Simulation of electron postacceleration in a two-stage laser wakefield accelerator

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I. INTRODUCTION
In recent years, experiments on the self-modulated laser wakefield (SMLW) accelerator [1] have been conducted at various laboratories [2][3][4][5][6][7][8]. In these experiments, usually a high-power laser pulse is focused on a gas jet. The laser pulse is sufficiently strong to create a plasma inside the gas jet and to excite a high-amplitude plasma wakefield as a result of self-modulation. Subsequently, electrons from the plasma are trapped and accelerated in this wakefield. The electron bunches produced in these experiments have been characterized by measurements of the energy distribution, the amount of charge, and the angular distribution. Typically, the energy distribution f͑g͒ of SMLW bunches is exponential, f͑g͒~exp͑2gmc 2 ͞T e ͒, with an effective electron temperature T e in the MeV range, where g is the Lorentz factor of the electrons [2][3][4][5][6][7][8]. The bunches contain high charge: up to 8 nC has been reported [6,7]. The opening angles are usually small and decrease with increasing energy: in several experiments, a tightly focused "hot core" of high energy electrons has been found [3,8]. Numerical simulations of the acceleration process have been performed with particle-in-cell (PIC) codes [9,10]. These simulation results show that, depending on the laser and plasma parameters involved, there are different mechanisms that lead to electron trapping and acceleration. These include direct laser acceleration [9], wave breaking of the Raman forward scattered plasma wave [10], and coupling to Raman backward scattering [11].
In this paper, we propose and investigate a novel twostage laser wakefield accelerator (LWFA). The first stage is a SMLW accelerator, in which electrons are self-trapped and accelerated from the gas jet plasma. The SMLW electron bunch, with the bulk of the electrons at low energy, is then injected into the second stage, which is a channel-guided resonant LWFA. This second acceleration stage, which we call the postacceleration, is studied in detail with fluid and particle simulations, with and without the effects of self-consistent beam loading.
The SMLW and resonant LWFA regimes are characterized primarily by the plasma density. For the SMLW, the plasma density is relatively high (.10 19 cm 23 , but still below the critical density), such that the laser pulse extends over several plasma waves l p and the laser power P exceeds the critical power for relativistic self-focusing P c [12]. Here l p 2p͞v p ; v p ͑4pn e e 2 ͞m͒ 1͞2 is the plasma frequency, n e is the plasma density, P c ͓GW͔ Ӎ 17͑l p ͞l͒ 2 , P ͓GW͔ Ӎ 21.5͑a 0 r 0 ͞l͒ 2 , r 0 is the spot size of the radial laser profile (assumed Gaussian), a 0 Ӎ 8.5 3 10 29 l ͓mm͔ ͑I ͓W͞cm 2 ͔͒ 1͞2 is the laser strength parameter, and l is the laser wavelength. In this regime, the laser pulse is highly unstable to self-modulation [13][14][15], which drives a plasma wave to sufficiently high amplitudes such that it traps and accelerates electrons from the background plasma.
For the resonant LWFA a lower density (ϳ10 18 cm 23 ͒ is used such that the laser pulse is approximately equal to the plasma wavelength and the plasma wave is driven in a controlled manner, which allows for a controlled acceleration process. However, for the resonant LWFA it is usually not possible to trap background electrons from the plasma and instead the accelerated electron bunch must be externally injected. In this paper we propose to inject the SMLW bunches into a resonant LWFA to exploit the advantages of both schemes. Because the resonant LWFA involves a much larger dephasing distance, and consequently much higher final electron energies, some form of laser pulse guiding [12] is needed. In this report a parabolic plasma channel is assumed to provide this guiding. Such channels have been produced in experiments [16][17][18]. In the conventional view of resonant LWFA [1], it is assumed that high energy, high quality bunches are produced by first injecting (from an external source) low energy, high quality electron bunches. To maintain low energy spread, the injected bunch must be short compared to the plasma wavelength and it must be injected at the proper wakefield phase [19,20]. In the two-stage LWFA considered here, the injected bunch (produced in the first stage by the SMLW) is nonideal in the sense that the initial energy spread of the bunch is large and the bunch is not short compared to the plasma wave of the second stage. Nevertheless, during the postacceleration process, a large fraction (40%) of the injected bunch is trapped and accelerated in the wakefield, resulting in a high energy (50 MeV) bunch with somewhat reduced energy spread (60%). Such a high energy bunch is useful for various applications in which a small energy spread is not essential, such as in nuclear activation for novel isotope production.
