Céline S. Hue, Yang Wan, Eitan Y. Levine, Victor Malka. Control of electron beam current, charge, and energy spread using density downramp injection in laser wakefield accelerators[J]. Matter and Radiation at Extremes, 2023, 8(2): 024401
- Matter and Radiation at Extremes
- Vol. 8, Issue 2, 024401 (2023)
Abstract
I. INTRODUCTION
Laser wakefield acceleration (LWFA) is one of the most promising accelerator technologies, offering acceleration gradients more than three orders of magnitude higher than conventional accelerators.1–3 In LWFA,1 plasma electrons are pushed outward by the ponderomotive force of the intense laser pulse, forming a plasma wave that travels together with the laser pulse at relativistic speeds and is therefore suitable for compact and efficient acceleration of trapped electrons. Numerous injection methods, such as self-injection,4–8 ionization injection,9–11 and density downramp injection,12–15 have been proposed, explored, and demonstrated numerically and experimentally. Over the past few decades, electron injection using a sharp plasma density transition profile13–15 has been demonstrated to be effective for the generation of high-quality electron beams in LWFA, with plasma electrons being trapped owing to the wave breaking induced by the longitudinal expansion of the plasma wave structure as it propagates along the density downramp. This approach can be easily implemented by inserting a sharp blade on top of the gas nozzle13,14 and has been used recently to demonstrate free-electron lasing with a high-quality LWFA electron beam of 10–50 pC.16
Density downramp injection has been thoroughly studied in the framework of beam-driven plasma wakefield acceleration (PWFA).17–19 In LWFA, however, the nonlinear laser propagation in the plasma20,21 makes analysis of the injection mechanism a much more subtle task. This process has been investigated by several groups22–24 through numerical scans of the downramp parameters. However, very few studies have focused on the current profile of the injected electron beam and its correlation with the beam charge and energy properties, although this is of critical importance for many applications, such as the high-efficiency energy gain in a staged LWFA-PWFA scheme that has recently been experimentally demonstrated.15,25
In this article, an approach for tailoring the beam current I by tuning the parameters of the plasma density profile (nhigh, nlow, L, and Posd) as shown in Fig. 1 is studied by means of numerical simulations performed with the quasi-3D particle-in-cell (PIC) code FBPIC,26 where the algorithm is based on Fourier–Bessel decomposition of fields on a set of 2D radial grids. It is also explained how the beam current plays a decisive role in tailoring other important beam parameters (charge Q, beam energy E, and energy spread δE).
Figure 1.Plasma density profile used in the study, where the laser pulse propagates along the positive
The laser parameters considered here are typical of laser chains delivering a few tens of terawatts in a few tens of femtoseconds. Here, we took those of the HIGGINS laser system at the Weizmann Institute of Science (WIS),27 with a pulse duration of 30 fs, a focal spot waist of 18 µm, and a normalized vector potential a0 of 2.2, corresponding to an on-target laser energy of 1.6 J. The simulations were performed in a cylindrical geometry with two azimuthal modes. We chose a mesh resolution of Δz = 38 nm and Δr = 90 nm in the longitudinal and radial directions, respectively. The results presented here were obtained with 45 particles per mesh cell. In the simulations, the plasma was set to be pre-ionized, which allowed us to study solely the impact of the plasma profile on the injected beam parameters. For all the simulations described in this article, nlow was fixed as 2.5 × 1018 cm−3 to ensure that the drive laser could be self-guided.28
The remainder of this article is organized as follows. We first show in Sec. II how one can tailor the beam current distribution I(ξ) by changing the plasma density parameters, where ξ = vgt − z ≈ ct − z represents the phase inside the wakefield, vg the laser group velocity in the plasma, and c the speed of light in vacuum. In Sec. III, we discuss the dependence of the beam energy evolution on the beam current I(ξ), and we also report on the achievement of electron beams with energies of hundreds of MeV and charges of hundreds of picocoulombs with relatively good beam quality. In Sec. IV, we explain the maximum limit on the injected beam charge and its effect on the final beam energy and energy spread. We present our conclusions and perspectives in Sec. V.
