## 2 Design of the Integrated Optical Prism  
Figure 1a is the schematic diagram of the principle of two-dimensional grating ruler [5]. When a laser beam is irradiated vertically on a two-dimensional grating, the \(\pm 1\) order diffractive light will be generated along the two grating lines of the two-dimensional grating. The total diffraction light is four beams, denoted as \(U_{X+1}\), \(U_{X-1}\), \(U_{Y+1}\), \(U_{Y-1}\) (\(U_{Y+1}\), \(U_{Y-1}\) are not shown in this Figure for clarity). The angle between the diffraction light and the normal line of the grating plane is equal to the diffraction angle of the grating. As shown in Figure 1a, the beam emitted by the laser source is vertically irradiated on the reference grating and the scale grating after the beam is divided by the beam splitter. The interference of \(\pm 1\) order diffractive beams from the scale grating and reference grating occurs after the beams combining in the splitter. The interference fringes are formed and can be detected by photoelectric detectors for its light intensity, and finally four interference signals are obtained. After that, the measured displacement can be solved according to the detected interference signal through the corresponding data processing algorithm. As illustrated in previous research [5; 6; 13], further phase delay and amplitude division of these signals are required so that the direct current components of the interference signals can be eliminated and the motion direction can be distinguished. Thus, more optics are added into this optical layout and this results the light path increasing to more than 100 mm [14]. This makes paralleling these diffraction beams necessary. As mentioned in the first section of this paper, Figure 1b-d illustrate these three present optical designs. As shown in Figure 1b, a collimation lens was employed. The diffraction beams were refracted. This is suitable for large grating period, generally a 10 micrometer order. Under such a condition, the diffraction angle from gratings are relatively small, and the collimation effect can be ensured [6]. However, a large grating period will result in a relatively poor resolution. A high resolution requires a small grating period. Correspondingly a large diffraction angle makes using the collimation difficult. The second trial is that of the triangular prism refraction method, as shown in Figure 1c. Similarly, a triangular prism can be replaced with a mirror. Both methods need the diffractive light deflection mirror to be precisely calibrated so that the reference and the scale grating diffractive light will be able to interfere after transmission to the detector. It was found in the trial production of the actual test prototype [5; 12], the process of assembly and adjusting the diffractive light deflection prism is extremely complicated and difficult for the mechanical assembly stress and the influence of such factors as poor precision stability of the optical path. In order to avoid the problem that diffractive light deflection prisms need to be adjusted separately, a structure using transmission grating for diffraction light deflection is proposed, as shown in Figure 1d [13; 15; 16]. In this structure, the transmission grating is actually four one-dimensional transmission gratings combined. This scheme greatly reduces the difficulty of assembling and debugging optical circuits by employing the integrating transmission grating, which is used as a diffractive light deflection device. However, limiting the transmission grating diffraction efficiency, the plan will greatly reduce the intensity of the diffractive light. And it is not conducive to improve the signal-to-noise ratio (SNR) in subsequent signal processing, which leads to reduce the measuring accuracy of the grating ruler system. In order to overcome the above problems, an integrated grating diffractive light deflection prism design is proposed.