## 2 Theory and Derivation
### Numerical Modeling of the Coupled-MTF
The second type of coupling error case is shown in Figure 5, where the coupling error between the image elements of the hexagonally arranged fiber-optic imaging bundle and the area array CCD causes a single CCD image element to receive the output optical signal from four adjacent fiber-optic image elements. As above, the light-receiving area of the adjacent four fibers in the hexagonal fiber imaging bundle has the following relationship:  
\[ \begin{array}{c}S_{i,j}=\frac{1}{2}(R-\Delta i)\left[\left(\frac{\sqrt{3}}{2 }R-\Delta j\right)+\sqrt{r^{2}-(R-\Delta i)^{2}}\right]+\\ \frac{1}{2}\left(\frac{\sqrt{3}}{2}R-\Delta j\right)\left[\left(R-\Delta i \right)+\sqrt{r^{2}-\left(\frac{\sqrt{3}}{2}R-\Delta j\right)^{2}}\right]+\\ \frac{r^{2}}{2}\left(\frac{3}{2}\pi-\arccos\frac{\frac{\sqrt{3}}{2}R-\Delta j} {r}-\arccos\frac{R-\Delta i}{r}\right)\end{array}  \tag{8}\]  
\[S_{i,j+1}=\sqrt{r^{2}-\left(\frac{\sqrt{3}}{2}R-\Delta j\right)^{2}}\left( \frac{\sqrt{3}}{2}R-\Delta j\right)+r^{2}\left(\pi-\arccos\frac{\frac{\sqrt{3 }}{2}R-\Delta j}{r}\right)-S_{i,j} \tag{9}\]  
\[S_{i+1,j+1}=\sqrt{r^{2}-\Delta i^{2}}\times\Delta i\times r^{2}\times\left( \pi-\arccos\frac{\Delta i}{r}\right)-S_{i,j} \tag{10}\]  
\[S_{i+1,j}=\pi r^{2}-S_{i,j}-S_{i,j+1}-S_{i+1,j+1} \tag{11}\]  
For the case of the second type of coupling error, the expression of the output signal intensity of the first row and column of the image element in the area array CCD is as follows:  
\[I_{coupling,i,j}=\frac{S_{i,j}}{\pi r^{2}}I_{out,i,j}+\frac{S_{i,j+1}}{\pi r^{2 }}I_{out,i,j+1}+\frac{S_{i+1,j}}{\pi r^{2}}I_{out,i+1,j}+\frac{S_{i+1,j+1}}{ \pi r^{2}}I_{out,i+1,j+1} \tag{12}\]  
According to the definition of \(MTF\), when the light intensity of the input signal is cosine-distributed, the \(MTF\) of the system is equal to the ratio of the light intensity modulation of the outgoing signal to the light intensity modulation of the incoming signal, as shown in Equation (13). That is, the response characteristics of the two-stage coupled discrete system for the input signal with cosine distribution of light intensity are reflected by the ratio of modulation. \[MTF(f,\delta)=\frac{C_{o}(f,\delta)}{C_{i}(f,\delta)} \tag{13}\]  
According to Equation (1), the optical intensity modulation system of the incident signal is \(C_{i}(f,\delta)\), \(C_{o}(f,\delta)\), which is the optical intensity modulation system of the outgoing signal, and according to (9) and (14), the two-dimensional discrete sampling cases with different coupling errors, the optical intensity distribution of the outgoing optical signal can be obtained. Then, according to the definition of the modulation system, the optical intensity modulation system of the outgoing optical signal can be obtained, and finally, according to (2), (3), (9), and (14), the response function of the coupled discrete sampling system, composed of a hexagonally arranged fiber-optic imaging bundle and an area array CCD for a signal with a cosine distribution of light intensity, can be obtained, i.e., the coupled-MTF of the system. \[MTF_{coupling}(f,n,\delta)=\frac{\frac{1}{N}\sum\limits_{i=1}^{N}I_{coupling,i, j,\text{max}}(f,n,\delta)-\frac{1}{M}\sum\limits_{i=1}^{M}I_{coupling,i,j,\text{min}}(f,n, \delta)}{\frac{1}{N}\sum\limits_{i=1}^{N}I_{coupling,i,j,\text{max}}(f,n, \delta)+\frac{1}{M}\sum\limits_{i=1}^{M}I_{coupling,i,j,\text{min}}(f,n, \delta)} \tag{14}\]in which \(n\) is the number of fibers in the imaging beam, \(N\) is the number of image elements with the output signal greater than the average value, \(M\) is the number of image elements with the output signal less than the average value, and \(I_{coupling,i,j,\max}(f,n,\delta)\) and \(I_{coupling,i,j,\min}(f,n,\delta)\) are the maximum and minimum values of the output signal intensity of the first \(i\) row and the first \(j\) column of the image elements, respectively. The MTF of the front telescope and coupling objective is set to a constant value, and the optical emphasis of the input signal \(C_{i}(f,\delta)\) is set to 1.