# Effect of Fiber Optic Plate on Centroid Locating Accuracy of Monocentric Imager
## 4 Simulation Analysis and Results
### FOP in-Plane Displacements: Rotation and Translation
#### 4.2.2 Translation  
As the FOP fiber diameter and the image sensor pixel spacing are not same in size and shape, the relative initial position is random. Since FOP is arranged periodically, the lowest CLA that the monocentric imager can withstand can be obtained by simulating all CLESDs at relative initial positions in a cycle. Figure 4 shows that optical fibers are periodically arranged in a hexagonal shape. The translation period in the \(x\) direction is \(T_{x}=2R\), and the translation period in the \(y\) direction is \(T_{y}=2\sqrt{3}R\). The centroid position of the imaging spot where FOP is shifted for one cycle is the same as the centroid position where FOP does not shift.  
In view of the two different FOP arrangements shown in Figure 7, we perform translation simulations on the two FOP schemes. We also set the image sensor pixel spacing to 10 \(\upmu\)m. The FOP is translated in the \(x\) direction with the translation step value of 1 \(\upmu\)m, and the FOP is translated in the \(y\) direction with the translation step value of \(\sqrt{3}\)\(\upmu\)m. Every time the FOP was translated by one step, we performed one PTOIS simulation in MATLAB. We could obtain the CLESD when the FOP had been translated for one period in the \(x\) and \(y\) directions relative to the image sensor. The simulation results are shown in Figures 8 and 9. We simplified the data and display them in Table 1, in order to obtain the maximum and minimum CLESD values of each FOP scheme more intuitively.  
When FOP scheme one is selected, the fiber diameter is 4 \(\upmu\)m and the fill factor is 0.7. Due to the in-plane displacement of the FOP, the lowest CLA that the monocentric imager can withstand has been reduced from 0.031 pixel to 0.038 pixel with an accuracy loss of 0.007 pixel. When FOP scheme two is selected, the fiber diameter is 6 \(\upmu\)m and the fill factor is 0.7. Due to the in-plane displacement of the FOP, the lowest CLA that the monocentric imager can withstand has been reduced from 0.037 to 0.045 pixels with an accuracy loss of 0.008 pixel. If the FOP is controlled to be arranged in the way shown in Figure 7a during processing, the lowest CLA of FOP scheme one is reduced from 0.031 to 0.034 pixels with an accuracy loss of 0.003 pixels, and the lowest CLA of FOP scheme two is reduced from 0.037 to 0.042 pixels with an accuracy loss of 0.005 pixels. Only considering FOP translation, compared with FOP scheme two, FOP scheme one can achieve a higher CLA value of 0.034 pixels.  
Figure 7: Two FOP arrangements. (**a**) FOP was rotated counterclockwise by 0, \(\pi/3\), \(2\pi/3\) or \(\pi\). (**b**) FOP was rotated counterclockwise by \(\pi/6\), \(\pi/2\) or \(5\pi/6\).  
Figure 8: The FOP was translated for one period in the \(x^{\prime}\) and \(y^{\prime}\) directions. The FOP translation period in the \(x^{\prime}\) direction was \((T_{x}=2R)\), and the FOP translation period in the \(y^{\prime}\) direction was \(\left(T_{y}=2\sqrt{3}R\right)\). The connection between the FOP translation and the CLESD was established. (**a**) FOP scheme one: \(d_{f}\) = 4 \(\upmu\)m; \(K=0.7\); angle of rotation: No rotation. (**b**) FOP scheme two: \(d_{f}\) = 6 \(\upmu\)m; \(K=0.7\); angle of rotation: No rotation.  
Figure 9: The FOP was translated for one period in the \(x^{\prime}\) and \(y^{\prime}\) directions. The FOP translation period in the \(x^{\prime}\) direction was \((T_{x}=2R)\), and the FOP translation period in the \(y^{\prime}\) direction was \(\left(T_{y}=2\sqrt{3}R\right)\). The connection between the FOP translation and the CLESD was established. (**a**) FOP scheme one: \(d_{f}\) = 4 \(\upmu\)m; \(K=0.7\); angle of rotation: \(\pi/6\). (**b**) FOP scheme two: \(d_{f}\) = 6 \(\upmu\)m; \(K=0.7\); angle of rotation: \(\pi/6\).