# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 5 Conclusions  
A study was made on the sensitivity of interferograms formed using structured reference beams with a flat and cylindrical wavefront. The maximum sensitivity of cylindrical interferograms is achieved by detecting medium and strong aberrations. In this case, the cylindrical interferogram is more sensitive than the linear one by 0.03. The average sensitivity for detecting strong aberrations, regardless of the tilt angle \(\alpha\), is 0.025.  
The increased sensitivity and information content of the proposed interferograms compared to classical (linear) ones based on a flat wavefront is confirmed by a decrease in the RMSE values and an expanded range of reference beam parameters, which allows more accurate tuning of the wavefront sensor to a specific range of the level of wave aberration.  
The use of reference beams with cylindrical wavefronts is proposed to improve the recognition of aberrations from interferograms using convolutional neural networks. The sensitivity of interferograms (the change in the aberrated interferogram relative to the reference one) when using a cylindrical reference beam increases by at least 10% compared to a flat reference beam for radially asymmetric types of aberrations, and the mean absolute error of aberration recognition decreases from 0.0068 to 0.0047.  
A comparative analysis between the use of a flat and cylindrical reference beam for wave aberration recognition from interferograms using neural networks showed that the average absolute error is reduced by more than 30%.  
It should be noted that a cylindrical interferogram with a reference beam tilt angle \(\alpha\) in the range from 2\(\pi\) to 6\(\pi\) turned out to be 15% more efficient than a high-frequency conical interferogram. This conclusion was made on the basis of data on the mean absolute recognition error [48] of conical interferograms.  
Considering that optical interferometry is already widely used in many applications, such as precision measurements, astrophysics, seismology, quantum informatics, biomedical imaging, and digital contouring and deformation analysis in mechanics, an extension of the method with the help of various reference beams in combination with intelligent data analysis will preserve the advantages of the wavefront sensor and eliminate some of the disadvantages, depending on the specific application task.  
Conceptualization, P.A.K. and S.N.K.; methodology, P.A.K. and S.N.K.; software, A.P.D., P.A.K. and A.V.C.; validation, A.V.C. and A.P.D.; formal analysis, P.A.K. and S.N.K.; investigation, A.V.C. and A.O.G.; resources, S.N.K. and N.V.P.; data curation, A.P.D., P.A.K., A.O.G. and A.V.C.; writing--original draft preparation, P.A.K., N.V.P. and S.N.K.; writing--review and editing, P.A.K., N.V.P. and S.N.K.; visualization, P.A.K., A.V.C. and A.P.D.; supervision, N.V.P. and S.N.K.; project administration, N.V.P. and S.N.K.; funding acquisition, N.V.P. and S.N.K. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Russian Science Foundation under Grant No. 22-12-00041 (numerical calculations), by the Ministry of Science and Higher Education of the Russian Federation under the FSRC \"Crystallography and Photonics\" of the Russian Academy of Sciences (state task No. 007-GZ/Ch3363/26) (theoretical research), and by the Ministry of Education and Science of the Russian Federation (Passport No. 2019-0903) (experimental research).  
Figure 16: Average values of information content parameter of cylindrical and linear interferograms to wavefront aberrations.