## 5 Conclusions  
For the improvement of the accuracy of pitch measurement of a diffraction grating based on the laser diffraction with an optical frequency comb laser source, major modifications have been made to the optical setup and signal processing. Instead of employing a Fabry-Perot etalon for the separation of the optical modes in the spectrum of the captured first-order diffracted beams, a weight of mass method has been employed in the signal processing to find out the peak wavelength in the spectrum for the calculation of the grating pitch. In addition, a mathematical model of the pitch measurement in the proposed method with the employment of the optical frequency comb laser has been established while considering misalignments of optical components in the setup. Experiments have been carried out to compensate for the misalignments of the optical components considered in the mathematical model. In addition, the influence of the non-uniform spectrum of the mode-locked femtosecond laser source employed in the developed setup has successfully been reduced through the normalization operations of the light spectra obtained by the detector unit in the setup. Furthermore, by using the developed mathematical model, the theoretical analysis of the uncertainty in the pitch measurement by the proposed method has been carried out based on the guide to the expression of uncertainty in measurement (GUM). Through the theoretical evaluation based on GUM, an expanded uncertainty \(U\) of the pitch measurement of a diffraction grating with a nominal pitch density of/1200 mm by using the information from a specific mode has been evaluated to be 81.30 pm (\(k=2.95\%\) confidence). The experimental results were found to be within the uncertainty value estimated in the above analysis. By using \(N\) optical modes observed in experiments, an expanded uncertainty of the pitch measurement by the proposed method can be reduced to \(81.30/\sqrt{\mathrm{N}}\) pm. These results of the theoretical investigation and experiments have demonstrated the feasibility of the proposed pitch measurement with an optical frequency comb laser.  
Conceptualization, W.G. and Y.S.; methodology, D.W.S., L.Q., Y.C., Y.S. and W.G.; software, D.W.S., L.Q. and Y.S.; validation, Y.S., W.G. and H.M.; formal Analysis, D.W.S., L.Q., Y.S. and W.G.; investigation, D.W.S., L.Q., Y.C. and Y.S.; resources, Y.S. and W.G.; data curation, Y.S. and W.G; Writing--Original Draft Preparation, D.W.S. and Y.S.; Writing--Review and Editing, W.G., Y.S. and E.M.; Visualization, W.G. and Y.S.; Supervision, W.G.; Project Administration, W.G.; Funding Acquisition, W.G., Y.S. and H.M. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the Japan Society for the Promotion of Science (JSPS) 20H00211.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
[PERSON] would like to thank the Graduate Program for Integration of Mechanical Systems (GP-Mech) of Tohoku University, Japan.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript or in the decision to publish the results.