Laser wakefield acceleration (LWFA)^1–4 is the most compact technique for producing ultrashort (fs) GeV-class electron beams.^5–7 LWFA is based on exciting a plasma wave in an underdense plasma by the ponderomotive force of an intense laser pulse (with normalized vector potential a₀ ≥ 1). A strong electrostatic field in the plasma wave then accelerates the electrons.⁸

To successfully perform LWFA, it is crucial to create the correct plasma density profile.⁹ For example, a longitudinal plasma density downramp can be used to improve the stability of the electron beam. This scheme is, at the same time, very efficient and easy to implement.^10,11 Superior beam quality is obtained by restricting the electron injection to the accelerating phase in the density downramp region. Furthermore, in the recently growing field of LWFA at reduced laser energies of few mJ (≤10 mJ, needed to increase the repetition rate to ≥1 kHz),^12,13 it is crucial to achieve a much higher plasma density for electron acceleration, while at the same time keeping the quantity of gas released in the vacuum chamber as small as possible.

Gas targets can be designed using analytical models and simulations.^14–16 Then, to characterize the particle number density in a gas jet expanding in vacuum, interferometric methods are typically used. Characterization of the particle number density in a gas jet expanding in vacuum by means of interferometric measurements has been the subject of numerous investigations focusing on different gases, nozzles, backing pressures, and interferometric methods.^17–19

In the work reported here, two different types of gas target were characterized. First, a de Laval-shape structured supersonic nozzle, already successfully used in LWFA experiments,²⁰ was studied in different configurations to exploit all of its capabilities in view of future experiments. Then, a cylindrical subsonic capillary, built in house, was investigated in different schemes as a gas target to be used for LWFA driven by a few-mJ laser driver.

The gas targets were designed using fluid-dynamics simulations and experimental interferometric measurements, as in previous work.¹⁵ The code used was ANSYS Fluent, which solves the Navier–Stokes equations with the finite volume method. The two-equation eddy-viscosity turbulence model k-ω SST was used, as already described in the literature for similar cases.²¹ This model combines k-ω and k-ε turbulence models such that the former is used near the boundary layer and the latter in the free shear flow. This model is very robust and widely used. The results presented here were obtained from 3D or 2D simulations, depending on the gas target geometry. The simulations were run in the Eclipse cluster at the ELI-Beamlines Project. Eclipse is a 84-node cluster equipped with a shared central storage and an Infiniband QDR interconnection network rated at 40 gigabits per second. Each node contains 16 cores (Haswell, E5-2630 v3 @ 2.4 GHz) and 113 GB of available memory. Parallelization was carried out using the native capabilities of ANSYS for multiprocessing and multithreading. The maximum number of processes allowed by the software was 200, which corresponded to 12 nodes on the cluster. 3D simulations usually took 8 hours, and 2D simulations 3 minutes on 12 nodes. In the case of cylindrical and de Laval-shape axisymmetric gas targets, 2D axisymmetric simulations were performed. For non-axisymmetric geometries, 3D simulations were performed. The gas target density profiles were measured by Mach–Zehnder interferometry using a diode-pumped solid state (DPSS) green laser (532 nm). To reconstruct the 3D density profile of the supersonic nozzle, the interferometric images were acquired from two different angles to allow tomographic reconstruction. The density profile was retrieved using the multiplicative algebraic reconstruction technique (MART). Because of the cylindrical symmetry of the density profile of the capillary, Abel inversion was used for its reconstruction. Measurement of the 3D density profile was required for two purposes: to provide a better comparison with the 3D simulations and to reconstruct the density profile in the central region of the nozzle (i.e., to avoid averaging over the whole transverse nozzle profile). All the density profiles presented in this work refer to the central region of the nozzle, which is where the laser propagates in LWFA experiments.

Two different gas targets were characterized: a supersonic 6.8 mm × 1.26 mm rectangular de Laval nozzle and a subsonic Ti:sapphire capillary of 300 µm inner diameter. The supersonic nozzle is typically used with high-power lasers (≥10 TW) owing to the longer laser propagation, while the smaller subsonic capillary was designed for kHz mJ lasers.

