# Plasma Expansion Characterization of a Vacuum Arc Thruster with Stereo Imaging
## 2 Materials and Methods
### Data Acquisition and Processing  
The following describes the process of the acquisition and processing of the images and data. The operating frequency of the PJP was set to 1 Hz. The maximum resolution of the camera, 2000 \(\times\) 2000 pixel, was used for all recordings. An exposure time of 1.5 us was set for all measurements. The recording of the ignition was time-synchronized with a variable time delay, which considered the temporal offset between the ignition of the thruster and the start of the camera recording. The measurements were done with and without a magnetic nozzle. The delay values listed in Table 1 were used. Since the maximum discharge current was measured after 10 us in the studies of other project participants, the plume expansion was sampled up to this point. One hundred images were recorded for all set values. In the following, the acquisition and evaluation process is described using the example of acquisition and recording from the setup without a magnetic nozzle. The procedure for measurements with a magnetic nozzle was identical. Figure 5: Schematic principle of the used stereo-optical measurement system: (**a**) schematic layout of camera and mirrors; (**b**) scheme of the divided CMOS camera chip. Figure 6: Recorded image after adjustment of the system. The white line in the middle of the image symbolizes the separation of the camera chip according to Figure 5b. The recording in Figure 7 displays the illustration of the discharge from two directions of view on the divided camera chip. The left one corresponds to the left direction of view in Figure 5 and the right one to the right direction of view. From this recording, the scale factor for the later calculation can be determined by using the observed size of the anode, too. For this purpose, the dynamic range of the recorded image was adjusted in such a way that the thickness of the anode could be determined in pixels. With the known dimensions of the anode, a scaling factor in \(\frac{mm}{Px}\) could be calculated. For further processing, a region of interest (ROI) from both observations was cropped from the camera image to reduce the calculation effort. Both sections had an identical size. The snippets were positioned manually, so that the lower border of the ROIs overlaid with the upper border of the cathode. The ROIs were placed horizontally, so that the cathode was centered. To be able to calculate the area of the plasma, the shape and expansion of the plume must be determined. The expansion of the plasma plume was calculated by extracting the contours from both extracted ROIs. In Figure 8, two contours are displayed. On the left side, the sector from the left direction of observation is shown, and on the right side, correspondingly from the right direction of observation. The contours of the plasma plume were sliced using a well-known algorithm called \"marching cubes\". The areas of the sectors using the previously determined scale factor could be calculated. To calculate a 3D model, both sectors were used to calculate a cross-sectional image from both perspectives. \[ \begin{pmatrix}x_{1}\\ \vdots\\ x_{m}\end{pmatrix} * \begin{pmatrix}y_{1}&\dots&y_{n}\end{pmatrix} = \begin{pmatrix}x_ {1}*y_{1}&\dots&x_{1}*y_{n}\\ \vdots&\ddots&\vdots\\ x_{m}*y_{1}&\dots&x_{m}*y_{n}\end{pmatrix}  \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Value** & **Unit** \\ \hline
10 & ns \\
500 & ns \\
750 & ns \\
1 & us \\
10 & us \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Used time delay values for the measurements. Figure 8: Assigned contours of the cropped ROIs from Figure 7 for calculating the plume area: (**a**) left direction of view; (**b**) right direction of view. Figure 7: Recorded image of the ignition from the PJP without magnetic nozzle with 1 μs time delay. On the left side of the camera chip, the left direction of view is mapped, and on the right side, the corresponding right direction of view.