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Abstract
Attosecond electron bunches have wide application prospects in free-electron laser injection, attosecond X/γ-ray generation, ultrafast physics, etc. Nowadays, there is one notable challenge in the generation of high-quality attosecond electron bunch, i.e., how to enhance the electron bunch density. Using theoretical analysis and three-dimensional particle-in-cell simulations, we discovered that a relativistic vortex laser pulse interacting with near-critical density plasma can not only effectively concentrate the attosecond electron bunches to over critical density, but also control the duration and density of the electron bunches by tuning the intensity and carrier-envelope phase of the drive laser. It is demonstrated that this method can efficiently produce attosecond electron bunches with a density up to 300 times of the original plasma density, peak divergence angle of less than 0.5$^\circ$, and duration of less than 67 attoseconds. Furthermore, by using near-critical density plasma instead of solid targets, our scheme is potential for the generation of high-repetition-frequency attosecond electron bunches, thus reducing the requirements for experiments, such as the beam alignment or target supporter.
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1. Introduction
Relativistic attosecond electron bunches have broad prospects in the fields of free electron laser injection [1–3], attosecond X/$\mathrm {\gamma }$-ray generation [4–7], ultrafast physics [8,9], etc. Especially, with the rapid development of laser technology, the laser power and intensity have been quickly raised, which provides new opportunities for the generation of high-quality attosecond electron bunches via the interaction between relativistic laser pulses and plasmas. In the past decades, extensive theoretical and experimental researches have been conducted, and various schemes of laser-plasma interaction have been proposed. Currently, the schemes for the generation of laser-driven attosecond electron pulses can be mainly divided into three types. The first type is based on the laser wakefield acceleration (LWFA) [10–17]. Thanks to the advantage of the laser wakefield, the obtained attosecond electron bunches via this method are characterized by high beam energy [10], low energy spread [11], and low divergence angle [12]. For example, in 2010, Luttikhof et al. [13] injected a single electron beam into the laser wakefield and successfully separated it into several attosecond electron bunches. The generated attosecond electron bunches have a duration of about 630 $\textrm{as}$ with a spacing of 800 $\textrm{nm}$, a peak energy of 655 $\textrm{MeV}$, and an energy dispersion of about 10%. However, it is also challenging to raise the electron density or obtain isolated attosecond electron bunches in the wakefield. Furthermore, this method has a high requirement for electron injection location. Apart from the LWFA scheme, we can also utilize relativistic laser irradiation on solid nanofilms to obtain attosecond electron bunches [18–23]. In this type of scheme, we can obtain isolated, high-charge attosecond electron pulses [18,19]. For example, Kulagin et al. [18] conducted a two-dimensional (2D) numerical simulation in 2007 and obtained isolated attosecond electron pulses with more than 10 $\textrm{nC}$ charge via the interaction of a solid nanofilm and a relativistic laser pulse with a steeply rising front. However, the generated attosecond electron pulses generally have significant divergence angle [20,21]. Additionally, this scheme has very high demands for the quality of the laser pulse. On one hand, the laser intensity should be so large that the light pressure can overcome the electric charge separation field between electrons and ions. On the other hand, the laser pulse should have steep rising edges to generate isolated electron pulses [19,22,23].
Aside from the above two type of schemes, we can also obtain high-quality attosecond electron pulses by utilizing vacuum heating mechanism [24–30]. When a relativistic $p$-polarized laser pulse is obliquely incident at a certain angle on a solid target with a sharp density gradient, the plasma wave will be excited on the target surface. If the plasma wave breaks, some electrons will be ejected into the vacuum and accelerated by the laser field, ultimately forming attosecond electron bunches [24,25]. Currently, various schemes have been proposed to use solid targets of different shapes, such as cone targets and channel targets [26–30], to generate high-quality attosecond electron pulses. However, since these schemes require a high density gradient at the boundary of the solid target, the laser contrast ratio should be high enough. In previous studies, the normal Gaussian laser pulse is usually utilized. Recently, the Gaussian laser pulse with high-order mode has attracted researchers’ attention, especially the Laguerre-Gaussian (LG) laser pulse, due to its spiral equiphase surface, hollow intensity distribution and carrying orbital angular momentum (OAM) [31–38]. For example, Hu et al. [31,32] proposed a scheme to generate high-quality attosecond electron pulses via the interaction of LG laser pulse with the nanofiber target in 2018. The attosecond electron pulses obtained via this scheme exhibit high collimation, low divergence and large beam charge. However, since the experimental requirements for this scheme are quite harsh, such as focusing the LG laser pulse to the front surface of the nanofiber target, it is not easy to implement the scheme in experiments.
