Effect of driver charge on wakefield characteristics in a plasma accelerator probed by femtosecond shadowgraphy

Figure 2(a) illustrates typical electron spectra, sorted by peak energy, recorded with LWFA stage in operation and laser blocker foil in position, but the second gas jet turned off. These beams have a mean energy of 328 ± 29 MeV and an FWHM-energy bandwidth of 44 ± 18 MeV and the integrated charge within this bandwidth is 318 ± 71 pC. Here, the values of uncertainty are given as the standard deviation of the whole set of reference shots, i.e. these values represent the shot-to-shot jitter. The power spectrum of transition radiation, emitted as the beam passes through a thin metal foil, was measured by a multi-octave broadband spectrometer and allows for reconstruction of the temporal profile of the electron bunch [15, 28]. This results in a FWHM-bunch duration of about 14.8 ± 1.6 fs, corresponding to an estimated peak-current of about 22.6 kA. Furthermore, the electron beam size at the exit of LWFA stage is inferred from the betatron x-ray spectral-shape, yielding a root-means-square (rms) beam size of about 1 μm [18].

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Small variations of the laser and plasma parameters lead to a shot-to-shot jitter of the electron beam properties. Especially the charge and the energy of LWFA accelerated beams are coupled due to the beam loading effect [16, 17, 29], i.e. electron beams of higher charge strongly flatten the accelerating field which results in a reduction of the beam energy. By plotting the beam charge as a function of the peak energy as depicted in figure 2(b), the charge–energy correlation can be extracted and is linearly fitted to Q(E_peak) ≈ (−2.02 ± 0.24) pC MeV⁻¹ ⋅ E_peak + (981 ± 80) pC. As the energy and charge can also be influenced by other parameters, the data points are scattered around the fit. For the whole experimental run, the nominal parameters of the laser and plasma for the LWFA stage are kept fixed.

The driver beam interaction and plasma dynamics within the PWFA stage differ depending on whether the electron beam enters a plasma environment, generated by the pre-ionization laser, or a neutral gas, which will be ionized by the space charge field of the driver beam itself (self-ionized regime). Both cases were investigated during the experiment and the resulting plasma waves, profiles along the axis of the waves and the CWT analysis, are shown in figures 3(a)–(c) for the self-ionized case and in (e)–(g) for the pre-ionized case. Both are recorded at the center of the PWFA stage (1.5 mm after the driver beam enters the PWFA stage), operated at a plasma density of n_p = 4.5 × 10¹⁸ cm⁻³.

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Here, a distinct difference on the wakefield structure is observed between the two cases. In the self-ionized regime, a narrow plasma channel inside the neutral gas background along the driver beam propagation axis is seen, in which a few clear periods of plasma wave oscillation located at the channel front side are visible (see figure 3(a)). Such a plasma wave, however, quickly damps out and vanishes to below the detection limit, at the latest after the ninth cavity. In contrast, in the pre-ionized regime, the associated shadowgram exemplified in figure 3(e) exhibits a pronounced plasma wakefield structure that extends significantly further out and can have up to 25 subsequent, well-pronounced cavities. Additionally, a cone like feature trailing the plasma wave after 1 ps can be identified, which is attributed to the transverse ion motion induced by the wakefield, as also reported in [12]. Furthermore, the wavelet analysis, presented in figures 3(c) and (g), yields a plasma wavelength of about 15.8 μm, fairly constant along the wakefield and almost identical for both cases. This value agrees well with the nominal plasma wavelength, ${\lambda }_{\mathrm{p}}\enspace (\mu \text{m})\simeq \,3.3\times 1{0}^{10}/\sqrt{{n}_{\mathrm{p}}\enspace ({\mathrm{c}\mathrm{m}}^{-3})}\simeq \ 15.6\enspace \mu$m [30], for the plasma density of 4.5 × 10¹⁸ cm⁻³, independently measured using tomographic interferometry [31].

