The use of an array of the focal spots' displacements uniformly distributed over the beam aperture makes it possible to reconstruct the entire wavefront. During the experiments, the set of frames, representing the displacements of the coordinates of the focal spots along \(X\) and \(Y\) axes, respectively, was recorded. With a sampling rate of 2 kHz and a recording time of 10 s, the total number of the stored frames was equal to 20,000. This ten-second recording made it possible to achieve a resolution along the frequency axis of 0.1 Hz [16]. The stored coordinates of the focal spots allowed for the reconstruction of the wavefront for each frame (hartmannogram). To reconstruct the wavefront, the modal method described in [17] was applied. According to [18,19], Zernike polynomials are orthogonal in the unit circle. In our work, we used Wyant indices of Zernike polynomials [20] (Z1 and Z2--tilts, Z3--defocus, Z4 and Z5--astigmatism, Z6 and Z7--coma, Z8--spherical, etc.). A total of 24 polynomials was used for wavefront decomposition. The wavefront \(\Phi(x,y)\) of laser radiation falling on the WFS can be presented as a linear combination of polynomials:  
\[\Phi(x,y)=\sum_{i}^{m}c_{i}Z_{i}(x,y) \tag{1}\]  
where \((x,y)\)--focal spot coordinates,  
\(m\)--the number of polynomials involved in the expansion, strictly speaking, this number should be equal to infinity,  
Figure 1: Experimental setup layout. \(c_{i}\)--coefficients of Zernike polynomials,  
\(Zi(x,y)\)--the value of Zernike polynomial at the given point \((x,y)\). It is assumed that all physical coordinates are normalized to the unit circle. The focal spots' shifts measured by the sensor can also be written as:  
\[\Delta S_{x}=f\frac{\partial\Phi(x,y)}{\partial x}=\sum_{i}^{m}c_{i}X_{i}(x,y) \tag{2}\]  
\[\Delta S_{y}=f\frac{\partial\Phi(x,y)}{\partial y}=\sum_{i}^{m}c_{i}Y_{i}(x,y) \tag{3}\]  
where  
\[X_{i}=f\frac{\partial Z_{i}(x,y)}{\partial x},\ \ Y_{i}=f\frac{\partial Z_{i}(x,y)}{ \partial y} \tag{4}\]  
\(f\)--the distance between the lens array and the camera sensor of the WFS. The expansion coefficients \(c_{i}\) are calculated from the minimization of the functional:  
\[\chi^{2}=\sum_{j}^{n}\left\{\left[\Delta S_{xj}-\sum_{i}^{m}c_{i}X_{i}(x_{j}, y_{j})\right]^{2}+\left[\Delta S_{yj}-\sum_{i}^{m}c_{i}Y_{i}\big{(}x_{j},y_{j} \big{)}\right]^{2}\right\} \tag{5}\]  
where \(n\)--the focal spot number  
by solving the system of equations:  
\[\frac{\partial\chi^{2}}{\partial c_{i}}=0 \tag{6}\]  
in matrix presentation:  
\[Mc=S \tag{7}\]  
where  
\[M=X^{T}X+Y^{T}Y \tag{8}\]  
\[X=\left[ \begin{array}{ccc}X_{1}(x_{1},y_{1})&\ldots&X_{m}(x_{1},y_{1})\\ \ldots&\ldots&\ldots\\ X_{1}(x_{n},y_{n})&\ldots&X_{m}(x_{n},y_{n})\end{array} \right],\ Y=\left[ \begin{array}{ccc}Y_{1}(x_{1},y_{1})&\ldots&Y_{m}(x_{1},y_{1})\\ \ldots&\ldots&\ldots\\ Y_{1}(x_{n},y_{n})&\ldots&Y_{m}(x_{n},y_{n})\end{array} \right] \tag{9}\]  
\[c=\left[ \begin{array}{c}c_{1}\\ \ldots\\ c_{m}\end{array} \right] \tag{10}\]  
\[S=X^{T}s_{x}+Y^{T}s_{y} \tag{11}\]  
\[s_{x}=\left[ \begin{array}{c}\Delta S_{x_{1}}\\ \ldots\\ \Delta S_{x_{n}}\end{array} \right],\ s_{y}=\left[ \begin{array}{c}\Delta S_ {y_{1}}\\ \ldots\\ \Delta S_{y_{n}}\end{array} \right] \tag{12}\]  
\(X^{T}\) and \(Y^{T}\) are transposed matrices of \(X\) and \(Y\). Thus, the expansion coefficients \(c_{i}\), which give an idea of the magnitude of various types of wavefront aberrations, are calculated from  
\[c=M^{-1}S \tag{13}\]  
using the singular value decomposition of the matrix \(M\).