Laser-Pulse and Electron-Bunch Plasma Wakefield Accelerator

While the parameter space for the driver bunch and the laser pulse is very wide, we will concentrate on exploring multi-GeV acceleration using sub-PW laser pulses with $\lambda_{0}=0.8{\rm\mu m}$ and sub-GeV driver bunches. Multi-GeV acceleration requires tenuous plasma. Therefore, we chose $n_{0}=4\times 10^{17}$ cm^-3 corresponding to $c/\omega_{p}\approx 8.4{\rm\mu m}$ and $P_{\rm crit}\approx 75$TW. To ensure self-focusing, the laser power is chosen to be $P=380$TW. Further, the Gaussian spot size and duration are chosen to be $w=29{\rm\mu m}$ and $\tau_{FWHM}=68$fs, respectively. These laser parameters corresponds to the normalized vector potential $a_{0}=3.7$.

The charge $Q_{b}=1.25$nC of a Gaussian electron driver bunch was chosen to correspond to the normalized charge $\bar{Q}\approx 2.6$ My_Driver_2017 to produce a fully-formed plasma bubble with $a_{\parallel}>a_{\parallel}^{\rm WB}$. The transverse size $w_{b}=8.4\mu$m and duration $\tau_{b}=56$fs of the bunch were chosen to approximately match $w_{b}\sim c/\omega_{p}$ and $\tau_{b}\sim 2\omega_{p}^{-1}$, resulting in the peak current $I_{b}=22.3$kA: sufficient to channel the laser pulse because $I_{b}>I_{A}$  BeamChannel_Shvets_PhysRevE97 . Driver electrons started with the initial energy $\gamma_{b}mc^{2}=0.65{\rm GeV}$ and the energy spread of $0.5{\rm MeV}$. Therefore, the total energy of the electron bunch is $U^{\rm bunch}\approx 0.8{\rm J}$, i.e. just $3\%$ of the laser pulse energy $U^{\rm laser}\approx 27{\rm J}$.

Simulating centimeters-long propagation of both laser pulse and driver bunch using conventional particle-in-cell (PIC) codes presents a unique computational challenge because of the high temporal resolution requirements imposed by the DLA mechanism and because of the equally high spatial resolution requirements imposed by the Courant stability condition. In order to capture the laser-electron interactions with sufficient accuracy, a longitudinal spatial step $\Delta z$ and time step $c\Delta t$ close to $\lambda_{L}/50\sim\lambda_{L}/100$ are needed DLA_require_Alex ; My_DLA_2019 . Therefore, we have selected to carry out fully three-dimensional quasi-static particle-in-cell (QPIC) simulations using an in-house developed code WAND-PIC WAND_PIC . WAND-PIC uses a quasi-static approach Mora_1997 ; Lotov_PiC_2003 ; QuickPiC_2006 ; HiPACE_2014 ; My_Driver_2017 to model the motion of the bubble-forming plasma particles and of the laser pulse envelope (amplitude and phase). This approach reduces the dimensionality of the simulation by one. It also reduces the required longitudinal resolution ($\xi=z-ct$) in the moving reference frame to $\delta\xi\sim c/\omega_{p}$ scale. Although the longitudinal resolution has been reduced, the nonlinearity of the wakefield, especially in the back of the bubble, is captured by using adaptive grid size in $\xi$ direction, i.e., the step size $\delta\xi$ is adjusted at every step based on the fastest longitudinal velocity of the plasma particles. On the other hand, full equations of motion are used to model the interaction between driver electrons and the high-frequency laser fields. The code accurately models resonant interactions between the bunch electrons executing betatron undulations and the laser pulse using the sub-cycling method, (see the comparison of WAND-PIC and a full-PIC code in My_DLA_2019 ). Such interactions are at the heart of the DLA, and their extreme sensitivity to the phase velocity of the laser field necessitates high temporal resolution.

