Proton Acceleration in Underdense Plasma by Ultraintense Laguerre-Gaussian Laser Pulse

Three-dimensional particle-in-cell simulation is used to investigate the witness proton acceleration in underdense plasma with a short intense Laguerre-Gaussian (LG) laser pulse. Driven by the LG10 laser pulse, a special bubble with an electron pillar on the axis is formed, in which protons can be well-confined by the generated transversal focusing field and accelerated by the longitudinal wakefield. The risk of scattering prior to acceleration with a Gaussian laser pulse in underdense plasma is avoided, and protons are accelerated stably to much higher energy. In simulation, a proton beam has been accelerated to 7 GeV from 1 GeV in underdense tritium plasma driven by a 2.14x1022 W/cm2 LG10 laser pulse.

Simulations and theoretical analyses have shown that high-quality proton beam with Giga-electron volts (GeV, 10 9 eV) can be generated from the interaction between a short laser pulse and a thin foil [2,4,[9][10][11]14]. Considering its huge potential applications, some promising works are on progress. Increasing the energy of accelerated protons by RPA is difficult because of decreasing acceleration gradient after the accelerated proton velocity approaches the light speed. In addition, many undesirable effects, such as multidimensional instabilities [15][16][17], occur during acceleration with long interaction time, even if the laser pulse can be well controlled. Therefore, protons of energy beyond 5 GeV are rather hard to be obtained through the RPA mechanism.
On the other hand, tens of GeV or even TeV high-quality proton beam can be generated in the so called sequential radiation pressure and bubble acceleration regime [18][19][20][21][22] by using a 10 23 W/cm 2 laser pulse with two dimensional (2D) particle in cell (PIC) simulations. The plasma channel can help to form the twin bubble structure, and thus proton energy can be over 10 GeV [22]. This promising approach can compete with the conventional accelerator in obtaining the ultra-high energy protons when the laser technology meets the requirement in the near future and the same encouraging results as the electron acceleration in bubble is expected for protons, so it is necessary to study the detailed process.
We know that, to get high energy protons, there are two key stages, namely, the trapping and the continual acceleration. The former can be reached by RPA like in the sequential radiation pressure and bubble acceleration regime. It has been found that a quasi-monoenergetic plasma bunch of high energy density can be obtained by irradiating a currently available short laser pulse irradiating on a small hemispheric shell target [23], which can also be used in the trapping process in the bubble. For obtaining ion energy higher than 5 GeV, the second stage, continuous acceleration, is more critical and challenging. In the regular bubble, electrons can be well-confined on the acceleration axis because of the transverse focusing field and continuously accelerated in the laser propagation direction. However for protons, it is an opposite situation that protons will be dispersed by the radial electric field and thus difficult to be further accelerated without additional compensation method. We note the recent work has reported special donut wakefields driven by a LG laser pulse for positron and electron acceleration [24]. For positron acceleration, the laser pulse should be short enough to avoid affecting the positron because the acceleration and the laser fields coexist at the same space. However, the proton mass is much larger, so maybe it is more appropriate to use such a donut wakefield to accelerate protons because of the limited effect of the laser field on them.
In this paper, the well-confined acceleration of externally injected protons in the wakefield driven by Laguerre-Gaussian (LG) laser pulse is studied by using three-dimensional (3D) PIC simulations. Trapping conditions and acceleration results are analyzed. It is found that, different from the regular bubble induced by a Gaussian laser pulse, a special bubble with an electron pillar on the axis will be formed when a LG laser pulse irradiates in underdense plasma. This structure provides an efficient and strong focusing force for protons in the transverse direction; thus, protons can be accelerated continuously for a long time. The 3D PIC simulations confirm that a proton beam with initial energy of 1 GeV has been accelerated to 7 GeV in the underdense plasma with a density of 2.4×10 20 /cm 3 . This acceleration is stimulated using a LG10 laser pulse with power similar to the Super-Gaussian (SG) laser pulse with an intensity 2.14×10 22 W/cm 2 .

