# Effect of Fiber Optic Plate on Centroid Locating Accuracy of Monocentric Imager
## 3 Materials and Methods
### Fiber Position Calibration  
We note that the position of a space target imaging to the FOP incident end at different times is different. The signal received by each fiber optic is also different. The final image will also be different. So we need to mark every fiber position to locate each fiber in the FOP. Fibers are arranged as shown in Figure 4. We set the position where the center of a certain optical fiber coincides with the center of a certain image sensor pixel as the original coordinate \((x_{0},y_{0})\) and observe each optical fiber center coordinate. We set the fiber radius to \(R\), the fiber core radius to \(r\), the coordinate unit spacing in the \(y_{f}\) direction to \(\sqrt{3}R\) and the coordinate unit spacing in the \(x_{f}\) direction to \(R\).  
Figure 4: Fibers arranged in dense hexagonal shapes. By establishing the rectangular coordinate \(O_{f}\)-\(x_{f}y_{f}\), each fiber can be located. \(T_{x}\) is an FOP translation period in the \(x_{f}\) direction, and \(T_{y}\) is an FOP translation period in the \(y_{f}\) direction.\(T_{x}=2R;T_{y}=2\sqrt{3}R\).  
Figure 3: The relationship between defocus degrees of imaging spots and CLESD. When the defocus degree was \(0.8(\lambda)\), the CLA was highest with a value of \(0.002\) pixels.  
Figure 4 shows that when \(x_{f}\) and \(y_{f}\) are both even or odd, that is, when \(x_{f}+y_{f}\) is an even number, \((x_{f},y_{f})\) is the fiber center coordinate. We set the coordinate of the spherical focal surface of the monocentric imager as \((x,y)\) (see Figure 1). The light at the spherical focal surface can be received by the FOP, when \(x,y\) satisfies the following formula:  
\[\left(x-(x_{0}+Rx_{f})\right)^{2}+\left(y-(y_{0}+\sqrt{3}Ry_{f})\right)^{2}<r ^{2},\text{ with }\left(x_{f}+y_{f}\right)\text{ is even,} \tag{5}\]  
Since optical fiber is regarded as the smallest pixel unit, the light intensity of the exit face of the fiber will be equal to the average of the light intensity falling on the entrance face [11]. For brevity, we make the following substitutions: \(x^{\prime}=x_{0}+Rx_{f};y^{\prime}=y_{0}+\sqrt{3}Ry_{f}\). Thus, the light intensity of the exit face of the fiber, when its center is at the point \(x^{\prime},y^{\prime}\) is given by  
\[\begin{split} I^{\prime}\big{(}x^{\prime},y^{\prime}\big{)}& =\frac{1}{\pi r^{2}}\int_{-\infty}^{+\infty}I(x,y)e\big{(}x^{ \prime}-x,y^{\prime}-y\big{)}dxdy\\ &\text{with }\left\{e\big{(}x^{\prime}-x,y^{\prime}-y\big{)} =1\text{ for }\big{(}x^{\prime}-x\big{)}^{2}+\big{(}y^{\prime}-y\big{)}^{2}<r^{2} \right.\\ &\text{ = 0 elsewhere.}\end{split} \tag{6}\]  
where \(I^{\prime}(x^{\prime},y^{\prime})\) is the light intensity of the exit face of the fiber, and \(I(x,y)\) is the light intensity on the spherical focal surface. The uniform light emitted from the FOP exit end face is received by the image sensor with the pixels arranged in a square lattice. The intensity value of the image sensor pixel with coordinate \((x^{\prime\prime},y^{\prime\prime})\) is expressed as [12]:  
\[I^{\prime\prime}(x^{\prime\prime},y^{\prime\prime})=\sum\
olimits_{x^{\prime \prime}=x^{\prime\prime}-\frac{pixel}{2}}^{x^{\prime\prime}+\frac{pixel}{2}} \sum\
olimits_{y^{\prime\prime}=y^{\prime\prime}-\frac{pixel}{2}}^{y^{\prime \prime}+\frac{pixel}{2}}I^{\prime}\big{(}x^{\prime},y^{\prime}\big{)} \tag{7}\]  
So far, all formula derivations required for PTOIS simulation of the monocentric imager with FOPRIT scheme have been given.