In this paper, we present a novel scheme to generate attosecond electron bunches of tunable density and duration by using near-critical density (NCD) plasma instead of solid targets. Our scheme is potential for the generation of high-repetition-frequency attosecond electron bunches and reduces the requirements for experimental conditions, such as the beam alignment and target supporter. It is found that when a left-handed circularly-polarized LG laser pulse irradiates NCD hydrogen plasma, electron bunches are accelerated forward, compressed longitudinally and confined transversely. Ultimately, attosecond electron bunches with a density up to 300 times of the original plasma density, peak divergence angle of less than 0.5$^\circ$, and duration of less than 67 attoseconds can be produced. We also discovered that we can not only effectively concentrate the electron bunches, but also control the duration and density of the attosecond electron bunches by adjusting the laser intensity and carrier-envelope phase.
2. Model
In this work, we use the fully three-dimensional (3D) particle-in-cell (PIC) code EPOCH [39,40] to investigate the interaction between the LG laser pulse and NCD plasma. We also verify the simulation results using the fully 3D PIC code Virtual Laser-Plasma Laboratory (VLPL) [41,42]. The size of the simulation box is $18\lambda _{0}\times 15\lambda _{0}\times 15\lambda _{0}\left ( x\times y\times z\right )$, where $\lambda _{0} = 1\,\mathrm{\mu}\textrm{m}$ is the laser wavelength and $T_{0} = \lambda _{0} / c$ is the laser cycle. The spatial step $\Delta {x} = 0.01\lambda _{0}$, $\Delta {y} = \Delta {z} = 0.05\lambda _{0}$, respectively. The time step $\Delta {t} = 0.005T_{0}$. The total simulation time is set to be $200T_{0}$. A moving window is employed, which moves along the $+x$ axis at the speed of light in vacuum since $t=12T_{0}$. The boundary of the field and particles in the $x$, $y$ and $z$ directions are all set to be open boundary conditions. In the simulations, we use NCD hydrogen plasma with a density of $0.1n_{c}$, where $n_{c} = 1.12 \times 10^{21}\,\textrm{cm}^{-3}$ is the critical density corresponding to the laser wavelength $\lambda _{0} = 1\,\mathrm{\mu}\textrm{m}$. The hydrogen target is located between $3\lambda _{0}$ and $16\lambda _{0}$ in the $x$ direction. The size of the target in the $y$ and $z$ directions are $15\lambda _{0}$. Initially, each cell contains 8 macro-electrons and 8 macro-protons. In experiments, a hydrogen gas jet can be utilized to provide the NCD target as required in our scheme [43–49]. Alternatively, we may also use a high-repetition-rate tape drive system [50–53] to rapidly replace aerogel targets [54–57] to meet experimental requirements while avoiding the thermalization effects.
At $t = 0$, the left-handed circularly-polarized LG laser pulse is incident from the left boundary of the simulation box and propagates along the positive direction of the $x$ axis. The amplitude of the dimensionless laser electric fields can be written as
As in cylindrical coordinates, $E _ { r } = E _ {y} \cos \varphi + E _ { z } \sin \varphi$, $E_{\varphi }=-E_{y}\sin \varphi +E_{z}\cos \varphi$, and $E_{x}=-({i} / {k_{0}})({\partial E_{y}} / {\partial y}+{\partial E_{z}}/{\partial z})$, $\vec {B}=-(i / \omega )\nabla \times \vec {E}$, so for a left-handed circularly polarized LG laser with (1, 0) mode, we can obtain the electromagnetic field distribution of LG laser in the cylindrical coordinate system as follows:
In the simulation, we take the laser dimensionless parameter $a_{0} = 20$, the beam waist radius $\sigma _{0} = 3\lambda _{0}$, and the position of beam waist $x = 13\lambda _{0}$. The laser temporal profile is set to be $G(\eta ) = \cos ^{2}(\pi \eta /(2\tau ))$, where the duration $2\tau = 6T_{0}$ ($-\tau \le \eta \le \tau$), and $\eta = (x - ct) / c$. In addition, due to limitations in the divergence, density, and duration of electron bunches produced by the vortex laser pulses with large topological charges [58–60], we finally set $l = 1$ and $p = 0$ in the simulations. The demanded intense few-cycle LG laser pulse can be attainable by utilizing multi-pass cells [61], or hollow-core fibers [62] to post-compress relativistic LG laser pulses.
