9 GeV energy gain in a beam-driven plasma wakefield accelerator

The final spectrum of the beam was diagnosed with a magnetic imaging spectrometer [1, 12, 25] consisting of a quadrupole magnet doublet and a strong dipole magnet that dispersed the beam in the vertical (y) plane. The quadrupole magnet strengths were set to image particles of a specific energy from the exit plane of the plasma source to a set of diagnostic screens at the end of the beamline. In order to fully characterize the outgoing beams' spectrum in aggregate, the quadrupoles were adjusted between sub-sets of shots to image different energies from the nominal object plane (i.e. the plasma exit location). In the data analyzed here, 100 shots each were taken with the spectrometer set to image energies corresponding to E₀, E₀  +  1.5 GeV, E₀  +  3.0 GeV, E₀  +  4.5 GeV, E₀  +  6.0 GeV, and E₀  +  7.5 GeV, where E₀ is the initial beam energy of 20.35 GeV. The magnification of the beam in the dimension transverse to the dispersive plane (x) was 8.1, while the magnification of the beam in the dispersive plane (y) was set to the smaller value of 1.0 in order to optimize the spectral resolution. These constraints required that the imaged energy in the y-plane from the plasma exit location to the screens be on the order of 0.5–1 GeV higher than that of the x-plane, depending on the imaging condition and the screen location.

The transverse projection of the dispersed beam was recorded at two different screens located at a distance L  =  9.57 m and L  =  10.9 m downstream of the dipole magnet, corresponding to a dispersion value D₀ for electrons at the energy E₀ of 54.8 mm and 59.5 mm, respectively. The dispersion ${{D}_{0}}=\Delta {{\theta}_{0}}\cdot L$, and $\Delta {{\theta}_{0}}=5.73$ mrad is the deflection angle for electrons at the energy E₀. The first screen was a phosphorescent sheet of Lanex material affixed to the downstream side of a 1 mm thick converter foil made of tungsten, viewed by a 16-bit scientific CMOS camera. The second screen was comprised of a silicon wafer reflecting Cherenkov light emitted by the passage of the electron beam through a 10 cm air gap immediately preceded by another silicon wafer used to block the Cherenkov light emitted further upstream [25]. The Cherenkov light was collected by a 16-bit scientific CMOS camera. The Cherenkov screen was roughly ten times less sensitive than the Lanex screen, but it covered a wider field of view and was insensitive to synchrotron, betatron, and Bremsstrahlung radiation backgrounds, which can be problematic under certain experimental conditions (though these effects were not a significant source of background in this study). The Cherenkov screen was used as the primary diagnostic for this study, while the Lanex screen was used as a secondary check for the major spectral features in the beam. The spectral features in the data were analyzed in two different ways for every shot: one method focused on the spectral charge density peak, and the other focused on the centroid energy value of all the accelerated charge in a given shot.

The spectral charge density (charge within a given energy bandwidth) peak (see figure 1(a)) was defined as the energy at which the projected spectral charge density was the highest for any point in the region above 21.5 GeV, which roughly delineates the upper reach of the slightly accelerated tail of the drive beam. The energy spread for the spectral charge density peak was calculated as the root-mean-square value for the distribution of charge above the spectral peak energy (also shown in figure 1(a)). The upper limit of the domain of the root-mean-square calculation was the point below which 95% of all observed charge above the spectral peak was located. To estimate the accelerated charge at the spectral peak, we summed all the charge located within two times the rms energy spread of the spectral peak. The contrast ratio of the peak was calculated as the ratio between the peak charge density and the charge density of the trough between E₀ and the spectral peak. In order to be considered for this analysis, the contrast ratio was required to be larger than 1.5 for a given shot. This removed 385 shots from the complete 600 shot data set. The spectral peak-based approach to the analysis is important because it places more emphasis on what would be considered the more 'usable' portion of the beam by mostly excluding the lower energy tail of the accelerated charge. On average, the accelerated charge about the peak accounts for roughly 84% of that considered when using the centroid energy calculation, described below. As an example, the spectral peak and the rms energy spread are shown for a shot with 9 GeV peak energy gain in figure 1(a) with solid and dashed black lines, respectively.

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The centroid (median) energy value was calculated using the entire range of energy extending above the spectral charge density trough. The energy spread about the centroid was calculated as the root-mean-square value over a domain that extended outward from the centroid until 95% of all charge observed above the spectral charge density trough was included, keeping the lower limit bounded at the spectral trough energy. The total accelerated charge about the centroid energy was calculated as the sum of charge located within two times the rms energy spread of the centroid energy, again with a lower bound at the spectral charge density trough. Because there was often a low-charge density tail that was accelerated to a lesser extent along with the mass of charge at the spectral peak, the centroid energy value tends to be slightly lower than the spectral charge density peak energy value, though the two track one another closely, giving energy gain values that agree to within 15% on average (see figure 2). The centroid energy analysis is valuable because it allows us to consider the entirety of the accelerated charge, which is needed to extract causal relationships in the final spectrum related to wake loading effects, as discussed in the section 3.

