-{\nabla }_{\perp }{H}_{e,i}=0$. (6)

This last equation means $P_{c\perp e,i}=\text{const}$. We can choose this constant to be zero without loss of generality, which means that initially all particles at a given (x, t) have the same perpendicular momentum $p_{\perp e,i}=\pm a_{\perp }\left(x,t\right)$. The Hamiltonian is now written:

$H_{e,i}=\frac{1}{M_{e,i}}\left({\left(1+M_{e,i}^2{p}_{xe,i}^2+M_{e,i}^2{a}_{\perp }^2\left(x,t\right)\right)}^{1/2}-1\right)\mp \phi \left(x,t\right)$. (7)

The 4D distribution function $F_{e,i}\left(x,p_{xe,i},p_{\perp e,i},t\right)$ can now be reduced to a 2D distribution function $f_{e,i}\left(x,p_{xe,i},t\right)$:

$F_{e,i}\left(x,p_{xe,i},p_{\perp e,i},t\right)=f_{e,i}\left(x,p_{xe,i},t\right)\delta \left(p_{\perp e,i}\mp a_{\perp }\right)$. (8)

$f_{e,i}\left(x,p_{xe,i},t\right)$ verify the relation:

$\frac{\text{d}{f}_{e,i}}{\text{d}t}=\frac{\partial f_{e,i}}{\partial t}+\frac{\partial H_{e,i}}{\partial p_{xe,i}}\frac{\partial f_{e,i}}{\partial x}-\frac{\partial H_{e,i}}{\partial x}\frac{\partial f_{e,i}}{\partial p_{xe,i}}=0$. (9)

Which gives the following Vlasov equations for the electrons and the ions::

$\frac{\partial f_{e,i}}{\partial t}+M_{e,i}\frac{p_{xe,i}}{{\gamma }_{e,i}}\frac{\partial f_{e,i}}{\partial x}+\left(\mp E_x-\frac{M_{e,i}}{2{\gamma }_{e,i}}\frac{\partial a_{\perp }^2}{\partial x}\right)\frac{\partial f_{e,i}}{\partial p_{xe,i}}=0$. (10)

where ${\gamma }_{e,i}={\left(1+{\left(M_{e,i}{p}_{xe,i}\right)}^2+{\left(M_{e,i}{a}_{\perp }\right)}^2\right)}^{1/2}$.

$E_x=-\frac{\partial \phi }{\partial x}$ and $E_{\perp }=-\frac{\partial a_{\perp }}{\partial t}$ (11)

and Poisson’s equation is given by:

$\frac{{\partial }^2\phi }{\partial x^2}={\displaystyle \int f_e\left(x,p_{xe}\right)\text{d}{p}_{xe}}-{\displaystyle \int f_i\left(x,p_{xi}\right)\text{d}{p}_{xi}}$ (12)

An alternative way, used in the present work, is to calculate $E_x^{}$ from Ampère’s equation: $\frac{\partial E_x}{\partial t}=-J_x$,

The transverse electromagnetic fields $E_y,B_z$ and $E_z,B_y$ for the circularly polarized wave obey Maxwell’s equations. With $E^{\pm }=E_y\pm B_z$ and $F^{\pm }=E_z\pm B_y$, we write these equations in the following form:

$\left(\frac{\partial }{\partial t}\pm \frac{\partial }{\partial x}\right)E^{\pm }=-J_y;\left(\frac{\partial }{\partial t}\mp \frac{\partial }{\partial x}\right)F^{\pm }=-J_z$ (13)

Which are integrated along their vacuum characteristic x = t [21] - [29] . In our normalized units we have the following expressions for the normal current densities:

$J_{\perp }=J_{\perp e}+J_{\perp i}\text{;}{J}_{\perp e,i}=-a_{\perp }{M}_{e,i}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{f_{e,i}}{{\gamma }_{e,i}}\text{d}{p}_{xe,i}}$. (14)

We use a method of characteristics for the numerical solution of Equation (10), as discussed in [21] - [31] , for the numerical solution of hyperbolic type partial differential equations These methods are Eulerian methods which use a computational mesh to discretize the equations on a fixed grid, and have been successfully applied to different important problems in plasma physics (see for instance the articles [21] - [31] ).

