# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 2 Materials and Methods
### Optical Setup  
Our work employed the experimental setup (Figure 3) based on a Mach-Zehnder interferometer with the digital micromirror device (DMD) and the Fourier-plane filtered 4f-system [49; 50] in one of its optical paths. The reference beam in the second optical path of the setup was utilized to monitor the phase distribution of the optical field formed by the DMD and 4f-system. Each of the optical paths of the interferometer was blocked by an opaque screen (shutter) according to the conditions of the experiment. The DMD device allows only binary modulation. In our experimental setup, the Texas Instruments Light Crafter DLP6500 FYE binary modulator (1920 \(\times\) 1080 pixels with a size of 7.56 \(\upmu\)m) operated by the DLPC900 controller was used. Each micromirror is located on a corresponding cell of the CMOS matrix and can be in one of two positions, +\(\beta\) or -\(\beta\), which correspond to logical 1 and 0, respectively. Despite the binary limitation, it is possible to achieve independent amplitude-phase modulation by conjugating the DMD with an input plane of the 4f-system with spatial Fourier filtering. We used the method of obtaining independent amplitude and phase modulation from binary Lee holograms [55] to generate the patterns. This independent wavefront modulation of the amplitude and phase greatly expands the possibilities of the experiment. The standard two-arm interferometer scheme is a straightforward option, where the object arm contains a DMD for shaping the wavefront with predefined aberrations, and the reference beam containing a flat wavefront is directed to the matrix photodetector by the beam splitter. Another option is to use a single-arm  
Figure 3: Schemes experimental setup: LS is laser source diode 532 nm, BE is beam expander, M\({}_{1-2}\) are mirrors, BS\({}_{1-2}\) are beam splitters 30R/70T and 50R/50T, L\({}_{1-2}\) are lenses with focal distance 200 and 100 mm, SF is the spatial filter, Shutter is the moveable screen, CMOS is the matrix of the camera Mind Vision MV-UB130 GM with 1280 \(\times\) 960 pixels with the size of 3.75 \(\upmu\)m.  
scheme for forming interferograms with predefined aberrations and the reference beam, which simplifies the system and increases experiment accuracy and vibration resistance. In the single-arm scheme, two beams with different amplitude and phase configurations can be implemented simultaneously using the same pattern on the DMD, in which the interference field of these beams is encoded. If one of these beams has a different wavefront tilt from the other, they can be spatially separated in the Fourier plane of the 4f-system. Figure 4 shows the Fourier spectra for the instances where the field is generated with the quadrafoil aberration (\(c_{44}=1\)) (a) and its interference field with the plane tilted wave \(\alpha=10\pi\) (b) and \(\alpha=50\pi\) (c). The separation of diffraction suborders (Figure 4b,c) proves the formation of the interference pattern in a single-arm scheme is physically identical to the classical two-arm interference. The presence of wavefronts with quadrafoil aberration and linear tilt in the diffraction suborders was confirmed in the two-arm scheme with sequential blocking of each of them. In our experimental setup, interferograms for the training of the neural network were formed only in the objective beam (the single-arm scheme), and the reference beam was used to validate the correct phase distribution, as shown in Section 3.2. The amplitude-phase distributions with a given type and level of aberrations were encoded into binary patterns as the sum of the complex fields of the wavefront aberration and the reference beam: \(E_{w}(x,y)=E_{B}(x,y)\). In order to experimentally confirm the coincidence between the interferograms synthesized in the single-arm scheme and the classical two-arm scheme with further quality estimation, the following experiment was performed.