Assuming a Gaussian beam distribution, the space-charge field of the driver reaches a maximum value of ${E}_{r}^{\mathrm{max}}\simeq \,-0.45\,{E}_{0}\,(2{I}_{\mathrm{b}}/{I}_{\mathrm{A}})/{k}_{\mathrm{p}}{\sigma }_{r}$, where I_b is the driver peak current and E₀ the non-relativistic wave-breaking field, at distance r ≃ 1.585 σ_r from the propagation axis [21]. There is no direct way to extract the transverse size σ_r of the driver beam while propagating inside the PWFA stage, as the beam refocusing dominates the dynamics. However, since the beam size [18] and the divergence at the exit from the LWFA stage are measured, the beam size at the entrance of the PWFA can be estimated. The divergence of the beam after the LWFA stage was measured at the electron spectrometer to be 1.81 ± 0.17 mrad. However, before entering the PWFA stage, the beam has to pass through the laser blocker foil. While fully reflecting the spent LWFA driver pulse, the high-intensity laser–foil interaction causes a current filamentation instability within the foil, leading to a significant divergence increase to the beam upon foil transition [25]. The divergence of the beam after the blocker foil increases to 4.2 ± 0.8 mrad. Accordingly the transverse beam size at the entrance of the PWFA stage is estimated to be σ_r ≈ 3.3 μm representing the upper limit of the size within the PWFA stage. Using this value, the space-charge field amounts to 186 GeV m⁻¹ (considering I_b = 22.6 kA), far above the ionization threshold for hydrogen of about 33.8 GeV m⁻¹ [21]. This implies that the peak current of the driver is sufficiently high to fully ionize the hydrogen gas target even in the self-ionized regime, which is confirmed by consistent measurement of plasma wavelengths between the two cases.

The structure of the observed plasma waves differ strongly between the self- and the pre-ionized regime, which can be explained as a consequence of the limited size of the generated plasma column. Without pre-ionization, a plasma channel with a narrow width of approximately the same order of magnitude as the transverse diameter of the electron driver beam is produced, i.e. a radius of ∼7 μm from figure 3(a), and sustained over ps time scale. Immediately after ionization, electrons are expelled from the beam propagation path to reach a maximum distance from the axis which is given by the blowout radius of ${r}_{\mathrm{m}}\approx 2\sqrt{2\times {I}_{\mathrm{b}}/{I}_{\mathrm{A}}}/{k}_{\mathrm{p}}\,\geqslant 7\enspace \mu$m [21]. Therefore, some electrons can escape into the neutral gas region, such that they no longer experience the attracting force from the ion channel and thus do not return to the axis [32, 33]. Electrons within the reach of the ionic attracting force move back and cross the propagation axis at various longitudinal positions, depending on their ionization location and thus with different oscillation phase and frequency, as can be seen in simulation figure 3(d). As a consequence, a less narrow electron sheath is formed which smears out after a few cavities. The strength of the accelerating field at the rear side of the first cavity is weaker and it has been observed that this results in lower witness bunch energies compared to the pre-ionized case [13]. That is because the pre-generated plasma channel is much wider as determined by the width of the pre-ionization laser. Hence, the expelled plasma electrons, independently of the initial location, oscillate collectively and, thus, return back to the beam propagation axis at a defined longitudinal position, which leads to a sustainable plasma wave generation, as shown in figure 3(h).

The variations of driver beam properties as described in section 3.1 can be utilized to explore the impact of the driver beam charge and energy on the generation of plasma wakefields. For this purpose, we compare the shadowgraphy images, taken in the pre-ionized regime, for different spectral-charge distributions of the remaining driver beam extracted from the electron spectrometer. Three sample shots are illustrated in figure 4(a) for the shadowgrams and (b) for the associated remaining driver. All shots are taken using the same nominal density of 3 × 10¹⁸ cm⁻³ for the PWFA stage which corresponds to a nominal plasma wavelength λ_p ≈ 19.3 μm. In all three images, the trailing plasma cavities have approximately the expected length while the first cavity is significantly elongated. This elongation gets stronger from top to bottom, whereas the most obvious change in the corresponding electron spectra is a decrease in energy.

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According to theoretical work [19], the elongation of the first cavity should mainly depend on the charge density of the driver at the observation point. However, in experiment, the initial driver charge cannot be directly measured as driver–plasma interaction in the PWFA stage leads to beam degradation. This becomes visible in the energy spectra (see figure 4(b)), where the decelerated electrons redistribute over a large energy range from the initial energy down to zero-MeV energy. As a result, the measured charge within FWHM of the high energy distribution drops on average to 21% of the reference shots. This implies that the initial charge reconstruction based on measured remaining charge is inaccurate. This is exacerbated by the fact that drivers containing initially more charge generate more strongly fields within the plasma cavity, causing them to decelerate stronger than less charged drivers.