The numerical scheme to advance Equation (10) from time $t_n$ to $t_{n+1}$ necessitates the knowledge of the electromagnetic field $E^{\pm }$ and $F^{\pm }$ at time $t_{n+1/2}$. This is done using a centered scheme where we integrate Equation (13) exactly along the characteristics with $\Delta x=\Delta t$, to calculate $E^{\pm n+1/2}$ and $F^{\pm n+1/2}$ as follows:

$E^{\pm }\left(x\pm \Delta t,t_{n+1/2}\right)=E^{\pm }\left(x,t_{n-1/2}\right)-\Delta t{J}_y\left(x\pm \Delta t/2,t_n\right)$ (15)

with $J_y\left(x\pm \Delta t/2,t_n\right)=\frac{J_y\left(x\pm \Delta x,t_n\right)+J_y\left(x,t_n\right)}{2}$

A similar equation can be written for $F^{\pm n+1/2}$ with J_z. From Equation (11), we also have $a_{\perp }^{n+1}=a_{\perp }^n-\Delta t{E}_{\perp }^{n+1/2}$, from which we calculate $a_{\perp }^{n+1/2}=\left(a_{\perp }^{n+1}+a_{\perp }^n\right)/2$. We also have:

$J_x^n=M_i{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{p_{xi}}{{\gamma }_i}{f}_i^n\text{d}{p}_{xi}}-M_e{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{p_{xe}}{{\gamma }_e}{f}_e^n\text{d}{p}_{xe}}$ (16)

We calculate $E_x^{n+1/2}$ from Ampère’s equation: $\frac{\partial E_x}{\partial t}=-J_x$, from which $E_x^{n+1/2}=E_x^{n-1/2}-\Delta t{J}_x^n$. Now given $f_{e,i}^n$ at mesh points (we stress here that the subscript i denotes the ion distribution function), we calculate the new value $f_{e,i}^{n+1}$ at mesh points from the relation [21] - [31] :

$f_{e,i}^{n+1}\left(x,p_{xe,i}\right)=f_{e,i}^n\left(x-2{\Delta }_{xe,i},p_{xe,i}-2{\Delta }_{p_{xe,i}}\right)$. (17)

${\Delta }_{xe,i}$ and ${\Delta }_{p_{xe,i}}$ are calculated from the solution of the characteristics equations for Equation (10), which are given by:

$\begin{array}{l}\frac{\text{d}x}{\text{d}t}=M_{e,i}\frac{p_{xe,i}}{{\gamma }_{e,i}}\\ \frac{\text{d}{p}_{xe,i}}{\text{d}t}=\mp E_x-\frac{M_{e,i}}{2{\gamma }_{e,i}}\frac{\partial a_{\perp }^2}{\partial x}\end{array}$ (18)

This calculation is effected as follows. We rewrite Equation (17) in the vectorial form:

$f_{e,i}^{n+1}\left(X_{e,i}\right)=f_{e,i}^n\left(X_{e,i}-2{\Delta }_X{}_{e,i}\right)$. (19)

$X_{e,i}$ is the two dimensional vector $X_{e,i}=\left(x,p_{xe,i}\right)$, and ${\Delta }_X{}_{e,i}=\left({\Delta }_{xe,i},{\Delta }_{p_{xe,i}}\right)$ is the two dimensional vector calculated from the implicit relation [21] - [31] :

${\Delta }_X{}_{e,i}=\frac{\Delta t}{2}{V}_{e,i}\left(X_{e,i}-{\Delta }_X{}_{e,i},t_{n+1/2}\right)$. (20)

