## 3 Design and Development of the Optical Setup and Verification Experiment  
A schematic of the setup employed in the following experiments is shown in Figure 5a. A fiber-based mode-locked femtosecond laser with a pulse repetition frequency \(\
u_{\text{rep}}\) of 100 MHz was employed as the laser source. The laser beam from the laser source was guided into the setup by using an optical fiber. The laser beam from the edge of the fiber was collimated by a collimating lens, and was then projected onto a grating mounted on a spindle. It should be noted that a half-wave plate was placed just after the collimating lens in the optical path of the laser beam so that the polarization direction of the laser beam could be controlled. A pair of a quarter-wave plate and a polarizing beam splitter was employed as an optical isolator to prevent the diffracted beam from going back to the laser source. The diffracted laser beams, whose propagating directions were changed by the polarizing beam splitters, were captured by a detector unit composed of an objective lens and a single-mode fiber with a mode-field diameter of approximately 10.8 \(\upmu\)m. One end of the single-mode fiber was placed at the focal plane of the objective lens, while the other was connected to a commercial optical spectrum analyzer (AQ6370D, Yokogawa Co., Ltd., Tokyo, Japan) so that the captured diffracted laser beams could be analyzed in the optical frequency domain. Figure 5b shows a photograph of the setup employed in the following experiments. The detection of the angular position of the grating in the developed setup could strongly affect the pitch measurement by the proposed method. Therefore, at the beginning  
Figure 4: Influence of misalignments in the setup; (**a**) Influence of the angular misalignment of the measurement laser beam about the \(Z\)-axis (\(\alpha\)); (**b**) Influence of the angular misalignment of the measurement laser beam about the \(X\)-axis (\(\beta\)). Figure 5: Experimental setup for measurement of the pitch of a reflective-type diffraction grating: (**a**) Schematic of the setup; (**b**) A photograph of the setup. of the following experiments, the stability of the reading of the rotary encoder embedded in the spindle in the developed setup was evaluated. A variation of the rotary encoder reading was observed while the spindle was kept stationary at the positive and negative first-order Littrow configurations. Figure 6a,b show the variations of the rotary encoder reading at the angular positions \(\theta\) = 28.08\({}^{\circ}\) and \(-\)28.08\({}^{\circ}\), respectively, in a period of two hours. A sampling frequency of the encoder reading was set to be 10 Hz. Standard deviations of the rotary encoder reading were evaluated to be 0.06054 arc-second and 0.06475 arc-second for the positive and negative Littrow configurations, respectively. From these results, the spindle was confirmed to have positioning stability within the specification of the spindle (1 arc-second). Alignments of the optical setup can be carried out by using the zeroth and first-order diffracted beams. Figure 7a shows a schematic of the setup. A retroreflector, which has the ability to reflect the laser beam to its source, is placed in front of the scale grating, while a beam profiler is placed at the focal plane of the objective lens. The reflected measurement laser beam is then focused on the detector plane at a point P\({}_{\text{M}}\) in the beam profiler, and its \(Y\)-position is recorded. After that, the retroreflector is removed from the setup where the angular position of the spindle is set to \(\theta\) = 0\({}^{\circ}\); this operation makes the zeroth-order diffracted beam from the scale grating be focused on the beam profiler at a point P\({}_{0}\). Meanwhile, with the existence of the angular misalignment \(\alpha\), the focused laser beam is made to shift along the \(Y\)-direction.