1. Introduction

The generation of fast electron beams in relativistic laser-solid interaction has been a topic of intensive theoretical and experimental studies in last decades. Dense and collimated fast electron beams at different energy ranges are required for a variety of applications. For example, the concept of fast ignition in inertial confinement fusion uses a collimated fast electron beam to heat the ignition spot in the pre-compressed fuel core [1]. A variety of mechanisms for fast electron generation have been proposed, such as resonance absorption [2], vacuum heating [3], acceleration by laser ponderomotive force and J×B force [4], stochastic heating and acceleration [5], and direct laser acceleration of electrons in a laser self-focusing channel [6]. Obviously, these mechanisms work under different laser and plasma parameters and interaction configurations.

In this paper, we propose a mechanism of electron acceleration along the front solid-target surface when a laser pulse is incident obliquely at large angles. Our simulations show that large quasistatic magnetic and electric fields are generated near the target surface. These two fields will confine some electrons at the target surface, where the electron trajectories show typical betatron oscillations. At large angles of incidence such as over 60°, the reflected laser pulse is able to intersect with the betatron oscillation trajectories of these confined electrons. Some of them are accelerated significantly in a way similar to that in the laser self-focusing channel [6].

2. Numerical modeling

We have conducted two-dimensional (2D) particle-in-cell (PIC) simulation. The simulation parameters are as follows. The target is 5λ ₀ thick and 60λ ₀ long, which is tilted with respective to the simulation box to allow for the oblique incidence of a laser pulse. The linearly polarized laser pulse obliquely irradiates the target from the left side of the simulation box with an incident angle α = 70° . The focal spot diameter is 10λ ₀ with a Gaussian profile, where λ ₀ is the laser wave length in vacuum. The temporal profile of the laser pulse is a = a₀ sin²(πt/T) where a is the vector potential normalized by mω ₀ C/e . In the simulation, we take the peak amplitude a ₀ = 2.0 , and the pulse duration is T = 60π/μ ₀ = 30T ₀ with ω ₀ the laser frequency and T ₀ the laser oscillation period. The total size of the simulation box is 62.5λ ₀×60λ ₀, where X is the pulse propagation direction and Y is the polarized direction. The laser field is in p-polarization.

 

Fig. 1 Simulation geometry and selected electron trajectories along the target surface. The inset frame shows the initial target density profile along the normal to the target surface.

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Figure 1 shows the simulation geometry. Simulation results show there is a bunch of electrons emitted along the target surface. Some selected trajectories of these fast electrons are shown in the figure. Typically they oscillate for a few periods along the surface before leaving the interaction region.

To analyze this kind of electron motion we present snapshots of the electric and magnetic fields time-averaged over one laser cycle near the front target surface in Fig. 2. It is obvious that localized strong quasistatic electric and magnetic fields have been generated. At the time 40T₀ when the peak of the laser pulse arrives at the focus at the position (X,Y)=(25λ ₀,3Cλ ₀), all these quasistatic fields are located within the laser focus region. Note that the magnetic field is unipolar at the front surface with its peak located inside the solid target, as also discussed in Ref [7]. While the electric fields have two peaks, one inside and another outside the target. Another interesting point is that the quasistatic surface magnetic field still increases with time even after the pulse has been fully reflected, meanwhile its peak moves forward along the target surface, as shown in Fig. 2(d). In a simulation when the target is not long enough, the surface magnetic field is found to move up to the end of the target surface and then appears at the rear surface of the target. With the adopted parameters, the maximum mω ₀ c/e , appears at the time 60T₀. The quasistatic magnetic field results from the loop electric current formed by the return currents of cold electrons and the fast electron currents due to the vacuum heating and J×B heating [3,4,7], which produce a large number of fast electrons at moderate energies. The presence of these two quasistatic fields will produce significant effects on the emitted electrons at high energies.

 

Fig. 2 Distributions of the quasistatic electric field along the Y-direction (a), X-direction (b), and magnetic field along the Z-direction (c) after the laser pulse has propagated for 40 laser periods from the left incident border. (d) The magnetic field at the time of 60 laser periods. The white dashed lines represent the target front surface.

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3. Theory model for surface electron acceleration along the target surface

 

Fig. 3 Configuration of electron betatron oscillation and acceleration along the target surface.

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Let us consider an electron moving in the combined fields of reflected laser pulse and the quasistatic fields as shown in Fig. 3. For simplicity, we adopt a 2D (x-y) planar geometry with the target surface along x-direction and the target normal along y-direction. According to the simulation results shown in Fig. 2, we assume a simplified quasistatic field distribution $E_y^s$ = (κ_Ey/ λ ₀)(mω ₀ C/e), κ_E > 0 for the static electric component and $B_z^s$ = -(κ_By/λ ₀)(mω ₀ C/e) > 0, κ_B > 0 for y > 0 and $B_z^s$ = 0,κ_B > 0 for y ≥ 0 for the static magnetic component. As compared with quasistatic fields inside a laser self-focusing channel given in Ref. [6], in our case the quasistatic magnetic field is always positive and tends to reflect electrons out of the target surface while the quasistatic electric field tends to drag them back. We assume the reflected laser is a planar electro-magnetic wave with field $E_x^l$ = -E^l cos α, $E_y^l$ = E^l sinα, $B_z^l$ = E^l /v_ph and E¹ = E ₀ cos ω ₀ [t-(xsin α + ycos α)/v_ph ], where V_ph ≈ c is the phase velocity of the laser pulse and α is the incident angle of the laser pulse. The equation of electron motion:

