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Abstract
Generation and acceleration of energetic positrons based on laser plasma have attracted intense attention due to their potential applications in medical physics, high energy physics, astrophysics and nuclear physics. However, such compact positron sources face a series of challenges including the beam dispersion, dephasing and unstability. Here, we propose a scheme that couples the all-optical generation of electron-positron pairs and rapid acceleration of copious positrons in the terahertz (THz) field. In the scheme, nanocoulomb-scale electrons are first captured in the wakefield and accelerated to 2.5 GeV. Then these energetic electrons emit strong THz radiation when they go through an aluminum foil. Subsequently, abundant γ photons and positrons are generated during the collision of GeV electron beam and the scattering laser. Due to the strong longitudinal acceleration field and the transvers confining field of the emitted THz wave, the positrons can be efficiently accelerated to 800 MeV, with the peak beam brilliance of 2.26 × 1012 s−1 mm−2 mrad−2 eV−1. This can arouse potential research interests from PW-class laser facilities together with a GeV electron beamline.
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1. Introduction
All charged particles have their corresponding antiparticles, with the same weight but opposite physical charges. When the charged particles encounter their antiparticles, they annihilate and release energy in the form of high-energy photons, making them stars in high energy physics, medicine and even science fiction movies. In 1931, Dirac predicted the existence of "anti-electron" [1], and in the following year, Anderson conducted an experimental demonstration and named it "positron" [2]. From Schwinger’s calculation, the vacuum would become unstable to electron-positron pair creation when the electric field reaches $E_s=1.32\times 10^{18}$V/m, corresponding to the intensity of $10^{29}\,\textrm{W}/\textrm{cm}^{2}$ [3]. In the case of laser-laser collisions, the required intensity to create pairs can be reduced to $10^{26}\,\textrm{W}/\textrm{cm}^{2}$ [4], which is still beyond the capabilities of current laboratories around the world. Fortunately, the field intensity is not a Lorentz-invariant, when relativistic electrons or photons collide with the Coulomb field of heavy nucleus, electron-positron pairs can be generated via either the Trident process [5] or Bethe-Heitler (BH) process [6–8]. In 1934, Breit and Wheeler theorized that electron-positron pairs could be generated by $\gamma$-photon collision, known as linear Breit-Wheeler (BW) process [9]. After nearly 90 years of effort, recent experiments conducted on the Relativistic Heavy Ion Collider (RHIC) have finally demonstrated that electron-positron pairs can be generated via linear BW process [10]. Apart from pure linear BW process, positrons can also be generated efficiently by collisions between $\gamma$-photons and strong laser fields via the so-called multi-photon BW process. The famous SLAC-E144 experiment, conducted in 1996, provided experimental evidence of the multi-photon BW process, with $106\pm 14$ positrons detected in collisions between a 46.6 GeV electron bunch and terawatt laser pulses [11–13].
With the advancement of high power lasers [14–16], especially the construction of 100 PW-scale laser facilities [17], electron-positron pair generation schemes based on BW process are gaining increasing attention. In the past decade, various schemes based on laser-plasma interaction have been proposed to generate abundant postirons [18–33]. Though significant progresses have been achieved in producing dense and copious positrons in a plasma-based or all-optical manner, the quality of the obtained positron beams can hardly meet the requirements of some applications, such as the study of nonlinear QED physics, precision measurements of QCD process as well as searches for physics beyond standard model [34]. Thus, the post acceleration of the generated positrons becomes crucial for high energy researches mentioned above, especially the design of compact $e^{+}e^{-}$ colliders [35]. It is well known that, by injecting PW-scale laser pulses into a capillary discharge waveguide, electrons can be accelerated up to 8 GeV through wakefield acceleration [36]. However, this path is not so promising for positrons since the accelerating and focusing area in the bubble is too narrow to capture abundant positrons [37]. Recently, several methods to improve the accelerating wakefield for positrons have been proposed, such as plasma channel [38], vortex drive laser [39] and particle drive beam [40]. These solutions succeed in providing an accelerating field and focusing field for positrons in the wakefield, but all require high quality drive beams. Especially, the monoenergetic injecting positron beams are currently only available on large-scale accelerators. In addition to wakefield acceleration, the pre-generated positrons can also be accelerated through other mechanisms, such as sheath field acceleration [41], direct laser acceleration [42] and coherent transition radiation (CTR) acceleration [8]. These methods are designed to couple the positron generation and acceleration process in one laser-plasma configuration, and to ensure the qualified positrons can be injected into an accelerator structure. The main difficulties lie in controlling the positron acceleration process, such as positron confining and short acceleration phase. Recently, terahertz (THz)-driven electron acceleration has attracted lots of research interests as a novel acceleration concept that benefits from both conventional radio-frequency accelerator and laser-based accelerator [43–47]. On the one hand, the THz wavelength is much longer than the lasers that work in visible or infrared range, promising a longer acceleration phase. On the other hand, coherent THz waves can provide a much larger acceleration gradient than radio-frequency accelerators. For example, researchers have demonstrated that terahertz-driven electron acceleration and manipulation can be achieved through the use of dielectrically lined waveguides and temporally long, narrow-band, multicycle THz pulses. This advancement opens up the possibility of prolonging interaction lengths and heightening electron energies by cascading multiple acceleration stages, utilizing the terahertz energy while preserving the integrity of the initial pulse [46]. These advantages not only benefit the electron acceleration, but also work for the positron acceleration process since the CTR waves have ring-shape transverse intensity distributions which can help confine the positrons. Therefore, a THz-driven positron acceleration scheme may improve the positron beam acceleration and might potentially benefit the design of compact $e^{+}e^{-}$ colliders in the future.
2. Overview of the scheme and simulation parameters
Here we demonstrate a scheme that couples the positron generation via the multi-photon BW process and positron acceleration driven by THz radiation in a laser-plasma configuration. This scheme can be realized on facilities that are capable of providing either $2\times 10$ PW laser beamlines or combination of a 10 PW laser and a GeV-level electron beamline. Our scheme can be divided into four stages, and the schematic diagram is illustrated in Fig. 1. In the first stage, a 10 PW-scale laser pulse is incident into a plasma channel, capturing abundant electrons and then accelerating them to about 2.5 GeV via the wakefield acceleration. Next, the laser pulse ionizes the aluminum foil and is reflected off the laser axis. While the energetic electrons transmit through the foil target, they excite intense CTR and then collide with a scattering laser pulse, generating abundant electron-positron pairs. In the final stage, copious positrons are well confined by the transverse electric field of the THz wave and accelerated swiftly by the longitudinal electric field $E_x$ at the same time, resulting in a 800 MeV positron beam with extremely high brilliance and density.
Fig. 1. Schematic diagram of generation and post-acceleration of positrons via THz wave. The scheme can be divided into four stages: (I) Drive laser interacts with gas target, acquiring GeV electron bunch via wakefield acceleration. (II) Hot electrons transmit through Aluminum foil, emitting coherent transition radiation (CTR). (III) The GeV electrons collide with the scattering laser, emitting abundant $\gamma$-photons and producing copious electron-positron pairs. (IV) Positrons experience phase-locked acceleration by CTR.
The whole process has been well demonstrated by multi-dimensional particle-in-cell (PIC) simulations which were carried out using the relativistic PIC code EPOCH. In order to balance the computation load, both the 2D and 3D simulations were carried out in two parts. The first simulation corresponds to stage (I), and the size of the 2D simulation box is $100\lambda _0 (x)\times 50\lambda _0 (y)$ with $2000\times 1000$ grid cells. Sixteen macro-particles of electrons and ions are placed in one cell and the window moves in x direction at the speed of light $c$. In the second simulation, which corresponds to stage (II-IV), the box size is $150\lambda _0 (x)\times 400\lambda _0 (y)$ with $1500\times 1200$ grid cells, and the window moves in x direction at the speed of light $c$. Eight macro-particles of each species are placed in one cell and absorbing boundary conditions are employed for both particles and EM fields. In the corresponding 3D simulations, the first box size is $60\lambda _0 (x)\times 50\lambda _0 (y)\times 50\lambda _0 (z)$ with $1500\times 300\times 300$ grid cells, and the second box size is $150\lambda _0 (x)\times 40\lambda _0 (y)\times 40\lambda _0 (z)$ with $600\times 200\times 200$ grid cells. Considering the use of moving windows, the full simulation length in x direction would be 1500 $\mu m$ and 2100 $\mu m$ for the two simulation parts, respectively. In both 2D and 3D simulations, the drive laser is linearly polarized with the dimensionless intensity of $a_0=50$ and the duration of $\tau =9T_0$, while the scattering laser intensity is $a_0=100$ and duration is $\tau =6T_0$. Here, $T_0$ is the laser period and the wavelength of both lasers are $\lambda =1 \mu m$. In order to acquire high-quality energetic electron bunches for gamma ray emission, the well-developed laser-driven wakefield acceleration has been applied. In the first stage, we inject a linearly-polarized Gaussian laser pulse into a plasma channel. The electron density distribution of the channel is set as $n_e=n_0+15r^2/(\pi r_0^2)$, and the resulting similarity parameter $S=n_0/(a_0 n_c)=0.0002\ll 1$, corresponding to the case of relativistic underdense plasmas. Here, $n_c=m_e \omega _0/4\pi e^2\approx 1.1\times 10^{21} cm^{-3}$ is the critical density at the laser wavelength of $1 \mu m$, $m_e$ is the electron mass, $e=-1.6\times 10^{-19}\ C$ is the electron charge, $r_0=10\mu m$, $\omega _0$ is the laser angular frequency and $n_0=0.01 n_c$.
