## 2 Requirements for Overall Structure and Beam Adjustment Accuracy of the Golay3 System
### Theoretical Analyses and Numerical Calculation of the Cophasing Errors  
Cophasing errors are mainly piston errors and tilt errors [14], which will cause a serious decline in the image quality. To control the cophasing errors of the sparse-aperture imaging system, we must specifically analyze the influence of the errors. _MTF_ is a good method to specify the image quality of an optical system and we can start from the pupil function. The total pupil of the sparse-aperture system is a combination of multiple sub-pupils. If the pupil function is expressed with \(P(x,y)\), when there is no cophasing error, the total pupil can be expressed as:  
\[P_{all}(x,y)=\sum_{i=1}^{n}P_{i}(x-x_{i},y-y_{i})=P(x,y)*\sum_{i=1}^{n}\delta (x-x_{i},y-y_{i}) \tag{1}\]  
where \(P(x,y)=circ(2\sqrt{x^{2}+y^{2}})d^{-1}\) is the entrance pupil, \(d\) is the pupil diameter, \(circ(\cdot)\) is the circular domain function, \((x_{i},y_{i})\) is the central coordinate of the subaperture, \(n\) is the total number of subapertures, \({}^{*}\) represents the convolution operation, \(P_{all}(x,y)\) is the total pupil, \(P_{i}(x-x_{i},y-y_{i})\) is a pupil that contains position information, \(\delta(x-x_{i},y-y_{i})\) is the position function of the pupil, and \((x_{i},y_{i})\) is the central coordinates of the \(i\)th sub-pupil. According to the Fourier transform, the relationship between the point spread function (_PSF_), the complex amplitude \(A(\xi,\eta)\) and pupil function \(P(x,y)\) is described as follows:  
\[PSF(x,y)=\left|A\left(\xi,\eta\right)\right|^{2}=\left|F\left\{P(x,y)*\sum_{i= 1}^{n}\delta(x-x_{i},y-y_{i})\right\}\right|^{2} \tag{2}\]  
The modulation transfer function (_MTF_) of the sparse-aperture system can be derived from the _PSF_:  
\[MTF(\xi,\eta)=\left|F\left\{PSF\left(x,y\right)\right\}\right|=MTF_{sub}+\frac {MTF_{sub}}{n}*\sum_{i}\sum_{j}\delta\left(\xi-\frac{x_{i}-x_{j}}{\lambda f},\eta-\frac{y_{i}-y_{j}}{\lambda f}\right) \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline
**Item** & **Parameters** \\ \hline Subaperture size & 200 mm \\ Equivalent aperture & 570 mm \\ Focal length & 6000 mm \\ Field of view angle & 0.2\({}^{\circ}\) \\ FSM mirror & 40 mm \\ Point source experiment & Three-beam interference \\ Image source experiment & The piston error \(\leq\) 55 nm, The tilt error \(\leq\) 0.25 \(\upmu\)rad \\ \hline \end{tabular}
\end{table} 
Table 1: Main system parameters and experimental requirements. When the system contains both piston and tilt errors, the pupil function can be expressed as:  
\[P_{all}(x,y)=P(x,y)*\sum_{i=1}^{n}\delta(x-x_{i},y-y_{i})\exp\frac{2\pi i}{ \lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\} \tag{4}\]  
where \(\exp\frac{2\pi i}{\lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\}\) represents cophasing errors and \(p_{n}\) is the piston error of the \(n\)th subaperture relative to the reference subaperture. \(\alpha_{n}\) and \(\beta_{n}\) are the slope of the \(n\)th subaperture along the \(X\)-axis and \(Y\)-axis of the ideal plane, respectively. Using the above relationship, the \(MTF\) of the system can be calculated through a numerical calculation, as shown in Figure 2.  
For the Golay3 three-aperture system, the change in the \(MTF\) of one subaperture is assigned different cophasing errors, which can be simulated in MATLAB [15]. In numerical calculations, the subaperture size is \(d=200\) mm and the working wavelength is \(\lambda=535\) nm. The coordinate positions \((x,y)\) of the three subapertures can be calculated by the designed filling factor of 0.369.