## 4 Uncertainty Analysis
From Equation (10), the sensitivity coefficients \(c_{Y\text{retro}}\), \(c_{0}\) and \(c_{Y}\) can be derived as follows:  
\[c_{Yretro}=\frac{\partial\alpha}{\partial Y_{retro}}=\frac{\partial}{\partial Y _{retro}}\bigg{[}\text{arctan}\bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}\bigg{]} =\frac{1}{F}\cdot\frac{1}{1+\bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}^{2}} \tag{12}\]  
\[c_{0}=\frac{\partial\alpha}{\partial Y_{0}}=\frac{\partial}{\partial Y_{0}} \bigg{[}\text{arctan}\bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}\bigg{]}=\frac{- 1}{F}\cdot\frac{1}{1+\bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}^{2}} \tag{13}\]  
\[c_{F}=\frac{\partial\alpha}{\partial F}=\frac{\partial}{\partial F}\bigg{[} \text{arctan}\bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}\bigg{]}=\frac{1}{1+ \bigg{(}\frac{Y_{retro}-Y_{0}}{F}\bigg{)}^{2}}\cdot\frac{Y_{0}-Y_{retro}}{F^{2}} \tag{14}\]  
By applying mean values of \(Y_{\text{retro}}\) (\(-186.160\)\(\upmu\)m), \(Y_{0}\) (\(-187.303\)\(\upmu\)m) and a nominal value of \(F\) of 31.1 mm, \(c_{Y\text{retro}}\), \(c_{0}\) and \(c_{Y}\) were evaluated as shown in Table 1. It should be noted that the misalignment of the image sensor with respect to the objective lens was considered to be within \(\pm 0.5\) mm. By evaluating the contribution of each uncertainty source, \(u(\alpha)\) was evaluated to be \(3.862\times 10^{-3}\) arcsecond. (b) Angular misalignment \(\beta\) of the detector unit about the \(X\)-axis with respect to the \(Z\)-axis (\(u(\beta)\))  
The contribution of \(\beta\) was then estimated. As shown in Figure 11b, \(\Sigma\,|\,P_{\text{pos}\_i}-P_{\text{neg}\_i}\,|\,\) corresponding to the change of \(\beta\) in a step of \(0.0001^{\circ}\) was calculated, then \(\beta\) which minimize \(\Sigma\,|\,P_{\text{pos}\_i}-P_{\text{neg}\_i}\,|\,\) was selected by taking the extremum of the data curve obtained by sixth-order polynomial approximation fitting. However, as shown in Figure 11c, \(P_{\text{pos}\_i}\) and \(P_{\text{neg}\_i}\) did not perfectly match by this fitting process, and \(\,|\,P_{\text{pos}\_i}-P_{\text{neg}\_i}\,|\,\) at each angular position of the spindle could contribute to the uncertainty \(u(\beta)\) of evaluating \(\beta\) by polynomial approximation fitting. The deviation of \(\beta_{i}\), which was obtained by the \(i\)th positive and negative Littrow angles was evaluated as shown in Figure 13. A standard deviation of \(\,|\,\beta\)-\(\beta_{1}\,|\,\) was evaluated to be 1.875 arc-second. Since, 100 sets of \(P_{\text{pos}\_i}\) and \(P_{\text{neg}\_i}\,|\,\) were employed, by applying a divisor of \(\sqrt{100}\), \(u(\beta)\) was evaluated to be 0.1875 arc-second. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Sources of Uncertainty** & **Symbol** & **Type** & **Uncertainty** & **Probability** & **Divisor** & **Standard** & **Sensitivity** & **Coefficient** & \(|\,\)\(e\,|\)\(1\,\)-\(\,|\)\(\,|\) \\ \hline Repeatability of the & & & & & & & \(u\) & \(c\) & \\ detection of \(Y_{\text{retro}}\) & & & & & & & & & \\ Repeatability of the & & & & & & & & & \\ detection of \(Y_{0}\) & & & & & & & & & \\ image sensor with respect & & & & & & & & \\ to the objective lens & & & & & & & & & \\ \hline \multicolumn{8}{c}{\(u(\alpha)\)} & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Uncertainty of the angular misalignment \(\alpha\). 3. Uncertainty of the detection of the angle \(\theta_{i}\) of the \(i\)th-mode positive or negative first-order diffracted beam (\(u(\theta_{i})\))  
In the proposed method, the \(i\)th-mode first-order Littrow angle \(\theta_{i}\) was obtained from the readings of the optical rotary encoder embedded to the precision air-bearing spindle. Therefore, the resolution, stability and uncertainty of the reading of the optical rotary encoder could contribute to the uncertainty \(u(\theta_{i})\) of detecting the \(i\)th-mode first-order Littrow angle of the positive or negative first-order diffracted beam.