3. Electron wakefield acceleration and strong terahertz pulse generation
As the drive laser pulse propagates in the plasma channel, a typical solitary cavity structure, so called "bubble", is recognized as shown in Fig. 2(a). The bubble runs with the group velocity of the laser pulse, and it traps electrons from the background. Then, these trapped electrons are effectively accelerated via the wakefield acceleration mechanism in the following $2000T_0$ by keeping an intact structure, resulting in an quasi-monoenergetic GeV electron beam, as shown in Fig. 2(b). Figure 2(c) presents the longitudinal momentum distribution and angle distribution of electrons at $t=2000T_0$. It is shown that the energy peak at around 3 GeV in Fig. 2(b) corresponds to the injected and trapped electron bunch, which has a narrow divergence angle within 5 degrees. For better illustration of the electron wakefield acceleration process, the time-evolution of the electron kinetic energy distribution is presented in Fig. 3. Here, the maximum center energy of injected electron bunch can be estimated by [48]
where $P$ is the power of the incident laser pulse in the unit of gigawatt. The number of electrons trapped by the wakefield can be estimated by where $k_0=2\pi /\lambda _0$ is the wavenumber of the laser pulse and $\epsilon _0$ is the vacuum permittivity. It is easy to get $E_{max} \approx 2.96$ GeV and $N_0\approx 17.6$ nC, which are both consisitent with the PIC simulation results of $E_{max}\approx 3.1$ GeV and $N_0\approx 20.1$ nC. Furthermore, the energy conversion efficiency from laser to electrons can be calculated by $\eta = N_0 E_{max}/(P\tau )\approx 17.66\%$, which is in good agreement with the theoretical prediction of $20\%$ [48] as well as a previous simulation result of $22\%$ [49].Fig. 2. (a) The drive laser interacts with the gas target, and electrons are captured and accelerated in the wakefield. (b) Energe spectrum of the trapped electrons at three time points. (c) Electron distribution in $x-p_x$ space at $t=2000T_0$, while the inset shows the angle distribution of electrons.
Fig. 3. Electron kinetic energy distribution at (a) t=25 $T_0$, (b) t=500 $T_0$, (c) t=1000 $T_0$, (d) t=2000 $T_0$.
In the following stage, the drive laser hits on a $5\ \mu m$ 45-degree-oblique aluminum foil and is reflected off the laser axis. From the PIC results with ionisation module open, it is demonstrated that most of the Al atoms are ionised into trivalent ions. When the energetic electrons pass through the ionised target, they emit strong CTR as shown in Fig. 4(a). By applying a 0.1-10 THz band-pass filter to the orginal signal, the noises are cut out and the THz signal appears as shown by Fig. 4(b). Compared with the original signal, the high-frequency noises are filtered and the THz signal remains with a maximum intensity of 2.5 TV/m.
Fig. 4. (a) Electric field of CTR in $x-y$ plane at $t=600T_0$. (b) Filtered signal corresponding to (a). (c) CTR field acquired from theoretical calculation. The black lines in (a), (b) and (c) illustrate the 1D Ey field along the red dashed line. (d) Power spectrum of the CTR signal in (a). Power(e) and (f) are the transformed signal in $k_x-k_y$ space corresponding to (b) and (c), respectively. The slope of red dashed lines in (e) and (f) represent the transmission angle of CTR.