For the postacceleration simulations, we use a model distribution for the injected SMLW bunch. This model distribution does not directly use data from SMLW experiments or simulations, but it is constructed in such a way that it contains a number of features found in most experiments and simulations. Using this model for the injected bunch, the postacceleration process is studied in detail via simulations based on a 2D code that combines a particle description for the electron bunch with a fluid model for the wakefields, including all beam loading effects. In the Appendix, a brief description of this code is given.
This paper is organized as follows: In Sec. II, a description of the injected bunch model is given. In Sec. III, beam loading effects are discussed in detail. Section IV contains the simulation results of the postacceleration process. In Sec. V we mention nuclear activation experiments as a possible application for accelerated SMLW bunches. Conclusions are offered in Sec. VI.

II. DESCRIPTION OF INJECTED BUNCH
For the effective electron temperature T e of the energy distribution a value of 3.4 MeV has been chosen. The longitudinal bunch distribution is shown in Fig. 1, which is a plot of energy versus t 2 z͞c. A number of features found in PIC simulations [10] have been included in this distribution. The simulations typically show a microbunching with a period slightly larger than the plasma wavelength. This is due to both the increase of wavelength in the nonlinear  regime and the effect of beam loading (electrons trapped in wakefield buckets modify the wakefield). Also visible in Fig. 1 is a correlation between energy and t 2 z͞c, i.e., electrons from the first few wakefield buckets have gained more energy than those trapped in other buckets. This is another beam loading effect also found in PIC simulations. In Fig. 2, a color contour plot of the bunch density as a function of x, t 2 z͞c is given. The bunch width w is varied in our simulations. The charge of the bunch is varied proportional to w 2 (this will be explained in Sec. IV) such that the peak density is constant (at about 25% of the plasma density n e0 on the axis of the plasma channel). In this plot, the microbunching is visible again.
A plot of the assumed opening angle versus energy is given in Fig. 3. This distribution reflects the experimentally observed correlation between the opening angle and energy (smaller opening angle for high-energy electrons).

III. BEAM LOADING EFFECTS
To illustrate the importance of beam loading effects, two runs with the same initial electron distribution are compared: one with beam loading for which all bunch wakefields are included, and a test particle run without beam loading for which no bunch wakefields are calculated. The width of the electron bunch is w 28 mm; the charge is 2 nC for the case with beam loading (test particle run is effectively the zero bunch charge limit). Other parameters are plasma density on axis n e0 3.4 3 10 18 cm 23 , laser power P 40 TW, laser spot size r 0 38 mm, and pulse duration 30 fs, which imply a peak laser strength of a 0 Ӎ 0.9. The pulse profiles in the axial and transverse directions are assumed to be Gaussian. Furthermore, the laser pulse propagates in a parabolic plasma channel with the channel depth given by the matched beam condition Dn 1͞pr e r 2 0 Ӎ 7.8 3 10 16 cm 23 , where r e is the classical electron radius.