3. Results
Figure 1 mainly shows the simulation results of the interaction between the LG laser pulse and NCD plasma at $t = 14T_{0}$. We can see that a cavity structure is generated, which is very similar with the bubble structure generated in the laser wakefield. However, dense electrons are located at the front of the cavitation in our scheme, which are marked by the white circle in Fig. 1. As the laser pulse propagates in plasma, the size of the cavity gradually increases. After $t = 18T_{0}$, the electrons at the top of the cavitation are pushed away from the NCD plasma. These electrons are continuously accelerated, and concentrated into high-density attosecond electron bunches as shown in Fig. 1.
Fig. 1. The interaction of LG laser pulse with NCD plasma at $t = 14T_{0}$. The blue curve in the $xoy$ plane in the bottom shows the electron density distribution along the optical axis, while the density distribution across the cross section at $x = 12.3\,\mathrm{\mu}\textrm{m}$ is shown in the $yoz$ plane.
Figure 2(a-c) shows the density distribution of electron bunches at $t = 40T_{0}$, $60T_{0}$, and $150T_{0}$, respectively. It can be seen that after leaving the NCD hydrogen plasma, electrons evolve into stable disk-like electron bunches of attosecond duration. Electron bunches continue to move forward for more than 100 fs at a speed close to the speed of light in vacuum. Figure 2(g) shows the evolution of the maximum electron bunch density. From this figure, it can be noted that at $t = 40T_{0}$, the density of electron bunches can reach 240 times of the original hydrogen plasma density. At $t = 45T_{0}$, the electron bunch density further increases to about $300n_{0}$. Especially, we can see that between $t = 45T_{0}$ and $t = 60T_{0}$, the electron density stays around $300n_{0}$, which is several orders of magnitude higher than traditional short electron bunches. After that, the electron bunch density gradually decreases, but after $t = 100T_{0}$, it keeps almost unchanged at $20\sim 30n_{0}$. These high-density attosecond electron bunches can be used to generate ultra-brilliant radiation sources via coherent transition radiation [63,64] or inverse Compton scattering [65,66].
Fig. 2. (a-c) Electron density and (d-f) divergence angle distribution at $t = 40T_{0}$, $60T_{0}$, and $150T_{0}$, respectively. (g) The electron density evolution over the simulation time. (h) The evolution of the full width at half maximum of the electron divergence angle over the simulation time. (i) The electron energy spectrum at $t = 40T_{0}$, $60T_{0}$, and $150T_{0}$, respectively.
Figure 2(d-f) shows the angular distribution of electron bunches at $t = 40T_{0}$, $60T_{0}$, and $150T_{0}$, respectively, where the electron divergence angle $\theta = \arctan (p_{\bot }/p_{x})$, and the electron transverse momentum $p_{\bot } = \sqrt {p_{y}^{2} + p_{z}^{2}}$. We can see that the electron divergence angle is very small and its collimation is getting better and better when the attosecond electron bunches propagate forward. It is shown that the peak divergence angle of the electron bunches are 0.7$^\circ$, 0.5$^\circ$ and 0.2$^\circ$ at $t = 40T_{0}$, $60T_{0}$ and $150T_{0}$, respectively. The corresponding full widths at half maximum (FWHM) of the divergence angle are 2.0$^\circ$, 0.9$^\circ$ and 0.4$^\circ$, respectively. Figure 2(h) shows the evolution of the FWHM of electron divergence angle over time. It can be seen more clearly that after $t = 20T_{0}$, the divergence angle of attosecond electron bunches is always at a very low level. With time flying, the electron divergence angle gradually decreases. These results are consistent with the observations from Fig. 2(d-f), which represents good collimation of the attosecond electron beam. Figure 2(i) shows the electron energy spectrum at $t = 40T_{0}$, $60T_{0}$, and $150T_{0}$, respectively. We can see that at $t = 40T_{0}$, the electron bunches are monoenergetic, with a relative energy spread of about 27%. However, as time going by, the maximum energy increases but the peak electron energy gradually decreases.
