## 4 Simulations and Experiments  
### Simulations  
To quantitatively describe the alignment tolerance, simulation codes for beam tracing were programmed with MATLAB. In the mathematical model, the O-x-y-z coordinate, optical interfaces, and vectors of incident beams were defined. The range of grating and apertures of the RRs are supposed to be infinite in the model, so that the influence and errors caused by out of clear apertures are not discussed. In Figure 6, four crucial structural parameters of the spatially separated heterodyne grating interferometer are marked. Specifically, A is the separating distance of the incident beams, B represents the lengths of diffracted beams from planar grating to the surfaces of RRs, C is the offset between the input and the returned output beams of RRs, and D is the beam diameter. Since the divergence of beams for interferometers are generally small (within a couple of mrad/hundreds of arcsec), only the Gaussian distribution of the intensity is considered while the divergence is ignored, which implies that the Gaussian beam seems to be a column in the simulation. The diameter D is assumed as the beam waist of the Gaussian beam. Therefore, the optical structure showing in Figure 1 could be digitized as A = 7 mm, B = 30 mm, C = 7 mm, and D = 3 mm. The influence of the stand-off error and three misalignment angles to such a structure are simulated; these results are demonstrated in Figure 7.  
Figure 7 shows that the deviations between the actual and the ideal centers of the spots were continuously enlarged as the errors increased. The blue lines represent the positions of the blue spot in Figure 6. Similarly, the red solid lines depict the deviations of the red diffracted beam in Figure 6, and the red dashed lines describe the spot positions after reflection by the beam-combined BS. Because the roll and pitch angles created the planar movements of the beam centers, their curves in the x-O-y  
Figure 5: Geometrical analysis of the roll and pitch movements and their influences on the measurement beam caused by spots deviation. (**a**) Roll movement; (**b**) Pitch movement. plane with contour lines of the angles are portrayed in Figure 7a,c. Blue numbers and red numbers represent the angles of the blue lines and dashed red lines, respectively. However, only one-dimensional movements of the beams in the x-direction were caused by the yaw angle and the stand-off error; Figure 7b,d depicts the relationship in curves with an explanation \(\Delta y=0\) mm. It is worth mentioning that the simulation results agree with the qualitative analysis in Chapter 3. Besides, the beam-combined BS placed at an angle of 45\({}^{\circ}\) could reduce the separation caused by the stand-off error and roll angle, but also deteriorate the influence of yaw and pitch angles. The work-off distances of the reflected red beams and the blue beams can be extracted from the distances at the same angles or errors between the corresponding lines. Figure 6: Mathematical model and the crucial structural parameters of the spatially separated heterodyne grating interferometer and the interfering intensity of the Gaussian beams. Figure 7: The results of beam tracing with the variation of misalignment angles and the stand-off error shown by the curves of the beam centers. (**a**) Roll angle \(\alpha\). (**b**) Yaw angle \(\beta\). (**c**) Pitch angle \(\gamma\). (**d**) Stand-off error \(d\). These curves share the legend shown in figure (**d**). In another simulation, the relationships of four specific structural parameters from \(A\) to \(D\) and their influence to interfering beams were investigated. Taking the pitch movement as an example, the simulation above has proved that the deviation of the two spots will decrease the overlapped region of the two Gaussian beams, and the larger the work-off distance \(L\) is, the lower the interfering intensity will be.