II. TAILORING THE BEAM CURRENT DISTRIBUTION WITH DIFFERENT PLASMA PROFILES
In this section, we show how one can tailor the beam current distribution I(ξ) by controlling the profiles of the plasma parameters nhigh, L, and Posd. These parameters can be easily tuned experimentally (for example, by changing the positions of blade or string-like obstacles13,14,29 along the path of the gas jet). Through a series of simulations, it is found that the laser intensity across the density downramp and accordingly the wakefield strength play decisive roles in shaping I(ξ).
For a clearer explanation, we first describe the simulation results presented in Fig. 2, where the notions of nonsaturated injection and saturated injection are introduced to facilitate further analysis of different injection processes and injected current distributions. We then explain in detail how the wakefield intensity influences the injection by tracking the injected particles to their initial positions in the plasma before they are influenced by the laser, similar to the discussion in Ref. 18. Finally, we investigate the relationship between the downramp parameters and the injected beam current and total charge.
Figure 2.Left: beam current distribution at the plasma exit for
A. Saturated and nonsaturated injection
The results of four simulations with different downramp positions Posd are reported in Fig. 2, where the plasma density before the downramp was fixed as nhigh = 5 × 1018 cm−3, and the downramp length for all cases was L = 40 µm. The plot on the left of Fig. 2 shows the injected electron beam current distributions at the exit of the plasma and (as dashed lines in the inset) the normalized laser vector potential a0 in the plasma density region of interest.
For larger values of Posd, the laser intensity across the downramp region increases, and the shape of the injected electron beam current I(ξ) changes from triangle-like (the blue and orange lines in the left-hand plot in Fig. 2) to rectangle-like (the green line), and then to a piecewise function of two segments consisting of a peak and a constant segment (the red line).
As mentioned at the beginning of this section, the behavior of the beam current distribution depends mainly on the wakefield strength in the plasma density downramp. The higher the laser intensity, the stronger is the wakefield and the easier is the injection [a higher beam current I(ξ)]. Figures 2(a)–2(d) show the simulated plasma charge densities of the wakefields at positions in the middle of the corresponding density downramp. It can be seen that the bubble structure becomes clearer as Posd increases, which signifies a higher wakefield strength as well as system nonlinearity.
Here, for simplicity, we term the injection process where the wakefield reaches full blowout saturated injection, which results in the piecewise current shape (the red line in the left-hand plot in Fig. 2). The injection process without a full blowout wakefield is termed nonsaturated injection, which results in the triangle-like current shape (the blue and orange lines in Fig. 2). It should be noted that with the laser evolution and the change in plasma density across the downramp, the injection evolves from a nonsaturated process to a saturated one, corresponding to a rectangle-like current shape (the green line in Fig. 2). The underlying mechanisms are explained in detail in Sec. II B.
B. Distribution of initial positions of injected electrons
For a better understanding of how the laser and wakefield intensity influence the injection, we apply a similar approach to that adopted in Ref. 18 to investigate the initial positions (zi, ri) of the injected electrons before they are affected by the laser, using three different simulations corresponding to the orange, green, and red lines in the left-hand plot in Fig. 2. The case corresponding to the blue line in this plot is not shown here, since it shares the same physics as the case corresponding to the orange line. Electrons from the injected beam are traced back to their original positions in the plasma, and their distributions in the zi–ri plane are shown in Figs. 3(a)–3(c), where the downramp starts from 200, 300, and 400 µm, and ends at 240, 340, and 440 µm, respectively. The evolutions of a0 in these regions are plotted as dashed black lines, from which it can be seen that a0 increases for higher Posd. The distributions of linear charge density (the charge per unit length across zi) over the longitudinal coordinate z, which we denote here by λi(zi), are plotted as red solid lines.
Figure 3.(a)–(c) Particle distributions of injected particles in the
For complete saturated injection, as shown in Fig. 3(c) with a larger value of Posd, owing to nonlinear processes (such as relativistic self-focusing20 or self-steepening effects21 that increase with propagation distance), the blowout/bubble regime is valid throughout the downramp region, and the injected particles cover the whole downramp region (from 400 to 440 µm for the case considered here). A small proportion of the particles before the downramp region (zi < 400 µm) are also involved in the injection process. An accumulation of charge is observed at the beginning of the downramp, and this results in a charge density peak, probably due to the rapid changes in wakefield structure, which causes extra particles to be injected, smoothing the peak to a lower value with a slightly rounded beginning of the density downramp. A constant λi(zi) is found just after the peak, which can be explained by the negligible evolution of a0, and accordingly the wakefield strength, across the plasma downramp. Actually, the supposition that a fixed driver intensity results in a relatively constant λi(zi) is also supported by a study of the downramp injection in PWFA,18 which indicates that only particles initially situated around a thin sheet near a fixed transverse position ri ≃ κrm are able to be injected, where rm is the maximum transverse radius of the bubble and κ ≃ 1 is a nondimensional value.
