## 1 Introduction  
An optical encoder, which is mainly composed of an optical sensor head and a scale grating, is one of the most commonly used sensors for precision positioning in production engineering [1], due to the advantages of non-contact measurement, a high resolution, and a wide bandwidth [2]. In the case of incremental type one-axis encoders, a diffraction grating having periodic line pattern structures is employed as a scale for measurement of relative translational displacement between the scale grating and an optical sensor head [3]. With the employment of a planar scale grating having two-dimensional grating pattern structures [4], the dimension of displacement/position measurement can be expanded to two or more [5, 6]. Among the components in a multi-axis planar encoder system [5], a planar scale grating having periodical structures along the \(X\)- and \(Y\)-directions is one of the key components that influence the accuracy of measurement, since the period of interference signal in the optical head, which will be interpolated and used to calculate the displacement, is directly influenced by the pitch of the scale grating, in principle [4]. Moreover, the \(Z\)-directional out-of-flatness of the scale grating could also be a source of measurement uncertainty in the planar encoder system. The \(X\)- and \(Y\)-directional pitch deviations, as well as the out-of-flatness of the scale grating, are thus required to be evaluated and calibrated over the entire surface of a scale grating for high precision measurement applications [7]. Line scale comparator [8; 9] and scanning probe microscopes (SPMs) [10; 11] are available as conventional solutions to evaluate the pitch deviations of a planar scale granting. The line scale comparator has long been used in the enterprises that manufacture linear encoders and national standard institutes for testing one-axis linear scale gratings [8; 9]. However, it is too expensive to construct a measurement system for two-dimensional scale grating that is based on the line scale comparator. Another drawback of the line scale comparator is that the out-of-flatness of a planar scale grating cannot be measured in principle. Meanwhile, SPMs can measure both the grating pitch deviations and the out-of-flatness of a planar scale grating. However, on the other hand, a limited scanning area and a measurement throughput of SPMs [11] prevent them being applied for the evaluation of planar scale grating, especially for the condition of laboratory-level, it may take hours and days to conduct the measurement of a large size planar scale grating over its whole area. In responding to the background described above, the authors' group has proposed a method to simultaneously evaluate both the \(X\)- and \(Y\)-directional pitch deviations and the \(Z\)-directional out-of-flatness of a planar scale grating by utilizing a Fizeau interferometer in the previous work [7]. In the propose method, wavefront of the zeroth-order diffracted beam and wavefronts of positive and negative first-order diffracted beams from a scale grating obtained in Littrow configurations were utilized to perform a non-contact evaluation of a planar scale grating over the entire area. The feasibility of proposed method has been verified in experiments [7]. In addition, the proposed method has been improved so that the self-calibration of the Fizeau interferometer and the planar scale grating can be conducted simultaneously while considering the out-of-flatness of the reference optical flat and the change of the coordinate system in the Littrow configuration [12]. As the first step of the research, a preliminary verification of the improved method has been performed successfully through the simulation in both the noise-free and noisy cases [12].