We give the energy equation of electrons in the electromagnetic fields as follows
where $P_{E_{x}}=-ev_{x}E_{x}$ is the power of the axial electric field force, and $P_{E_{\bot }}=-ev_{y}E_{y}-ev_{z}E_{z}$ is the power of the transverse electric field force. In order to analyze the electron acceleration in detail, we track some typical electrons according to the density distribution at $t = 80T_{0}$ and record $P_{E_x}$, $P_{E_\bot }$ and Lorentz factor $\gamma$. To distinctly represent the evolution of $P_{E_x}$, $P_{E_\bot }$ and Lorentz factor $\gamma$ over time, we also plot the distribution of a typical electron in Fig. 3. According to the power of axial and transverse force, the electron acceleration process is divided into three stages, as shown in Fig. 3(a) and Fig. 3(b). In stage I, $P_{E_x} > 0$ and $P_{E_\bot } > 0$. Electrons are simultaneously accelerated in the axial and transverse direction and obtain an initial velocity. Then due to the low electron velocity, electrons gradually slide to the deceleration phase of axis electric field force and enter into the stage II. Fortunately, since the power of transverse and axial force are in contrast, namely $P_{E_\bot } > 0$ and $P_{E_x} < 0$, these electrons continue to move forward along with the laser pulse and the Lorentz factor $\gamma$ maintains a small value. As electrons slide to the acceleration phase of axis electric field force again, the stage III of electron acceleration begins. In this stage, electrons are effectively accelerated due to the rapid growth of $P_{E_x}$. The ultimate $\gamma$ is more than 100. Since the axial electric field $E_x$ varies with $r$ and $x$, it is notable that the evolution of $P_{E_x}$ is different when electrons are accelerated. Thus the energy dispersion of these electrons gradually increase. Along with the phase slipping of electrons, the $E_x$ field and $P_{E_x}$ gradually decrease. When electrons slide to the deceleration phase again, the Lorentz factor $\gamma$ decreases. Thus the peak energy gradually decreases, which is consistent with the evolution of the electron energy spectrum in Fig. 2(i). Figure 3(c) shows the distribution of the electron bunch in the phase space $(\eta _{x}, \eta _{\bot })$ during $t = 0 \sim 20 T_{0}$, representing the electron energy gain in the axial and transverse direction after going through stage I and II. Here, the energy gain of the axial electric field $\eta _{x}= \int _{t_{1}}^{t_{2}}P_{E_x}dt/(m_{0}c^{2})$, while the energy gain of the transverse electric field $\eta _{\bot }= \int _{t_{1}}^{t_{2}}P_{E_\bot }dt/(m_{0}c^{2})$. It can be seen that most electrons are located in the range of $\eta _{x} < 0$ and $\eta _{\bot } > 0$. This indicates that the electrons are accelerated by the transverse electric field while simultaneously decelerated by the axial electric field. Therefore, the Lorentz factor $\gamma = \eta _{x} + \eta _{\bot } + \gamma _0$ is maintained at a low level, which is consistent with the analysis above. Figure 3(d) shows the distribution of the electron bunch in the phase space $(\eta _{x}, \eta _{\bot })$ at stage III. It can be seen that most electrons are in the range of $100 < \eta _{x} < 300$, $-30 < \eta _{\bot } < 10$. The energy gain of the axial electric field is much larger than that of the transverse electric field. Therefore, the axial electric field $E_x$ is dominant in the electron acceleration in our scheme.Fig. 3. (a) Evolution of the power of the axial electric field force $P_{E_x}$ and the power of the transverse electric field force $P_{E_\bot }$ of the selected electron bunch and typical electron. (b) Evolution of the Lorentz factor $\gamma$ of the selected electron bunch and typical electron. The distribution of tracked electrons in $(\eta _{x}, \eta _{\bot })$ space at (c) $t = 0 \sim 20 T_{0}$ and (d) $t = 20 \sim 60 T_{0}$.