The key results of our simulations are shown in Fig. 2. The top row shows the color-coded electron density of the plasma at the propagation distances $z=0$mm, $12$mm, $28$mm, and $62$mm. The plasma bubble is clearly defined. The witness electron bunch is placed near the back of the bubble, where the accelerating field $E_{\parallel}=E_{z}$ (see Figs. 2 (e-h), blue-dotted line) is large. The longitudinal $(\xi,\Delta\gamma)$ phase spaces of the driver (dark-green dots) and witness (red dots) electrons are presented in Figs. 2 (e-h), where $\Delta\gamma$ is the change of the relativistic Lorentz factor and $\xi=z-ct$ is the longitudinal position in the moving frame. In this specific simulation, the initial energy of the witness beam is taken to be $\gamma_{w}mc^{2}=50$MeV. This particular choice is unimportant because future TeV-scale plasma-based accelerators will be segmented into at least a hundred independently-driven segments (stages). The initial energy $\gamma_{w}mc^{2}$ will depend on the segment number. Therefore, for all but the first acceleration segment, the following relationship between the energies of the driver and witness beams will be assumed: $\gamma_{w}\gg\gamma_{b}$. Therefore, we will be concentrating on the energy gain $(\Delta\gamma)$ of the witness beam, not on its absolute energy.

Below we introduce the three conceptual stages of LEPA that are differentiated from each other by the importance of the DLA for the driver bunch propagation, as well as the role played by the driver bunch in channeling the laser pulse and generating the plasma bubble. The first stage starts immediately at $z=0$. During this stage, the plasma bubble is produced mainly by the driver bunch, and its size is somewhat larger than that of the bubble produced by the laser pulse alone. Therefore, a higher accelerating gradient is generated in the back (accelerating) portion of the bubble. The first stage of LEPA (from $z_{0}=0$mm to $z_{1}=12$mm for this specific example) is characterized by highly efficient DLA: almost $50\%$ of the electrons from the driver bunch experience significant DLA. For example, we observe from Fig. 2 (f) that some of the driver bunch electrons in the front portion of the bubble ($(z-ct)>10k_{p}^{-1}$) have gained $\sim 500{\rm MeV}$ of energy. The bunch-averaged DLA gain for all driver electrons is $\sim 200$MeV. Owing to the DLA, electrons stay in the decelerating (front) portion of the bubble for a much longer time than they would have stayed without the laser pulse. In fact, in the absence of the DLA, the driver bunch would have lost most of its energy after less than $z\approx 10$mm of propagation through the plasma due to rapid self-deceleration by its own wakefield. The physics underlying the extended bunch propagation during Stage 1 of LEPA is described in detail in Sec. III.

The second stage of LEPA (from $z=z_{1}$ to $z_{2}=28$mm for this specific example) is characterized by the bunch electrons getting out of resonance with the laser pulse. The physics of falling out of resonance has been described elsewhere Farfield , and will not be described here. During Stage 2, driver electrons either get decelerated by the wakefield and slip back into the back portion of the bubble, or move out of the bubble entirely because of the increased amplitude of their betatron oscillations Farfield . As shown in Fig. 2 (c) and (g), the majority of driver bunch electrons have already slipped back into the decelerating portion of the wake at $z=z_{2}$. This has two effects, both of which are deleterious to the acceleration of a witness beam. First, the plasma bubble becomes smaller because it is primarily driven by the laser pulse. Second, the wake is actually depleted via the beam-loading effect produced by the driver bunch in the back portion of the plasma bubble.

The third stage of LEPA, during which the laser pulse continues to propagating in a self-guided regime until its complete depletion, takes place from $z=z_{2}$ to $z_{3}=62$mm: see Figs. 2 (d, h). The witness beam experiences the highest average acceleration gradient of the order $E_{\parallel}\approx 90{\rm GV/m}$ during the first two stages ($z_{0}<z<z_{2}$) because the driver bunch enhances the bubble created by the laser pulse. On the other hand, a smaller average acceleration gradient of approximately $E_{\parallel}\approx 58$GV/m is experienced by the witness beam during the third stage of LEPA ($z_{2}<z<z_{3}$). Overall, the witness beam gains $W_{\rm LEPA}\approx 4.5$GeV of energy in total over a distance of $L_{\rm LEPA}=z_{3}\approx 62$mm.

Understanding whether the above energy gain of $W_{\rm LEPA}$ is sufficient for justifying the hypothesis of synergy between the bunch-guided laser pulse and laser-accelerated bunch requires a quantitative comparison between energy gains by a witness bunch when identical laser pulse and electron driver are used separately. For example, one can envision two sequential plasma accelerator stages: a LWFA driven by a laser pulse alone, followed by a PWFA driven by an electron bunch; the laser, beam, and plasma parameters for both stages are listed in Sec. II.1. The results of these simulations are listed in Table I.

