## 3 Design and Development of the Optical Setup and Verification Experiment
Figure 11b shows the sum of \(\left|P_{\text{pos},i}-P_{\text{neg},j}\right|\) at each \(\beta\). By performing fitting by sixth-order function approximation and finding the extremum, \(\beta\) that minimizes \(\Sigma\left|P_{\text{pos},i}-P_{\text{neg},i}\right|\) was evaluated to be \(0.001212^{\circ}\) (\(4.363\) arc-seconds). Figure 11c shows \(P_{\text{pos},i}\) and \(P_{\text{neg},i}\) calculated based on Equations (4) and (5) under the condition of \(\beta\) = \(4.363\) arc-seconds. As can be seen in the figure, a good agreement can be found between \(P_{\text{pos},i}\) and \(P_{\text{neg},i}\) after the compensation of the influence of \(\beta\). Figure 10: Spectra obtained at each angular position of the scale grating: (**a**) with the positive first-order diffracted beams; (**b**) with the negative first-order diffracted beams. It should be noted that the optical frequency comb laser employed in this paper had a non-uniform spectrum, as can be seen in Figure 9. The non-uniform spectrum of the light source could affect the calculation of the peak wavelength based on the weight-of-mass method, resulting in the degradation of the uncertainty in pitch measurement. Compensation of the influence of the non-uniform spectrum of the mode-locked femtosecond laser beam employed in the setup was thus carried out by using the observed spectrum of the zeroth-order diffracted beam shown in Figure 5:0 In the compensation process, the spectrum of the first-order diffracted beam \(I_{\pm 1}(\lambda_{i})\) obtained at each angular position of the spindle was divided by the spectrum of the zeroth-order diffracted beam \(I_{0}(\lambda_{i})\) to obtain the normalized spectrum \(I_{\mathrm{norm}}(\lambda_{i})\). Figure 12a,b shows the grating period \(P_{\mathrm{neg\_i}}\) calculated by using the spectra \(I_{1}(\lambda_{i})\) and \(I_{\mathrm{norm}}(\lambda_{i})\) of 100 optical modes observed in experiments, respectively. By the compensation process, the deviation of the calculated grating period \(P_{\mathrm{neg\_i}}\) obtained from the spectra in the wavelength range of 770 nm to 800 nm was reduced from 94.76 pm to 74.34 pm. This result demonstrated the effectiveness of the compensation process for the reduction of the uncertainty of pitch measurement. It should be noted that a slight difference can still be observed between \(P_{\mathrm{pos\_i}}\) and \(P_{\mathrm{neg\_i}}\); The possible root causes contain the influence of the chromatic aberration of the optical components in the setup, as well as the instability of the spectrum of the mode-locked femtosecond laser beam, and further detailed investigation is required for the better understanding of the results. Figure 5:1 Grating pitch calculated from the peak wavelength obtained at each angular position of the scale grating: (**a**) Grating pitches \(P_{\mathrm{pos\_i}}\) and \(P_{\mathrm{neg\_i}}\) calculated by using the positive and negative first-order diffracted beams without the compensation of \(\beta\) (\(\beta\) = 0); (**b**) Variation of \(\Sigma\,\)\(1P_{\mathrm{pos\_i}}-P_{\mathrm{neg\_i}}\)\({}^{1}\) with the change in \(\beta\); (**c**) \(P_{\mathrm{pos\_i}}\) and \(P_{\mathrm{neg\_i}}\) after the compensation of \(\beta\) (\(\beta\) = 4.363 arc-seconds).