Simulation and Analysis
The proposed method is demonstrated with 3D PIC simulation code (VORPAL) [25]. The simulation box is 60 μm( ) 100 μm( ) 100 μm( ) x y z   , which corresponds to a moving window with 600 300 300  cells and one particle per cell. T is the laser period and  is the azimuthal angle within the range of   0 2 . The laser electric field is used with power similar to that of the SG laser pulse with peak intensity of 2.14×10 22 W/cm 2 , which corresponds to its normalized amplitude t  , the laser pulse enters the simulation box from the left boundary. Here, the slow-moving massive background tritium ions allow the formation of a stable electron bubble with a large space-charge field [26]. The witness proton beam emitted from is of the size of 3μm1 μm1 μm  and with total charge of 160 pC.
The background electrons are expelled by an intense laser pulse through the pondermotive force, which is proportional to ▽I . In a regular bubble driven by a Gaussian laser pulse, the transverse field will disperse protons located near the x axis and push them to the bubble sheath wall. However, in the case of LG laser pulse, the LG10 laser pulse has a hollow-structure electric field as shown in Fig. 1, in which the field on axis is zero. Therefore, electrons expelled outward by the pondermotive force of the laser pulse in transversal direction form the outer bubble sheath, whereas the electrons expelled inward form the inner bubble sheath. That is, a bubble structure with an electron pillar on the x axis will be formed when it propagates in the underdense plasma because of its transverse donut-like shaped intensity [24,27]. The electron pillar on the x axis of the special bubble structure will result in a focusing field for protons around the inner electron pillar. If the transverse laser ponderomotive force is sufficiently intense to compress the inner electron pillar to an electron thread with high density, the wakefield can trap and accelerate protons for a long time. In this case, this structure provides the focusing force for protons in transverse direction and the accelerating force in longitudinal direction. With the parameters in this paper, a clear special bubble structure with high density electron thread is formed, as shown in Fig. 2. The charge density of the electron thread is higher than the background ion density. Also there is a high density electron bunch trapped in the rear of the special bubble, which will not be described because it is beyond the scope of the present context.  . Simultaneously, the witness proton beam is located in the black-dashed box and confined by the focusing field. The focusing field is similar to the twin bubbles in previous study [22], but with different generation schemes. In the present study, the structure is axial symmetry decided by the laser pulse mode and can propagate stably for a long distance for proton acceleration. As expected, protons are well-confined transversely on the axis in the special bubble driven by the LG10 laser pulse, as shown in Fig. 4(a) with the momentum distribution at t=1.12 ps (most protons are located in a bucket within the radius of r=1μm in transverse direction), and accelerated stably to t=2.13 ps when they start dephasing (surpassing the bubble front). Eventually, the witness protons gain the peak energy of 7 GeV as shown in Fig. 4(b). In the case of a SG laser pulse, keeping other simulation parameters unchanged, the protons diffuse gradually and cannot be accelerated stably, Fig. 4(c) shows that the proton beam filling with the simulation box (out of the bubble). Finally, the peak energy stops at 2.2 GeV.

Discussion
There are two important aspects for the present acceleration approach. First, we should note the electron pillar in the special bubble structure, which determines the longitudinal acceleration field for the witness proton beam, is crucial for the acceleration. If the electron pillar is too thick, the wakefield around the witness protons will be too weak to accelerate the protons because the charge separation field is almost neutralized. Therefore, laser intensity, laser spot size, and plasma density should be adjusted to be appropriate to make the inner electron pillar thin enough to make the longitudinal acceleration field intense enough. Assuming the bubble is spherical-like, its radius (the distance between the bubble wall and the electron thread) can be estimated by balancing the transverse laser ponderomotive force on a single electron and the transverse charge separation field force [28]. The transverse ponderomotive force of the CP laser pulse is   For the ultraintense CP laser pulse interacting with the underdense plasma, the high density electron sheet in front of the laser pulse expelled by the intense light pressure is overdense and the laser pulse is reflected. By balancing the momentum of the electron layer with the light pressure and considering the Doppler effect [29], course, the higher initial energy, the shorter acceleration time before they are dephased and the less energy they gain. Therefore, the maximum potential of the bubble play a key role for the trapping and acceleration. Fig. 5(b) shows that the maximum potential ( max  ) changes with the laser amplitude ( 0 a ) for different underdense plasma densities. That is to say, in order to obtain the high max  , we may choose high laser intensity and low plasma density. One point should be noted in this case that the bubble velocity will be higher and so the required potential for trapping will be higher from Eqs. (3) and (4).
The analysis shows that the LG laser pulse generates the focusing force in transverse direction, which is crucial for the proton acceleration in the wakefield. Although the acceleration field around the x axis is slightly weakened, the protons with an appropriate initial energy which is related to the maximum potential of the acceleration field still can be trapped and accelerated stably, and get nearly ten orders energy gain.

Conclusion
In conclusion, the witness proton acceleration in the wakefield driven by a LG10 laser pulse has been studied. Confining the protons near the acceleration axis is important for the continuous acceleration in bubble regime. By using the LG10 laser pulse, a special bubble with a high density electron thread on the axis has been found, in which an enough intense acceleration field in longitudinal direction and a focusing field in transverse direction for the protons coexist. The 3D PIC simulation results show that protons can be well-confined near the axis and accelerated stably for a long time. The energy of protons is increased to 7 GeV from 1 GeV.
One point should be noted that even though the initial radius of the witness proton bunch is much larger than that of the on-axis electron sheath, the proton bunch will be focused to the axis because of the existence of the transverse focusing field and the simulation results verifies it (not presented in this paper). Additionally, obtainable high energy protons with the currently available laser level is realistic because it is easier to realize, and our study show that the present scheme still works for the lower intensity, such as a0 around 20, but requires witness protons with higher initial energy. Moreover, generation of such intense LG10 laser pulse is critical for the present scheme. Fortunately, a potential approach has been proposed recently [30].