## 2 Measurement Principle  
Figure 1 demonstrates the measurement of the \(Z\)-directional out-of-flatness of a planar scale grating \(e_{Z}(x,y)\) using the typical function of the Fizeau interferomter. Assume that the field-of-view (FOV) of the Fizeau interferometer is larger than the size of the grating employed for measurement. As can be seen in the figure, the wavefront of the zeroth-order diffracted beam from the grating superimposed on the wavefront of the reflected beam from the reference optical flat. Consequently, the zeroth-order phase output \(I_{0}(x,y)\) from the interferometer can be expressed by the following equation [7]:  
\[I_{0}(x,y)=\frac{4\pi[e_{Z}(x,y)-e_{R}(x,y)]}{\lambda} \tag{1}\]where \(\lambda\) represents the light wavelength of the laser source and \(e_{R}(x,y)\) represents the out-of-flatness of the reference optical flat in the Fizeau interferometer. The out-of-flatness of the grating \(e_{Z}(x,y)\) can be obtained by the following equation according to Equation (1):  
\[e_{Z}(x,y)=\frac{\lambda}{4\pi}I_{0}(x,y)+e_{R}(x,y) \tag{2}\]  
As can be seen in Equation (2), the result of \(e_{Z}(x,y)\) contains the out-of-flatness of the reference flat \(e_{R}(x,y)\). Since the calibration of the absolute flatness of the reference flat is quite complicated in practice [14], in this paper, the mean (average or expectation) of \(e_{R}(x,y)\) is taken to be zero and its uncertainty being the specified peak-to-valley (PV) value, which is typically less than \(\lambda/20\) over the whole field of view (FOV) of a commercial Fizeau interferometer. For the evaluation of \(X\)-directional pitch deviation \(e_{X}(x,y)\) and \(Y\)- directional pitch deviation \(e_{Y}(x,y)\) of the planar scale grating, measurement of the wavefronts of the \(X\) and \(Y\)-directional positive and negative first-order diffracted beams in Littrow configurations is needed. Figure 2 shows an example of the Littrow configuration for the measurement of the wavefront of \(X\)-directional positive first-order diffracted beam, where the grating is tilted clockwise, so that the \(X\)-directional positive first-order diffracted beam is back-reflected directly into the direction of the incident laser beam and superimposed with the reference beam in the Fizeau interferometer. Assume that the actual inclination angle in measurement \(\theta_{1}\) can be represented by \(\theta_{1}=\theta/2+\varepsilon_{1}\), where \(\theta=\arcsin(\lambda/g)\) is the first-order diffraction angle, \(\varepsilon_{1}\) is the misalignment of \(\theta\), and \(g\) is the nominal pitch of the grating along the \(X\)-direction. The \(X\)-directional pitch deviation \(e_{X}(x,y)\) of the planar scale grating causes a phase shift in the wavefront of the diffracted beam [7]. As depicted in Figure 2, the coordinate system has changed after setting the grating in Littrow configuration. Since the measurement results should correspond  
Figure 1: Measurement of zeroth-order diffracted wavefront using Fizeau interferometer (positive first-order diffracted wavefront is not shown for the sake of simplicity). to the original grating coordinate system, the \(X\)-directional positive first-order phase output \(I_{X+1}(x,y)\) that was obtained in the Fizeau measurement can be described, as follows:  
\[I_{X+1}(x,y)=2\pi\frac{e_{X}(x/\alpha_{1},y)}{g}+\frac{4\pi}{\lambda}[\alpha_{1 }e_{Z}(x/\alpha_{1},y)-e_{R}(x,y)] \tag{3}\]  
where \(\alpha_{1}=\cos\theta_{1}\). In the same manner, the wavefront of the \(X\)-directional negative first-order diffracted beam can be measured by tilting the scale grating counterclockwise with an opposite inclination angle, the \(X\)-directional negative first-order phase output \(I_{X-1}(x,y)\) can be obtained, as follows:  
\[I_{X-1}(x,y)=-2\pi\frac{e_{X}(x/\alpha_{2},y)}{g}+\frac{4\pi}{\lambda}[\alpha _{2}e_{Z}(x/\alpha_{2},y)-e_{R}(x,y)] \tag{4}\]  
where \(\alpha_{2}=\cos\theta_{2}\).