For Posd = 200 µm, as shown in Fig. 3(a), where the wakefield is still weak, only nonsaturated injection occurs, and it does so at a position closer to the rear of the downramp where a0 is strong enough to trigger injection, corresponding to a triangle-like shape of λi(zi). For Posd = 300 µm, the wakefield strength is slightly below the blowout regime, and nearly saturated injection occurs approaching the end of the downramp, with λi(zi) exhibiting a trapezoidal shape. In these two cases, particles with smaller initial transverse positions (nearer to the axis) are also involved in the injection process. These particles are accelerated and injected inside the wakefield without going along the sheath of bubbles in the plasma wake.
The beam charge density distributions in the ξ–zi plane, together with the beam current shapes, are shown in Figs. 3(d)–3(f), corresponding to Posd = 200, 300, and 400 µm, respectively. Along almost the whole bunch duration, a nearly linear correlation is observed between the initial longitudinal position zi and the injected phase position ξ. This near-linearity indicates that the dependence of λi(zi) on zi has a similar shape to the dependence of the final beam current I(ξ) on ξ, as can be seen by comparing Figs. 3(a)–3(c) with Figs. 3(d)–3(f), respectively. This correspondence in the case of PWFA is explained in Ref. 18. It should be mentioned that the nearly one-to-one mapping is also valid for LWFA under the condition that there is little evolution of a0 across the density downramp, which holds for the cases reported here.
C. Relationship between plasma density parameters and injected beam current
To investigate the effect of the density gradient on the beam parameters, we performed another series of simulations for different downramp lengths L. Figure 4 shows the beam current distributions for four different values of the density gradient and four different downramp positions Posd. For cases with nonsaturated injection, the steeper the downramp (i.e., the smaller the value of L), the easier the injection becomes. When the wakefield across the plasma density downramp is relatively weak (corresponding to small Posd), no injection at all occurs for large L. From the results shown here, no injection is observed for L = 60 and 75 µm when Posd is set as 100 µm.
Figure 4.Beam sliced currents for fixed
For cases with saturated injection, the value of the constant part of the current profile is barely influenced by the downramp length L, as can be seen by comparing the red lines in Fig. 4 near the tail of the injected beam. Further evidence for this is provided by the simulation results shown in Fig. 4(b), where similar beam currents near the tail of the bunch are observed for different values of the downramp steepness.
Simulations (not reported here) were also performed with different values of nhigh but the same nlow. For the full blowout situations across the downramp region, the injected beam is observed to have a longer duration with higher plasma density before the downramp, which potentially increases the beam charge Q. This has also been observed in Ref. 24, and the reason is that the bubble size in the high-density region is smaller for higher nhigh, as a consequence of which the head of the injected electron bunch moves closer to the laser driver, while its tail stays at the rear of the bubble. However, Q is also limited by the acceleration capacity of the structure after the downramp, as discussed in Sec. IV.
It is interesting to calculate the total injected charge Q = (1/c)∫beamI(ξ) dξ produced by each set of plasma parameters. Figure 5 shows the variation of the beam charge Q with the downramp length L for four downramp positions Posd, similar to Fig. 2. The majority of the bunches are injected without reaching saturation for smaller Posd, corresponding to the blue, orange, and green lines here. A decrease in Q is observed with decreasing downramp steepness when the value of L is relatively small. The charge Q increases after a certain value of L, because the end of the downramp sees an increase in laser intensity a0, which results in more injected particles. For cases with saturated injection, corresponding to the red line here, Q decreases at larger L. The data represented by the large dots correspond to the cases with optimal energy performance, which will be discussed in Sec. III.
