# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 1 Introduction  
The task of measuring wavefront distortions is often encountered in optics, for example, in the design of ground-based astronomical telescopes, augmented reality systems, optical communication systems, industrial laser technology, medicine, etc. [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. Almost always, the measurement of wavefront distortions is performed in order to compensate them, in particular, by means of adaptive or active optics [11; 12; 13; 14]. The wavefront sensor is one of the main elements of the adaptive laser radiation correction system. Its task is to measure the aberrations of the wavefront and transmit the results of these measurements to the processing device. The main causes of wavefront aberrations are atmospheric turbulence [15], non-ideal shapes of the optical elements of the system, errors in system adjustment, etc. Today, there is a wide variety of wavefront sensors, including Shack-Hartman [16; 17], pyramidal [18; 19], fiber [20; 21; 22], and other sensors. One of the types of correlation sensors [23] are multiorder diffractive optical elements matched with the basis of Zernike polynomials [24; 25; 26]. Interferometry is one of the most accurate techniques for measuring wavefront aberrations [27]. However, it has its well-known limitations, such as the complexity of decoding interferograms and the sensitivity of the measuring equipment to vibrations. Traditionally, a flat wavefront has been used as the reference beam in interferometric sensors, as it allows for a simple algorithm based on the Fourier transform to be used for phase reconstruction [28; 29]. Another area closely related to interferometry, where fringe demodulation techniques are also widely used, is fringe projection. Without using the interference principle for data recording, but at the same time actively using often identical data processing algorithms, these methods provide reliable information about the shape of objects. One of the current trends in further research aimed at improving these techniques is the optimization of the reference wave structure. Thus, it has recently been theoretically shown [30] that the optimal reference wave structure can be achieved by using circular-arc-shapedfringes centered at the epipole, providing the best phase sensitivities over the entire fringe pattern. In addition, it has been shown that the use of circular fringes in fringe projection profilometry is favorable since their central location serves as a good benchmark for shape reconstruction [31; 32; 33; 34]. In addition, hexagonal stripes [35], gray-level coded patterns [36; 37], and others are used. Returning to interferometry, it should be noted that in solving a number of problems, not only real but also virtual reference waves can be used [38; 39]. Recent advancements in data mining and neural networks have enabled the extraction and recognition of the wavefront from both the intensity distribution [40; 41; 42; 43; 44; 45] and real [47] and emulated [48] interference patterns. The current manuscript is a continuation of previous work [48], where neural networks were used to recognize the wavefront from conical interferograms. In that study, we utilized a system based on a digital micromirror device (DMD) and a technique for independent amplitude and phase modulation [49; 50] to simulate the desired interference pattern with aberrations. In this work, two reference beams with a flat and cylindrical wavefront are considered to assess the sensitivity of the proposed interferograms. The use of reference beams with cylindrical wavefronts in the formation of interferograms is studied in order to improve the recognition of aberrations using a convolutional neural network. Notably, the better the interferogram reflects the features of various aberrations, the more efficiently the recognition algorithm will work. In addition, it is necessary that the patterns corresponding to different types and levels of aberrations differ markedly. This possibility can be implemented in interference methods by choosing the type of reference beam.