# A Study of Current Controlled Discharge in a Nitrogen Filled Tube
## Appendix A Optical Diagnostics of the Discharge  
Based on the analysis of interference patterns from the Mach-Zehnder interferometer, the discharge electron density in the discharge column is estimated. The refractive index n of the plasma is given by the instantaneous discharge electron density Ne:  
\[n=\sqrt{1-\frac{N_{e}}{N_{c}}}\doteq 1-\frac{1}{2}\frac{N_{e}}{N_{c}}, \tag{1}\]where \(N_{c}\) is the critical electron density for an electromagnetic wave with a angular frequency \(\omega=2.356\times 10^{15}\,\mathrm{s}^{-1}\) (\(\lambda=800\,\mathrm{nm}\)):  
\[N_{c}=\frac{\epsilon m_{e}\omega^{2}}{e^{2}}=1.744\times 10^{21}\,\mathrm{cm}^{-3}. \tag{10}\]  
We assume that the waves in both arms of the interferometer propagate along the z-axis. The period of the recorded interference pattern is given by the set angle \(\alpha\) between the beams passing through the different arms of the interferometer. The ideal shape of the recorded interference patterns is proportional to the function  
\[1+\cos\Bigl{[}\pi\frac{\alpha x}{\lambda}+\Phi(x,y)\Bigr{]}. \tag{11}\]  
The phase difference \(\Phi(x,y)\), which is recorded after passing through the arms of the Mach-Zehnder interferometer, is determined by the integral  
\[\Phi(x,y)=\frac{\pi}{\lambda N_{c}}\int_{0}^{d}N_{c}(x,y,z)dz, \tag{12}\]  
or  
\[\Phi(x,y)=\pi\frac{d}{\lambda}\frac{\overline{N_{c}}}{N_{c}}, \tag{13}\]  
if the mean value of electron density \(\overline{N_{c}}\) along the path is introduced. In our case, the angle \(\alpha\) between the beams was set to about \(2.4\,\mathrm{mrad}\) (about 30 strips per tube diameter). The phase shift \(\Phi\) would significantly affect the interference pattern if its value is comparable to \(2\pi\). We recorded the interference pattern and determined the phase difference as follows. We have used a code for an interferogram evaluation based on Fourier transform [26]. An interferogram can be represented in a mathematical formula as:  
\[g(x,y)=a(x,y)+b(x,y)\cdot\cos[2\pi f_{0}x+\Phi(x,y)]. \tag{14}\]  
Functions \(a(x,y)\) and \(b(x,y)\) denote a fluctuating background and an amplitude, respectively, \(f_{0}\), is a spatial frequency of a fringe pattern and \(\Phi(x,y)\) represents a phase shift which can be extracted by method proposed by [PERSON] et al. [26]. The crucial method assumption of slowly varying functions \(a(x,y),b(x,y),\Phi(x,y)\) was fulfilled. For each measurement, a reference and signal interferogram was recorded and evaluated. The reference phase was subtracted from the signal one and unwarped. The diagnostic beam passed through the tube filled with gas/plasma. However, there was no region of plasma free medium. For this reason, we assumed from acquired data that the amount of electron density was negligible near to the tube walls and set to zero value. This assumption could cause an inaccuracy in determination of an absolute value of electron density. This might be neglected, because from our measurement, the maximal phase shift tended to be localized in the tube center. To obtain the electron density spatial distribution, it had to be assumed that the plasma in a tube is cylindrically symmetrical. On the other hand, we can consider the electron density to be averaged, as was expressed in the text earlier. The average electron density can be calculated as  
\[\overline{N_{c}}(x,y)=N_{c}\frac{\lambda_{0}}{d}\frac{\Phi(x,y)}{\pi}. \tag{15}\]  
The acquired dependencies of the phase shift \(\Phi(x,y)\) in repeated shots under the identical initial and boundary conditions are seen in Figure 11. They are not identical. The stochastic nature of the results may be interpreted either by the stochastic nature of the electron distribution in the discharge or by the inaccuracy of the interference pattern data recording and subsequent data processing. We performed reference measurement of \(\Phi(x,y)\) in a case without discharge in the tube. In such an ideal case, the phase shift \(\Phi(x,y)\) should be zero. During repeated shots, we received a random function \(\Phi(x,y)\) with amplitudes \(\pm 0.5\) rad.