## 1 Introduction  
Nowadays, development in modern technology and industry requires more precise displacement measurement and nanopositioning sensors. Taking semiconductor manufacturing as an example, because of the decreasing node size, measuring the spatial position and attitude of a wafer stage in lithography will be a challenge for high accuracy and stability [1]. The trend towards a stable sub-nanometer accuracy measurement propels grating interferometer, a substitution for a laser interferometer, to be the solution of displacement measurement in a lithography machine. Advantages of the grating interferometer mainly have two aspects. One is the resistance to ambient disturbance, especially to the fluctuation of the air refractive index, which makes it convenient to achieve accuracy in the sub-nanometer scale. Referring to the work of [PERSON] et al., the measuring uncertainty of a laser interferometer is mainly caused by air the refractive index error and wavelength error, while the influences of a grating interferometer are manufacturing errors and thermal expansion of the scale [2]. Compared with those influencing factors in a laser interferometer, the manufacturing errors and thermal expansion of the grating are easier to control by the use of zero thermal expansion materials and the development of a fabrication technique of gratings. Recently, [PERSON] et al. achieved linear displacement measurement with 68.6 pm resolution by using eightfold optical construction equipping a 1780 lines/mm high-density grating [3]. Another advantage is that the utilization of optical components such as large planar gratings makes grating interferometer easy to implement an additional measuring dimension. Considering a measured rigid body, its spatial degree of freedoms (DOF) include three translational DOFs and three rotational DOFs. The grating interferometer has been widely used in multi-DOF metrology systems. For instance, [PERSON] et al. proposed a 3-DOF homodyne laser encoder using grating interferometry and quadrant detectors (QD) to measure movement along the x-direction and rotation around the x-axis and y-axis; the angular measuring ranges were \(\pm 100\) arcsec [4]. [PERSON] et al. designed a similar 6-DOF optical encoder with \(\pm 200\) arcsec angular measuring ranges [5]. [PERSON] et al. reported researches on 6-DOF homodyne surface encoders in a multi-probe all-interferometric configuration and a grating-QD combination, the angular measuring ranges of which were over 60 arcsec [6; 7]. The limitation of these ranges mentioned above could be attributed to angular mismatches--the beam misalignment caused by the rotations of the grating. As a common method to attain a high fold factor, the double-diffraction structure also has the feature of effectively enhancing the alignment tolerance (also written head-to-scale tolerance in some references) [8; 9; 10]. [PERSON] et al. proposed a 2-DOF double-diffracted heterodyne grating interferometer in common optical path configuration. However, the specific alignment tolerance was not mentioned [9]. Because the output beams are parallel with the incident beams, the double-diffraction structure is capable of turning the angular mismatches into the position mismatches--the deviations of output beams. However, it is worth emphasizing that the position mismatches will still cause intensity loss of interfering beams so that further analyses still need to be investigated. Currently, as a newly proposed concept, spatially separated heterodyne grating interferometer, which focuses on eliminating the periodic nonlinear errors from several to tens of nanometers, has become an effective approach to sub-nanometer scale measurement accuracy [11; 12; 13]. Compared with a typical common optical path grating interferometer, being spatially separated means it has two parallel modulated laser beams with frequency differences.