## 2 Materials and Methods
### Evaluating of Electron Concentration in Non-thermal Plasma Jet Generated at Pulse Excitation  
As we have already mentioned, the problem of plasma diagnostics can be solved by different methods. However, it would be prudent to have a sufficiently reliable and flexible method for estimating the concentration of plasma particles, and digital holographic interferometry is a proper method for this task. The results of the application of this method are preliminary and demonstrate the principal possibility of digital holographic interferometry for the investigation of a non-thermal plasma jet generated at pulse excitation. The investigation of plasma by holographic methods is based on the study of a phase (transparent) object, in which the phase changes of the object field depend on the spatial localization of the object and its refractive index [17]. The obtained values of the plasma refractive index allow us to calculate its electron concentration [18]. It is well known in holographic interferometry that phase difference is related to the corresponding refractive index by the following expression  
\[\Delta\varphi(x,y)=\frac{2\pi}{\lambda}\int\limits_{l_{1}}^{l_{2}}[n(x,y,z)- n_{0}]dz \tag{1}\]  
where \(n(x,y,z)\) is the refractive index of the medium (plasma), and \(n_{0}\) is a refractive index of the medium against which the comparison is being made [19]. The integration limits determine the length of the path along which the probing radiation propagates. In conventional (analog) holographic interferometry, fringe analysis is carried out with the accuracy of phase detection associated with changes in the optical path up to \(\lambda/10\)[20]. However, in digital holographic interferometry, holograms are recorded digitally, where the phase detection accuracy is determined by the dynamic range of the detector, with the 8-, 10- or 16-bit resolutions of an analog-to-digital converter. Thus, with 8-bit, it is possible to increase the phase detection sensitivity up to \(\lambda/256\)[11]. Digital holographic interferometry is based on a comparison of the phases of at least two optical wavefronts recorded at different moments in time as digital holograms. The result of the phase comparison between the wavefronts is represented as a numerical phase difference map [21]. A digitally recorded hologram is the distribution of the interference patterns from a superposition of reference and object beams on the detector. The correct acquisition of the hologram can be provided if the condition for the sampling theorem has been met [22]. This theorem determines the angle between the reference and object beam  
\[\alpha_{max}=\frac{\lambda}{2\Delta x}\]  
where \(\Delta x\)--is the pixel size of the detector. The evaluation of the holograms and phase reconstruction were performed using the Fourier transform method [23]. If holograms are recorded at time moments t\({}_{1}\) and t\({}_{2}\), corresponding to different object states, it is possible to calculate the phase difference after the reconstruction routine. Result intensity recorded on the camera sensor is:  
\[I(x,y)=[E_{R}\exp(-i\varphi_{R}(x,y))+E_{o}\exp(-i\varphi_{o}(x,y))]\times[E_ {R}\exp(-i\varphi_{R}(x,y))E_{o}\exp(-i\varphi_{o}(x,y))]^{*} \tag{2}\]  
where \(E_{R}\)\(uE_{O}\)--are the amplitudes of the reference and object waves, respectively. Applying double Fourier transform, followed by filtering and inverse double Fourier transform for different states of the object, allows one to obtain the phase difference function:  
\[\Delta\varphi=arctg[tg(\varphi_{1}-\varphi_{2})]=arctg\left[\frac{Im_{1} \times Re_{2}-Im_{2}\times Re_{1}}{Im_{1}\times Im_{2}+Re_{1}\times Re_{2}}\right] \tag{3}\]  
which describes the change in the state of the object [24]. Thus, for example, if an object is subjected to mechanical loading, the phase difference will be related to the value and direction of displacements of the object's surface points [25].