Color contours of F x , F z after an acceleration distance of 2 mm are shown in Figs  Comparison of the color contours shows the influence of the bunch on the fields. The amplitude of the longitudinal force behind the bunch is considerably lower for the 2 nC bunch (beam loading case, Fig. 4), indicating that the bunch is taking energy from the plasma wave [21,22]. Behind the bunch (t 2 z͞c . 350 fs) the on-axis accelerating field has been completely canceled by the bunch wakefield. Depending on the position of the electrons in the wakefield, the bunch could also increase the wake amplitude by giving energy to the plasma wave, but inside the bunch, wakefields from the bunch are always decelerating, i.e., the bunch always lowers the value of F z . The amplitude of the transverse force behind the bunch is considerably larger for the 2 nC bunch, indicating that the bunch induces strong focusing forces. It is important to note that these forces are always focusing inside the bunch, which is due to a magnetic self-pinching effect known as plasma lensing [23,24].  The influence of the bunch wakefields on the electron distribution can be seen from Figs. 6 and 7, which show snapshots of energy versus t 2 z͞c and real space after an acceleration distance of 2 mm. Electrons in the 2 nC bunch have much lower energy than the test electrons, and the difference increases with t 2 z͞c. This is due to the asymmetry of the bunch wakefields: particles in the head of the bunch experience only the laser wakefield, and particles in the tail experience both the laser wakefield and the cumulative decelerating wakefield due to all preceding electrons in that bunch.
In the real space snapshot, the distribution of the 2 nC bunch is seen to be rather different from the test particle distribution. Close to the axis, the distribution of the 2 nC bunch is shifted back with respect to the test particle distribution. This is due to the wakefields of the first microbunches, which cause a phase shift of the wakefield close to the axis, resulting in curvature of the wave fronts, as seen in Fig. 4. The 2 nC distribution also has off-axis lobes, giving a typical Y shape to the microbunches.

IV. SIMULATION RESULTS
Our simulation results show that the postacceleration process consists of two steps. In the first step, a large portion of the bunch is blown out radially, and only a fraction of the electrons gets trapped in the laser wakefield. The main reason for the blowout to occur is that the spacing of the electron bunch does not match the wave of the wake-field (this is the wave of the low density LWFA plasma, whereas the bunch is modulated at the plasma wave of the high density SMLW plasma). As a result of this, a part of the bunch is injected in a defocusing phase and will be lost. In the second step, the trapped electrons are focused and accelerated.
A formal definition of trapping can be given using the separatrix of a wakefield Hamiltonian [25]. In our case however, this separatrix would be difficult to define because of beam loading effects. Instead we use the following heuristic definition of trapping: a simulation particle is considered to be trapped if it is inside the channel at less than 1.5 laser spot size from the propagation axis. We expect this definition to coincide with the formal one in the second step of the acceleration, after the radial blowout  of electrons in defocusing regions. Only trapped particles are taken into account for calculating dynamical quantities such as the energy spread and the transverse emittance. The difference between the acceleration steps is illustrated in Figs. 8 and 9. These figures show snapshots of real space for the 2 nC bunch at propagation distances of 0.2 and 3.0 mm, respectively. Simulation parameters are the same as in Sec. III. Figure 8 shows a typical snapshot of the first step, in which a lot of electrons are being expelled from the channel (the figure shows the region of trapped particles). This is in contrast with a typical snapshot of the second step (Fig. 9), in which the remaining trapped electrons are well within the channel. In this snapshot, the typical Y shape of the microbunches is visible.
