# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 4 Examples of Gas Jet Density Measurements
### Shock Front Characterization in an LWFA  
For the purpose of density down-ramp injection in the linear regime of LWFA, a razor blade is inserted laterally into the gas jet to create a supersonic shock front [4]. A typical phase projection of the shock front is given in Figure 16. Two images next to each other are shown, one with positive and one with negative signal values. This is an artifact due to the working principle of the Wollaston prism, since it produces two parts in the interferograms; one represents the reference (unperturbed)  
Figure 15: ML-EM reconstructed on-axis density of the solenoid valve without blade. (**Left**) Dashed lines represent the estimate for the respective backing pressure. The data points with error bars are from the measurements. The resonant density is reached with backing pressures between 30 and 35 bar in the region of interaction for the Ti:Sa pulse (\(h=2.6\)–\(2.8\) mm). (**Right**) The density appears to increase for pressures up to 37.5 bar. For higher pressures, stagnation and even regression are observed. and the other one the signal (cf. Section 2.1). From the projection, it can already be noted that the density gradient is decreasing with increasing vertical distance above the nozzle. The tomographic data obtained with the rotational setup, cf. Figure 7, are analyzed via the ML-EM reconstruction algorithm explained in Section 3.2.1. Figure 17 depicts the reconstructed density distribution of the shock front at 35 bar backing pressure. To study and optimize the ramp properties, this measurement is carried out for different backing pressures \(P_{b}\) and blade positions \(L_{b}\) from the center up to the edge of the gas flow (edge: \(L_{b}=-3.2\) mm). The ML-EM tomography is computed at several distances from the nozzle \(h\). The range of these parameters is summarized in Table 1. In order to evaluate the characteristics of the shock front numerically, the following quantities are defined:  
* Height \(h_{s}\) of the shock front: density difference between ramped and undisturbed distribution at the ramp,
* Ramp factor \(r\): ramped peak density divided by undisturbed peak density,
* \(w_{1}\): half-width (left) defined by the ramp peak density,
* \(w_{2}\): half-width (right) defined by the height \(h_{s}\). For a better understanding, the parameters \(w_{1}\) and \(w_{2}\) are indicated in Figure 18, showing the density of a shock front, as well as the undisturbed distribution along the \(z\)-direction. The main effect of the backing pressure is the central density, which is plotted in Figure 15. The ramp characteristics are governed by the parameters \(L_{b}\), \(h\) and \(y\), and their respective effects are summarized in Figure 19 at a fixed backing pressure of 35 bar. A sharp density down-ramp is desirable to achieve a small electron energy spread in an LWFA ([2; 26; 27; 28]). The length of the down-ramp is quantified by \(w_{2}\), which is increasing with distance from the blade. A slight dependency of \(w_{1}\) on the blade position \(L_{b}\) is observed with a local minimum at \(L_{b}=-2.8\) mm. Another important quantity is the ramp factor \(r\), which is mainly determined by \(L_{b}\) and \(y\). It is larger when the blade is positioned closer to the center of the gas flow, as the ramp is created in a region of higher undisturbed density. The measurements indicate that the ramp factor also depends on the horizontal position \(y\). In particular, \(r\) has a local maximum 0.5-0.6 mm displaced from the center. This is understandable by looking at the profiles of a two-dimensional Gaussian. When the profile is off-centered, the distribution is flatter. This means that for a centered profile, the density is rising more after the ramp, which results in a lower ramp factor. Figure 17: **Left**: ML-EM Reconstructed density distribution in a plane perpendicular to the gas flow at distance \(h=2.6\) mm from the nozzle, i.e., 1 mm from the blade. **Right**: Density profile along \(z\) direction for \(y=0.55\) mm.