The Littrow angle \(\theta_{m,i}\) of the \(i\)th mode in the \(m\)th-order diffracted beam can be expressed by the following equation:  
\[P_{m,i}=\frac{m}{2 sint\theta_{(m,i)}}\times\frac{c}{\
u_{i}}=\frac{m}{2 sint \theta_{(m,i)}}\times\frac{c}{i\times\
u_{rep}+\
u_{CEO}} \tag{3}\]  
where \(\
u_{i}\) is the optical frequency of the \(i\)th mode, which can be represented by using the pulse repetition frequency \(\
u_{rep}\) and the carrier offset frequency \(\
u_{CEO}\) as \(\
u_{i}=i\cdot\
u_{rep}+\
u_{CEO}\). Since a lot of Littrow angles can be obtained in the same diffraction order, the improvement of the measurement uncertainty with the averaging effect can be expected. It should be noted that the diffracted laser beams should be observed in the optical frequency domain so that the light wavelength of each mode in the Littrow configuration can be determined. In the previous work by the authors [17], a Fabry-Perot etalon with a high free-spectral range was employed so that each mode could be distinguished in the optical frequency domain. In this paper, a major modification is made to the optical setup; instead of separating the modes with the Fabry-Perot etalon, a single-mode optical fiber was placed on the optical axis to specify the light wavelength mode corresponding to the Littrow configuration from the overlapped neighboring modes as shown in Figure 2. When a group of the diffracted beams was made incident to the optical fiber, the beams propagated through the core of the optical fiber by reflection on the boundary surface with the shelter part named clad. Due to the loss of the intensity from the reflection, the intensity spectrum consequently shows Gaussian distribution in which the light mode made parallelly incident to the optical axis shows the highest intensity. The spectrum of propagated beams within the mode field diameter (MFD), determined by the fall of the intensity to \(\mathrm{e}^{-2}\), was then detected by the optical spectrum analyzer, and the peak wavelength was determined by the weight of mass method. This modification enables the setup to fully utilize the information of the modes in the group of the diffracted beams. Figure 3a shows a three-dimensional drawing of the setup shown in Figure 1b. With the employment of a polarizing beam splitter (PBS) and a quarter-wave plate (QWP), the diffracted beams can be guided to the detector unit without going back to the laser source. The detector unit consists of an objective lens and a single-mode fiber, one side of  
Figure 2: Peak wavelength detection for measurement of the grating pitch. which is connected to an optical spectrum analyzer (OSA). In the setup, the normal of the grating is aligned to be parallel with the \(X\)-axis when the angular position of the spindle is set to \(\theta=0^{\circ}\). However, the measurement laser beam and the detector unit have angular misalignments with respect to the optical axis as shown in Figure 3b. Thus, variable \(\alpha\) is given to the angular misalignment of the measurement laser beam about the \(Z\)-axis with respect to the \(X\)-axis, and variable \(\beta\) is given to that of the optical axis of the detector unit about the \(X\)-axis with respect to the \(Z\)-axis.