# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 4 Examples of Gas Jet Density Measurements
### Solenoid Gas Jet for LWFA  
A disadvantage of piezo valves is that the valve closing mechanism can only sustain backing pressures \(\leq\)10 bar. Therefore, such valves are not suitable for high density applications. For the purpose of LWFA characterization, high-Z neutral atom densities are required, which are larger by one order of magnitude than the peak density, which the piezo valve can produce. The Parker solenoid valve (Figure 6A) is expected to provide densities exceeding \(1\times 10^{18}\) cm\({}^{-3}\) at a distance of 2.6 mm from the nozzle, when operated with backing pressures around 35 bar (Figure 3).  
Due to geometric limitations in our setup (blade holder), the focused laser beam in the LWFA setup cannot interact with the gas closer to the nozzle than 2.6 mm. The plasma frequency \(\omega_{p}\), which is  
Figure 14: The central density in the gas jet reduces by a factor of 20 after expanding 25 mm.  
Figure 13: Density measurements of the piezo valve used for SwissFEL instrumentation. The reconstruction is computed via an Abel inversion. The subplot indicates the FWHM of the gas density distribution as a function of the distance from the throat.  
determined by the plasma density, has to be matched to the laser pulse duration in order to excite the plasma wake efficiently. In the experiment, the plasma frequency can be controlled by the density and ultimately by the backing pressure applied to the valve. The central density measurement of the solenoid gas jet (without blade) for argon is summarized in Figure 15. From this plot, one can infer the required backing pressure to create the resonant electron density at a certain distance from the nozzle. For a laser pulse of \(\tau=15\) fs duration (FWHM), the required argon density is \(1.75\times 10^{18}\) cm\({}^{-3}\), assuming that the gas is ionized eight-fold (all valence electrons) [14]. Figure 15 also contains the density estimates from Section 2.2. Good qualitative agreement between the model and the measurement is observed. However, the measured densities are systematically higher than the estimate. A reason may be the fact that the estimate is based on the straight streamline model, which assumes that the expansion angle of the gas flow is equal to the opening angle of the nozzle [13]. However, the measured expansion angle, defined by a linear fit to the FWHM of the gas distribution, is around \(52^{\circ}\), which is significantly smaller than the opening angle of the conical nozzle (\(90\ ^{\circ}\)). This would result in a higher particle density and could explain the discrepancy between the used model and the measurement results. Figure 15 (right) depicts the argon density with respect to the backing pressure applied to the valve at fixed vertical distances \(h\). It is observed that the argon density is increasing for backing pressures up to 37.5 bar. For even higher pressures, the data indicate a stagnation and even a regression of the density. This can be explained by the working principle of the solenoid valve. When operating at higher backing pressures, a larger force is needed to lift the puppet out of the seal. Therefore, the valve may not open properly with too high backing pressures. This can then result in a smaller density. To overcome the force due to the backing pressure, a high-voltage pulse (burst) is applied to the valve, the duration of which can be set internally. For the measurements shown here, the burst duration was set to 220 \(\upmu\)s.