In order to investigate the acceleration, focusing and collimation of attosecond electron bunches in detail, we give electron trajectories and their projections in the $yoz$ plane from $t=0$ to $t=42T_{0}$, as shown in Fig. 4(a) and (b). It can be seen that most electrons initially originate from $x = 10\lambda _{0}$. At stage I and II, these electrons first rotate counterclockwise in the vortex laser fields. At the same time, the distance between the electrons and the center axis $r$ gradually increases. Fortunately, the evolution of electrons turns clockwise at a critical point. The radial coordinate $r$ gradually decreases with the electrons movement. The electrons finish focusing at around $x = 15\lambda _{0}$. At stage III, they move forward and keep collimated. Actually, even at $t = 200T_{0}$, the electron trajectories still maintain good collimation, which is quite consistent with the result of low divergence angle of the electron beam in Fig. 2(f). To further study the electron dynamics, we analyze the forces acting on a randomly-selected electron in detail. Figure 4(c) shows the directions of the electron velocity and force along the trajectories. Here $r$ is the distance between the electron and the center axis, $\vec {F}_{r}$ is the electron radial force, $\vec {F}_{\varphi }$ is the electron azimuthal force, and $\vec {v}$ is the instantaneous transversal velocity, which is consist of the radial velocity $\vec {v}_{r}$ and azimuthal velocity $\vec {v}_{\varphi }$.
Fig. 4. (a) The trajectories of tracked electrons from $t=0$ to $t=42T_{0}$. (b) Projections of electron trajectories in the $yoz$ plane. Here the colorbar represents the electrons’ energy. (c) The directions of the electron velocity and force suffered by the electron along the trajectories.
In the cylindrical coordinate system, the force $\vec {F}$ can be expressed as
Fig. 5. (a) Evolution of the electron axial force $F_{x}$, the electric field components $F_{E_x}$, the axial Lorentz force $F_{L_x}$, and the electron axial momentum $p_{x}$. (b) Evolution of the radial force $F_{r}$, the centripetal force component $p_{\varphi }^{2} / \gamma m_{0}r$ and the change rate of radial momentum $dp_{r}/dt$. (c) The evolution of the axial, radial and azimuthal velocity $v_{x}$, $v_{r}$, $v_{\varphi }$ and the radial coordinate $r$.
In stage I, electrons are pre-accelerated axially and transversally by the laser field and comove forward with the laser pulse with an axial velocity $v_{x} = 0.9 c$. Meanwhile, electrons obtain initial azimuthal velocity $v_{\varphi }$ and rotate around the optics axis. In the radial direction, electrons are subjected to the radial force $F_{r}$ and centripetal force component $p_{\varphi }^{2} / \gamma m_{0}r$. The radial coordinate $r$ gradually increases. Afterwards, these electrons enter into the stage II and begin to gather together. In the axial direction, since $F_{E_x} < 0$ and $F_{L_x} > 0$, the axial momentum $p_{x}$ and velocity $v_x$ first rises slowly and then gradually decreases, as shown in Fig. 5(a) and (c). Due to $\gamma = p_x/m_e c +1$, the Lorentz factor $\gamma$ also fluctuates with the axial momentum $p_{x}$. Since the $\gamma$ factor in this stage is very low, it is easy to focus electrons along the radial direction.
