## 4 Experiments
### Results and Discussions
During the period of experiment, the environment effects, such as the air pressure, relative humidity, and temperature, could induce a drift in air refractive index as well as the laser wavelength. However, the refractive index drift and laser wavelength fluctuation in the whole measurement can be small under a well-controlled experimental environment with a limited testing time, which could be roughly estimated to be several picometers [24] and the caused combined errors are negligibly small. Therefore, the environmental effects and laser wavelength fluctuation are not problems in this experiment and its uncertainty is omitted in Equation (12). In addition, the angular displacement errors of the scale grating about the \(X\)- and \(Y\)-axes during the experiment were small, thus the misalignment angles due to the rotation error with respect to the \(X\)-and \(Y\)-axes do not contribute significantly to the overall combined uncertainty. Moreover, since this paper has focused on the form errors and PV uncertainties evaluation of the planar scale grating, the rotational error about the \(Z\)-axis is not a problem either in this case; the phase shifts information in the diffracted wavefronts will not change and the rotation error about the \(Z\)-axis is believed to be very small with careful adjustments. Figure 4:5 Measurement and fitted results of the \(X\)- and \(Y\)-directional pitch deviations. (**a**) \(X\)-directional pitch deviation of polynomial fitted results; (**b**) \(Y\)-directional pitch deviation of polynomial fitted results. Since the tested location of the planar scale grating was near the center of the FOV of the Fizeau interferometer, the PV value of the reference optical flat was expected to be far better than \(\lambda/20\) in the experiment. In the uncertainty analysis, the standard uncertainty of the out-of-flatness of the reference optical flat is thereby taken to be \(\sigma[\varepsilon_{R}(x,y)]=\lambda/300\), according to the size of the analyzed region of the grating under test. The standard uncertainty of the grating pitch was taken as \(u(g)=100\) nm, which was obtained by the specification sheet of the grating provided by the grating manufacturer, in addition, the uncertainty was taken as the same in the \(X\)- and \(Y\)-directions approximately, since the evaluated values were roughly the same. The standard uncertainty induced by the mechanical adjustment errors of the manual tilt stage was taken as \(u(\varepsilon)=0.1^{\circ}\), according to the specification of the tilt stage. The rectangular distribution was assumed for the mechanical adjustment error according to the scale interval of the vernier of the manual tilt stage. For the sake of simplicity, the uncertainty components were considered to be the same in the \(X\)- and \(Y\)-directions. The mean values as well as the uncertainties of the measured phase outputs \(I_{0}(x,y)\), \(I_{X\pm 1}(x,y)\) and \(I_{Y\pm 1}(x,y)\) were estimated by the data from six attempts. The standard uncertainty of each pixel in the analyzed area was calculated followed by the Type A standard uncertainty evaluation approach. The maximum value was adopted as the repeatability (1\(\sigma\)) of the corresponding phase output. The uncertainty of the regression coefficients matrices contains multiple values and the detailed results are presented in Appendix A. Figure 12 shows the obtained error maps as well as the corresponding uncertainty maps of the planar scale grating over the analyzed area. Figure 12a-c show the error maps of the \(Z\)-directional out-of-flatness, \(X\)- and \(Y\)-directional pitch deviation of the planar scale grating, respectively. Figure 12d-f present the corresponding standard uncertainty maps of the form errors according to Equations (28) and (29). The finally evaluated PV uncertainties in the above three errors were 4.21 nm, 1.60 nm, and 10.72 nm (with a confidence interval of approximately 95%), respectively.