As a single particle with large velocity $v_e$ transports thorough a solid target, the transition radiation field can be estimated by [50]
where $\gamma =1/\sqrt {1-(v_{e/c})^{2}}$, $r=\sqrt {y^{2}+z^{2}}$, $R=\sqrt {x^{2+}y^{2+}z^{2}}$ and $\theta$ is the angle between the observation line and the particle propagation direction. In the above formulas, where $\gamma \gg 1$, the Coulomb fields are ignored since they decay as $1/\gamma ^2$. In our scheme, the THz band wavelength ($0-3000\mu m$) is larger than the size of the electron bunch, so the transition radiation can be considered as coherent [50]. Thus, the total CTR field can be calculated by summing the radiation field of all the contributing electrons as indicated in Fig. 4(c), which well reproduces the numerical results in Fig. 4(b). It is worth noting that both Eq. (3a) and (3b) contain $\delta (ct-R)$, which mean the radiation field of a single particle contributes only in a infinitely thin sphere surface. By the superposition of all electrons in the bunch, the radiation frequency can be estimated by $f_{CTR}\approx c/(\delta _x+\delta _r)$. Here, $\delta _x\approx 32 \mu m$ and $\delta _r\approx 15 \mu m$ represent the longitudinal and transverse size of the electron bunch, respectively. The corresponding frequency of the radiation is then estimated to be around 6.3 THz. Figure 4(d) shows the power spectrum of the radiation by applying maximum entropy spectral estimation to the original signal. A clear peak exhibit at 6 THz, which is in good agreement of the estimation. It turns out that 85% of the signal is within the THz band (0.1-10 THz) and the Coulomb field together with some high-frequency noises induced at the boundary take up only 15%. In order to investigate the transmitting feature of the radiation, we present the filtered signal from the PIC simulations and the theoretical signal in $k_x-k_y$ space, as shown in Fig. 4(e) and Fig. 4(f), respectively. Here, $k_x$ and $k_y$ are the wave vectors’ module in $x$ and $y$ direction, respectively, and they are normalized by $k_0=2\pi /\lambda _e$ with $\lambda _e$ being the length of the electron bunch. It is known that the slope of the 2D spectrum signal represent the transmitting angle of the radiation which should be $\theta _{CTR}\approx 1/\gamma$ [50]. Considering $\gamma \gg 1$ and $\theta _{CTR}\to 0$ in our scheme, the radiation direction can be indicated by the red dashed lines in Fig. 4(e) and Fig. 4(f). It is demonstrated that the main signals of both PIC and theoretical radiation are along the red line, corresponding to an almost horizontal transimitting direction. It is also shown that 9% of the electron energies is converted to the THz radiation, hence the conversion efficiency from the drive laser to THz radiation is $17.66\%\times 9\%\approx 1.5\%$. This value is already much higher than that from the THz sources based on solid plasma in experiments, which are often lower than $1\%$ [51,52].4. Positron generation and phase locked acceleration driven by THz wave
After the energetic electrons pass through the aluminum foil, they collide head on with the scattering laser coming from the right side of the simulation box, triggering intense nonlinear Compton scattering (NCS), so that abundant $\gamma$-photons are radiated as indicated in Fig. 5(a). Here, the collision zone is marked by the gray square between $x=80 \mu m$ and $x=100\ \mu m$. Here, the intensity of the NCS process can be characterized by the quantum invariant $\chi _e=(\gamma _e/E_s)\sqrt {(\vec {\beta }\times c\vec {B}+\vec {E})^2-(\vec {B}\cdot \vec {E})^2}$, where $\beta =v_e/c$ is the normalized electron velocity, $\gamma _e=1/\sqrt {(1-\beta ^2 )}$, B and E are the magnetic field and electric field of the laser pulse, respectively. Figure 5(b) presents the $\chi _e$ distribution in the collision zone along the x axis at $t=75T_0$. It is shown that the maximum $\chi _e \approx 2$ at the collision center in our configuration, promising the generation of abundant $\gamma$-photons. After the collision, at $t=150T_0$, the photon distribution in $x-p_x$ phase space is shown in Fig. 5(c). We see the photon momentum in the x direction is as high as $4000m_e c$. The inset of Fig. 5(c) shows the angle distribution of the generated $\gamma$-photons, and the curve presents the full width at half maximum (FWHM) of 4 degrees. Thus, the corresponding brightness of the $\gamma$-photon bunch can be calculated by $(1.99\times 10^{13} photons/0.1\%bandwidth)/(33 fs \cdot (0.7 mm\cdot mrad)^2)\approx 1.23\times 10^{27}\ photons s^{-1} mm^{-2} mrad ^{-2} (0.1\%bandwidth)^{-1}$.