At the beginning of the electron focusing, since the electron density is low, the Coulomb repulsive force can be neglected. By substituting the expression of the laser angular magnetic field $B_{\varphi }$, the radial force on the electron can be represented by
where $\beta _{x} = v_{x} / c$. It is shown that the radial electric field $E_{r}$, the axial magnetic field $B_{x}$, the axis velocity $v_x$, and the angular velocity $v_{\varphi }$ play a key role in the electron focusing process. According to Eq. (3) and Eq. (7), the phase difference of the radial electric field $E_{r}$ and the axis electric field $E_x$ is $-\pi /2$. Since $E_x >0$ at this stage, the radial electric field $E_{r} >0$ at the first half of stage II and $E_{r} <0$ at the second half of stage II. Furthermore, the axial magnetic field $B_{x}$ fluctuates with the radial electric field $E_{r}$ when $r < \sigma$ according to Eq. (8). Thus, the electrons are subjected to inward radial force at the first half of stage II, namely $F_{r} < 0$. We notice that the evolution of electrons turns to clockwise from counterclockwise during electron focusing. The angular velocity $v_{\varphi }$ becomes negative. As the quick increase of $v_{\varphi }$, the radial force $F_{r}$ is also negative at the second half of stage II. The centripetal force component $p_{\varphi }^{2}/(\gamma m_{0}r)$ is also much less than $F_{r}$, as shown in Fig. 5(b). Thus the change rate of radial momentum $dp_{r}/dt$ is always negative. The radial velocity $v_{r}$ gradually decreases to 0, and then increases in the radially inward direction. Correspondingly, the radial coordinate $r$ first increases and then rapidly decreases. Thus the electrons get together toward the optics axis. The density of attosecond electron bunch rapidly increases. At the same time, the coulomb repulsion force also become strong. Since $v_{\varphi }$ gradually rises, the centripetal force component $p_{\varphi }^{2}/(\gamma m_{0}r)$ also rapidly increases. When the sum of the Coulomb repulsion force and the centripetal force component $p_{\varphi }^{2}/(\gamma m_{0}r)$ are equal to $F_{r}$, the radial velocity $v_{r}$ reaches the maximum. Then the radial velocity $v_{r}$ rapidly decreases to 0, and the electron density reaches the maximum. The electron focusing process ceases.After that, these electrons enter into the stage III. Electrons are effectively accelerated by the axial electric field $E_{x}$. The electron energy significantly increases. Since the axial velocity $v_x$ gets approaching to $c$, electrons can remain in this acceleration phase for a long time, which helps to further increase the electron energy. Since the Lorentz factor $\gamma$ and the moving mass of electrons $\gamma m_{0}$ rapidly rises. It is hard to repel electrons along the radial direction. Since the axial velocity $v_x$ gets close to $c$, the angular velocity $v_{\varphi }$ gradually decreases, as shown in Fig. 5(c). This causes the rapid drop of the centripetal force component $p_{\varphi }^{2}/(\gamma m_{0}r)$ after the electron focusing process. In addition, we notice that the Coulomb repulsion force is proportional to $1/ \gamma$. With the rapid increase of electron energy, the coulomb repulsion force also greatly decreases. At the same time, since $v_{\varphi } B_{x}$ is always positive, the direction of the radial force is always inward. The radial velocity $v_r$ gradually approaches 0 after the electron focusing. Electrons are concentrated near the optics axis and can maintain a high density for a long time. As electrons are accelerated continuously, the electron divergence angle $\theta$ gradually decreases. This is the reason why the electron beam always maintains good collimation after focusing.
