## 3 Experimental Setup
### Data Evaluation of the \(H_{\beta}\) Line  
To extract the \(H_{\beta}\) line, a pure nitrogen spectrum and a nitrogen-hydrogen spectrum are recorded, consecutively. In both cases, an average image is first calculated from all recorded images (at least 100) for further processing, and subsequently, each measured pixel line is corrected with the according pixel line of the radiance calibration measurement. Afterwards, due to the fact that the hydrogen line is superimposed on the nitrogen spectrum, as shown in Figure 5, each nitrogen image is multiplied by a variable scaling factor and subtracted from the nitrogen-hydrogen image to obtain an image containing only the hydrogen line. The scaling factor averaged about \(0.85\pm 0.7\) and was adjusted for each parameter set.  
As briefly mentioned in the previous section, the shape of the recorded hydrogen line is determined by a convolution of several broadening mechanisms, the most significant being: broadening due to instrumental profile, Doppler, van der Waals (short vdW) and Stark broadening. The parameter evaluation is easiest if an assumption can be made that the Stark broadening is dominant and described by a Lorentzian curve. Then, the measured data can be approximated with a Voigt profile, i.e., the convolution of a Gaussian and a Lorentzian profile, where the width of the Gaussian component is taken as the width of the instrumental profile, and the width of the Lorentzian component is taken as the Stark broadening. The additional, weaker influence of the Doppler and vdW components can be estimated by formulas given in [52; 54]. For the expected heavy particle temperatures, the Doppler broadening is more pronounced with \(\Delta\lambda_{Dop}\cong 0.03\) nm than the vdW broadening with \(\Delta\lambda_{vdW}\cong 0.01\) nm. Nevertheless, the width of the instrumental profile of the spectroscopic setup (depicted in Figure 4) will increase only by less than 5% when convoluted with the above \(\Delta\lambda\). Thereafter, the Doppler and vdW contributions are neglected in the evaluation.  
After the nitrogen background is subtracted, prior to the fitting of a Voigt profile to the resulting data, the local intensity distribution of the hydrogen line is calculated using the Abel transform, assuming that the line is optically thin and axially symmetric. The Voigt profile is recommended for the \(H_{\beta}\) line [53; 54]. The assumption is generally valid if the effluent plasma jet is nearly homogeneous or at least axially symmetric, which is only satisfied if no time resolution is considered. On a short time scale, especially in the region where  
Figure 5: The left plot shows an \(N_{2}\) and an \(N_{2}/H_{2}\) spectrum after correction with the radiance calibration standard, recorded with similar camera settings at nozzle axis (\(r=0\) mm) for an axial distance of \(d=2\) mm. The plasma generator was operated at 60 kHz and the experimental setup was triggered at peak current of \(\sim\)1.0 A (power setting of 100%). The hydrogen \(\beta\)-line shown in the right plot is obtained after subtracting the pure nitrogen spectrum (multiplied by a variable scaling factor) from the hydrogen-containing measurement.  
the discharge is blown out of the nozzle, an assumption of rotational symmetry can only be made locally, i.e., for the region in direct proximity of the nozzle axis. The assumption is further supported by the fact that average images are used for parameter evaluation. Hence, only a radius range of 0.6 mm from the generator axis is analyzed. Thereafter, the local \(H_{\beta}\) profiles are fitted by a Voigt profile, where the Lorentzian component is varied to achieve the best possible fit of the measured data. With the so determined FWHA, the electron number density is finally calculated using the approximation Formula (1). Furthermore, three measurements are made and evaluated for each parameter set, so that the results discussed in Section 4 represent an average of these measurements.