# High-Precision Manufacture and Alignment of Image Slicer Based on Thin Glass Bonding
## 4 Error Analysis
### Eccentricity Error
#### 4.1.3 Stack Error  
Stacking error is introduced during the slice and image slicer sticking processes (Section 3, processes II and IV). The accuracy of the double stacking process affects the final image slicer accuracy. By analyzing the stacking error, we can improve the sensitivity of the error analysis and ensure better manufacturing through accuracy allocation. The errors of the stack are divided into translation error and rotation error. We define the ideal position \((X_{n},Y_{n},Z_{n})\) of each slice at sphere fabrication as the origin. A shift in the XYZ direction directly affects the XYZ eccentricity, whose corresponding error values are \(\Delta X_{n}\), \(\Delta Y_{n}\), \(\Delta Z_{n}\). The combination of the rotation error caused by the thickness of the adhesive layer complicates XYZ eccentricity. In the following discussion, the rotation error is divided into pitch, yaw, and roll errors which are analyzed separately. The errors in pitch, yaw, and roll affect the errors of the YZ, XZ, and XY eccentricities, respectively. The effect of the YZ eccentricity error, caused by the pitch angle, can be analyzed by axial translation and rotation. The pitch error is divided into two situations: slice working surface contact, shown in Figure 10a, and slice back surface contact, shown in Figure 10b. The contact pitch errors for both cases are shown in Figure 10c,d. In Figure 10c, \(nH\) represents the \(n\)th channel of the image slicer. The pitch error can be analyzed by translation and rotation. The behavior of the channel slice with pitch error can be obtained by rotating the ideal channel slice around the \(X\)-axis on the sphere center. The distance translation between the rotational and actual positions is the translation error in the XZ direction. Using the same method, we can obtain the translation error for the slice back surface contact, as shown in Figure 10d. The auxiliary angle \(\theta_{\rm f8}\) is given by:  
\[\theta_{\rm f8}=\sin^{-1}\frac{nH}{R} \tag{1}\]  
The angle error \(\Delta A_{8}\) of the pitch results in the eccentricity errors \(\Delta Y_{8}\) and \(\Delta Z_{8}\) at the Y and Z direction. For the slice working surface contact (Side 1 contact, Side 2 separate), the Y eccentricity error \(\Delta Y_{8-1}\) is given by:  
\[\Delta Y_{8-1}=R\sin(\Delta A_{8-1}+\theta_{\rm f8})-nH\approx R\Delta A_{8-1} \tag{2}\]  
The Z eccentricity error \(\Delta Z_{8-1}\) should satisfy  
\[\Delta Z_{8-1}=R(\cos(\Delta A_{8-1}+\theta_{\rm f8})-\cos(\theta_{\rm f8})) \ll\Delta Y_{8-1} \tag{3}\]  
For the slice back surface contact (Side 2 contact, Side 1 separate), the error value can be obtained by translating the above situation. The Y eccentricity error \(\Delta Y_{8-2}\) is given by:  
\[\Delta Y_{8-2}=R\sin(\Delta A_{8-2}+\theta_{\rm f8})+W\sin\Delta A_{8-2}-nH \approx R\Delta A_{8-2} \tag{4}\]  
The Z eccentricity error \(\Delta Z_{8-2}\) satisfies  
\[\Delta Z_{8-2}=R(\cos(\Delta A_{8-2}+\theta_{\rm f8})-\cos(\theta_{\rm f8})) +W(1-\cos\Delta A_{8-2})\ll\Delta Y_{8-2} \tag{5}\]  
The effect of the XZ eccentricity error, caused by the yaw angle, can be analyzed by axial translation and rotation. An abridged general view of the yaw error is given in Figure 11a. The error analysis process is illustrated in Figure 11b. The slicer rotates from the ideal position to the rotational position around the Y-axis. The sphere center remains unchanged. The XZ direction displacement of the slicer from the rotational position to the actual position constitutes the eccentricity error in the XZ direction. The auxiliary angle \(\theta_{\rm f9}\) used to analyze this error satisfies  
\[\theta_{\rm f9}=\tan^{-1}\frac{|D|+\frac{L}{2}}{\sqrt{R^{2}-(nH)^{2}+W}}+ \Delta A_{9} \tag{6}\]Figure 11: Yaw error of stack in the top view: (**a**) A schematic diagram of the yaw error. (**b**) Schematic diagram of the yaw error analysis. The slicer rotates from ideal position to rotational position around \(Z\)-axis and then translates to translation position.