Figure 5.Beam charge
III. CONTROL OF BEAM ENERGY EVOLUTION AND OPTIMIZATION OF BEAM PARAMETERS
The results of a typical simulation of the production of monoenergetic beams with energies of hundreds of MeV for initial plasma density parameters nhigh = 5 × 1018 cm−3, Posd = 300 µm, and L = 40 µm are presented here. The evolution of the electron beam energy is shown in Fig. 6(a), where the energy spectrum for the injected bunch with respect to the propagation distance is presented as a color map. The mean energy of the beam is plotted as a red solid line (with values given on the left vertical axis) and the absolute energy spread of the beam δE is plotted as a green dashed line (with values given on the right vertical axis). The current I(ξ) of the injected beam in this simulation corresponds to the green line in the left-hand plot in Fig. 2.
Figure 6.(a) Evolution of beam energy with propagation distance for a simulation with
As the beam propagates, its energy spread reaches a minimum value before increasing again. To understand the reason for this energy evolution, the electron beam density distributions in longitudinal phase space (the z–E plane) at different propagation distances, as well as the acceleration force, are plotted in Figs. 6(b)–6(d). It can be seen from the very thin beam profiles in longitudinal phase space that the large energy spreads are due to energy chirping, whereas the sliced energy spreads remain low throughout the propagation. The beam chirps negatively (the tail has a higher energy than the head of the bunch) at the beginning of the simulation and then flattens during its propagation before taking a nearly horizontal shape (with the head and tail of the beam holding nearly the same energy). The nearly horizontal shape is where the minimum energy spread is found. The chirp then turns positive and increases continuously until the end of the simulation, and the energy spread increases correspondingly. Here, we define the optimized propagation distance to be the zone where the minimum energy spread (also the minimum relative energy spread, because for longer propagation distance the beam energy does not increase significantly in this group of simulations) is obtained. Here, we focus on three elements that characterize the evolution of the beam energy: (i) the minimum energy spread min(δE); (ii) the beam mean energy, at which the minimum energy spread reaches E(min(δE)); (iii) the rate d(δE)/dz at which δE increases after its minimum.
The evolution of the beam profile in longitudinal phase space is a result of the evolution of the acceleration force Facc(z) to which the beam is subjected at different longitudinal positions. This is a combined effect of the laser wakefield and the beam’s self-loaded wakefield. Shortly after the bunch has been injected, the acceleration force to which the tail of the beam is subjected is stronger than that at its front, which results in the negative chirp. As the wakefield structure propagates in the plasma, the acceleration force weakens owing to laser depletion. Accordingly, the beam’s self-loaded wakefield, which exerts a decelerating effect, becomes more pronounced at the rear of the beam. As a result, the head of the bunch is subjected to a stronger acceleration field than the tail. The wakefield structures shortly after the downramp and near a propagation distance of 3.5 mm are illustrated in Figs. 6(e) and 6(f), respectively. It can be seen that the wakefield contributed by the laser (the fine periodic structure before the beam head in the figure) becomes much weaker after a few millimeters of propagation, and thus the beam’s self-loaded wakefield starts to have a stronger effect.
As explained in Ref. 30, one can shape the acceleration force by modifying the beam current I(ξ). Since the beam current profile I(ξ) can be tailored by tuning the plasma density parameters as mentioned in Sec. II, the beam energy performance can be optimized by tuning the four main parameters of the plasma profile: nhigh, nlow, L, and Posd.
We performed a series of simulations to map the relationship between the plasma parameters and the evolution of the beam energy. The first group of simulations corresponds to the green line in Fig. 5, for which nhigh = 5 × 1018 cm−3 and the downramp position is Posd = 300 µm. Only the downramp length L varies. The evolution of the energy spread δE with propagation is plotted for several cases in this group in Fig. 7(a). The profiles of the energy evolution for the whole beam are plotted in Fig. 7(b). The black dots outline the zones of interest where the minimum energy spreads min(δE) are reached. The plasma density profiles and the evolution of a0 are illustrated in Fig. 7(c).
Figure 7.(a) and (b) Evolution of
The beam currents I(ξ) for three cases in this group are plotted in Fig. 7(d). Note that according to the analysis in Sec. II, there is a transition of the injection process from nonsaturated to saturated. With a steeper downramp (smaller L), the current behavior changes from a decreasing trend to a slightly increasing trend. As L continues to increase, an accumulation of particles is again found at the head of the bunch.
