Under the grazing incidence of a relativistic intense laser pulse onto a solid target, two-dimensional particle-in-cell simulations show that intense quasistatic magnetic and electric fields are generated near the front target surface during the interaction. Some electrons are confined in these quasistatic fields and move along the target surface with betatron oscillations. When this oscillating frequency is close to the laser frequency in the particle frame, these electrons can be accelerated significantly in the reflected laser field, similar to the inverse free-electron-laser acceleration. An analytical model for this surface betatron acceleration is proposed.
© 2006 Optical Society of America
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 . A variety of mechanisms for fast electron generation have been proposed, such as resonance absorption , vacuum heating , acceleration by laser ponderomotive force and J×B force , stochastic heating and acceleration , and direct laser acceleration of electrons in a laser self-focusing channel . 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 .
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.
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 . 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.
3. Theory model for surface electron acceleration along the target surface
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 = (κEy/ λ 0)(mω 0 C/e), κE > 0 for the static electric component and = -(κBy/λ 0)(mω 0 C/e) > 0, κB > 0 for y > 0 and = 0,κB > 0 for y ≥ 0 for the static magnetic component. As compared with quasistatic fields inside a laser self-focusing channel given in Ref. , 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 = -El cos α, = El sinα, = El /vph and E1 = E 0 cos ω 0 [t-(xsin α + ycos α)/vph ], where Vph ≈ 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 pz = 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 -( + /c]-1/2 and the frequency ωβ is given by
As we can see that the electrons will oscillate along the target surface with the betatron frequency ωβ 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 px 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 ωβ = (ωβ+ + ωβ- )/2 . Therefore, with the two quasistatic fields, electrons can be confined at the target surface while moving along it if the transverse momentum py 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 , electron acceleration can occur during the electron oscillation along the target surface. By taking the relativistic transformation to a moving frame with the velocity Vxêx , it is easy to find that in the new frame electrons just oscillate with the frequency ωβ′ = ωβ (1 - /c 2)-1/2 and the frequency of the laser fields is ω′0 = ω0 (1 - Vx sin α/ c)(1 - /c 2)-1/2. When ω′β = ω′0 , i.e. (ωβ = ω0 (1 - Vx sin (α/ c) the betatron motion of electrons in the two quasistatic fields El . For Vx ≈ c , we can get +(κE + κB )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).
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. . 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.
The authors would like to thank Prof. K. Mima and Dr. T. Nakamura at ILE , Osaka University, as well as our colleagues Xiao-Hui Yuan, Miao-Hua Xu, and Zhi-Yuan Zheng in the institute for helpful discussions. This work was supported in part by the NSFC (Grant No. 60321003, 10335020, 10425416, 10374115, and 10390161), the National High-Tech ICF program, the Knowledge Innovation Program, CAS, and the JSPS-CAS Core University Program on Plasma and Nuclear Fusion.
References and links
1. M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, “Ignition and high gain with ultrapowerful lasers,” Phys. Plasmas 1, 1626–1634 (1994). [CrossRef]
2. K. G. Estabrook, E. J. Valeo, and W. L. Kruer, “Two-dimensional relativistic simulations of resonance absorption,” Phys. Fluids18, 1151–1159 (1975); D. W. Forslund, J. M. Kindel, Kenneth Lee, E. L. Lindman, and R. L. Morse, “Theory and simulation of resonant absorption in a hot plasma,” Phys. Rev. A11, 679–683 (1975). [CrossRef]
4. W. L. Kruer and K. Estabrook, “J×B heating by very intense laser light,” Phys. Fluids 28, 430–432 (1985). [CrossRef]
5. Z. M. Sheng, K. Mima, Y. Sentoku, M.S. Jovanovic, T. Taguchi, J. Zhang, and J. Meyer-ter-Vehn, “Stochastic Heating and Acceleration of Electrons in Colliding Laser Fields in Plasma,” Phys. Rev. Lett. 88, 055004 (2002). [CrossRef]
6. A. Pukhov, Z. M. Sheng, and J. Meyer-ter-Vehn, “Particle acceleration in relativistic laser channels,” Phys. Plasmas 6, 2847–2854 (1999). [CrossRef]
7. Y. Sentoku, K. Mima, H. Ruhl, Y. Toyama, R. Kodama, and T. E. Cowan, “Laser light and hot electron micro focusing using a conical target,” Phys. Plasmas, 11, 3083–3087 (2004); T. Nakamura, S. Kato, H. Nagatomo, and K. Mima, “Surface-Magnetic-Field and Fast-Electron Current-Layer Formation by Ultraintense Laser Irradiation,” Phys. Rev. Lett.93, 265002 (2004). [CrossRef]