## 4 Experiments
### Results and Discussions
In the same manner, the \(Y^{\prime}\)-directional deviation (\(e_{Y}(x,y^{\prime})\)) was also evaluated as shown in Figure 9. The PV of the \(Y^{\prime}\)-directional pitch deviation from the measured and fitted results were evaluated to be 79.68 nm and 68.52 nm, respectively, as shown in Figure 9. It should be noted that the PV values of the central \(XY\) axes cross-sections from the \(X^{\prime}\)- and \(Y^{\prime}\)-directional deviations were almost within 10 nm, as shown in Figures 8 and 10; when considering the existence of the high frequency components in the original measurement data, the actual fitting error may be even smaller. Therefore, the fitted results could express the measured forms well with the existence of noise even though the PV value was small, which is acceptable for the precision requirement of the proposed evaluation methods. Figure 5: Measured and fitted results of the \(Z\)-directional out-of-flatness. (**a**) Measured \(Z\)-directional out-of-flatness; (**b**) \(Z\)-directional out-of-flatness fitted by a polynomial. PV: peak-to-valley. Figure 6: Central cross-sections of the \(Z\)-directional out-of-flatness shown in Figure 5. (**a**) Central cross-section along the \(X\)-direction; (**b**) Central cross-section along the \(Y\)-direction. Figure 8: Central cross-sections of the \(X^{\prime}\)-directional pitch deviation shown in Figure 4:0 (**a**) Central cross-section along the \(X\)-direction; (**b**) Central cross-section along the \(Y\)-direction. Figure 4:1 Central cross-sections of the \(Y^{\prime}\)-directional pitch deviation shown in Figure 4:2 (**a**) Central cross-section along the \(X\)-direction; (**b**) Central cross-section along the \(Y\)-direction. Figure 4:3 Measured and fitted results of the \(X^{\prime}\)-directional pitch deviation. (**a**) Measured \(X^{\prime}\)-directional pitch deviation (**b**) \(X^{\prime}\)-directional pitch deviation fitted by a polynomial. Figure 4:4 Measured and fitted results of the \(Y^{\prime}\)-directional pitch deviation. (**a**) Measured \(Y^{\prime}\)-directional pitch deviation (**b**) \(Y^{\prime}\)-directional pitch deviation fitted by a polynomial. The error maps of the \(X\)- and \(Y\)- directional pitch deviations were finally obtained based on the fitted results of \(e_{X}(x^{\prime}\),\(y)\) and \(e_{Y}(x\),\(y^{\prime})\), as shown in Figure 11a,b. The PV values of the fitted \(X\)- and \(Y\)-directional pitch deviations were evaluated to be 39.88 nm and 67.61 nm, respectively. It should be noted that the differences of the PV values in the fitted pitch deviation errors between the original \(X\)-, \(Y\)-directional results shown in Figure 11 and the tilted \(X^{\prime}\)-,\(Y^{\prime}\)-directional results shown in Figures 7b and 9b were evaluated to be 0.01 nm and 0.91 nm, respectively, which were negligibly small, since the required Littrow angle was small and the newly calculated regression coefficients were similar to the original coefficients according to Equation (30). In addition, the analyzed area has slightly shifted after changing the coordinate system in the Littrow configuration back to the original grating coordinate system. It could be readily explained by the cosine impact from the inclination angle, which would induce a shift of the analyzed area with a distance determined by the cosine value of the inclination angle. However, since the influence of the inclination angle is already considered in the regression coefficients calculation of the polynomial fitting and the uncertainty evaluation process for the new area according to Equations (30) and (31), the finally obtained PV values and the PV uncertainties will thus not be overestimated after changing the coordinate system. Table 2 shows the uncertainty budget. Since the complete model is quite complex, only the major uncertainty sources were taken into consideration, as discussed in Section 3.