# Piston Error Measurement for Segmented Telescopes Based on a Hybrid Artificial Neural Network
## 2 Piston Error Detection Method Based on Hybrid Neural Network
### Theoretical Relation between System MTF and Piston Error  
For a segmented telescope with a mask set on the exit-pupil plane and the observation target as a point source with broadband illumination, there is a determined theoretical relationship between the system \(MTF\) and the piston errors of the sub-mirrors based on Fourier optics. [PERSON] et al. have deduced the formula in detail and presented their work in paper [15]. Based on their work, we further deduced the theoretical relationship and briefly introduce the derivation process. For the segmented system shown in Figure 2, the GPF is  
\[G(\varepsilon,\eta)=A(\varepsilon,\eta)\left[circ\left(\frac{\varepsilon-b/2, \eta}{d/2}\right)\cdot e^{i\phi_{1}}+circ\left(\frac{\varepsilon+b/2,\eta}{d/ 2}\right)\cdot e^{i\phi_{2}}\right], \tag{5}\]  
where \(A(\varepsilon,\eta)\) is the binary shape function of the hexagon segment, \(circ()\) stands for circle function, the phase difference between the two segments is \(\Delta\phi=\phi_{1}-\phi_{2}=\frac{2\pi}{\lambda}2p\), \(\lambda\) is the observation wavelength, and \(p\) is the piston error between the two sub-mirrors. When the input light centered at \(\lambda_{0}\) has a broad spectrum \(\Delta\lambda\), based on Equations (3) and (4), the system \(PSF\) can be written as  
\[PSF(x,y,\lambda_{0},\Delta\lambda)=\frac{\Delta\lambda}{n}\sum_{t=1}^{n}2\left( \frac{d}{2}\right)^{2}\frac{I_{1}{}^{2}(\pi d\sqrt{x^{2}+y^{2}})}{x^{2}+y^{2} }\left[1+\cos\left(\frac{2\pi}{\lambda_{t}}2p-2\pi xb\right)\right], \tag{6}\]  
where \(I_{1}\{\}\) is first order Bessel function and \(\Delta\lambda\) is equally divided into \(n\) intervals since a differential summation approximation is used to replace the integral calculation. The complex \(OTF\) of input broadband light is the 2D Fourier transform of the \(PSF\) in Equation (6), which is shown as\[OTF(f_{x},f_{y},\lambda_{0},\Delta\lambda) =\Im\{PSF(x,y,\lambda_{0},\Delta\lambda)\}\] \[=\frac{\Delta\lambda}{n}\sum\limits_{t=1}^{n}\left[ \begin{array}{ c}2 OTF_{sub}(f_{x},f_{y})+OTF_{sub}(f_{x}+\frac{b}{\lambda_{t}f},f_{y})e^{-i\frac{2\pi}{ \lambda_{t}}2p}+\ldots\\ OTF_{sub}(f_{x}-\frac{b}{\lambda_{t}f},f_{y})e^{i\frac{2\pi}{\lambda_{t}}2p} \end{array} \right],\] (7) where \((f_{x},f_{y})\) is the spatial frequency in the \(x\) and \(y\) direction, respectively, and \(OTF_{sub}(f_{x},f_{y})\) is the \(OTF\) of a single circle aperture diffraction system given by \[OTF_{sub}(f_{x},f_{y})=\left\{ \begin{array}{c}\frac{2}{\pi}\left[\arccos \left(\frac{\rho}{2\rho_{0}}\right)-\frac{\rho}{2\rho_{0}}\sqrt{1-\left(\frac{ \rho}{2\rho_{0}}\right)^{2}}\right],\\ 0\end{array} \right.\qquad\qquad\rho\leq 2\rho_{0}\] (8) among which \(\rho=\sqrt{f_{x}^{2}+f_{y}^{2}}\) is radial coordinate on the frequency plane, \(\rho_{0}=\frac{d}{2\lambda f}\) is the system cut-off frequency, and \(f\) is focal length of the imaging lens. Performing modulus operation to \(OTF\), the \(MTF\) of the system can be obtained. Based on Equation (7), the sidelobes of \(MTF\) can be extracted out and shown as \[MTF_{sidelobe}(f_{x},f_{y},\lambda)=\frac{\Delta\lambda}{n}\Bigg{|}\sum\limits_ {t=1}^{n}\left[OTF_{sub}(f_{x}+\frac{b}{\lambda_{t}f},f_{y})e^{-i\frac{2\pi}{ \lambda_{t}}2p}\right]\Bigg{|}.\] (9) Since the value of the \(MTF\) central peak is \(1\), the peak height value of the \(MTF\) sidelobe with piston error is \[MTF_{sidelobe}=\frac{1}{nN}\Bigg{|}\sum\limits_{t=1}^{n}e^{-i\frac{2\pi}{ \lambda_{t}}2p}\Bigg{|}. \tag{10}\]  
We can see that the peak height of the \(MTF\) sidelobe is only related to the number of sub-pupils of the segmented telescope, the input wavelength, and the piston error between segments. So, the modulus of the \(MTF\) sidelobe can be easily calculated when the piston error is known. Figure 4 shows the values of the \(MTF\) sidelobes calculated from Equation (10) aimed for the two sub-mirrors segmented system described in Section 2.1.