## 1 Introduction  
Planar encoders play a key role in precision positioning of linear stages due to their high resolution, high robustness, and relatively low cost [1; 2; 3; 4; 5; 6; 7; 8; 9]. The key component in the planar encoder is a two-dimensional (2D) planar scale grating, on which typically surface relief square holes or sinusoidal hills with a period spacing from submicron to several microns are arrayed along two orthogonal directions. Laser interference lithography (LIL), engine machining, and imprinting are three typical methods for fabrication of the scale grating. Compared with engine machining or imprinting, LIL is more convenient and cost-effective because only a coherent laser source is required and neither sophisticated precision positioning system nor expensive photolithography equipment and mask preparation are required. For fabrication of a 2D planar grating, according to the exposure times, the LIL process can be separated into two categories, double exposure and single exposure. For the two-beam two-exposure, it requires twice exposure procedures and the second time exposure requires rotation of the substrate by 90 degrees [5; 10]. The exposure can be based on an amplitude-division type two-beam LIL, called two beam two exposures (TBTE) or a wavefront-division type Lloyd's mirror type LIL, called one beam two exposures (OBTE) [11]. TBTE is good at fabrication of gratings with a relatively large area up to 100 mm \(\times\) 100 mm, while OBTE can be easily handled with a relatively simple and stable optical system. However, a big challenge existing in the method of the two-exposure process is that the grating structures generated in the first exposure will be influenced by the background light in the second exposure. Thus the depths of the grating structures will be different in two orthogonal directions. As a result, diffraction efficiencies of both the negative and positive diffraction beams will not be consistent. This is a critical problem for the encoder system since it directly affects the accuracy and the detecting limit of the interference signals. Therefore, efforts for solving this problem using a single exposure for 2D scale grating have been done these years. There are two trends for this motivation, multibeam LIL and one beam LIL. Multibeam LIL includes, but is not limited to, triple-beam and quadruple-beam interference technique. This technology can achieve the goal of 2D grating patterns with a single exposure. Some researchers invented a novel nanofabrication technique, called four-beam interference lithography, by using a two-dimensional grating [12]. Nevertheless, it cannot be neglected that the multibeam interference lithography has complicated optical structure easily to be disturbed by external environment. Furthermore, the optical path difference (OPD) is also difficult to be manipulated. In contrast, one beam LIL, represented by two-axis Lloyd's mirror interferometer and a corner-like reflector interferometer, allows fabricating a crossed 2D grating with a single exposure [13; 14; 15; 16; 17; 18]. For one beam LIL, one more proposal called the two-axis Lloyd's mirror interferometer with polarization modulation technique, proposed in 2014, has the advantages of large grating area, easy handling, and stable grating patterns [13]. However, optimization of the theoretical inclination in grating shape requires sophisticated in situ detection system for exposure and development. The inclination will influence the diffraction efficiency in a certain degree when it is directly used as a photoresist grating, though this influence might be reduced when a further etched process. On the other hand, the corner-like reflector interferometer, also called orthogonal type two-axis Lloyd's mirror interferometer, also proposed in 2014, has the advantages on grating shape in depth direction, although the complicated interference patterns on the grating substrate require deep modification for application as a 2D scale grating [14].