Fig. 5. (a) Energy density distribution of energetic electrons and transverse electric field distribution of the scattering laser at $t=75 T_0$. (b) Distribution of quantum parameter $\chi _e$ along the x axis at $t=75T_0$. (c) Electron distribution in $x-p_x$ phase space at $t=150 T_0$ while the inset presents the angle distribution of electrons. (d) The same to (a) but for $\gamma$-photons at $t=75 T_0$. (e) Distribution of quantum parameter $\chi _{\gamma }$ along the x axis at $t=75T_0$. (f) The same to (c) but for $\gamma$-photons. The gray squares in (a) and (d) indicate the collision zones.
These $\gamma$-photons continue to propagate forward and interact with the scattering laser, initiating the BW process and generating prolific electron-positron pairs, as shown in Fig. 5(d). In this process, the emissivity of positrons can be described by the quantum parameter $\chi _{\gamma }=\hbar \omega _0|\vec {E}_{\bot }+\vec {k}\times c\vec {B}|/(2m_ec^2E_s)$, where $\hbar$ is reduced Planck constant, $\vec {E}_{\bot }$ is the electric field perpendicular to the velocity of $\gamma$-photons, $\vec {k}=2\pi \vec {v}/(\lambda _{ph} c)$ is the wave vector of the emitted photon, $\lambda _{ph}$ is the wavelength of the emitted $\gamma$-rays and $\vec {v}$ is the speed vector of the photon. It is shown that $\chi _\gamma$ approaches 1 in the collision zone, as shown in Fig. 5(e), guaranteeing the creation of copious positrons. Figure 5(f) presents the $x-p_x$ phase space distribution and angle distribution of the positrons at $t=150T_0$, indicating small energy spread and narrow angle distribution. Here, the geometric emittance of the positron bunch in y direction is $\epsilon _y\approx 0.058\ mm\cdot mrad$ and the brilliance of the positron bunch is $2.26\times 10^{12} \ s^{-1} mm^{-2} mrad^{-2} eV^{-1}$, which is five orders of magnitude larger than the reactor-based positron source NEPOMUC [53]. Such bright positron beams could possibly benefit the design of laser-based electron-positron colliders and the investigation of laboratory astrophysics.
In such a head-on collision configuration, the two quantum invariants $\chi _e$ and $\chi _{\gamma }$ can be maximized, ensuring the generation of copious $\gamma$-photons and positrons. Meanwhile, the scattering laser will be scattered off the laser axis by the oblique Al target, protecting the laser system by reducing backward reflection. Then, the generated positrons continue to propagate forward with the CTR wave. Figure 6(a) presents the radial electric field $E_r$ that positrons suffer in the y-z plane, where the red arrows represent the direction of $E_r$ field and the colored rings are the contour lines of the field intensity. The result is sliced from a 3D simulation at $t=75 T_0$, illustrating that the positrons are effieciently pushed toward the laser axis by the radial field of the THz radiation. Meanwhile, the longitudinal field of the THz wave accelerates the positrons along the x-axis with the maximum acceleration gradient of $0.45$ TV/m at $t=75 T_0$, as shown in Fig. 6(b). This value is in consistent with the theoretical result of $E_x\approx 0.61$ TV/m, estimated from Eq. (3b). Figure 6(c) illustrates the positron density distribution in the $x-y$ plane and the corresponding $E_r$ and $E_x$ field distribution at $t=75 T_0$. As we can see, the positrons are confined near the laser axis and located in the acceleration phase of the $E_x$ field, i.e., phase-locked acceleration driven by the THz radiation. Since the THz radiation was driven by the wakefield-accelerated electrons from the first stage, the positrons can be accelerated from 300 MeV to 614 MeV within $1000 T_0$, as shown in Fig. 6(d). Therefore, the average acceleration gradient for positrons is $0.31$ TV/m, which is comparable to the wakefield acceleration gradients for electrons. Moreover, the dephasing condition can be written as $(c-v(t))t=(c-v_0-E_x(t) et/m_e)t=\lambda _{THz}$, $v_0$ denotes the original velocity of the positrons. By solving the above equation, we can obtain the dephasing time of approximately 5.3 ps, corresponding to the acceleration length of about 1600 $\mu m$, which is consistent with the simulation results.