In the case of the same plasma density for LWFA and PWFA listed in Sec. II.1, and a low-charge accelerated witness beam, the results are listed in the first three columns, top row of Table I. Laser-beam synergy is confirmed by observing that $W_{\rm LEPA}>W_{\rm LWFA}+W_{\rm PWFA}$ in the case when the depletion (also known as loading) of the plasma wake by the witness bunch can be neglected. This result is quite remarkable given that the driver bunch can significantly deplete the plasma wake after most of its electrons slip into the back (accelerating) portion of the plasma bubble. The synergy between the driver bunch and the laser pulse is even more pronounced in the case of a moderately-charged witness beam with a total charge of $q=0.18$nC (bottom row of Table I): the energy gain in a LEPA scheme exceeds the sum of energy gains in sequential LWFA and PWFA schemes by over $60\%$ while producing smaller energy spreads.

We note that while $W_{\rm PWFA}$ is limited by the transformer ratio, it is possible to increase the energy gain of a LWFA by keeping $U^{\rm laser}$ the same while reducing the pulse duration and increasing the plasma density in the LWFA stage (see Sec. IV for details). The values of thus optimized $W_{\rm LWFA}$ without (with) witness beam loading are listed in top (bottom) rows, the fourth column of Table I. Such optimization of the LWFA stage does not change our conclusion: the synergy between the driver beam and the laser pulse enables an overall increase of the energy gained by the witness bunch: $W_{\rm LEPA}>W_{\rm LWFA}+W_{\rm PWFA}$ with and without beam loading. The key to understanding such synergy lies in analyzing the physics responsible for the extension of the laser-bunch propagation in LEPA described in the following Sec. III. Specifically, we demonstrate the extension (i) of the driver beam propagation by the DLA mechanism (see Sec. III.1), and (ii) of the laser pulse propagation by the bunch channeling (see Sec. III.2) during Stage 1 of LEPA. Further details of the comparison between LEPA, LWFA, and PWFA scheme are presented in Sec. IV.

The DLA mechanism can counter the decelerating field and, in fact, accelerate bunch electrons at a rate of $d\gamma/dt\approx e\bar{E}_{\parallel}/mc^{2}$. The maximum energy achieved by the electron while in resonance with the laser field has been estimated as $\epsilon_{\rm max}\approx mc^{2}(\omega_{p}/\omega_{0})^{2}(E_{0}/\bar{E}_{\parallel})^{4}/130$ Farfield . The approximate equation for betatron resonance of a laser pulse with an electron with the energy $\epsilon_{\rm res}\equiv\gamma_{\rm res}mc^{2}$ undergoing a transverse undulation with the betatron frequency $\omega_{\beta}\approx\omega_{p}/\sqrt{2\gamma_{\rm res}}$ and time-averaged transverse momentum $p_{\perp}$ is given by Zhang_PPCF16

$$\omega_{\beta}(\gamma_{\rm res})=\omega_{0}\left(\frac{1+p_{\perp}^{2}/m^{2}c^{2}}{2\gamma_{\rm res}^{2}}+\frac{1}{2\gamma_{\rm ph}^{2}}\right),$$

(1)

where the relationship between the laser wavenumber $k_{0}$ and frequency $\omega_{0}$ is expressed as $\omega_{0}^{2}=c^{2}k_{0}^{2}\left(1+\gamma_{\rm ph}^{-2}\right)$.

A typical driver electron with a large initial momentum $\gamma_{b}\gg\gamma_{\rm res}$ undergoes the following sequence of energy exchanges with the wake and the laser field. Initially, most electrons are decelerated by the wakefield to the energy comparable to $\epsilon_{\rm res}$ because of the resonance condition given by Eq.(1) is not initially satisfied. Such deceleration takes place during the $0<z<z_{\rm dec}$ interval, where $z_{\rm dec}\approx 8$mm. Note that $z_{\rm dec}\ll L_{\rm dec}$ because the bunch is decelerated by the combined wakefield: its own, and that of the laser pulse.

