## 2 Optical Configuration and Mechanical Fixture of the Proposed Grating Interferometer  
Based on the concept of spatially separated heterodyne grating interferometer reported in our previous study [12], the optical configuration of the proposed grating interferometer is shown in Figure 1. Considering the real sizes of all components, several changes and adjustments were made to form a compact layout.  
From a typical separately modulated heterodyne laser source, beams at different frequencies were transferred into the optical reading head via polarization maintaining optical fibers. Instead of a parallel distribution, two fiber collimators (FC1 and FC2) in vertical distribution guaranteed enough space for mounting and adjusting. The beams were divided into two parts, the reference part and measurement part, in the beam splitter (BS1). Then, the reference beam was combined in the upper part of the beam splitter (BS2), reflected at the surface of the mirror (M1), entered into the fiber coupler (FCr), and transferred to the corresponding photodetector by a multi-mode optical fiber. Another part of the beams reflected at the mirror (M2) to change directions for a compact layout and diffracted by the linear grating (G) at a normal incident angle. In the diffraction, called the first diffraction here, each of the beams generated a pair of \(\pm 1^{\text{st}}\) order diffracted beams following their diffraction angles. Choosing a \(+1^{\text{st}}\) order and a \(-1^{\text{st}}\) order diffracted beam with different modulated frequencies \(f_{1}\) and \(f_{2}\) propagating symmetrically, two corner-cube retro-reflectors (RR1 and RR2) were replaced to cause a U-turn of the chosen beams. Then, following the grating equation, these beams entered the grating at oblique incident angles and were diffracted again. To be unambiguous, this diffraction was called the second diffraction. Obviously, only two of all the diffracted beams spread away from the grating vertically, with fourfold Doppler frequency shifts. It is worth emphasizing that the retro-reflectors guarantee that the selected diffracted beams of the second diffraction are parallel with the incident beams of the first diffraction when unexpected tip and tilt of grating occur. After being reflected in proper order by mirrors M2 and M3, the measurement beam was generated in the lower part of BS2. Similar to the reference beam, it entered the fiber coupler (Fcm) and transferred to its photodetector by another multi-mode optical fiber. Displacement information, contained in Doppler frequency shifts, was detected by photodetectors and extracted from electrical signals by the processing board.  
Compared to our previous study [13], the biggest improvement in optical configuration was substituting a polarized beam splitter (PBS) and half wave plate by the beam splitter BS1 and the mirror M3. The advantages were that not only that less beam splitter was used, but also that it was a polarized-insensitive optical configuration. In particular, it could be configured into a polarized or non-polarized interferometer by using a PBS or a BS at the place of BS1, decreasing the requirement of the laser source.  
The mechanical fixture is also displayed in Figure 1. As shown, a pedestal with holes for mounting is designed for supporting all optical and mechanical components. Two optical fiber collimators were fixed on matched kinematic mounts fastening onto the pedestal. The parallelism of beams was adjusted to several arc-seconds by kinematic mounts so that the assembling error could be omitted in simulations and experiments below. Optical components, including beam splitters, mirrors and corner-cube retro-reflectors, were glued on the pedestal by ultraviolet curable adhesive. Moreover, two optical fiber couplers mounted on custom mechanical components (red and orange columns) were also glued onto the pedestal. The grating interferometer assembly was convenient for mounting on moving stages or linear guide rails.