# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 3 Results
### Experimental Formation of Interferograms  
Tables 3 and 4 show examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations. Since conical interferograms showed [48] a significant difference in sensitivity for high-frequency and low-frequency types, both types were used in the analysis of experimental patterns. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aberration Type \(Z_{mn}(x,y)\)** & **Linear \(I_{p=1}(x,y)\)** & **Cylindrical \(I_{p=2}(x,y)\)** \\
**Phase \(W(x,y)\)** & **Interferogram** & **Interferogram** \\ \((c_{mn}=0.5)\) & \((c_{mn}=0.1,\,c_{mn}=0.5)\) & \((c_{mn}=0.1,\,c_{mn}=0.5)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Interferogram patterns (\(\alpha=2\pi,\,\beta=0\)) for aberrations as separate Zernike functions. As can be seen from the interferograms given in Table 4, both types demonstrate significant differences in patterns (distortion of interference lines) at high levels of aberrations (\(c_{nm}~{}=~{}0.5\)) and are highly consistent with the results obtained in numerical simulation. The interferograms were obtained in the single-arm scheme (see Figure 3, with the reference beam arm blocked by the shutter). The interference of the two fields was encoded in the pattern calculation, which was supplied to the DMD modulator. This approach simplifies the registration of complex distributions and avoids the influence of vibrations. In order to confirm the physical analogue of the amplitude-phase field distribution for the single-arm scheme, the second arm of the experimental setup was engaged, and the interference distributions of the interferogram and the tilted plane wave were recorded. Table 5 shows the model and reconstructed phase distributions, the cylindrical interferograms for the angle \(\alpha=6\pi\) with different types of aberration superposition. Since a complex interference field of two beams containing amplitude and phase distributions is encoded on the DMD to form interferograms, a two-arm interferometer scheme was used to control the resulting phase distributions. A set of digital interferograms was recorded for each DMD pattern containing aberration for further averaging. Processing of the digital hologram series was performed as follows: The phase was reconstructed from the set of holograms for one DMD pattern by the local least squares method. The phase distributions were multiplied by a binary mask to ensure that noise was excluded in further processing. The phase difference between the fields recorded with and without the introduced aberration was then calculated. Next, two stages of averaging the phase difference distributions were performed to eliminate the reconstruction error associated with external vibrations. The first stage of averaging consisted of calculating the mean value for each distribution and subtracting it from the corresponding phase. The second step was to calculate the arithmetic mean of the entire set of phase distributions for one pattern. Table 5 shows the model and reconstructed phase distributions, the cylindrical interferograms for the angle \(\alpha=6\pi\) with different types of aberration superposition. The reconstructed phase distributions in Table 5 obtained from experimental interferograms with a two-arm scheme match with the simulated phase distributions with the error of the object wavefront reconstruction method and distortions induced by imperfections of the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Images of Linear Interferograms**} & \multicolumn{2}{c}{**Images of Cylindrical Interferograms**} \\ \hline \(2\pi\) & \(4\pi\) & \(6\pi\) & \(2\pi\) & \(4\pi\) & \(6\pi\) \\  & & \(0.32Z_{20}(x,y)+0.2Z_{44}(x,y)\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations. experimental scheme such as beam quality and interference fringes period instability.