In principle, the amount of trapped charge can be increased by increasing the laser power. We have simulated a range of peak powers P between 10 and 100 TW by changing the spot size of the laser pulse proportional to P 1͞2 between 19 and 60 mm and keeping the amplitude constant (at 0.9 in standard dimensionless units). In the linear wakefield regime (amplitude ø1), this scaling would result in a laser wakefield potential with the same amplitude and a width proportional to P 1͞2 , which yields an acceptance (amount of charge that can be accelerated) proportional to P. To check whether the scaling holds for our parameters, the injected charge has been taken proportional to P by scaling the bunch width w proportional to P 1͞2 and keeping the peak density constant. In the linear wakefield regime, this would result in a bunch wakefield potential with the same scaling as the laser wakefield potential, as long as the bunch width w is large compared to l p (quasi-1D regime). For small w, the amplitude of the bunch wakefield is reduced as compared to the 1D case [26]. Other parameters are not varied and taken as in Sec. III. In particular, the opening angles are not varied, so the transverse emittance also scales with P 1͞2 . Figures 10-12 show trapped charge, normalized emittance, and average energy after an acceleration distance of 2.7 mm, as a function of peak power. The trapped charge scales linearly with P, indicating that the trapped fraction is constant. The emittance at 2.7 mm is proportional to P 1͞2 and therefore proportional to the initial emittance. This means that the transverse dynamics is qualitatively the same. The average energy shows a slight decrease with P, which is a result of the reduction of the decelerating bunch wakefield for narrow bunches. These results show that the influence of wakefield nonlinearities is small for the parameters of our simulations, in spite of a relatively high wakefield amplitude. More details of the acceleration process are given in Fig. 13, which shows the trapped charge as a function of the propagation distance, for both the 40 TW case and the 90 TW case. In the first step (0-1.5 mm), the trapped fraction is seen to drop quickly to about 40%. In the second step, the trapped fraction remains constant. For the 90 TW case, a slight decrease of the trapped fraction is seen to occur around 3.5 mm, indicating that there is a small amount of additional particle loss.
The plot of normalized transverse emittance as a function of propagation distance, shown in Fig. 14, also illustrates the difference between the steps. In the first step, the emittance rapidly increases, reaches a maximum, and decreases again. This behavior is caused by radial blowout of electrons: while still inside the trapping region, these electrons cause the emittance to grow rapidly. Once they have left the trapping region, they are not counted anymore, and therefore the emittance drops again. In the second step, when the bunch is much more stable inside the channel, the emittance evolves very slowly. For the 90 TW case, Fig. 14 shows additional emittance increase, due to the minor additional particle loss (as seen in Fig. 13).
A plot of average energy as a function of acceleration distance is given in Fig. 15. The energy increase is due to both acceleration and loss of low-energy particles. Figure 16 shows the relative energy spread as a function of propagation distance. The 40 and 90 TW cases are seen to give almost the same result. A minimum in energy spread of about 60% occurs at 1.7 mm acceleration distance.
In Fig. 17 the energy distributions of trapped particles are shown, both at injection (green curve) and after 1.7 mm acceleration for the 40 TW case (red curve) and the 90 TW case (blue curve). The distribution is seen to change from exponential to highly nonthermal with a peak around 30 MeV. Above about 15 MeV, the amount of charge in a given energy bin is higher for the accelerated bunches. Around 30 MeV, the difference is already 2 orders of magnitude. The blue and red curves almost overlap, demonstrating that in both cases the acceleration dynamics is almost the same. The overlap is a result of normalizing  the spectra: for the 90 TW case, the bunch charge is of course 9͞4 times higher.

V. NUCLEAR ACTIVATION EXPERIMENTS
Nuclear activation experiments are a possible application for SMLW generated electron bunches [6,27]. In these experiments, the SMLW bunches interact with a suitable target to produce bremsstrahlung photons. The photons may be used to perform activation experiments involving (g, n) and (g, multi-n) reactions. The use of SMLW bunches is not particularly efficient because typically only a small fraction of the electrons has sufficiently high energy to produce the energetic photons necessary for the (g, n) and (g, multi-n) reactions. The yield Y in an activation experiment can be calculated from the scattering cross section s as where dN g ͞dv is the photon number spectrum produced by bremsstrahlung. Note that the scattering cross section s usually has a single peak around the resonant energy, typically a few 10 s of MeV. This suggests that the yield could