# Piston Error Measurement for Segmented Telescopes with an Artificial Neural Network
## 2 Theoretical Relation between the Piston Error and Amplitudes of MTF Sidelobes
### Establishing the Theoretical Relationship Based on Fourier Optics Principle
According to the OTF calculation method of the diffraction limited system, the expression for the OTF of the segmented system can be given by:  
\[OTF(f_{x},f_{y})=2 OTF_{sub}(f_{x},f_{y})+OTF_{sub}(f_{x}+\frac{b}{\lambda f},f_ {y})e^{-i\frac{2\pi}{\lambda}2p}+OTF_{sub}(f_{x}-\frac{b}{\lambda f},f_{y})e^{ i\frac{2\pi}{\lambda}2p}, \tag{6}\]  
Figure 1: The schematic diagram of the telescope composed of two hexagonal segments with sparse circles configuration. where \(OTF_{sub}(f_{x},f_{y})\) is the OTF of a single circle aperture diffraction system, which is given by:  
\[OTF_{sub}(f_{x},f_{y})=\left\{ \begin{array}{cc}\frac{2}{\pi}\left[\arccos\left( \frac{\rho}{\rho_{0}}\right)-\frac{\rho}{\rho_{0}}\sqrt{1-\left(\frac{\rho}{ \rho_{0}}\right)^{2}}\right],&\rho\leq 2\rho_{0}\\ 0&,others\end{array} \right. \tag{7}\]  
where \(\rho=\sqrt{f_{x}^{2}+f_{y}^{2}}\) is the radial coordinate on the frequency plane, \(\rho_{0}=\frac{d}{2\lambda f}\) is the system cut-off frequency, \(f\) is focal length of the imaging lens. Equation (6) shows that the OTF of the segmented telescope composed of two hexagonal segments with mask includes three parts: the central part and two sidelobes. The central part's spatial frequency is \((0,0)\) and its value is the maximum modules of the sidelobe without piston error multiplied by the number of sub-pupils. Since the OTF is normalized, the value of the central part is always 1, so the maximum height of the sidelobe part without piston error is \(1/N\), where \(N\) is the number of sub-pupils, and \(N=2\) is the telescope composed of two segments. The spatial frequencies of the two sidelobes are \((\pm b/\lambda f,0)\) respectively, and they are symmetrically distributed about the central part. Based on the Fourier optics, we can get the MTF of the system by doing a modulus operation for Equation (6), which is shown as:  
\[MTF(f_{x},f_{y})=\left|OTF(f_{x},f_{y})\right| \tag{8}\] \[=\left|2 OTF_{sub}(f_{x},f_{y})\right|+\left|OTF_{sub}(f_{x}+\frac {b}{\lambda f},f_{y})e^{-i\frac{2\pi}{\lambda}2p}\right|+\left|OTF_{sub}(f_{x }-\frac{b}{\lambda f},f_{y})e^{i\frac{2\pi}{\lambda}2p}\right|.\]  
As we can see, only the value of the MTF sidelobe is modulated by the piston error. We take one sidelobe out marked as \(MTF_{sidelobe}\), which is shown as:  
\[MTF_{sidelobe}(f_{x},f_{y})=\left|OTF_{sub}(f_{x}+\frac{b}{\lambda f^{\prime} },f_{y})e^{-i\frac{2\pi}{\lambda}2p}\right|. \tag{9}\]  
Then the peak height value of the sidelobe with piston error marked as \(MTF_{sidelobe-peak}\) is obtained by:  
\[MTF_{sidelobe-peak}=\frac{1}{N}\left|e^{-i\frac{2\pi}{\lambda}2p}\right|. \tag{10}\]  
From Equation (10), we can see that the peak height of the MTF sidelobe is only related to the number of sub-pupils, input wavelength and piston error. The effective detection range of the piston error is limited for a single wavelength due to the \(2\pi\) ambiguity. Therefore, we need a broadband input light value to expand the piston error detection range. For a broadband input light centered at \(\lambda_{0}\) with the bandwidth \(\Delta\lambda\), the PSF of the system is the sum of all PSFs at different monochromatic wavelengths which is given by:  
\[PSF_{broad}(u,v,\lambda)=\int_{\lambda_{0}-\frac{\Delta\lambda}{2}}^{ \lambda_{0}+\frac{\Delta\lambda}{2}}PSF(u,v,\lambda)S(\lambda)d\lambda \tag{11}\] \[=\int_{\lambda_{0}-\frac{\Delta\lambda}{2}}^{\lambda_{0}+\frac{ \Delta\lambda}{2}}2\left(\frac{d}{2}\right)^{2}\frac{\lambda_{1}^{2}(\pi d\sqrt {u^{2}+v^{2}})}{u^{2}+v^{2}}\big{[}1+\cos\left(\frac{2\pi}{\lambda}2p-2\pi ub \right)\big{]}S(\lambda)d\lambda,\]  
where \(S(\lambda)\) is PSF weight of different wavelengths, assuming \(S(\lambda)=1\). Since the integral is difficult to calculate, a differential summation approximation is used.