# 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
The \(\Delta\lambda\) is divided into \(n\) intervals equally, then the PSF can be rewritten as:  
\[PSF_{broad}(u,v,\lambda)=\frac{\Delta\lambda}{n}\sum_{t=1}^{n}2\left(\frac{d} {2}\right)^{2}\frac{\lambda_{1}^{2}(\pi d\sqrt{u^{2}+v^{2}})}{u^{2}+v^{2}} \bigg{[}1+\cos\left(\frac{2\pi}{\lambda_{t}}2p-2\pi ub\right)\bigg{]}. \tag{12}\]  
By performing Fourier transform for the \(PSF_{broad}\), the corresponding OTF of the system in broadband light marked as \(OTF_{broad}\) is given by:\[OTF_{broad}(f_{x},f_{y},\lambda)=\frac{\Delta\lambda}{n}\sum_{t=1}^{n}\Bigg{[} \begin{array}{c}2 OTF_{sub}(f_{x},f_{y})+OTF_{sub}(f_{x}+\frac{b}{\lambda_{t}f },f_{y})e^{-i\frac{2\pi}{\lambda_{t}}2p}+\dots\\ OTF_{sub}(f_{x}-\frac{b}{\lambda_{t}f},f_{y})e^{i\frac{2\pi}{\lambda_{t}}2p} \end{array} \Bigg{]}. \tag{13}\]  
As we know, performing a modulus operation for the OTF, we can get the MTF of the system and the piston error is only related to the MTF sidelobes, thus the MTF sidelobe marked as \(MTF_{sidelobe-broad}\) is extracted out from the modulus of the \(OTF_{broad}\) and shown as:  
\[MTF_{sidelobe-broad}(f_{x},f_{y},\lambda)=\frac{\Delta\lambda}{n}\Bigg{|} \sum_{t=1}^{n}\Bigg{[}OTF_{sub}(f_{x}+\frac{b}{\lambda_{t}f},f_{y})e^{-i\frac {2\pi}{\lambda_{t}}2p}\Bigg{]}\Bigg{|}. \tag{14}\]  
In fact, for different wavelengths, the position coordinates of the MTF sidelobes are different. However, the coordinate difference introduced by different wavelengths is very small when the bandwidth is much smaller than the central wavelength [26], so the position coordinate \((f_{x}+\frac{b}{\lambda_{t}f},f_{y})\) with different wavelengths can be approximated as \((f_{x}+\frac{b}{\lambda_{t}f},f_{y})\). The effect of different wavelengths is mainly reflected in the e-index part. Thus Equation (14) can be rewritten as:  
\[MTF_{sidelobe-broad}(f_{x},f_{y},\lambda)=\frac{\Delta\lambda}{n}\bigg{|}OTF_{ sub}(f_{x}+\frac{b}{\lambda_{0}f},f_{y})\bigg{|}\Bigg{|}\sum_{t=1}^{n}e^{-i\frac{2 \pi}{\lambda_{t}}2p}\Bigg{|}. \tag{15}\]  
For the broadband input light value, when the \(OTF_{broad}\) is normalized, the value of the central peak is 1. From Equation (13) we can see that the peak height of the MTF sidelobe without piston error, namely \(\big{|}OTF_{sub}(f_{x},f_{y})\big{|}\), is equal to \(\frac{1}{N\Delta\lambda}\) (\(N=2\) for Equation (13)), thus the peak height value of the MTF sidelobe with piston error is  
\[MTF_{sidelobe-broad-peak}=\frac{1}{nN}\Bigg{|}\sum_{t=1}^{n}e^{-i\frac{2\pi}{ \lambda_{t}}2p}\Bigg{|}. \tag{16}\]  
Based on the above derivation, we obtained a more universally applicable and precise theoretical conclusion shown in Equation (16). We can see that the peak height of the MTF sidelobe is only related to the number of sub-pupils of the segmented telescope, input wavelength and piston error between segments based on the coherence measurement of a star image when using broadband input light. So the amplitudes of the MTF sidelobe can be easily calculated when the piston error is known. However, on the other hand, due to the complexity and high nonlinearity of Equation (16), the piston error cannot be directly solved when the peak height of the MTF sidelobe is measured.