Nevertheless, the initial charge of the driver beam is still encoded in the spent driver peak energy via the beam loading effect described earlier above in section 3.1. This correlation allows us to estimate the initial charge using the linear function presented in figure 2(b). Since the decelerating field in the PWFA is not constant along the length of the driver bunch, the deceleration occurs at different rate, depending on where the electrons are located with respect to the decelerating field. The front part of the beam can maintain its energy until the end of the plasma, as confirmed by simulations presented in section 3.4.

Figure 5 shows the length of the first cavity normalized to the nominal plasma wavelength versus the derived initial FWHM beam charge. Here, the elongation increases significantly with increasing driver charge. The error in the estimated bunch charge represents the uncertainty of the indirect determination method as illustrated by the light green band in figure 2(b). The uncertainty of the fit itself would lead to a slightly different horizontal axis, but would not effect the relative correlation between the data points and is therefore not shown. The vertical error is assumed to be two pixels for the measurement of the cavity sizes.

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The correlation of cavity elongation and charge is valid for the full range of densities where data was taken, so between 3.0 × 10¹⁸ cm⁻³ and 6.1 × 10¹⁸ cm⁻³, corresponding to linear plasma wavelengths of 19.3 μm down to 13.5 μm. As the duration, and thus longitudinal size (length) of a single bunch, is set from the LWFA stage, the change in density also changes the ratio of bunch length relative to the plasma wavelength. Because the values of elongation for a certain value of initial driver charge hardly differ at different densities, this means the influence of this longitudinal ratio in this setup is much smaller compared to that of charge. This clearly shows that $\frac{{n}_{\mathrm{b}}}{{n}_{\mathrm{p}}}\geqslant 1$ still holds for such high plasma densities. Assuming a cylindrical bunch, the average density of a bunch n_b with the properties described in section 3.1 is estimated to be n_b = 1.2 × 10¹⁹ cm⁻³ and thus larger than the plasma densities used in experiment, as required to enter the blowout regime.

Experimental findings of a charge-dependent elongation are supported by three-dimensional particle-in-cell (PIC) simulations performed with PIConGPU [34], starting right after the laser blocker foil. While most of simulated driver and plasma parameters are deduced from the measured experimental quantities, the bunch charge is stepwise increased to numerically access the strong-beam interaction regime. All other input parameters such as transverse and longitudinal rms size, energy and energy spread are fixed and remain the same for all three simulations. It is assumed that the driver bunch features an uncorrelated phase-space when leaving the LWFA stage and enters the fully ionized second stage with a finite divergence, i.e. defocused phase-space distribution. As the plasma cavity starts to develop in the density up-ramp, the focusing field pinches most of the beam, resulting in a compact driver for the wakefield excitation (see supplemental material). Figure 6(a) presents the simulated plasma waves excited by different energy-FWHM-integrated driver bunch charges: 228 pC (top), 152 pC (middle) and 76 pC (bottom). One can clearly see that for the lowest driver charge the plasma wave exhibits a regular structure from the head to the trailing part with a wavelength matched to the linear theory, i.e. ${\lambda }_{\mathrm{p}}=2\pi c\sqrt{{{\epsilon}}_{0}{m}_{\mathrm{e}}/{e}^{2}{n}_{\mathrm{e}}}$ with c denoting the speed of light, ₀ the vacuum permittivity, m_e the electron mass and e the elementary charge. When the driver charge increases, the first cavity elongates and simultaneously the accelerating field strength at the cavity rear-side becomes stronger, exceeding 500 GV m⁻¹ for the 228 pC driver. Due to the stronger field induced by drivers of higher charge, the plasma electrons are pushed further out, resulting in a larger radial and longitudinal size of the cavity as they need longer time for returning back to the axis. This increased radius of the positively charged ion background of the cavity leads to a stronger electric field inside the plasma cavity. This observation is consistent with previous theoretical predictions [19, 21, 35], which all show an increase in field strength for increasing driver peak current. The simulations qualitatively resemble the experimental findings, demonstrating that such a dense bunch is capable of driving the full blowout and the first cavity length significantly increases for high charge driver beams.

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Figure 6(b) shows the corresponding simulated electron spectrometer. Comparing the spectrum of the initial bunch (top) and after the interaction (middle) a strong deceleration and charge loss as in experiment was observed, which is consistent with the experimental observations discussed in section 3.3. Even though these two effects become stronger for higher charge driver (see supplementary chapter 4), the high-energy peak stays at approximately the same value, justifying our approach of reconstructing the initial charge via the peak energy.