## 3 Grating-Groove Error Correction  
Based on the above analysis results, the main factors affecting the grating spectral performance index are the grating ruling machine itself, worm gear error, screw error, and the random error generated during the grating movement.  
In order to eliminate the ghost lines and stray light errors that affect the grating performance index during the grating ruling process, the research group further developed the worm gear vice, as shown in Figure 11. The worm gear we selected reached the international Grade 3 manufacturing accuracy. In the 750 mm stroke range, the cumulative error of the selected screw and nut does not exceed 4 \(\upmu\)m, and the pitch period error does not exceed 0.2 \(\upmu\)m, as shown in Figure 12. At the same time, the active control system is used to control the grating groove error, to improve the performance index of the ruled grating.  
Figure 11: The physical diagram of worm gear vice.  
Figure 10: The relative intensity distribution of diffracted light.  
The active control system of the grating ruling machine, as shown in Figure 13, the device mainly consists of a blank bearing seat, an interferometer, a piezoelectric ceramic lifting tool (z-axis), a reference mirror, and a piezoelectric ceramic control tool (x-axis).  
We built a random error measurement optical path, as shown in Figure 14. The light emitted by the laser passes through the beam splitter 8. One beam is used as beam 1 to measure the random error caused by the ruling system, and the other beam is used as beam 2 to measure the random error caused by the indexing system [25].  
Figure 14: Optical path diagram of measuring system. 1. Laser; 2. inner carriage; 3. slider; 4. quartz guide rail; 5. ground glass; 6. interferometer A; 7. tool system; 8. beam splitter; 9. interferometer B; 10. interferometer C; 11. measuring mirror.  
Figure 12: Machining completed for the preferred screw.  
Figure 13: Structural Diagram of the active tool control system.  
In optical path 1, one beam, \(f_{1}\), is incident on the reference glass 5 as the reference light, and the other beam, \(f_{2}\), is incident on the quartz guide 4 as the measurement beam to measure the ruling error caused by the change of the quartz guide face shape and the saddle slider pitch and twist error. Since the measurement mirror is a quartz guide, the reference surface of the quartz guide is not coated and its reflectivity is low. In order to get the highest contrast of the interference fringe after the reference light and the measurement light is coherent, the reference mirror is selected as a glass with a face shape accuracy better than \(\lambda/10\), while the other side will be polished as gross glass to reduce the influence of the reflected light on the reference beam and improve the contrast of the interference fringe. In optical path 2, the spectroscope divides the optical path into two beams, which enter interferometer 9 and interferometer 10, respectively. During the movement of the ruling carriage, the quartz guide is bent by the lateral force of the saddle-type slider. The instability ruling system causes the distance of the ruling carriage mounting plate relative to the quartz guide to change, making the ruling system produces random errors. The measurement signal and the reference signal are processed, and the number of pulses \(N_{1}\) is calculated by the counter. At the same time, during the one-way feed motion of the grating ruling machine table, the number of pulses \(N_{2}\) is calculated by the counter after processing the measurement signal and reference signal due to the non-parallelism of the guide rail and the random error factor in the ruling process. Finally, the random error displacement \(L\) in the grating ruling process is calculated by Equation (22) as the random error correction amount.  
\[L=\frac{\lambda}{4}\int\limits_{0}^{t}\left(\Delta f_{1}+\Delta f_{2}\right) \times dt=(N_{1}+N_{2})\frac{\lambda}{4} \tag{22}\]