A rapid-expansion rectangular de Laval nozzle designed and manufactured by Smartshell Co., Ltd. was studied. The dimensions of the nozzle are schematically represented in Fig. 1. It has a 140 µm inner throat and a 6.8 mm × 1.26 mm exit. The nozzle’s lateral exit walls of length 1.26 mm are removable, allowing the longitudinal density profile to be tuned. The goal of this study was to characterize the density profile of this nozzle in a range of possible experimental configurations. Owing to the huge amount of possible combinations, different density profiles were studied by simulations. In a few cases, the simulated density profiles were verified by interferometric tomography, in order to benchmark the simulated profiles. The measured and simulated density profiles 0.15 mm and 1 mm above the nozzle are shown in Fig. 2. These data are for a nozzle orientation with the longest density profile (longitudinal). It can be seen that simulation and measurements are in good agreement at the edges of the profile, but there is a minor discrepancy in the central region. This disagreement is due to the fact that the tomographic reconstruction has been done using only two projections. In fact, the ratio between the peaks and the plateau in the central region is minimized in the center of the nozzle, while it increases closer to the edges. For this reason, since the interferometric measurements are integrated over the whole volume, they tend to underestimate the density in the central region. It can be seen that the density profiles 0.15 mm and 1 mm above the nozzle are very similar. The input pressure for both cases is the same: 30 bar of argon. After benchmarking of the simulations, the nozzle was studied under different conditions. First, the transverse density profile (1.26 mm) was numerically studied for different gases typically used in LWFA experiments (helium, nitrogen, and argon). The normalized plots of the density profiles taken at a height of 0.5 mm above the nozzle are shown in Fig. 3.

There is no significant difference between helium and argon, while in the case of nitrogen, two peaks appear at the edges of the profile. This is due to the different heat capacity ratios of the gases (γ = 1.66 for helium and argon, γ = 1.4 for nitrogen), which affects the propagation of shock waves inside the nozzle. This effect can be described by the so-called θ-β-M relation:¹⁴$$\text{tan}\theta =2\text{cot}\beta \left[\frac{M_1^2{\text{sin}}^2\beta -1}{M_1^2(\gamma +\text{cos}2\beta )+2}\right].$$(1)This equation relates the shock wave angle β to the wall curvature angle θ, the Mach number M₁ before the shock wave, and the heat capacity ratio γ of the gas (see Fig. 4). On using the specific nozzle parameters and Eq. (1), it turns out that the lower the value of γ, the lower is that of β. It should be noted that the Mach number of the outgoing flow also depends on γ, according to the following equation:¹⁴$$M_2^2=\frac{1}{{\text{sin}}^2(\beta -\Theta )}\frac{M_1^2{\text{sin}}^2\beta +[2/(\gamma -1)]}{[2\gamma /(\gamma -1)]M_1^2{\text{sin}}^2\beta -1}.$$(2)On using the specific nozzle parameters and Eq. (2), it turns out that the nitrogen flow has a Mach number lower than in the case of argon and helium. Despite the fact that this nozzle has been optimized for γ = 1.66, from the simulation with nitrogen (γ = 1.4), a density profile similar to those for helium and argon at a height of 0.5 mm above the nozzle is observed at a height of 1 mm.

The two density profiles already presented (longitudinal and transverse) were then compared with a third longitudinal profile obtained by removing one of the two 1.26 mm nozzle walls. This comparison was done for nitrogen gas at a height of 0.5 mm above the nozzle. The three density profiles are shown (normalized) in Fig. 5. In LWFA experiments, the laser propagates inside the density profiles from right to left. The transverse and longitudinal profiles each exhibit two peaks, one at the entrance and one at the exit of the gas target. Besides the length, these two profiles differ also in peak-to-plateau density ratio, which is 1.1 for the transverse profile and 1.4 for the longitudinal profile. By removing one of the flat walls, a density peak is removed, leading to a moderate reduction in the gas density toward the exit side. The possibility of having three different density profiles with the same nozzle is of great help in LWFA experiments. In fact, by using the transverse profile, it is possible to limit the acceleration length and reduce the electron beam energy. With the longitudinal profile, the acceleration length is then limited by the laser–plasma interaction itself (for TW laser systems), and the ability to play with the second density peak is useful for gaining additional insight into the process.

To gain more flexibility in the experimental environment, density profiles obtained by placing a sharp obstacle (blade) along the supersonic gas path were studied. This technique is routinely used in LWFA experiments to create a controlled peak inside the density profile. In this case, only simulation results will be reported. In fact, measurement of density profiles with such steep density gradients is quite a complex task.¹⁹ High magnification is needed to resolve the shock region, and, moreover, making measurements from different angles requires a minimalistic mount for the blade.