$V_{e,i}=\left(M_{e,i}\frac{p_{xe,i}}{{\gamma }_{e,i}},\mp E_x^{n+1/2}-\frac{M_{e,i}}{2{\gamma }_{e,i}}\frac{\partial {\left(a_{\perp }^{\left(n+1/2\right)}\right)}^2}{\partial x}\right)$. Equation (20) for ${\Delta }_X{}_{e,i}$ is implicit and is solved iteratively: ${\Delta }_{Xe,i}^{k+1}=\frac{\Delta t}{2}{V}_{e,i}\left(X_{e,i}-{\Delta }_{Xe,i}^k,t_{n+1/2}\right)$, where we start the iteration with ${\Delta }_{Xe,i}^0=0$ for iteration number k = 0. Two or three iterations are sufficient to get a good convergence. Then $f_{e,i}^{n+1}$ is calculated from $f_{e,i}^n$ in Equation (17) by calculating the shifted value using a two-dimensional interpolation in the two dimensional phase-space $\left(x,p_{xe,i}\right)$. Similarly in Equation (20), the shifted value was calculated at every iteration using a two-dimensional interpolation. In all cases, these two-dimensional interpolations are executed using a tensor product of cubic B-splines [21] - [23] . As mentioned, this code applies a numerical scheme based on a two-dimensional advection technique, of second order accuracy in time-step, where the value of the distribution function is advanced in time by interpolating in two dimensions along the characteristics using a tensor product of cubic B-splines. Cubic splines have been tested in several applications and shown to give accurate results [21] - [23] . See also [28] for the integration of the Vlasov equation in one and two dimensions.

The forward propagating circularly polarized laser beam enters the system at the left boundary (x = 0), where the forward propagating fields are $E^{+}=2E_0{P}_r\left(t\right)\cos \left(\tau \right)$ and $F^{-}=-2E_0{P}_r\left(t\right)\sin \left(\tau \right)$, as defined in Equation (13). Time is in units of ${\omega }^{-1}$. In these expressions, $\tau =t-1.5t_p$ and $t_p=24$ is the pulse duration at full width at half maximum of the beam intensity. In our normalized units $E_0=a_0$. For a Gaussian time dependence, the shape factor $P_r\left(t\right)$ is given by:

$P_r\left(t\right)=\exp \left(-2\ln \left(2\right){\left(\tau /t_p\right)}^2\right)$ (21)

so that the forward propagating laser beam behaves as a Gaussian pulse, which reaches its peak at $t=1.5t_p=36$.(The units of time and space can be easily translated in units of ${\omega }_{pe}^{-1}$ and $c/{\omega }_{pe}$ by multiplying them by a factor of 5, since ${\omega }_{pe}/\omega =5$, which corresponds to $n/n_{cr}=25$, since the laser frequency $\omega /{\omega }_p=1/\sqrt{n/n_{cr}}=0.2$).

A characteristic parameter of high power laser beams is the normalized vector potential or quiver momentum $\left|a_{\perp }\right|=\left|e{A}_{\perp }/M_{e}c\right|=a_0$, where $A_{\perp }$ is the vector potential of the wave. We chose for the amplitude of the vector potential. $a_0=25/\sqrt{2}$ For the circularly polarized wave $2a_0^2=I{\lambda }_0^2/1.368{\text{x10}}^{\text{18}}$, I being the intensity in W/cm², and ${\lambda }_0$ the laser wavelength in microns. We use N = 5500 grid points in space, and in momentum space we use 1600 grid points for the electrons and 4800 grid points for the ions (extrema of the electron momentum are ±4, and ±360 for the ion momentum). The Lorentz factor for the transverse oscillation of an electron in the field of the wave is ${\gamma }_0=\sqrt{1+a_0^2}=17.7$. Deuterium ions are used with $M_i/M_e=2\times \text{1836}$, where M_i is the deuterium mass. The initial distribution functions for electrons and ions are Maxwellians with temperatures T_e = 1 keV for the electrons and T_i = 0.1 keV for the deuterons. The length of the system is L = 22 in units of $c/{\omega }_{pe}$. From the parameters presented we have a grid spacing $\Delta x=0.004$, and we use a time step $\Delta t=\Delta x$. We have a vacuum region of length L_vac = 9.0 on each side of the plasma slab. The steep ramp in density at the plasma edge on each side of the flat top density of the slab target has a length of $L_{edge}=0.3$. The length of the central plasma slab, with flat top density of 1 (normalized to $25n_{cr}$), is $L_p=2.884$, for a total length of 22. In our units, the skin depth $c/{\omega }_{pe}=\frac{c}{\omega }\frac{\omega }{{\omega }_{pe}}=0.2\frac{c}{\omega }$. So the length of the large plasma slab is about $L_p\approx 15$ skin depth $c/{\omega }_{pe}$. In free space $\omega =k$ for the electromagnetic wave, and as ${\lambda }_0=2\pi$, the condition that ${\lambda }_0\gg L_{edge}$ is very well satisfied.