Next, most of the driver bunch electrons become resonant with the laser field and start gaining energy from the DLA process. As can be observed in Fig. 2 (f), at least half of the electrons regain most of their energy from the DLA process and stay in the front portion of the plasma bubble. Finally, as resonant electrons gain significant transverse and total energy, the resonant condition is no longer be satisfied. The total distance that traveled by a typical resonant electron before detuning away from the DLA resonance can be estimated as $L_{\rm DLA}\approx z_{\rm dec}+mc^{2}(\omega_{p}/\omega_{0})^{2}E_{0}^{4}/\left(130e\bar{E}_{\parallel}^{5}\right)$. Assuming that $e\bar{E}_{\parallel}/m\omega_{p}c\sim 1$  Lu_GeV ; Pukhov_Pheno_2004 ), the propagation distance of the DLA-assisted driver is longer than the self-stopping distance by the following factor: $L_{\rm DLA}/L_{\rm dec}\approx 1+(mc/\gamma_{b})(\omega_{0}/\omega_{p})^{2}a_{0}^{4}/130$. For the parameters listed in the caption of Fig. 2, we find that $L_{\rm DLA}\approx 2.7L_{\rm dec}$. Therefore, we estimate that $L_{\rm DLA}\approx 21.6$mm. This distance is in good agreement with the value of $z_{2}$ observed in Fig. 2 (g).

To further clarify the role of DLA in extending driver bunch propagation in LEPA, it is instructive to separately calculate the work done by the laser and by the wakefield on a given $j$’th electron: $W_{x}^{(j)}$ and $W_{z}^{(j)}$, respectively. The dimensionless total energy of each electron, $\gamma^{(j)}=\gamma_{b}+\left(W_{x}^{(j)}+W_{z}^{(j)}\right)/mc^{2}$, is color-coded in the $(W_{x},W_{z})$ space and plotted in Fig. 4 (a) for all driver bunch electrons at the propagation distance $z=z_{1}$ through the plasma. Fig. 4 (a) is convenient for classifying the electron population into two distinct resonant (”DLA”) and non-resonant (”non-DLA”) sub-populations. For convenience, we classify those with $W_{x}^{(j)}>200mc^{2}$ as DLA electrons (approximately $50\%$ of all driver electrons), and the remaining as non-DLA electrons (the superscript $j$ is dropped hereafter).

As we see from Fig. 4 (a), the DLA sub-population electrons have gained energy up to $W_{x}\approx 3,000mc^{2}$ directly from the electric field of the laser; the average DLA energy gain for those electrons is $\langle W_{x}\rangle\approx 860mc^{2}$. We further observe from Fig. 4 (a) that those DLA electrons which gained more energy from the laser field also lost more energy to wakefield, i.e. $W_{x}$ and $W_{z}$ are anti-correlated. This implies that the electrons with the largest $W_{x}$ stayed in the decelerating (front) region of the bubble for a longer time. Therefore, resonant DLA extended their propagation distance in the front portion of the bubble and potentially contributed to enhancing the wakefield strength and the size of the plasma bubble. On the other hand, the non-DLA electrons exhibit much smaller (yet still negative) energy exchange with the wakefield. This implies that they have rapidly slipped to the back of the bubble, where their positive energy gain from the wake partially offset their initial energy loss to the wake.

To further investigate the difference between DLA and non-DLA sub-populations, it is helpful to select one representative DLA (black star) and one non-DLA (blue circle) electron marked in Fig. 4 (a). The trajectories of the DLA and non-DLA electrons (black and blue lines, respectively) inside the plasma bubble are plotted in Fig. 4 (b). Although the non-DLA electron starts its motion at the head of the driver bunch, where the decelerating field is smaller, it stays in the decelerating field for a much shorter time compared to the DLA electron. For example, at $z=z_{1}$ the non-DLA electron has already slipped to the back of the bubble, whereas the DLA electron remains in the decelerating field. The energies and longitudinal positions of these two electrons inside are shown in Fig. 4 (c) as a blue circle and a black star.

The evolution of $W_{x}(z)$, $W_{z}(z)$, and $\gamma(z)$ for the representative DLA electron is plotted in Fig. 4 (d) as a function of the propagation distance. From $z=0$mm to $z=z_{1}$, the DLA electron gains $W_{x}=1.0$GeV from the laser pulse and loses $W_{z}=-1.05$GeV to the wakefield. Note that the gain $W_{x}>0$ only starts at $z=7.0$mm illustrating that this electron indeed gains energy from the laser at nearly twice the rate of its loss. Its further trajectory illustrates a rather typical fate of a DLA electron: it keeps propagating in the decelerating field of the bubble up to $z=26$mm, and then leaves the bubble. In contrast, its non-DLA counterpart in Fig. 4 (e) stays in the decelerating field of the bubble for up to $z=7.5$mm. This comparison indicates that the propagation distance of the DLA electron got in the leading portion of the plasma bubble is extended by $\Delta z=18$mm, in a good agreement with the earlier provided estimate. Having demonstrated that the DLA mechanism extends the electron bunch propagation, we now demonstrate how the presence of the electron bunch extends the propagation of the laser pulse itself.

