# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 3 Results  
### Numerical Simulation  
We consider the formation of a linear and cylindrical (cubic) interferogram with the reference beam tilt angle \(\alpha\) divisible by \(2\pi\), thus matching the frequency parameter of the interferogram with the value of the phase shift of the reference beam transmission function. Figures 1 and 2 show that the parameter \(\alpha\) becomes consistent with the phase shift at a multiplicity of \(2\pi\). Thus, the phase passes values from 0 to \(2\pi\)\(\alpha/\pi\) times, and, accordingly, the ideal interferogram (\(Z_{00}\)) in the resulting plane will be represented as \(\alpha/\pi\) full interference fringes. The interferograms for each of the aberrations up to the 4 th order were calculated. Table 1 shows the patterns of the considered interferograms at \(\alpha=2\pi\) for aberrations in the form of separate Zernike functions. As can be seen from the interferograms shown in Table 1, the cylindrical interferogram is less sensitive to weak aberrations, however, both types demonstrate significant differences in patterns (distortion of interference lines) at high levels of aberrations (\(c_{nm}~{}=~{}0.5\)), i.e., they can be successfully used to analyze aberrations in a situation where expansion in terms of Zernike functions leads to noticeable recognition errors [61]. Consider a cubic interferogram using the example of a coma-type aberration \(Z_{31}\) with an aberration value \(c_{nm}~{}=~{}0.5\) and different values of the tilt angle parameter of the reference cylindrical beam \(\alpha\) (Figure 7). It should be noted that the use of a reference cylindrical beam, in contrast to a flat beam in interferograms, leads to a complication of the intensity distribution pattern with an increase in the alpha \(\alpha\). This consideration is detailed in the Section 4. It is assumed that interferograms using a structured reference beam corresponding to a cylindrical (cubic) carrier make it possible to form a more complex pattern of intensity distribution (especially in the peripheral part) in the resulting plane of the optical system under study and make it possible to more accurately detect and interpret aberrations in analyzed wavefronts, including by training neural networks. For each type of interferogram, a data set was prepared corresponding to the superposition of two aberrations. The value of \(c_{nm}\) in the range from 0 to 1 and the type for the superposition of two aberrations were chosen randomly. Examples of generated linear and cylindrical interferograms for various types and levels of aberrations with a variable parameter \(\alpha\) are presented in Table 2.  
Table 2 shows that the tilt angle of the flat reference beam \(\alpha\) significantly affects the structure of the linear interferogram pattern. However, the image of a cylindrical interferogram retains its structure, with noticeable changes only in the peripheral part. This \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Images of Linear Interferograms**} & \multicolumn{3}{c}{**Images of Cylindrical Interferograms**} \\ \hline \(2\pi\) & \(4\pi\) & \(6\pi\) & \(2\pi\) & \(4\pi\) & \(6\pi\) \\  & & \(0.3Z_{20}(x,y)+0.2Z_{44}(x,y)\) & & \\  & & \(0.4Z_{22}(x,y)+0.1Z_{33}(x,y)\) & & \\  & & \(0.3Z_{42}(x,y)+0.2Z_{20}(x,y)\) & & \\ \end{tabular}
\end{table} 
Table 2: Examples of generated linear and cylindrical interferograms for various types and levels of aberrations with a variable parameter \(\alpha\). Figure 7: Patterns of cubic interferograms depending on different tilt angles \(\alpha\) ((**a**) is \(2\pi\); (**b**) is \(4\pi\); (**c**) is \(6\pi\); (**d**) is \(8\pi\); (**e**) is \(10\pi\)) of the reference cylindrical beam for coma-type aberration \(Z_{31}\) with aberration value \(c_{nm}~{}=~{}0.5\). property of the proposed cylindrical interferogram gives an advantage in deciphering and unambiguously interpreting aberrations in the analyzed wavefront.