We first present in Figure 1 some of the results obtained for the density profiles against distance. The electron and ion density profiles at times t = 24, 40, 43, 50, 58, and 77 (time is in units of ${\omega }^{-1}$) are given in Figure 1 (full curve for the electrons, broken curves for the ions). The electric field (dashed-dotted curve) is also plotted divided by a factor of 5, to have it on the same figure. We concentrate these plots in the region where the plasma slab is located, since otherwise on both sides we have vacuum. In Figure 1 at t = 24 electrons are pushed and accelerated at the wave-front plasma-edge interface under the effect of the radiation pressure and form a steep density spike. Ions (dashed curve) have moved very little, and their profile gives essentially the initial plasma slab profile. An electric field is formed at the edge due to the charge separation appearing there, and its penetration in the plasma at the sharp gradient at the surface of the target is of the order of about $0.2c/\omega$, which is the value of the skin depth in our units as we previously mentioned. In Figure 1 at t = 40 the steepening of the electrons is increased, due to the increase in the radiation pressure caused by the increase of the incident wave amplitude. The ion density profile is also increasing at the target surface.

And then, very rapidly, there is a huge jump in the ion density at the target edge, as shown in Figure 1 at t = 43, reaching a peak of about 27. We note from Equation (21) that the laser intensity reaches a peak at 3t_p/2 = 36 at the boundary x = 0. This peak has to travel a distance of about x = 9 to reach the target surface (see for instance the position of the edge in Figure 1 at t = 24). So the maximum laser intensity will reach the target surface at about t ≈ 45, i.e. about the time where we observe a strong acceleration of the ions at the target surface in Figure 1 at t = 43. In Figure 1 at t = 50 we see the electron peak separating from the ion peak, which decreases compared to the peak in Figure 1 at t = 43. In Figure 1 at t = 58 the electron peak seems to come back closer to the ion peak, and a shock-like structure of an expanding neutral plasma jet propagating to the right is formed. In Figure 1 at t = 77 we see to the right two electron peaks coinciding with the two ion peaks, and the structure of a neutral plasma jet propagating in the forward direction. The solitary-like structure of the neutral plasma propagating to the right is now very close to the position of the back edge position of the initial target at about x = 13. In Figure 1 at t = 77 the electron and ion peaks to the left at the target surface are decreasing, due to the decay of the intensity of the Gaussian incident laser pulse, which is now exercising negligible pressure at the target surface at the left (see Figure 5 at t = 77). At this stage this wave-front plasma-edge interface is not propagating to the right anymore, since the radiation pressure has vanished. We have in Figure 1 at t = 77 a neutral plasma jet population at the right, with two peaks, which is now emerging from the back of what was the target and travelling in the forward direction to the right. And at the left, a standing wave-front plasma-edge interface.

Figure 2 shows the phase-space contour plots of the ion distribution function at t = 24, 43 (during the rapid acceleration of the ions at the target surface presented in Figure 1 ) at t = 43), 50, and 77. The velocity of the free streaming jet at the right can be calculated from Figure 2 at t = 77, for instance, where we see the edge of the free streaming deuterium ions in the forward direction having a normalized momentum around 350. Hence $M_i\upsilon /M_{e}c\approx 350$, or $\upsilon /c\approx 350/\left(2\times \text{1836}\right)=0.095$. This corresponds to the energy of the deuterium ions of $M_i{\upsilon }^2/2\approx M_i{c}^2{\left(\upsilon /c\right)}^2/2\approx 938\times 0.009=8.46\text{MeV}$. The appearance of loops at the top in Figs. (2) at t = 50, t = 77 appear because slower ions (lower part of the loop), which were accelerated by the shock during the rise of the laser pulse, are caught up by faster ones (upper part of the loop), which were accelerated by the pulse when it was at its maximum. The rightmost parts of these loops correspond to the shock appearing at the right of the expanding neutral plasma edge, as we see for instance in Figure 1 at t = 77. These loops are close to what has been presented in [29] using PIC codes. See also the results in [30] [31] .

Figure 3 presents the phase-space contour plots of the electron distribution function, at the same time as the corresponding phase-space contour plots of the ion distribution function presented in Figure 2 . The steep edge at the target surface is due to the effect of the ponderomotive pressure pushing the electrons. In particular, in the initial phase, Figure 3 at t = 24 shows on the upper side of the contour small electron jets, a picture close to what has been reported in [32] , and in [33] for the study of wave breaking and electron acceleration using a Vlasov code (a more detailed presentation of this phase of wave breaking is presented in Figure 6 below).