This study was performed only along the transverse profile. The blade was positioned 3 mm above the nozzle and was 0.5 mm thick. The resulting profiles at different heights above the blade are shown in Fig. 6. The gas used was helium at 30 bar, and in this case the blade was longitudinally placed 0.9 mm from the center of the nozzle. An increase in the density peak is observed, without any effect on the plateau density. Moreover, the density profiles 0.3 mm and 0.5 mm above the blade are very similar. The blade position with respect to the nozzle axis and its relative angle were scanned. No significant differences were observed when the angle was changed. On the other hand, the density profile was strongly affected by the longitudinal position of the blade. The peak-to-plateau density ratio is plotted in Fig. 7(a) versus the distance from the center of the nozzle. The gradient length, defined as the length between the density peak and the plateau, also depends strongly on the blade position. This dependence is shown in Fig. 7(b). It can be seen that the sharpest gradients are obtained further from the center of the nozzle, at the cost of a reduced peak-to-plateau ratio.

A subsonic gas target was studied and designed for LWFA driven by few-mJ, kHz laser pulses, where a lower gas load is needed owing to the higher repetition rate. Owing to the short laser focusing geometry and the expected laser damage, the possibility of a cost-effective target is very attractive. For this reason, the proposed gas target consisted of a cylindrical Al₂O₃ capillary glued to a remotely controlled gas valve. In this case also, the density profile was studied under different conditions by computational fluid dynamics (CFD) simulations, and a few specific cases were validated through experimental measurements. The first operation performed was optimization of the capillary length and diameter by CFD simulations. It was observed that the gas density depended slightly on the capillary length, with a tenfold increase in length leading to a reduction in density by about 10%. At the same time, the diameter was observed to play a crucial role: the larger the diameter, the higher the density. Furthermore, no dependence of the density profile on the capillary diameter was found. The propagation of different gases (helium, nitrogen, and argon) was also studied. In contrast to the supersonic nozzle, no differences were observed among the profiles, although a 30% higher atomic density was obtained with nitrogen. For these reasons, we chose to fabricate the shortest possible capillary with the greatest diameter. The limit on length was set by the laser focusing geometry, while the limit on diameter was set by the gas load in the chamber. Thus, a capillary of length 13 mm and diameter 300 µm was fabricated. Figure 8 shows the simulated density profiles and those measured by interferometry. Good agreement between simulations and measurements can be seen. In this case, the density profile of the gas depends strongly on the height above the exhaust. This is due to the fact that the gas propagation is in the subsonic regime, which means that the gas exits the capillary with a larger opening angle than in the supersonic regime. The density gradient between the peak density and vacuum becomes sharper in regions closer to the exhaust. For this reason, in LWFA experiments, it is crucial to focus the laser as close as possible to the capillary exit.

To obtain a sharper density gradient, as in the previous case, a blade can be positioned along the gas path, and a doubling of the gas density and much sharper gradients were observed when this was done. The results are shown in Fig. 9. The sharpest gradients can be seen at a height above the blade of 100 µm or less. For this reason, a special mount for the blade had to be engineered to allow such close laser focusing.

In this work, we demonstrated, by benchmarking with experimental measurements, that our fluid-dynamic simulations can predict the density profiles of gas targets in both the supersonic and subsonic regimes. Therefore, these simulations can be used for designing new gas targets and for evaluating their performance in different configurations. The simulations show that there is a significant difference between subsonic and supersonic gas targets. The former are cost-effective and easy to handle, but at the cost of a lower density and a much wider entry gradient. In contrast, supersonic nozzles offer higher density, steeper gradients on the sides, and a flat shape of the top profile. It was found that the use of different gases affected the density profile in the supersonic nozzle case; this has to be taken into account during LWFA experiments. When a blade was placed along the gas path, a density peak about twice the plateau density was observed in both the supersonic and subsonic cases. No dependence on the blade thickness or angle was found, and the most important parameter turned out to be the distance of the blade from the center of the nozzle. Finally, it was observed that in the supersonic case, the density profile above the nozzle and above the blade did not depend on height up to about 0.5 mm. This is a good condition for experiments, since it allows easier laser focusing geometries to be used. In contrast, in the subsonic case, the density profile was strongly dependent on height. This means that in order to shoot the laser in the best density profile, the focus has to be placed very close to the capillary exit or just above the blade.