We assume p_z = 0 at beginning and it can remain in the interaction. Using the expression of the quasistatic fields given above we can rewrite the equation for the transverse motion (2) as:

where γ=[1 -($V_x^2$ + $V_y^2$/c]^-1/2 and the frequency ω_β is given by

As we can see that the electrons will oscillate along the target surface with the betatron frequency ω_β if there is no external laser fields. Note that the restoring force for the electrons oscillating along the target surface is different in the different regions of the target. Outside of the target, the force comes from the static electric field; While inside the target, both the quasistatic electric and magnetic fields contribute to restore electrons as shown in Eq. (5). In the simulations we found p_x is always positive and quite large, thus the oscillation period inside the target is shorter than outside it. As a whole, the oscillation frequency is ω_β = (ω_β+ + ω_β- )/2 . Therefore, with the two quasistatic fields, electrons can be confined at the target surface while moving along it if the transverse momentum p_y is not so large, as shown in Fig. 1. However, if the latter is large enough, the last term on the right hand side of Eq. (4) can partially cancel the linear restoring forces. In this case, electrons are not confined.

Similar to the process in the laser channel reported by Pukhov et al. before [6], electron acceleration can occur during the electron oscillation along the target surface. By taking the relativistic transformation to a moving frame with the velocity V_xê_x , it is easy to find that in the new frame electrons just oscillate with the frequency ω_β′ = ω_β (1 - $V_x^2$/c ²)^-1/2 and the frequency of the laser fields is ω′₀ = ω₀ (1 - V_x sin α/ c)(1 - $V_x^2$/c ₂)^-1/2. When ω′_β = ω′₀ , i.e. (ω_β = ω₀ (1 - V_x sin (α/ c) the betatron motion of electrons in the two quasistatic fields E_l . For V_x ≈ c , we can get ${\kappa }_E^{1/2}$ +(κE + κ_B )^1/2 ≈ (8πγ)^1/2 (1 - sin α). Once the resonance occurs, electrons can be accelerated continuously by the laser fields provided that they are located in a suitable phase of the laser fields. In this case, the phase felt by the electrons will not be changed or changed very slowly. The resonant condition suggests that to keep the electrons being continuously accelerated, the quasistatic fields should increase along the target surface. This does occur in our simulations as illustrated by Figs. 2(c) and 2(d).

 

Fig. 4. Selected two electron trajectories [labeled with (1) and (2)] and their energy changes along the trajectories (a) and with time (b). The color bar in (a) shows the relativistic factor of the electrons. Frames (c), (d), and (e) show snapshots of the laser field (Ey), the vertical component of the quasistatic electric filed (〈E _⊥〉), and the quasistatic magnetic field (<Bz>), respectively, together with the two electron trajectories.

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In Fig. 4(a) and 4(b) we plot the trajectories of two selected electrons in space and their energy evolution, where the color bar indicates the electron energy. The electron labeled by the thin solid line (1) is accelerated almost continuously along its oscillating trajectory. But the electron labeled by the thick solid line (2) is accelerated only at the earlier time. It experiences deceleration later. The oscillating trajectories (trajectories before positions P1 and P2) of both electrons resemble to those in the betatron acceleration inside a laser self-focusing channel proposed by Pukhov et al. [6]. These trajectories are found near the target surface where both the laser fields and quasistatic fields are strong. The betatron acceleration is found during this stage. After the first acceleration phase, electrons are energetic enough to get rid of the confinement of the quasistatic electric fields and move to a region where the quasistatic fields are weak. In this region, the electrons only interact with the reflected laser fields. Because the velocity of the electrons is approximately the same as the phase velocity of the reflected laser, they can be accelerated further. The laser phase felt by the electrons determines acceleration or deceleration of the electrons just as Fig. 4(a) shows. It is obvious the two acceleration processes are dominant in different regions. For the two electron trajectories given in Fig. 4, it appears that particle (1) interacts weakly with the quasistatic fields as shown in Figs. 4(c)-4(e) and it is accelerated dominantly by the laser pulse in the second stage, while particle (2) is accelerated dominantly in the betatron oscillation process. With the parameters we have adopted, in our simulation totally 27.8% of the out emitted electrons are along the target surface with the spreading angle about 20°. The maximum energy of the electrons accelerated through these two acceleration processes is larger than 10MeV.

4. Concluding remarks

A new mechanism for fast electron emission and also the acceleration along the target surface during the relativistic laser solid-target interaction has been studied both numerically and analytically. It is shown that the self-generated quasistatic electric and magnetic fields along the target surface can confine electrons, which move along the target surface with betatron oscillation. The betatron oscillation trajectories overlap with the reflected laser fields sufficiently when the incident angle is large. For some electrons their betatron frequencies are close to the laser frequency in the particle’s frame. Therefore they can be accelerated resonantly along the target surface. This surface betatron acceleration mechanism provides a possibility for controlling the emission of surface electron bunches, which will benefit for future wide applications.