Note in Figure 3 at t = 43 and t = 50, beyond the solitary structure at the sharp front edge of the target, the long tail remaining from the initial electron distribution function, which is slowly disappearing. At t = 77 there is no ponderomotive pressure of the wave at the target surface (see Figure 5(c) below), the plasma at this surface location at x = 11.5 in Figure 1 at t = 77 is not moving anymore, and the electron and ion densities at this location are decaying slowly, with electrons slightly expanding in the backward direction as seen in Figure 1 at t = 77. But beyond the target surface, the neutral plasma jet is propagating to the right in the forward direction, at a constant free-streaming speed as discussed above. Note in the right of Figure 3 at t = 77 the two peaks of the neutral plasma jet we see at the right of Figure 1 at t = 77.

We follow in Figure 4 at t = 86 the two bump free-streaming plasma jet structure at the right in Figure 3 at t = 77. We concentrate the phase-space plots in Figure 4(a) on the ions, in Figure 4(b) on the electrons, and in Figure 4(c) we present the corresponding density and electric field plots.

We have indeed a stable solitary-like structure with a double peak, of a neutral

plasma free-streaming at constant speed to the right. The electron amd ion density curves (full curve and dashed curve respectively) are identical. An estimation of this speed can be obtained by following the right boundary of the jet. In Figure 1 at t = 77 the right boundary is located at x = 13.4. In Figure 4(c) the right boundary is at x = 14.3 at t = 86. This corresponds to a speed of 0.1, very close to the value of 0.095 estimated above from Figure 2 at t = 77.

Figure 5 presents in the left figures the forward propagating wave $E^{+}$ (full curves) and the backward reflected wave $E^{-}$ (dashed curves), and in the right figures presents the corresponding results for the forward propagating wave $F^{-}$(full curves) and the backward propagating wave $F^{+}$ (dashed curves).

In Figure 5(a) the curves are calculated at t = 43 at the time of the rapid acceleration of the ions at the target surface observed in Figure 1 at t = 43, when the maximum of the wave at the target surface is very close to the peak of the pulse. From Equation (21), the laser intensity reaches a peak at 3t_p/2 = 36 at the boundary x = 0. This peak has to travel a distance of about x = 9 to reach the target surface (see Figure 1 at t = 24). So the maximum laser intensity will reach the target surface around t ≈ 45, i.e. about the time where we see the strong acceleration

of the ions at the target surface in Figure 1 at t = 43. The electromagnetic wave damps in the plasma over a distance of the order of the skin depth $c/{\omega }_{pe}$, equal to 0.2 in our normalized units. The strong increase of the ion and electron densities at the wave-front plasma-edge interface makes the plasma more opaque, with the steep plasma edge acting as a moving mirror for the incident light. Note that when $E^{+}$ is at a peak, the value of $F^{-}$ is zero and vice-versa, so the wave is always maintaining a pressure on the surface of the plasma., until the decay of the Gaussian pulse. The frequency of the backward reflected wave is slightly down-shifted by the moving reflecting plasma surface. We can check that in Figure 5 , the reflected waves (dashed curves) in vacuum have a wavelength (and a period) slightly bigger than the corresponding ones for the incident wave (full curves). Figure 5(b) presents the incident and reflected waves at t = 60, at the time of the acceleration we observe in Figure 1 at t = 77, and in Figure 5(c) at t = 77 the Gaussian pulse is decaying and the ponderomotive pressure of the wave at the target surface is negligible. The neutral two-peaks structure at the right of Figure 1 at t = 77 is now free streaming as we previously mentioned, with a constant speed of 0.1.

We finally present in Figure 6 a sequence showing the wave breaking at the front edge of the electron phase-space, as presented in Figure 3 at t = 24. This is due to the longitudinal field generated at the front edge of the plasma due to the electron-ion separation. We see in the sequence of Figure 6 how peaks are building up at the target surface in the electron phase-space, then the accelerated electrons are ejected forming this finger-like structure we observe. The sequence is presented at t = 15.5, 17, and 20.

