## 3 Uncertainty Evaluation  
Major sources of uncertainty in the evaluation of out-of-flatness \(e_{Z}(x,y)\) and pitch deviation \(e_{X}(x,y)\) of a planar scale grating by a Fizeau interferometer primarily arise from uncertainties in the measured phase outputs \(I_{0}(x,y)\) and \(I_{X\pm 1}(x,y)\). In addition, out-of-flatness \(e_{R}(x,y)\) of the reference flat in a Fizeau interferometer and uncertainties in the nominal pitch \(g\) of the planar scale grating under test, as well as the inclination angles \(\theta_{1}\) and \(\theta_{2}\), could also be major sources of uncertainty. According to the GUM (guide to the expression of uncertainty in measurement) [13], the uncertainties in \(I_{0}(x,y)\) and \(I_{X\pm 1}(x,y)\) attribute to the Type A evaluation, while those in \(e_{R}(x,y)\), \(g\), \(\theta_{1}\) and \(\theta_{2}\) attribute to the Type B evaluation. Through the uncertainty analysis, The \(E(\cdot)\), \(\sigma(\cdot)\) and \(\sigma^{2}(\cdot)\) are used to represent the mean, standard deviation, and variance of a random variable, respectively. The expanded uncertainty of a random variable is given by \(u(\cdot)=k\sigma(\cdot)\), where \(k\) denotes the coverage factor; \(k=2\) produces a confidence interval of approximately 95%; and, \(k=3\) produces an interval with a confidence of approximately 99%. As mentioned in Section 2, in this paper, the mean and \(3\sigma\) uncertainty of \(e_{R}(x,y)\) are treated to be zero and the specified PV value, respectively. The uncertainty in the inclination angles \(\theta_{1}\) and \(\theta_{2}\) contains two components; one is the uncertainty induced by the deviation of the nominal grating pitch with respect to its true value, and another is induced by the misalignments in measurement. In the uncertainty analysis, the means of \(\theta_{1}\) and \(\theta_{2}\) are taken to be \(\theta/2\) and \(-\theta/2\), respectively, and the uncertainties that are induced by the misalignments are assumed to be identical; namely, \(\sigma(\varepsilon_{1})=\sigma(\varepsilon_{2})\). Due to the independence between \(\theta\) and \(\varepsilon_{1}\) (\(\varepsilon_{2}\)), the means of \(\alpha_{1}\) and \(\alpha_{2}\) will be \(\cos(\theta/2)\), which is denoted as \(\alpha\) in the following (\(\alpha=\cos(\theta/2)\)). By taking the means for both sides of Equations (2) and (5), the following equations can be obtained:  
\[E[e_{Z}(x,y)]=\frac{\lambda}{4\pi}E[I_{0}(x,y)], \tag{9}\]  
\[E[e_{X}(x/\alpha,y)]=\frac{g}{4\pi}\{E[I_{X+1}(x,y)]-E[I_{X-1}(x,y)]\}, \tag{10}\]  
It should be pointed out that here has \(E[\alpha_{1}e_{Z}(x/\alpha_{1},y)]=E[\alpha_{2}e_{Z}(x/\alpha_{2},y)]\) mathematically. In practical experiments, the second term of the right side of Equation (5) could be minimized by making interference fringes of the Fizeau interferometer minimum though adjusting the tilt stage on which the scale grating is fixed. Although the influence from the out-of-flatness of the substrate surface cannot completely be eliminated by the differential operation, when considering the holographic grating with a good surface quality, the effect on the evaluation of the pitch deviation can be ignored by conducting the operation mentioned above. A straightforward approach to evaluate the uncertainty in the obtained \(e_{Z}(x,y)\) and \(e_{X}(x,y)\) is to generate a matrix, called the uncertainty matrix [15], which has the same size to \(e_{Z}(x,y)\) or \(e_{X}(x,y)\) and it contains the corresponding uncertainty of \(e_{Z}(x,y)\) or \(e_{X}(x,y)\) at every pixel position (\(x_{i}\),\(y_{j}\)) (\(l=1\), \(2\),...,\(M;j=1,2,\ldots,N\)). Taking the propagation of errors into consideration, the variances \(\sigma^{2}[e_{Z}(x,y)]\) and \(\sigma^{2}[e_{X}(x/\alpha,y)]\) can be represented, as follows:  
\[\sigma^{2}[e_{Z}(x,y)]=\bigg{(}\frac{\lambda}{4\pi}\bigg{)}^{2}\sigma^{2}[I_{0} (x,y)]+\sigma^{2}[e_{R}(x,y)], \tag{11}\]  
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