As the driver bunch gains energy and increases its propagation distance due to its DLA interaction with the laser pulse, it concurrently extends the propagation distance of the laser pulse. Specifically, during the first and second stages of LEPA, the bunch can sustain a deeper plasma channel that channels the laser pulse and mitigates laser diffraction BeamChannel_Shvets_PhysRevE97 . The channeling effect takes place in addition to the enlargement of the laser-produced plasma bubble by the bunch, and of the corresponding accelerating field at the back of the bubble.

To illustrate the channeling effect, we have compared laser propagation through the plasma with (LEPA) and without (LWFA) the driver bunch. The results are presented in Fig. 5. By comparing Figs. 5 (a) and (b), we observe that the spot size of the bunch-guided laser pulse in the LEPA scenario is considerably smaller than that of the self-guided laser pulse in the standard LWFA scheme. The diffraction of the laser pulse in the LWFA scheme results in a significant drop (by almost a factor $2$) of its peak on-axis normalized vector potential $a_{0}$. This should be contrasted with almost constant $a_{0}$ in the LEPA configuration, as shown in Fig. 5 (c). In fact, the laser has essentially diffracted by the time it propagated $z=40$mm into the plasma, and the peak $a_{0}$ drops below unity. In contrast, the laser pulse propagates up to $z=z_{3}\approx 62$mm in LEPA.

Relatively rapid diffraction of the self-guided laser owes to its moderate power: $P/P_{\rm crit}\approx 5$. Therefore, the maximum energy gain of a low-charge witness beam is only $W_{\rm LWFA}\approx 2.4$GeV. On the other hand, the laser pulse in LEPA is channeled by the bunch, thereby gaining an additional $\Delta z=24$mm of propagation while maintaining a much higher intensity (see Fig. 5 (c) for comparison). Crucially for the efficiency of a LWFA, the self-guided laser pulse diffracts before it loses just $\eta_{\rm LWFA}\approx 20\%$ of its total energy. This happens because a shallow plasma bubble created by a transversely-spread laser pulse is a poor absorber of the laser energy Mora_1997 . On the other hand, the bunch-driven laser pulse transfers $\eta_{\rm LEPA}\approx 70\%$ of its energy to the plasma, thereby creating a larger accelerating gradient and providing a larger energy gain of $\Delta W_{\rm LEPA}\approx 4.5$GeV to a low-charge witness beam.

Figure 5 (d) presents the comparison between laser energy absorption by the plasma in the LWFA (blue line) and LEPA (red line) schemes as a function of the propagation distance. The higher depletion of the laser pulse in the LEPA scheme explains the higher energy gain $W_{\rm LEPA}$ in comparison with the energy gain $W_{\rm LWFA}$ in a LWFA. Note that the ratio $W_{\rm LEPA}/W_{\rm LWFA}$of the gained energies in the LEPA and LWFA schemes is considerably smaller than the ratio $\eta_{\rm LEPA}/\eta_{\rm LWFA}$ of the extracted laser energies. Plasma wake depletion (”loading”) by the driver bunch electrons that eventually slip into the accelerating portion of the bubble account for this discrepancy.

Because additional driver energy ($\sim 3\%$ of the laser energy) is expended in the LEPA scheme, it is instructive to investigate how much additional energy can be imparted to a witness beam with finite charge $q$. We have carried out numerical simulations with WAND-PIC for all three acceleration schemes and included the beam loading of the wake by the witness beam. Note that beam loading by the driver bunch electrons that eventually slip to the back of the bubble has been already accounted for in the simulations presented in Figs.2, 3, 4. Therefore, in what follows we drop the direct reference to the witness beam when discussing beam loading.

In the anticipation that a significant portion of the driver bunch energy will be utilized to increase the energy content of the witness bunch, we have selected the witness bunch charge according to $q=Q_{b}\left(\gamma_{b}mc^{2}/W_{\rm LEPA}\right)$. Gaussian witness beam’s transverse and longitudinal sizes were chosen to be $w_{x}=w_{y}=4.2\mu$m and $w_{z}=2.8\mu$m, respectively. Note that the longitudinal density profile of a witness bunch can be optimized to reduce the energy spread Witness_Shaping ; Witness_Shaping_2 . Such optimization is beyond the scope of this work and will be the subject of future research. Throughout this paper, Gaussian density profiles are assumed for both driver and witness bunches.