We have presented in Figure 6 the finger-like structures in the electron distribution function due to a wave-breaking mechanism, because these structures are well resolved by the fine (x, p_x) computational mesh of the Vlasov code, a picture close to what has been reported in [32] , and in [33] for the study of wave breaking and electron acceleration using a Vlasov code. Our presentation of this phase of wave breaking we presented in Figure 6 contains more details because of the precision of the Vlasov code.

The detailed analysis presented above in this section for the formation and ejection at constant speed of the neutral structure of a plasma jet with two peaks at t = 77, when the Gaussian electromagnetic pulse has damped, and applies no more pressure on the neutral population advancing at a constant speed as mentioned in the discussion above, testify to the power of the Eulerian codes for the numerical resolution of the details of this problem. Similar neutral plasma jets are obtained in several simulations where the ponderomotive pressure of the wave plays an important role in the formation and acceleration of the neutral plasma jet. See, for instance Figs. (1) and (5) in [7] , and Fig. 1 in [34] .

Time t is normalized to the inverse electron plasma frequency ${\omega }_{pe}^{-1}$, velocity and momentum are normalized respectively to the velocity of light c and to M_ec, where M_e is the electron mass, length is normalized to $l_0=c{\omega }_{pe}^{-1}$, and the electric field is normalized to $E_0=M_e{\omega }_{pe}c/e$. The 1D relativistic Vlasov equations for the electrons and ions distribution functions $f_{e,i}\left(x,p_{xe,i},t\right)$ verify the relation [35] :

$\frac{\partial f_{e,i}}{\partial t}+\frac{p_{xe,i}}{m_{e,i}{\gamma }_{e,i}}\frac{\partial f_{e,i}}{\partial x}\mp \left(E_x+{\left(u_{\perp e,i}\times B_{\perp }\right)}_x\right)\cdot \frac{\partial f_{e,i}}{\partial p_{xe,i}}=0$ (22)

where ${\gamma }_{e,i}\approx {\left(1+{\left(p_{xe,i}/m_{e,i}\right)}^2\right)}^{1/2}$. The upper sign in Equation (22) is for the electron equation and the lower sign for the ion equation, and subscripts e and i denote electrons and ions, respectively. In our normalized units $m_e=1$, and $m_i=M_i/M_e=1836$ is the ratio of hydrogen ion to electron masses. We use a lower case letter for the normalized masses in this section, to differentiate from the upper case letter for the normalized masses for the deuterium used in Section 2. In this 1D model, the transverse velocity $u_{\perp e,i}\left(x,t\right)$ is assumed non-relativistic and is calculated from a transverse “fluid” equation, but the kinetic features of the plasma are fully preserved along the external magnetic field in the longitudinal direction x. The transverse momentum $p_{\perp e,i}\left(x,t\right)=m_{e,i}{u}_{\perp e,i}$ is obtained through the introduction of the generalized canonical momentum defined by $P_{\perp e,i}\left(x,t\right)=p_{\perp e,i}\left(x,t\right)\mp a_{\perp }\left(x,t\right)$ ( $A$ is the vector potential of the wave, and $a=eA/M_{e}c$ is the normalized vector potential), which obeys the ‘fluid’ equation [35] :

$\left(\frac{\partial }{\partial t}+u_{xe,i}\frac{\partial }{\partial x}\right)P_{\perp e,i}=P_{\perp e,i}\times {\Omega }_{ce,i}\pm a_{\perp }\times {\Omega }_{ce,i}$ (23)

With $u_{xe,i}={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{p_{xe,i}}{m_{e,i}{\gamma }_{e,i}}{f}_{e,i}\text{d}{p}_{xe,i}}$.

${\Omega }_{ce,i}={\Omega }_{ce,i}{e}_x$, where ${\Omega }_{ce,i}$ is the electron (ion) cyclotron frequency, and $e_x$ is the unit vector in the direction of the external magnetic field.