Fig. 6. (a) Radial electric field $E_r$ distribution in the $y-z$ plane sliced from a 3D simulation at $t=75T_0$. The red arrows represent the direction of $E_r$ vectors and the colored rings are the contour lines of the field intensity. (b) Longitudinal acceleration field $E_x$ in the $y-z$ plane at $t=75 T_0$. (c) Density distribution of positrons and the corresponding electric field $E_x$ along the dashed line at $t=75 T_0$. (d) Energy spectrum of the positrons from $t=85 T_0$ to $1800 T_0$.
5. Discussion and conclusion
One can see from Eq. (3a) and Eq. (3b) that the CTR field intensity is closely related to the electrons’ charge and the bunch profile. Therefore, a series of PIC simulations are performed to check the influence of the drive bunch on the resultant positron yield and CTR field intensity. In these simulations, the electron energies are set to be 2.5 GeV, and the shape of the electron bunch is initiated by $n_e=n_1 exp[-(x-10^{-6})^2/\delta _x^2-r^2/\delta _r^2]$. Here, the maximum density of the electron bunch $n_1$ is the only variable, which results in different electron charges. It turns out that the longitudinal acceleration field $E_x$ increases almost linearly with the drive electron charges, as shown in Fig. 7(a). Despite the THz fields are not so relevant with the electrons’ energy in relativistic ocassions ($\gamma \gg 1$), we still need to enhance the electron energies to ensure a large value of the quantum parameters $\chi _e$ and $\chi _\gamma$ to generate abudant $\gamma -$photons and positrons. Since both the positron yield and acceleration field increase with the drive bunch charges, it is foreseeable that the average positron energy should also increase, as illustrated in Fig. 7(b). However, when a larger amount of electrons e.g., 200 nC are involved in the scheme, the energy spread of positrons would increase greatly since more positrons will fall out of the acceleration phase. For comparison, we also initialized the injecting electron bunch with the FACET II parameters [54], i.e., the total charge of 3 nC, the electron energy of 10 GeV and the bunch length of 20 $\mu$m, and the results are shown in Fig. 7(c). With the electron energy increased from 2.5 GeV to 10 GeV, the THz field intensity and the positron average energy remains at the same level, while the positron yield has increased two orders of magnitude, indicating that our scheme is also quite promising on both large accelerators and laser facilities. Especially, an oblique Al target could scatter both drive and scattering pulses off the laser axis, protecting the laser systems without sacrificing the cross-section of the QED processes, which may facilitate some potential experiments in the future.
Fig. 7. (a) $E_x$ field intensity of CTR and positron yield at different drive bunch charges ranging from 2 nC to 200 nC. (b) Positron energy spectrums at $t=1800 T_0$ using electron bunches with different charges ranging from 2 nC to 200 nC, and the corresponding average positron energy. (c) Positron yield and $E_x$ field intensity of CTR driving by electron bunches with FACET II parameters.
In summary, we propose and demonstrate an all-optical method that couples the generation and acceleration of positrons, with beam energy up to GeV, and the beam brilliance of $2.26\times 10^{12} s^{-1} mm^{-2} mrad^{-2} eV^{-1}$. Here, the positron acceleration is driven by the THz wave that is produced in the interaction of energetic electron bunches and an aluminum foil. Especially, the THz wave owns a transverse field that confines the positrons and a strong longitudinal acceleration field at the same time. As a result, pC-scale positrons can be stably accelerated to near GeV via phase-locked acceleration with the average acceleration field of $0.31$ TV/m. When it comes to experiments, the drive source could be either 10 PW scale lasers or nC-scale energetic electron beams from conventional accelerators, making this scheme achievable on both laser facilities that owns more than one $10$ PW beamlines, like SEL [55], and laboratories which can deliver energetic electrons and PW-scale laser pulses simultaneously, like SLAC [15].
Funding
National Key Research and Development Program of China (2018YFA0404802); National Natural Science Foundation of China (12004433, 12135009, 12275356); Hunan Provincial Innovation Foundation for Postgraduate (CX20200038, CX20220048).
Acknowledgments
We acknowledge the access to the EPOCH code developed by University of Warwick.
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.
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