## 3 Design and Development of the Optical Setup and Verification Experiment
According to the principle of laser autocollimation [22], the \(Y\)-directional spot displacement \(\Delta d\) on the beam profiler due to \(\alpha\) can be expressed by the following equation:  
\[\Delta d=F\tan\alpha \tag{6}\]  
Since the focal length \(F\) of the objective lens was known to be 31 mm, \(\alpha\) can be evaluated by detecting \(\Delta d\) with the beam profiler. Figure 6: Reading of the rotary encoder embedded to the air-bearing spindle: (**a**) Positive first-order Littrow configuration; (**b**) Negative First-order Littrow configuration. Figure 7: Alignment of the measurement laser beam by using a retroreflector and a beam profiler: (**a**) Setup with a retroreflector; (**b**,**c**) Procedure of the alignment. Figure 8 shows the variations of the \(Y\)-positions of the measurement laser beam and the zeroth-order diffracted beam focused on the beam profiler in a period of 20 s. As the target grating, a holographic-type grating with a line density of 1200/mm was employed. From the mean values, \(\Delta d\) was evaluated to be 1.143 \(\upmu\)m, and \(\alpha\) was thus evaluated to be 7.579 arc-seconds based on Equation (6). After the verification of the angular misalignment \(\alpha\), the alignment of the single-mode fiber in the fiber detector unit was carried out while the angular position of the spindle was kept stationary at \(\theta\) = 0\({}^{\circ}\). Figure 9 shows the spectrum of the zeroth-order diffracted beam captured by the detector unit. The alignment of the single-mode fiber was carried out so that the total laser power to be detected by the detector unit became maximum while observing the spectrum of the captured laser beam. As can be seen in the figure, a non-uniform spectrum of the optical frequency comb laser was successfully observed. Meanwhile, the detector unit had a small angular misalignment \(\beta\) about the \(X\)-axis even after the alignment; this can be evaluated in experiments by using the positive and negative first-order diffracted beams. Figure 10a,b show the variations of the spectra of the positive and negative first-order diffracted beams captured by the same detector unit when rotating the scale grating. In the figures, the spectra obtained at the angular position \(1\theta\)\(|\) ranging from 27.48\({}^{\circ}\) to 27.52\({}^{\circ}\) in a step of 0.01\({}^{\circ}\) are plotted. As can be seen in the figure, the spectra of the negative first-order diffracted beams were found to shift  
Figure 8: \(Y\)-position of the focused laser beam on the beam profiler during the alignment. Figure 9: The optical spectrum of the mode-locked femtosecond laser employed in this paper. toward the shorter light wavelength; this was mainly due to the influence of \(\beta\), regarding Equations (4) and (5). The grating pitch \(P\) can be evaluated by detecting the peak wavelength of the spectrum at each angular position of the spindle \(\theta\). In this paper, a weight-of-mass method based on the following equation was employed to detect the peak wavelength \(\lambda_{\text{center}}\) in a spectrum:  
\[\lambda_{\text{Center}}=\frac{\sum\left(I_{\lambda_{i}}\times\lambda_{i} \right)}{\sum I_{\lambda_{i}}} \tag{7}\]  
where \(\lambda_{i}\) is the \(i\)th sampled wavelength, and \(I_{\lambda i}\) is the light intensity of the light wavelength component \(\lambda_{i}\). Figure 11a shows the grating pitch calculated based on Equations (4) and (5) by the series of \(\lambda_{\text{center}}\) obtained at different angular positions \(\theta\) of the spindle. It should be noted that the parameter \(\beta\) was set to be \(0^{\circ}\). As can be seen in the figure, a discrepancy mainly due to the misalignment \(\beta\) was found between the grating pitches \(P_{\text{pos},i}\) and \(P_{\text{neg},i}\) obtained from the positive and negative first-order diffracted beams, respectively. To compensate for the influence of \(\beta\), calculations based on Equations (4) and (5) were repeated by changing \(\beta\) in a step of \(0.0001^{\circ}\).