We adopt the Coulomb gauge $\nabla \cdot a=0$, where the vector potential is in the perpendicular plane. The longitudinal electric field $E_x$ is calculated either from the gradient of a potential $E_x=-\partial \phi /\partial x$ or from Ampére’s law:

$\frac{\partial E_x}{\partial t}=-J_x,J_x={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{p_{xi}}{m_i{\gamma }_i}{f}_i\text{d}{p}_{xi}}-{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{p_{xe}}{m_e{\gamma }_e}{f}_e\text{d}{p}_{xe}}$ (24)

The transverse electric field $E_{\perp }$ is calculated from the relations:

$E_{\perp }=-\frac{\partial a_{\perp }}{\partial t}$ (25)

The transverse electromagnetic fields $E_y,B_z$ and $E_z,B_y$ for the circularly polarized wave obey Maxwell’s equations. Defining $E^{\pm }=E_y\pm B_z$ and $F^{\pm }=E_z\pm B_y$ we have the following equations:

$\left(\frac{\partial }{\partial t}\pm \frac{\partial }{\partial x}\right)E^{\pm }=-J_y;\left(\frac{\partial }{\partial t}\mp \frac{\partial }{\partial x}\right)F^{\pm }=-J_z$ (26)

which are integrated along their vacuum characteristics $x=\pm t$. In our normalized units we have the following expressions for the normal current densities:

$J_{\perp }=J_{\perp e}+J_{\perp i}\text{;}{J}_{\perp e,i}=\mp n_{e,i}\left(x,t\right)u_{\perp e,i}\left(x,t\right);n_{e,i}\left(x,t\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{f}_{e,i}\text{d}{p}_{xe,i}}$ (27)

The numerical scheme applies a direct solution method of the Vlasov equation as a partial differential equation in phase-space. In Equation (22) ${\gamma }_{e,i}\approx {\left(1+{\left(p_{xe,i}/m_{e,i}\right)}^2\right)}^{1/2}$ is a function $p_x$ only, which allows to apply a time-splitting scheme with the separation of $x$ and $p_x$ variables by a method of fractional steps, as discussed for Eulerian Vlasov codes in several publications [21] - [23] [35] - [38] .

Results

The normalized vector potential or quiver momentum for the laser beam is $a_{0p}$. A forward propagating circularly polarized laser beam of constant amplitude enters the system at the left boundary, where for the forward pump $E^{+}=E_y+B_z=2E_{0p}\cos \left({\omega }_{0p}t\right)$ and $F^{-}=E_z-B_y=2E_{0p}sin\left({\omega }_{0p}t\right)$ at x = 0, ${\omega }_{0p}=3.2$ (normalized to ${\omega }_{pe}$), and $a_{0p}=0.02$. In our normalized units $E_{0p}={\omega }_{0p}{a}_{0p}$. We use a system of length L = 600. The number of grid points in space is N = 120000, so $\Delta x=\Delta t=0.005$. The extrema of the momentum for the electrons are ±0.5, with 1000 grid points in velocity space, and for the ions these extrema are ±1.5 × 10⁻³ with 500 grid points in velocity space.

The pump precursor reaches the right boundary x = 600 at t = 600, since in our normalized units x = t. The seed pulse is injected at the right boundary x = 600 towards the backward direction just before the arrival of the pump, in the time interval 540 < t < 600, with $E^{-}=E_y-B_z=2E_{0s}{P}_r\left(t\right)\cos \left({\omega }_{0s}\left(t-540\right)\right)$, $F^{+}=E_z+B_y=-2E_{0s}{P}_r\left(t\right)sin\left({\omega }_{0s}\left(t-540\right)\right)$. The SBS produces a very small frequency downshift in the scattered wave ${\omega }_{0p}={\omega }_{0s}+{\omega }_{ion}$. We neglect this small downshift ${\omega }_{ion}\ll 1$, and set the seed frequency ${\omega }_{0s}\approx {\omega }_{0p}=3.2$, therefore $k_{0p}\approx k_{0s}$. The SBS is backward propagating, so the selection rule $k_{0p}=-k_{0s}+k_{ion}$ will result in an ion wave $k_{ion}\approx 2k_{0p}$. Also $a_{0p}=a_{0s}=0.02$ and $E_{0s}=E_{0p}$. The initial distribution functions for electrons and ions are Maxweellian with temperatures T_e = 0.3 keV and T_i = 0.02 keV. respectively. The shape factor P_r(t) for the injected seed pulse has a Gaussian time dependence $P_r\left(t\right)=\exp {\left(-0.5\left(t-570\right)/t_p\right)}^2$, for 540 < t < 600, with t_p = 10. The backward propagating seed pulse reaches its peak at t = 570, and has completely penetrated the plasma at t = 600, i.e. the time the pump precursor reaches the right boundary. The electron cyclotron frequency (normalized to the electron plasma frequency) is ${\omega }_{ce}=0.5$. From the linear dispersion relation $k^2={\omega }^2-1/\left(1\mp {\omega }_{ce}/\omega \right)$ the corresponding wavenumber for the left hand circularly polarized (LCP) wave is 3.062, and for the right hand circularly polarized wave (RCP) the wavenumber is 3.01.

