# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 2 Materials and Methods  
### Theoretical Foundations  
Consider the Zernike circle polynomials, which are the complete set of orthogonal functions on a circle of unit radius [51]:  
\[Z_{nm}(r,\varphi)\ =\ \sqrt{\frac{n+1}{\pi v_{0}^{2}}}R_{n}^{m}(r)\left\{ \begin{matrix} \cos(m\varphi)\\ \sin(m\varphi)\end{matrix} \right\}, \tag{1}\]  
where \(R_{n}^{m}(r)\) is the Zernike radial polynomials, \(m\) and \(n\) are integers,  
\[R_{n}^{m}(r)\ =\ \sum_{s\ =\ 0}^{(n-m)/2}\frac{(-1)^{s}(n-s)!}{s!((n+m)/2-s)!(( n+m)/2-s)!}\left(\frac{r}{r_{0}}\right)^{n-2s}, \tag{2}\]  
\(A_{n}\) is the normalizing factor:  
\[A_{n}\ =\ \sqrt{\frac{n+1}{\pi}}. \tag{3}\]  
Wavefront aberrations encountered in optical systems are usually described in terms of Zernike functions as follows:  
\[E_{W}(r,\varphi)\ =\ \exp[iW(r,\varphi)], \tag{4}\]  
\[W(r,\varphi)\ =\ 2\pi\sum_{n\ =\ 0}^{n_{\rm max}}\sum_{m\ =\ 0}^{n}c_{nm}Z_{nm}(r,\varphi), \tag{5}\]  
where \(c_{nm}\) is the weight of the wavefront aberration,  
The interference of two fields is registered by the intensity of the following distribution:  
\[I(x,y)\ =\ \left|E_{W}(x,y)+E_{B}(x,y)\right|^{2} \tag{6}\]where \(E_{W}(x,y)\) is analyzed field, e.g., aberrated wavefront, \(E_{B}(x,y)\) is the reference beam. As a reference beam, as a rule, some standard wavefronts are used (for example, flat, spherical, or vortex):  
\[E_{B}(x,y)\ =\ \exp\big{[}iB_{p}(x,y)\big{]} \tag{7}\]  
In this paper, we consider two types of reference beams: a flat tilted beam (Figure 1) \(B_{p\ =\ 1}(x,y)\ =\ \alpha x+\beta y\) (\(p\) is the reference beam type; \(\alpha\), \(\beta\) are parameters that determine the beam tilt angle), and a cylindrical beam (Figure 2) \(B_{p\ =\ 2}(x,y)\ =\ \alpha x^{3}\) (\(\alpha\) is the parameter that determines the angle of inclination of the cylindrical front to the optical axis). The choice of the cubic beam type is related to its wide use in recent years for wavefront coding in order to increase the depth of focus [51; 52; 53; 54]. Thus, in the case under consideration, the intensity of interferograms (6) is proportional to the following expression:  
\[I_{p}(x,y)\cong 1+\cos\big{[}W(x,y)-B_{p}(x,y)\big{]} \tag{8}\]  
Furthermore, the frequency of the interferogram fringes will mainly depend on the parameters of the reference beam, and the complexity of the pattern being formed is determined both by the type of reference beam and by the aberration of the wavefront. In order to quantitatively estimate the sensitivity of recognition of the types of interferograms under consideration, we introduce the parameter \(S\), which corresponds to the value of the root-mean-square error (RMSE) of the p-type interferogram formed by an aberrated wavefront \(I_{p}(x,y)\), from the interferogram corresponding to the absence of aberrations \(I_{0p}(x,y)\):  
\[S_{p}\ =\ \sqrt{\iint\big{[}I_{p}(x,y)-I_{0p}(x,y)\big{]}^{2}dxdy/\iint I_{0p} ^{2}(x,y)dxdy} \tag{9}\]  
Increasing the deviation for one type of interferogram compared to another should lead to an increase in recognition efficiency. This assumption is confirmed by the more stable recognition of interferograms with a conical reference wave compared to a flat reference wave [48]. For clarity, the value \(\Delta\) is introduced as the sensitivity of interferogram recognition by machine learning methods:  
\[\Delta\ =\ (S_{2}-S_{1})/c_{nm} \tag{10}\]  
corresponding to the difference in RMSE values of cylindrical (\(p\) = 2) and linear (\(p\) = 1) interferograms and average sensitivity:  
\[\overline{\Delta}\ =\ (\overline{S}_{2}-\overline{S}_{1})/c_{nm} \tag{11}\]  
\[A_{n}\ =\ \sqrt{\frac{n+1}{\pi}}. \tag{3}\]0 The transmission function of a flat reference beam (\(p\) = 1) for a linear interferogram depending on different beam angles \(\alpha\): (**a**) is \(2\pi\); (**b**) is \(4\pi\); (**c**) is \(6\pi\); (**d**) is \(8\pi\); (**e**) is \(10\pi\).