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Figures 1(a) and (b) show the energetically dispersed profile of the single shot from the data which had the highest spectral peak energy, 29.3 GeV, corresponding to an energy gain per unit length of 6.9 GeV m⁻¹. The amount of charge accelerated in the peak of this shot was $28.3\pm 2$ pC, and the energy spread about the peak was 5.0%. Figure 1(c) and d show the same shot as recorded on the Lanex screen, which gives a spectral peak value of 29.2 GeV, agreeing with the wide FOV Cherenkov screen to better than 3%. The centroid energy, energy spread, and accelerated charge values for this shot were 27.5 GeV, 8.0%, and $43\pm 5$ pC, respectively.

The projected spectral charge density of all 215 analyzed shots are shown in figure 2, where the charge density is represented on a log-scale by the color axis. The shots are sorted from left to right in order of ascending centroid energy, which is depicted by the cross-shaped markers, while the spectral peak of each shot is indicated by the star-shaped markers. The distributions of energy gain, energy spread, and accelerated charge values as calculated for the centroid energy and peak energy all roughly fall into normal distributions. A statistical summary of these quantities is presented in table 1.

The mean accelerating gradient for the spectral peak energy was 4.0 GeV m⁻¹, which is within 10% of the previously recorded value of 4.4 GeV m⁻¹ in the previous two-bunch PWFA experiments at SLAC in [1], indicating that the plasma wake is non-evolving over meter-scale distances. The mean total energy transferred from the drive beam via the wake to the accelerated charge in the the experimental results presented here was 0.6 J, which is greater than the energy transfer in the previous experiment by a factor of about five. This difference can be accounted for by the ratio of the length of the two plasma sources (3.6) and the ratio of accelerated charge (1.6), the combination of which would lead to a rough estimate of an improvement in energy transfer of about a factor of 5.8, which is within 16% of the experimental value reported here.

The variation of the incoming electron beam allowed us to sample a range of final outcomes. In particular, as the amount of accelerated charge was varied, we could observe a closely correlated change in the final spectrum. As the charge increased, the energy spread and the centroid energy decreased as shown in figures 3(b) and (c). The physics of wake loading implies this to be a causal relationship. For a non-linear plasma wake that is loaded with an ideal beam that perfectly flattens the wakefield, the product of the accelerated charge and the energy gain is conserved [23]. That we have a large energy spread in our final spectrum (see figures 3(a) and (c)) is an indicator that we are not perfectly flattening the plasma wake, though we still expect that the energy available for acceleration inside the wake remains the same from shot-to-shot, on average, and thus by loading it down with more charge we expect to see the mean energy gained per particle in a given shot to decrease in a roughly linear fashion.

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In figure 3 we have plotted three strong correlations that give evidence of wake-loading effects in the data. Figure 3(a) shows the increase of energy spread with the increase of centroid energy gain. This observation on its own gives a strong suggestion of wake loading, but there is no causation implied by the variables plotted alone. To see evidence of the cause for the correlation in figure 3(a), we must turn to figures 3(b) and (c). In both of these plots, the causal variable is the amount of weighted accelerated charge, ${{q}_{\text{acc}.,n}}={{q}_{\text{acc}.}}\cdot <{{E}_{\max.}}>/{{E}_{\max}}$. The accelerated charge tells us how much charge was sitting in the accelerating phase of the wake, while the maximum energy reach of a given shot ${{E}_{\max}}$ serves as a proxy for the strength of the wake. This rests on the assumption that the high energy tail consistently samples the highest field in the wake for every shot, and thus by measuring the highest energy observed in the tail, we can get an indirect handle on the strength of the wake in which it was accelerated for a particular shot. We use the maximum energy as a weighting factor for the accelerated charge in each shot to give a more uniform comparison of the wake loading effects across all the analyzed data.

The wake loading theory tells us that when the wake is well-flattened, the product of the accelerated charge and the energy gain of that charge is held constant [23]. Further, simple conservation of energy tells us that if we add more and more particles to the accelerating phase of the wake then the average amount of energy that can be gained by each must get smaller and smaller. What we observe in figure 3(b) shows exactly this. As the normalized accelerated charge is increased, the centroid energy is reduced, following a fairly linear trend.

Another consequence of the wake loading dynamics is that the energy spread is decreased as the wake is loaded more strongly. This is a behavior that can arise from under-loading the wake, which occurs when there is not sufficient charge to fully flatten the wakefield. When the charge is reduced, the wakefield becomes less flat and each longitudinal slice of the accelerated beam experiences a field strength that is different from the slice next to it.

Put together, the consequences of the wake-loading dependencies shown in figures 3(b) and (c) explain the observed correlation between the energy spread and the centroid energy gain shown in figure 3(a). These plots describe a picture where the plasma wake is more loaded down and better flattened when more charge is added to the accelerating phase of the wake, suggesting that the experiment was operated in a somewhat under-loaded regime. The fact that the minimum observed energy spread about the centroid energy was 2.43% (0.92% about the peak energy) already tells us that this was the case, as a perfectly loaded wake would lead to no increase in the energy spread beyond the intrinsic energy spread of the witness prior to acceleration in the PWFA.