Figure 7 shows at t = 1200 successively: in the first frame the incident pump wave $E^{+}$(full curve) and the growing backward seed pulse $E^{-}$ (dashed curve), in the second frame is the longitudinal electric field $E_x$, the third and fourth frames show the electron and ion density.

Figure 8 presents contour plots for the electron distribution function, and Figure 9 for the ion distribution function, at t = 1200. In Figure 10 , the wavenumber spectra associated with the forward wave $E^{+}$, the backward wave $E^{-}$, the longitudinal electric field $E_x$ and the ion density, are calculated by Fourier transform of the corresponding signal in the domain x = (75,403) at t = 1200, at a time where the backward seed pulse has reached the left boundary. The spectrum of $E^{+}$ in Figure 10 ) shows the peak at $k_{0p}=3.049$ (the linear dispersion relation calculates a wavenumber of 3.062 for the LCP wave for a frequency of 3.2 as mentioned above). The same mode is appearing in the wavenumber spectrum of the backward wave $E^{-}$.

The presence of these two modes will result in the appearance of a SBS mode at $k_{ion}\approx 2k_{0p}=6.089$, very close to the peak at 6.12 in the wavenumber spectrum of the ion density in Figure 10 . Note the harmonics of 6.12 at 12.25, 18.37 and 24.5. We note in the wavenumber spectrum of $E^{+}$ the presence of a forward scattered mode at 2.01, with the selection rule 3.049 = 2.01 + k, from which k = 1.039, very close to the peak at 1.05 in the spectrum of $E_x$. The corresponding frequencies we see in Figure 11 ) by taking the temporal Fourier transform of the signals registered at x = 90 in the time t = (1070, 1234) give a frequency of 3.2 for the pump $E^{+}$ and 2.186 for the forward scattered mode (the corresponding wavenumber calculated from the linear dispersion relation is 1.991, very close to the value of 2.01 we see in the wavenumber spectrum of $E^{+}$). They obey the selection rule $3.2=2.186+\omega$, from which a plasma mode at $\omega =1.014$, very close to the peak at 1.035 we see in the frequency spectrum of $E_x$ in Figure 11 . Also in the wavenumber spectrum of $E^{-}$ a backward scattered mode at $k_s=2.01$ is dominating the spectrum, such that $k_{0p}=-k_s+k_e$, from which $k_e=3.049+2.01=5.059$, very close to the value of 5.043 we see in the wavenumber spectrum of $E_x$. Note the harmonics of 5.043 at 10.086, 15.13 and 20.13. The forward scattered mode at 2.01 and the backscattered mode at −2.01 will couple through Brillouin scattering to give another SBS mode at $k_{ion}\approx 2\times 2.01=4.02$, very close to the value of 4.0 we see in the ion spectrum in Figure 10 ). So there are two SBS events, at 4.0 and 6.12. Other resonances are also

present. We see in the spectrum of $E^{+}$ the anti-Stokes at 4.103 obtained by the coupling of the plasma mode at 1.05 and the pump $E^{+}$ at the wavenumber 3.049, such that 3.049 + 1.05 = 4.099, very close to the value of 4.103 in the wavenumber spectrum of $E^{+}$, with the corresponding frequencies 3.2 + 1.035 = 4.235, very close to the value of 4.29 we see in the frequency spectrum of $E^{+}$ in Figure 11 ).

Again, accurate results have been presented in Section 3 for the problem of the amplification of seed pulses by Brillouin backscattering in magnetized plasmas. For more information, see [8] - [10] . The relevance to the amplification of whistler pulses along the Earth’s magnetic field, and along the magnetic field in other planetary and space plasmas situations, is an example of how space plasma can be used as a laboratory to test the plasma theory.