The minimum energy spread is very sensitive to the beam current, which takes different forms for different downramp gradients (different values of L). The value of min(δE) decreases at larger L and reaches its lowest value for the simulations in this group when L = 60 µm, as shown in Fig. 7(a). As L continues to increase, the beam energy obtained in the optimized zone for beam delivery, E(min(δE)), decreases [Fig. 7(b)]. Meanwhile, the speed at which δE evolves after the zone of delivery, d(δE)/(d)z, is higher, which makes the system more sensitive to experimental control.
We are also interested in the relationships between the achievable δE, E, and Q, which we consider to be the three main parameters characterizing the injected beam, and these are presented in Fig. 7(e), where the downramp length L increases in the direction of the arrow. For given values of E(min(δE)) on the horizontal axis and min(δE) on the vertical axis, the corresponding beam charges Q in picocoulombs are shown in the circles. It can be seen that the achievable minimum energy spread coincides with the attainable maximum energies [the right-most charges in Fig. 7(e) are also the lowest], which is favorable for the production of high-energy monoenergetic beams. Table I presents the final beam parameters of four cases in this group that are considered to be optimal in the context of obtaining high-energy monoenergetic beams, i.e., those with the smallest δE/E values.
| Plasma parameters | Final beam parameters | |||||||
|---|---|---|---|---|---|---|---|---|
| Simulation group | nhigh (cm−3) | nlow (cm−3) | L (μm) | Posd (μm) | Q (pC) | δE (MeV) | E (MeV) | δE/E (%) |
| 1 | 5 × 1018 | 2.5 × 1018 | 40 | 300 | 186.2 | 6.20 | 278.14 | 2.2 |
| 5 × 1018 | 2.5 × 1018 | 60 | 300 | 171.5 | 3.31 | 265.02 | 1.2 | |
| 5 × 1018 | 2.5 × 1018 | 75 | 300 | 171.1 | 4.32 | 255.03 | 1.7 | |
| 5 × 1018 | 2.5 × 1018 | 100 | 300 | 185.4 | 7.05 | 240.82 | 2.9 | |
| 2 | 5 × 1018 | 2.5 × 1018 | 100 | 190 | 28.0 | 10.22 | 993.2 | 1.03 |
| 3 | 5 × 1018 | 2.5 × 1018 | 10 | 400 | 390.1 | 7.70 | 144.5 | 5.3 |
Table 1. Plasma parameters and final beam parameters for the optimum cases that give the smallest δE/E for each group of simulations.
The second group of simulations corresponds to the orange line in Fig. 5, for which nhigh = 5 × 1018 cm−3, and Posd = 190 µm. The evolution of the beam parameters is plotted for some cases in this group in Fig. 8. Further investigations with longer propagation distances are performed for those simulations of interest in which δE is conserved or increases only slowly when 5 mm propagation is reached, corresponding to the cases with L = 75, 100, and 120 µm. As shown in Fig. 8(c), the relative energy spreads are only a few percent where the beam peak energies reach approximately their maximum, ∼1 GeV. Beam currents for five of these cases are plotted in Fig. 8(b). For this simulation group, the current shape is pulse-like. Similar to the first simulation group, the current moves further to the rear of the bubble with larger downramp length L, and the beam length decreases significantly. The energy spread is roughly conserved only for weak beam current and thus low charge Q. This conservation of energy spread is possible because of the short beam lengths and weak current strength, as a consequence of which the beam loading is not significant and the wakefield driven by the laser is dominant throughout the propagation. The high energy gain, on the other hand, is due to the fact that the drive laser reaches a self-guided condition and provides a stable wakefield for a long propagation distance.28 The beam charges Q for the cases of interest are around 30 pC. The parameters of the best case of this group are presented in Table I.
Figure 8.(a) Energy spread as a function of the propagation distance for six values of the downramp gradient. (b) Corresponding beam currents. (c) Relative energy spread and energy as functions of the propagation distance for a set of four values of the downramp gradient.