The results of the beam-loaded simulations are listed in the second row of Table. I. The average energy gain of the witness bunch in the LEPA scheme is $\langle W_{\rm LEPA}\rangle\approx 3.6$GeV, with a $\pm 0.25$GeV energy spread due to beam loading. The reduction from $W_{\rm LEPA}\approx 4.5$GeV is also due to beam loading. The average energy gain in the PWFA is also reduced: $\langle W_{\rm PWFA}\rangle\approx 1$ GeV. The largest energy gain reduction is found in the LWFA case: $\langle W_{\rm LWFA}\rangle\approx 1.2$GeV, which is only $50\%$ of the $W_{\rm LWFA}\approx 2.4$GeV energy gain in the absence of beam loading. We also listed the transverse emittance of the wittiness beam in the bottom row of Table. I. The witness beams in LEPA, PWFA, and PWFA schemes have comparable emittance.

This observation reveals another advantage in combining the laser pulse and an electron bunch in a LEPA: a steep bubble created by the two is less susceptible to depletion by a witness beam. On the other hand, the bubble created by the laser alone in a LWFA at moderate power is relatively shallow, and thus more susceptible to beam loading. Overall, the witness beam gains $\Delta W\equiv\left(\langle W_{\rm LEPA}\rangle-\langle W_{\rm LWFA}\rangle\right)\approx 2.4$GeV more energy per electron in the LEPA than in the LWFA scenario. Therefore, the excess energy gain of the witness bunch is $\Delta U_{\rm witt}=q\Delta W\approx 0.4$J. Remarkably, $\Delta U_{\rm witt}$ is only a factor $2$ smaller than $U^{\rm bunch}\approx 0.8{\rm J}$. This confirms our initial expectation of high energy utilization of the driver bunch by the witness beam.

In comparing energy gains of the witness bunch in the LEPA vis a vis LWFA and PWFA schemes, we have assumed that the same plasma density $n_{0}=4\times 10^{17}$ cm^-3 is used for all three schemes. In fact, it is reasonable to ask if the same laser energy $U^{\rm laser}\approx 27{\rm J}$ can be deployed more efficiently is a LWFA scheme by shortening the pulse and increasing the plasma density. Such an approach is based on increasing the ratio of the peak laser power $P$ (which increases inverse proportionally to laser pulse duration $\tau_{\rm FWHM}$) to critical laser power $P_{c}$ (which decreases proportionally to plasma density). A higher $P/P_{c}$ ratio contributes to longer laser pulse propagation length and overall efficiency of laser energy deposition into the plasma. Therefore, we have identified a set of ”optimal” laser-plasma parameters that result in the longest laser propagation and the largest energy gain of the witness beam. We refer to this scheme as LWFA-OPT.

Specifically, the plasma density was increased to $n_{0}^{\rm OPT}=1.5\times 10^{18}$cm^-3 and the laser pulse duration was reduced to $\tau_{\rm FWHM}^{\rm OPT}=49$fs, thereby increasing the laser power to $P=533{\rm TW}\approx 27P_{c}$ while preserving its total energy. The normalized vector potential $a_{0}=6$ is chosen based on the optimized matching conditions Lu_GeV . The laser duration is chosen to have a comparable dephasing and depletion lengths Lu_GeV . The results of the WAND-PIC simulation are plotted in Fig. 6. As one can observe from Fig. 6 (b), the laser pulse propagates up to $z=15$mm, at which point the depletion of the pulse and the dephasing of the witness beam were simultaneously achieved. Without the beam loading, the maximum possible gain of a witness beam is $W_{\rm LWFA}^{\rm OPT}\sim 3$GeV. For a realistic witness beam with a moderate charge $q=0.18$nC charge, the average energy gain is reduced to $\langle W_{\rm LWFA}^{\rm OPT}\rangle\sim 1.5$GeV, with the energy spread of $\Delta W=\pm 0.7$GeV. Therefore, the LEPA scheme still enables larger energy gain ($\langle W_{\rm LEPA}\rangle>\langle W_{\rm LWFA}^{\rm OPT}\rangle+\langle W_{\rm PWFA}\rangle$) and smaller energy spread.