*4.1. Scintillations of Amplitude and Emergence of Singular Points in the Wavefront of a Beam with a Phase Set by Zernike Polynomials*

The first block of results included in paragraph 3 of the current paper illustrates the fact that a singular wavefront can be obtained if the initial phase profile of a beam is set to be Zernike polynomials. To prove this fact, the beam with a given phase passed some distance in a non-aberrating medium, its amplitude and phase profiles were calculated in the plane of observations (the optical layout of the corresponding numerical experiment is shown in Figure 1), and singular points were registered by three different algorithms.

Figures 2–7 show that points of zero intensity appear in such a beam, the localization of optical vortices is illustrated in Figures 8 and 9. If a phase profile is formed by a sum of polynomials with equal coefficients, amplitude distribution is influenced mainly by the last polynomial entering the sum. This can be seen from the comparison of Figures 3 and 6. In the first case, the phase was set by trefoil, and in the second by the sum including polynomials from the first to trefoil. All polynomial coefficients were the same and equal to one. In both variants (Figures 3c and 6c) amplitude distributions have similar forms. The same conclusions can be drawn if we compare Figures 4c and 7c, corresponding amplitude profiles were obtained by setting the phase with coma and by sum including polynomials from the first to coma.

In the observation plane, we also survey the appearance of singular points in the wavefront of the beam. The application of two registration algorithms gives approximately the same results, i.e., close numbers of vortices and similar forms of their distribution (Figures 8a,b and 9a,b). A much smaller number of singular points were localized by the interferometric algorithm (Figures 8c and 9c) which can be explained by the low resolution by the applied technique. Nevertheless, in all situations the vortices were detected, which proves the development of singular points in a beam with an initially smooth phase profile.

Data illustrating the dependence of dislocation number on the size of the registration region, number of polynomials setting the phase profile, and the beam propagation distance are presented in Figures 10–12. As is shown in Figure 10, vortices do not appear if the phase is prescribed with the use of the first five polynomials. The singular points started to appear from the polynomial number six and higher. Comparing the form of curves in Figure 10, we can deduce that number of vortices increases with the increase in the region radius where we look for singular points.

The dependence of dislocation quantity on the distance passed by the beam is shown in Figure 11. The phase was set by polynomials. Characteristic features of all curves are oscillations and general decreases.

Approximately the same traits we observe if the phase is given by the screen simulating atmospheric turbulence (Figure 12). As in the previous graph, the curves oscillate, pass maximums, and go to zero.

#### *4.2. Approximation of the Phase Screen Formed by Zernike Polynomials*

In the previous part, we demonstrated that optical vortices appear in a wavefront if a beam passes the phase screen formed by Zernike polynomials. Let us consider how precisely we can approximate such a screen by a series of polynomials. Of course, the actual problem is the approximation of a screen simulating turbulent distortions, but to assess the precision of the method we simplified the project. The obtained results are presented in Tables 1–5. The accuracy of the method was characterized by the deviation of the given polynomial coefficients from coefficients calculated by the least-mean-square method [18]. We also compared the parameters of two beams, the phase of the first was set by the screen, and the phase of the second was obtained as the result of approximation. Both beams passed the same distance.

If the phase was set as a sum of nine polynomials and the basis of approximation was larger (Table 1, 12 polynomials in the basis) or the same (Table 2, 9 polynomials in the basis), high precision can be achieved even with small-scale (256 × 256 nodes) computational grids. In this case, values of all calculated coefficients coincided with values of given coefficients. Astigmatism is an exception, but in this case, the main influence on amplitude exerts not the magnitude but the difference between coefficients of two astigmatisms, and this difference was calculated correctly.

The precision of phase reconstruction decreases with the increase in polynomial number in the sum forming the screen (Tables 3 and 4). Unsatisfying results were obtained with the application of small-dimension grids (Table 3). A large difference was observed between values of coefficients (columns 1 and 2 of Table 3), as well as between parameters of the two beams (columns 8–13).

The accuracy of approximation can be increased with an increase in the grid dimensions (Table 4, the grid with 2048 × 2048 nodes), but notwithstanding the small difference between coefficients registered in this case (Table 4, columns 1–7) and coincidence of such integral parameters as effective radius and shift of gravity center along coordinate axes (columns 11–13), the phase of two beams differs by 22% and amplitude by 14% (columns 9 and 10).

Least-mean-square techniques also give incorrect results when the number of polynomials in the basis of approximation is smaller than in the phase screen. Corresponding data are given in Table 5 and in Figure 13. To reduce the influence of the last polynomial in the sequence forming the screen we decreased the magnitude of its coefficient from 1.0 to 0.2. However, even such lessening of its influence did not cause a coincidence in the results, the calculated parameters of the two beams were dissimilar (Table 5, columns 8–13), and the difference between their amplitude profiles can be seen by the naked eye (Figure 13).

## *4.3. Approximation of a Phase Screen Simulating Atmospheric Distortions*

In this problem, the same model was used as in the previous part, i.e., the phase of the beam was set by a screen, then this screen is approximated by a sum of polynomials, and obtained distribution was employed as an initial phase profile of a beam with Gaussian amplitude profile. After propagation on some distance parameters, two beams were compared. Specifically, we can compare the difference in the initial phase of two beams, their amplitude distributions in the plane of observations, PIBs, shifts of gravitation centers, and so on. Corresponding parameters were put in Tables 6–8 and presented in Figures 14–16.

Comparing images in Figure 14a,c we can see that the phase profiles of two beams are not unlike, though the positions of extremums (the brightest and darkest regions) in two pictures are slightly shifted. As a result, in the plane of observations, we register the same magnitudes of PIBs (Table 6, column 3) and the same shifts of gravity centers (columns 4 and 5). The largest difference is observed in the magnitudes of effective radii (column 6–8), consequently, amplitude distributions of two beams do not also coincide (Figure 14b,c).

With the increase in turbulent intensity characterized by Fried's coherence length, the main features of the problem remain the same, the phase profiles of the two beams are similar, the effective radii and amplitude distributions are different (Table 7 and Figure 15).

The increase in the solution accuracy cannot be achieved by increasing the grid dimensions up to 2048 × 2048 nodes (Figure 16 and Table 8). In this case, the difference between amplitude distributions is approximately 68% (Table 8, column 2), effective radii along axes OX and OY of the second beam were calculated erratically (columns 6–8 of the Table), and some errors appeared in the calculation of gravity center shifts (column 5). In general, even with the large inner scale, we could not obtain the exact approximation of the phase screen

simulating the influence of atmospheric turbulence. If the inner scale decreases, the quality of approximation decreases even further.