4. Discussion
4.1 Influence of laser parameters on the quality of the obtained electron bunches
We also conduct a series of simulations to study the influence of the laser intensity $a_{0}$ and carrier-envelop phase $\varphi _{\textrm{CEP}}$ on the quality of attosecond electron bunches. Figure 6 shows the changes of the electron beam quality for $a_{0} = 10, 15, 20, 25, 30$, respectively. Figure 6(a) displays the electron density distribution along the optics axis between $x = 57 \lambda _{0}$ and $x = 58 \lambda _{0}$ at $t = 60T_{0}$. We can clearly see that with the increase of the laser intensity, the overall structure, density, and duration of the electron bunch change significantly. When the laser intensity is relatively small, e.g., $a_{0} = 10$ or $15$, the electrons form an isolated sheet structure. There appears a single density peak in the density profile. However, the electron density is still low, and the duration of the electron bunch is also relatively large. Fortunately, as the laser intensity increases, the electron density increases but the duration decreases. As shown in Fig. 6(a), when the laser intensity is $a_{0} = 20$, the electron density profile exhibits a bimodal structure, indicating the presence of two electron slices. The density of the electron slices is as high as 307$n_0$, while the duration is just 67 as. It is worth noting that the spacing between electron slices obtained from other schemes is generally 1$\lambda _{0}$. In our scheme, the spacing is much less than 1$\lambda _{0}$. When the laser intensity $a_{0}$ becomes larger, e.g., $a_{0} = 25$ or $30$, some new changes of the structure, density, and duration of the electron bunches occur. The electron bunch exhibits a multi-slice structure with spacings of 100s attosecond scale. In addition, we found that the duration of the electron slices slightly increases and the density decreases in comparison with that of $a_{0} = 20$ case. Figure 6(b) shows the variation of the electron bunch density, divergence angle, and cutoff energy with the laser intensity $a_{0}$ at $t = 60T_{0}$. We can see that with the increase of $a_{0}$ ($a_{0} \le 20$), the density and cutoff energy of electron bunches increase significantly and the electron divergence angle decreases. However, when $a_{0} > 20$, the electron density decreases rapidly and the divergence angle becomes slightly higher than that of the $a_{0} = 20$ case. Furthermore, the cutoff energy is saturated at $a_{0} = 20$. Therefore, considering the electron density, peak divergence angle and cutoff energy, the optimal laser intensity should be $a_{0} = 20$.
Figure 7 shows the changes of the electron beam quality for different laser carrier-envelope phase $\varphi _{\textrm{CEP}} = 0,\,\pi /4,\,\pi /2,\,3\pi /4$ and $\pi$. Figure 7(a) displays the electron density distribution along the optics axis between $x = 57 \lambda _{0}$ and $x = 58 \lambda _{0}$ at $t = 60T_{0}$. When $\varphi _{\textrm{CEP}} = 0$, we obtain a double-electron-slice structure of high density and short duration. As $\varphi _{\textrm{CEP}}$ gradually increases, the density of the electron bunch decreases significantly, and the duration also rises. In addition, an isolated electron slice structure forms when $\varphi _{\textrm{CEP}} \ge \pi /2$. It is worth noting that there is a significant displacement of the position of the electron slices, since the variation of $\varphi _{\textrm{CEP}}$ changes the laser fields. Figure 7(b) shows the variation of the density, divergence angle and cutoff energy of electron bunch at $t = 60T_{0}$ with the laser carrier-envelope phase $\varphi _{\textrm{CEP}}$. As the carrier-envelope phase $\varphi _{\textrm{CEP}}$ increases, the electron density quickly decreases. Meanwhile, the divergence angle of the electron bunch gradually increases, especially when $\varphi _{\textrm{CEP}} > \pi /2$. Additionally, the cutoff energy of the electron bunch firstly decreases and then rises when $\varphi _{\textrm{CEP}} > \pi /2$. Therefore, we can conclude that when the laser carrier-envelope phase $\varphi _{\textrm{CEP}} = 0$, the quality of the electron bunch is the best. In addition, we also consider the influence of the laser pre-pulse and the central wavelength of the laser pulse. It is shown that the generated electron bunches still keep intact and maintain high density and good collimation in these cases, which demonstrates the robustness of our scheme (see Fig. S1 and Fig. S2 of Supplement 1).
Fig. 6. (a) Distribution of electron density along the optics axis between $x = 57 \lambda _{0}$ and $x = 58 \lambda _{0}$ when the laser intensity $a_{0} = 10, 15, 20, 25, 30$ at $t = 60T_{0}$; (b) Variation of the electron density, peak divergence angle and electron cutoff energy with the laser intensity $a_{0}$ at $t = 60T_{0}$.
Fig. 7. (a) Distribution of electron density along the optics axis between $x = 57 \lambda _{0}$ and $x = 58 \lambda _{0}$ when the laser carrier-envelope phase $\varphi _{\textrm{CEP}} = 0,\,\pi /4,\,\pi /2,\,3\pi /4$ and $\pi$; (b) Variation of the electron density, peak divergence angle and electron cutoff energy with carrier-envelope phase $\varphi _{\textrm{CEP}}$ at $t = 60T_{0}$.