be greatly improved if the bulk of the SMLW-bunch electrons is accelerated to higher energy by a resonant LWFA, as discussed in Sec. IV. Figure 17 indicates that the number of electrons available to produce photons at the resonant energy of the reaction cross section can be orders of magnitude higher for the postaccelerated beam. As a numerical example, consider activation of 63 Cu by the reactions (g, n) and (g, 2n), with the reaction energy thresholds 10.8 and 19.7 MeV, respectively [6]. Consider production of bremsstrahlung photons using a simplified model based on a thin target approximation (i.e., much less than the radiation length), such that a relativistic electron experiences only a single scattering event and multiple scattering and cascade effects are neglected. For a single highly relativistic electron, the bremsstrahlung pho-ton spectrum [28] is approximately ͓dN g ͞dv͔ 1 c g ͞v forhv , gmc 2 , where c g is a constant determined by the target properties. The bremsstrahlung spectrum produced by an electron bunch with energy distribution f͑g͒ is and therefore the yield produced by the bunch is We can compare the activation Y 1 produced by the SMLW electron bunch distribution f 1 and the activation Y 2 produced by the postaccelerated electron bunch f 2 shown in Fig. 17. Evaluating Eq. (3) for the distributions shown in Fig. 17 we find Y 2 ͞Y 1 Ӎ 40 for the reaction 63 Cu͑g, n͒, and Y 2 ͞Y 1 Ӎ 4.2 3 10 2 for the reaction 63 Cu͑g, 2n͒. As the numerical example shows, orders of magnitude improvement can be achieved in nuclear activation yields by postaccelerating a SMLW bunch using a two-stage laser wakefield accelerator. More detailed simulations using a Monte Carlo based nuclear interaction code [29] have been carried out to design optimized bremsstrahlung targets for enhancing the yield of specific nuclear reactions [30]. These calculations confirm that up to 4 orders of magnitude increase in activation yields can be obtained using beams from a channel guided laser wakefield accelerator as opposed to the SMLW bunch.

VI. CONCLUSIONS
We have proposed and investigated a novel two-stage LWFA accelerator consisting of a SMLW (stage 1) that generates a low-energy electron bunch, which is injected into a channel-guided resonant LWFA (stage 2) for acceleration to high energy. The postacceleration process (stage 2) was simulated in detail using a fluid model for the plasma and a particle method for the bunch electrons that includes the effects of beam loading. Because of the lower plasma density involved, resonant LWFA offers a larger dephasing energy; because the dephasing distance is longer, a plasma channel is needed for pulse guiding. We have simulated trapping of about 40% of the injected charge and acceleration to an average energy of about 50 MeV for a density of 3.4 3 10 18 cm 23 in a plasma channel of 3 mm. We have illustrated the importance of beam loading effects by comparing the run of a 2 nC bunch with a test particle run.
Our simulation results show that the postacceleration process consists of two steps: in the first step there is radial blowout of electrons that are injected in defocusing regions; in the second step the remaining electrons are focused and accelerated. The energy distribution changes from an exponential distribution to a highly nonthermal distribution with somewhat reduced energy spread (about 60%).
We have investigated a range of peak powers between 10 and 100 TW for the laser pulse by scaling the spot size between 19 and 60 mm and keeping the peak amplitude constant. Scaling bunch width and bunch charge accordingly, we have found that the dynamics was about the same for all simulations. From this we conclude that a quasi-1D linear scaling holds for our parameters, in spite of a relatively high wakefield amplitude and a relatively small bunch width for the low-power cases.
As a possible application for the bunch after postacceleration we have mentioned nuclear activation experiments, in which a small electron energy spread is not necessary. From our simulation results, the nuclear activation yield for accelerated bunches is estimated to be orders of magnitude higher than for the injected SMLW bunches.

APPENDIX: DESCRIPTION OF THE SIMULATION CODE
In our simulation code, a PIC model for the electron bunch is combined with the electron fluid description for the plasma response. It is a 2D slab geometry code, developed at Eindhoven University of Technology. It has been used to describe the laser wakefield acceleration of ultrashort electron bunches with high beam loading [19].
The quasistatic approximation [1,12,31] is used for calculating the wakefields. In the wakefield calculation, a nonevolving laser pulse is assumed, and therefore effects such as pump depletion, pulse broadening, and frequency redshifting are neglected. The hydrodynamic wakefield equations are fully nonlinear. In the absence of beam loading, the plasma response is given by the 2D slab version of the quasistatic equations given in Refs. [1,31]. All beam loading effects are taken into account by including the bunch density of accelerated electrons as a source term for the wakefields.