## 2 Requirements for Overall Structure and Beam Adjustment Accuracy of the Golay3 System
### Theoretical Analyses and Numerical Calculation of the Cophasing Errors
The \(MTF\) of the system can be obtained after the piston error of the subaperture is successively loaded with 0\(\lambda\), 0.25\(\lambda\), 0.5\(\lambda\), 0.75\(\lambda\) and 1\(\lambda\), as shown in Figure 3.  
When tilt errors loaded with 0.25 \(\upmu\)rad, 0.50 \(\upmu\)rad, 0.75 \(\upmu\)rad, and 1 \(\upmu\)rad are assigned to one subaperture, the \(MTF\)s are shown in Figure 4.  
When both the piston error (0.5\(\lambda\)) and the tilt error are applied to one subaperture, Figure 5 shows the \(MTF\) changing rule. Figure 4: \(MTF\) for different tilt errors. Figure 3: \(MTF\) for different piston errors. Figure 2: The calculation flow chart. The piston error is mainly caused by the optical path difference along the exit pupil plane axis and is only related to each sub-beam. The excessive piston error reduces the system resolution to a single-aperture level. The maximum allowable piston error is about \(\lambda/10\), and it will be 55 nm if the operating wavelength is 550 nm. The tilt error is mainly caused by the rotation of the pupil surface in the \(x\)- and \(y\)-axes, which upsets the parallelism of the sub-beam wave surface. The maximum allowable value of the tilt error is 0.25 \(\upmu\)rad. In addition to the point source, the image source is also simulated and analyzed. The objective image meets the convolution relationship, and the obtained image can be expressed as [16]:  
\[d(x,y)=o(x,y)*h(x,y)+n(x,y) \tag{5}\]  
where \(d(x,y)\) is the image acquired by the camera, \(o(x,y)\) is the ideal image, \(*\) represents the convolution, \(h(x,y)\) is the point spread function of the sparse-aperture system, and \(n(x,y)\) is the noise distribution. The image without any error can be obtained by a convolution operation using an object-image resolution test pattern in MATLAB software, as shown in Figure 6[17; 18], in which we mark the spatial frequency of a group of lines. The imaging results, when the piston errors of one aperture are 0, \(\lambda/4\), \(\lambda/2\), \(\lambda 3/4\) and \(\lambda\), respectively, are shown in Figure 7.  
When the piston error is 0, the imaging result is the same as the ideal imaging result. The imaging result becomes increasingly blurred when the piston error increases from 0 to  
\[P_{all}(x,y)=P(x,y)*\sum_{i=1}^{n}\delta(x-x_{i},y-y_{i})\exp\frac{2\pi i}{ \lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\} \tag{4}\]0 _MTF_ with both piston error and tilt error. \[P_{all}(x,y)=P(x,y)*\sum_{i=1}^{n}\delta(x-x_{i},y-y_{i})\exp\frac{2\pi i}{ \lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\} \tag{4}\]1 Simulated imaging without cophasing error. \[P_{all}(x,y)=P(x,y)*\sum_{i=1}^{n}\delta(x-x_{i},y-y_{i})\exp\frac{2\pi i}{ \lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\} \tag{4}\]2 Simulated imaging with different piston errors. \(\lambda/2\). The imaging result gradually becomes clear when the piston error increases from \(\lambda/2\) to \(\lambda\), therefore, the piston error is periodic. Similarly, the imaging results when the tilt errors of one aperture are respectively \(0.2\)\(\upmu\)rad, \(0.4\)\(\upmu\)rad, \(0.6\)\(\upmu\)rad, \(0.8\)\(\upmu\)rad and \(1.0\)\(\upmu\)rad, are shown in \[P_{all}(x,y)=P(x,y)*\sum_{i=1}^{n}\delta(x-x_{i},y-y_{i})\exp\frac{2\pi i}{ \lambda}\{p_{n}+[\alpha_{n}(x-x_{i})+\beta_{n}(y-y_{i})]\} \tag{4}\]3  
It can be seen from the figures that the larger the tilt error is, the farther the image of the subaperture deviates from the initial position. According to the analysis result, cophasing error control is the key to system design. The steering control structure of the beam in this system undertakes the steering convergence of beams received by subapertures and the relative phase adjustment between the beams, so it is a key part of the design and assembly of the entire system.