We now focus on the third group of simulations, which corresponds to the red line in Fig. 5. The energy evolution is presented for several cases in this group in Fig. 9. Only saturated injection occurs for this simulation group. The beams are barely accelerated after reaching 150 MeV. Similar to the other simulation groups, the optimized beam delivery zone is where the minimum energy spread is reached. Among all the simulations in this group, the most monoenergetic case is found to be that with the smallest L, for which the injected beam charge approaches 390 pC.
Figure 9.Energy spread and energy evolution for Pos
The simulations presented in this section have mostly used the same plasma densities nhigh and nlow. With different nhigh, as discussed in Sec. II, nearly identical beam currents can be obtained, which means similar beam energy evolutions.
IV. LIMITS ON ACCELERATION CAPACITY
The beam charge that can be accelerated is limited by the wakefield amplitude and depends on the laser evolution after the downramp and the plasma density nlow. We call this limit the deceleration limit.
When the beam current and accordingly the total beam charge are sufficiently high, the decelerating wakefield caused by the beam itself becomes so strong that the majority of particles in the beam are decelerated. The energy spectrum for a simulation that exceeds this deceleration limit is shown in Fig. 10(a), and the corresponding beam profiles in longitudinal phase space at two different propagation distances are illustrated in Figs. 10(b) and 10(c). The initial plasma parameters of this simulation are nhigh = 5 × 1018 cm−3, Posd = 435 µm, and L = 25 µm. The injected charge is 394.4 pC.
Figure 10.(a) Energy spectrum. (b) and (c) Longitudinal phase space at two different propagation distances. The initial plasma parameters are
Among all the performed simulations, the maximum injected charge obtained is 454.2 pC, for nhigh = 1 × 1019 cm−3, Posd = 225 µm, and L = 25 µm. The beam current is nearly flat, which makes the energy spread relatively stable during propagation, as shown in Fig. 11.
Figure 11.Energy spectrum for the simulation with initial plasma parameters
Increasing nhigh is not helpful in reaching a higher beam charge, because the bunch length cannot exceed the acceleration and focusing part of the bubble structure. One can expect that lowering nlow for a larger bubble structure should enable longer bunches to be accelerated. Further mappings of the achievable beam charge Q and plasma parameters are not presented here, because the beam energy performance for saturated injected beams is not optimal in the context of obtaining a monoenergetic and highly charged beam.
V. CONCLUSIONS
A systematic and comprehensive study on obtaining high-quality electron beams using the density downramp injection technique in a one-stage laser wakefield accelerator has been presented, and it has been shown how the plasma target parameters affect the beam performances.
It is understood that the beam current I(ξ) is of decisive importance not only for obtaining a desirable beam charge but also for the beam energy evolution. The current profile can be tailored by adjusting the position of the plasma downramp region, taking account of the local laser intensity. Different current profiles I(ξ) give rise to different energy evolution patterns. The achievable beam charge for a fixed nlow = 2.5 × 1018 cm−3 varies from 0 to 450 pC. The beams with the highest charge and variable current shapes can potentially be tuned and adapted to provide suitable driver beams for high-transformer-ratio PWFA.15,25
For injected charges on the scale of hundreds of picocoulombs, a minimum energy spread min(δE) is found as a result of the competition between the self-loaded decelerating field and the evolution of the accelerating wakefield driven by the laser during propagation. The optimum zones for beam delivery are those where min(δE) is reached. Beams with charges over 170 pC, energies in the 200 MeV range, and less than 2% relative energy spread are obtained, and these suit nicely the requirements for radiotherapy applications with very high-energy electrons (VHEE).31–33
For cases where relatively low charge is injected, conservation of the energy spread over long propagation distances is observed, which permits very high beam energies (∼1 GeV) to be reached with less than 2% energy spread. Such cases can be further tuned and adapted to free-electron laser applications that have recently been demonstrated.16
In the context of downramp injection, we have presented qualitative relationships between output beam parameters and input laser–plasma parameters that can provide guidance for future experiments conducted with the laser systems of a few tens of terawatts that are available in many laboratories. These are general relationships, and the determination of precise parameter values will require further studies to answer the varying requirements of different applications and available experimental conditions.
ACKNOWLEDGMENTS
Acknowledgment. This work was supported by the Fondation Jacques Toledano and the Schwartz-Reisman Center for Intense Laser Physics, and by ERC PoC Vherapy and EIC ebeam4therapy grants.
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