4.2 Simulation results with laser parameters from SULF equipment
In the past years, ones have maken significant advances in vortex laser generation and application. For example, Wang et al. [33] generated LG laser pulse with intensity up to $6.3 \times 10^{19}\,\textrm{W/cm}^{2}$ by using a high-reflectivity phase mirror in the SULF laser facility. Then in 2023, they further increased the intensity of the LG laser to $1.2 \times 10^{20}\,\textrm{W/cm}^{2}$, with the duration of $27\,\textrm{fs}$, and the laser power reaching $1\,\textrm{PW}$ [34]. This is the highest-intensity vortex laser pulse ever achieved experimentally. To confirm the feasibility of the proposed scheme, we conducted simulations with laser parameters from the SULF facility. In the simulations, we take the laser intensity $I_{0} = 1.38 \times 10^{20}\,\textrm{W/cm}^{2}$, the laser wavelength $\lambda _{0} = 0.8\,\mathrm{\mu}\textrm{m}$, the beam waist $\sigma _{0}=3\lambda _{0}$, the position of beam waist $x = 14\lambda _{0}$, and the duration $2\tau = 8T_{0} (T_{0}=\lambda _{0}/c)$. In addition, the density of the NCD hydrogen plasma is set to $0.08n_{c} (n_{c}=1.75\times 10^{21}\,\textrm{cm}^{-3})$ in the simulation, and the hydrogen target is located between $2\lambda _{0}$ and $18\lambda _{0}$ in the $x$ direction. All other parameters in the simulation keep fixed.
Fig. 8(a) shows the electron density distribution at $t = 40T_{0}$. A disk-like electron bunches of attosecond duration can be produced. The density of electron bunches reaches 50 times of the original plasma density. Figure 8(b) shows the influence of the carrier-envelop phase $\varphi _{\textrm{CEP}}$ on the electron density, peak divergence angle and electron cutoff energy at $t = 40T_{0}$. We can conclude that when the laser carrier-envelope phase $\varphi _{\textrm{CEP}} \in [0, \pi /4]$, the quality of the electron bunch is the best. Figure 8(c) shows the evolution of the maximum electron bunch density. It can be noted that the electron density gradually increases to $90n_{c}$ at $t = 30T_{0}$. After that, the density gradually decreases. Figure 8(d) shows the evolution of the divergence angle. We can see that the electron divergence angle is very small and its collimation is getting better and better when the attosecond electron bunches propagate forward. Therefore, we can still obtain electron bunches with high density, low divergence angle and attosecond duration. However, we admit that the density of electron bunches here is lower than that in the main text. The reason for this is that the intensity of the input laser pulse is much lower than that in the proposed scheme.
Fig. 8. (a) The electron density distribution at $t = 40T_{0}$ under the consideration of the pratical experimental conditions. (b) Variation of the electron density, peak divergence angle and electron cutoff energy with carrier-envelope phase $\varphi _{\textrm{CEP}}$ at $t = 40T_{0}$. (c) The electron density evolution over the simulation time. (d) The evolution of the full width at half maximum of the electron divergence angle over the simulation time.
5. Conclusion
In summary, we propose a scheme to generate dense attosecond electron bunches by the interaction of left-handed circularly-polarized LG laser pulse and NCD hydrogen plasma. Using 3D PIC simulations, we demonstrated the generation of the attosecond electron bunch with high density, low divergence angle, and short duration. It is demonstrated that we can obtain attosecond electron bunch with a density up to 300 times the original target density, peak divergence angle of less than 0.5$^\circ$, and duration less than 67 attoseconds. Fortunately, the electron divergence angle also continuously decreases over time. We also revealed the underlying physics for electron focusing by tracking typical electrons. It is discovered that we can also control the density, duration, divergence angle and cutoff energy of the attosecond electron bunch by adjusting the laser intensity and carrier-envelope phase. Furthermore, in comparison with previous schemes [31], our scheme is potential for generation of high-repetition-frequency attosecond electron bunches and reduces the requirements for experimental conditions, such as the beam alignment and target supporter. The generated attosecond electron bunches may find wide applications in the free electron laser, inertial confinement fusion and secondary particle and radiation generation.
Funding
the Innovation Foundation for Graduate Students (CX20220048); National Natural Science Foundation of China (12105362, 12135009, 12275356, 12375244).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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