## 4 Uncertainty Analysis
Regarding the contribution of each uncertainty source, standard uncertainties \(u_{pi}\) of the pitch \(P_{i}\) obtained by the \(i\)th positive or negative first-order Littrow angle can be estimated by the following equations:  
\[u_{Pi}=\sqrt{(c(\alpha)\cdot u(\alpha))^{2}+(c(\beta)\cdot u(\beta))^{2}+(c( \theta_{i})\cdot u(\theta_{i}))^{2}+(c(\lambda_{i})\cdot u(\lambda_{i}))^{2}+( c(n_{\text{air}})\cdot u(n_{\text{air}}))^{2}} \tag{15}\]  
where \(c(\alpha)\), \(c(\beta)\), \(c(\theta_{i})\), \(c(\lambda_{i})\) and \(c(n_{\text{air}})\) are the sensitivity coefficients that can be described by the following equations:  
\[c(\alpha)=\frac{\lambda_{\text{i}}\sin\left(\frac{\alpha-\beta}{2}\right)}{4n _{air}\cos^{2}\left(\frac{\alpha-\beta}{2}\right)\sin\left(\theta_{i}+\frac{ \beta}{2}\right)} \tag{16}\]  
\[c(\beta)=\frac{-\lambda_{\text{i}}\left[\cos\left(\theta_{i}+\frac{\beta}{2} \right)\cos\left(\frac{\alpha-\beta}{2}\right)+\sin\left(\theta_{i}+\frac{ \beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\right]}{4n_{air}\left[ \cos\left(\frac{\alpha-\beta}{2}\right)\sin\left(\theta_{i}+\frac{\beta}{2} \right)\right]^{2}} \tag{17}\]  
\[c(\theta_{i})=\frac{-\lambda_{\text{i}}\cos\left(\theta_{i}+\frac{\beta}{2} \right)}{2n_{air}\cos\left(\frac{\alpha-\beta}{2}\right)\sin^{2}\left(\theta_ {i}+\frac{\beta}{2}\right)} \tag{18}\]  
\[c(\lambda_{i})=\frac{1}{2n_{air}\cos\left(\frac{\alpha-\beta}{2}\right)\sin \left(\theta_{i}+\frac{\beta}{2}\right)} \tag{19}\]  
\[c(n_{air})=\frac{-\lambda_{i}}{2n_{air}^{2}\cos\left(\frac{\alpha-\beta}{2} \right)\sin\left(\theta_{i}+\frac{\beta}{2}\right)} \tag{20}\]  
In the above equations, \(\alpha\) is the angular misalignment of the measurement laser beam about the \(Z\)-axis, \(\beta\) is the angular misalignment value of the measurement laser beam about the \(X\)-axis, \(\theta_{i}\) is the average Littrow angle, \(\lambda_{i}\) is the central wavelength, and \(n_{\text{air}}\) is the refractive index in air. Table 3 summarizes the contribution of each uncertainty source. Based on Equation (15), \(u_{Pi}\) was evaluated to be 40.65 pm. An expanded uncertainty \(U\) of the pitch was thus evaluated to be 81.30 pm (\(k=2\), 95% confidence). From the results shown in Figure 12b, the pitch values were found to distribute in a range from 833.404 \(\pm\) 0.0372 nm; the experimental results were found to be within the uncertainty value estimated in the above analysis. It should be noted that \(U\) estimated in the above analysis is the expanded uncertainty of the pitch by a specific optical mode in the optical frequency comb laser. By using \(N\) optical modes (\(N=100\) in Figure 12) observed in experiments, an expanded uncertainty of the pitch measurement by the proposed method becomes 81.30/\(\sqrt{N}\) pm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sources of** & **Standard Uncertainty** & **Sensitivity Coefficient** & \(lc\) & \(lc\) \\
**Uncertainty** & **Symbol** & \(\boldsymbol{u}\) & \(\boldsymbol{c}\) & \(lc\) \\ \hline \(\alpha\) & \(u(\alpha)\) & \(3.862\times 10^{-3}\) arc-second & \(9.02\times 10^{-7}\) nm/arc-second & \(3.484\times 10^{-6}\) \\ \(\beta\) & \(u(\beta)\) & \(0.187\) arc-second & \(-0.217\) nm/arc-second & \(40.58\) \\ \(\theta_{i}\) & \(u(\theta_{i})\) & \(5.077\times 10^{-3}\) arc-second & \(-0.434\) nm/arc-second & \(2.203\) \\ \(\lambda_{i}\) & \(u(\lambda_{i})\) & \(7.100\times 10^{-4}\) nm & \(1.062\) & \(0.754\) \\ \(n_{\text{air}}\) & \(u(n_{\text{air}})\) & \(5.0\times 10^{-8}\) & \(-833.263\) nm & \(4.166\times 10^{-2}\) \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c} \hline \hline
**Combined uncertainty \(u_{Pi}\)** & **40.65 pm** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Combined uncertainty.