## 4 Experiments
### Results and Discussions  
The form errors of the tested grating were first fitted by the set of \(n\)-degree bivariate polynomials, as proposed in Section 3. The highest degree \(n\) of the \(x\) and \(y\) coordinates in the respective model was set to be identical, which was 5 for the fitting of the \(Z\)-directional out-of-flatness error map and was 4 for the fitting of the \(X\) and \(Y\)-directional pitch deviations error maps for the sake of clarity. The regression surface may be too sharp or wiggly in the higher degree case. In addition, the higher degree polynomials will have problems with numerical accuracy. Therefore, the highest degree of the \(n\)-degree bivariate polynomials for each fitting model was selected not to exceed 5. Before the least square approximation, the \(x\) and \(y\) coordinates were normalized by the \(Z\)-score method [23] for a better fitting performance. In addition, the calculated variances from Equations (11) and (12) were used as the weights in the weighted least squared fitting process for fitting the corresponding error forms to make the approximation more precise. The fitting performance characterized by the proposed criterion in Section 3 was summarized in Table 1. As can be seen in the table, the values of \(R^{2}\) and \(\chi_{v}\) of each model indicate that the set of polynomials well captured the error forms in the three directions. The small root-mean-square error (RMSE) values also show that the regression surface closely fits the measured form errors without the loss of precision. In the following, the comprehensive comparisons of the calibration results and polynomials fitting results of each directional form error are obtained. An area of 20 mm \(\times\) 20 mm over the scale grating located near the center of FOV was extracted for the evaluations. Figure 5a shows the out-of-flatness of the planar scale grating that was obtained in the experiment. The PV value was evaluated to be 85.80 nm. Figure 5b shows the polynomials fitting result of the measured out-of-flatness, as shown in Figure 5a. The PV value was evaluated to be 75.16 nm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Form Error Type** & \(\mathbf{R^{2}}\) & \(\boldsymbol{\chi_{v}}\) & **RMSE** & **Polynomials Degree** \\ \hline \(e_{z}(x,y)\) & 0.9844 & 0.9463 & 2.1599 & 5 \\ \(e_{x}(x^{\prime},y)\) & 0.9320 & 0.2905 & 2.7534 & 4 \\ \(e_{Y}(x,y)\) & 0.8608 & 0.1522 & 4.5063 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting performance of the polynomial models for the form errors of the grating. RMSE: root-mean-square error. Figure 4: Photographs of measurement of the wavefronts of the diffracted beams. (**a**) 0 th-order diffracted beam; (**b**) \(X\)-directional +1 st-order diffracted beam. To take a closer look at the fitting detail, the central cross-sections of the two results were extracted and compared with each other. Figure 6a,b show the \(X\)- and \(Y\)-directional central cross-sections of the out-of-flatness shown in Figure 5. The differences in the PV values between the measured result and the fitted results of the \(X\)- and \(Y\)-directional cross-sections were evaluated to be 13.71 nm and 6.64 nm, respectively. The \(X^{\prime}\)-directional pitch deviation was also evaluated. Figure 7 shows the measured \(X^{\prime}\)-directional deviation (\(e_{X}(x^{\prime}\),\(y)\)) obtained by analyzing the phase outputs of the positive and negative fist-order diffracted beams that were measured by the commercial Fizeau interferometer. Figure 8 shows the \(X\)- and \(Y\)-directional central cross-sections of the pitch deviation that are shown in Figure 7. The PV of the \(X^{\prime}\)-directional pitch deviation from the measured and fitted results were evaluated to be 53.06 nm and 39.89 nm, respectively.