# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 2 Materials and Methods
### Optical Setup
The interference fields of the plane tilted wave and complex wave containing the quadrafoil aberration synthesized in the single-arm scheme as well as the interference fields of the synthesized complex wave containing the quadrafoil aberration and the reference plane wave in the two-arm scheme were recorded. The local least squares method was applied to demodulate the interference fringes [56, 57]. Figure 5 demonstrates the target amplitude and phase distributions encoded in the DMD pattern, the interferograms for single-arm and two-arm schemes, and the corresponding reconstructed phases. The reconstructed phase distribution quality for the low carrier frequency interferograms (\(\alpha=10\pi\) and \(\alpha=14\pi\)) is lower than for the high-frequency case (\(\alpha=50\pi\)). This is due to the limitation of the reconstruction method, the main condition for it is that the object wave has smaller local variations than the reference wave. Therefore, in some cases, with a small angle of inclination of the reference wavefront to the optical axis of the interferogram, there may be a deterioration in the quality of the object wavefront reconstruction. Nevertheless, the low carrier frequency does not have a negative impact on the ability of the neural network to recognize aberrations. Figure 4: Beam separation principle in Fourier plane of 4f-system. The amplitude modulus distribution in logarithmic scale for a single quadrafoil aberration pattern with \(E_{B}=0\) (**a**) and its interference field with the tilted plane wave \(\alpha=10\pi\) (**b**) and \(\alpha=50\pi\) (**c**). Selectable diffraction orders and suborders are highlighted by the red rectangle. The reconstructed phase distributions in Figure 5 have less smooth phase variation in the center of the distributions compared to the target phase distribution. This effect is caused by the limitations on quantization levels induced by the Lee holography method and amplitude-phase modulation with DMD [50]. Root mean square error (RMSE) was used as a quality metric for reconstructed phase distributions. The table under Figure 5 shows RMSE values for the phase distributions obtained for single-arm and two-arm schemes with respect to the target phase distribution. The error for interferograms with \(\alpha=10\pi\) and \(\alpha=14\pi\) is higher than for interferograms with \(\alpha=50\pi\) due to the difficulty of demodulating such interference fringes. With the single-arm scheme, there was a decrease in RMSE of 0.12 rad in the case of a high-frequency interferogram and by 0.08 rad in the case of a low-frequency interferogram compared to the results obtained with the two-arm scheme. It can be explained by additional factors affecting the quality of reconstruction in the two-arm scheme, such as the quality of the reference wavefront, induced by imperfections of optical elements in the reference arm, and external vibrations, which cause changes in the angle between the reference and object waves. The quality of the phase distributions is higher due to the elimination of additional artifacts introduced by the second arm of the interferometer. In summary, the quality of the reconstructed object wavefront from interferograms in our scheme is affected by: (1) the quality of the wavefront; (2) the modulation error of the target distribution; and (3) the reconstruction error of the object wavefront. In the case of a two-arm scheme, the quality of the reference wavefront and vibrations, which can cause dynamic changes in the angle between the reference and object waves during registration, is also influenced. In addition, the formation of interference fields with low carrier frequency and their further demodulation is simpler from the experimental point of view in the single-arm scheme. Therefore, we experimentally verified the capability of the single-arm scheme to generate interference fields used to train a neural network for aberration recognition.