## 3 Experiments and Discussions
### Displacement Measurement with Wavelength Correction  
Figure 13 shows simultaneous displacement measurement results of the NMM-1 system and the MMI kit with a step interval of 2 mm in the measurement distance of 20 mm. The actual calibrated wavelength (referred to as \(\lambda_{cal}\)) was estimated by multiplying the nominal laser wavelength of 635 nm by the slope \(k_{p}\) of the fitted curve of the results in Figure 13.  
In the wavelength corrector kit, a transmission diffraction grating (Edmund optics, Barrington, USA) with a grating pitch of 1 \(\upmu\)m was selected. The optical components used in AC2 were the same as that in AC1. The performance investigations of wavelength correction were firstly conducted. Figure 14 shows the stability of AC2, which was employed to measure the drift of +1 st order diffraction angle. As seen from Figure 14, the maximum angle drift was measured as less than 1 arcsec. The laser wavelength variation caused by the variation of AC2 was evaluated to be 0.0037 nm by Equation (9). Figure 12: Comparison results: (**a**) residual of measured displacement by the NMM-1 and 3 DOFLM systems without wavelength correction; (**b**) standard deviation of five measurements. Figure 13: Comparison results of measured displacement by the NMM-1 and 3 DOFLM systems without wavelength correction. The angular measurement accuracy of AC2 was calibrated by the HP5529A system and the results are shown in Figure 15. It can be seen that the angle residual for the range of \(\pm\)100 arcsec was within \(\pm\)0.5 arcsec. It is verified that AC2 was acceptable for sub-arcsecond precision angular error measurement. The real-time laser diode wavelength was then evaluated based on Equation (9) according to the output of AC2. The wavelength measured by the self-wavelength correction is referred to as \(\lambda_{mean}\). The calibrated wavelength (\(\lambda_{cal}\)) was obtained from the ratio of the actual displacement of NMM-1 system and the measured total phase change of the MMI kit, as given by Equation (7). Figure 16 shows the relationship between the calibrated laser wavelength \(\lambda_{cal}\) and the variation of slope \(k_{p}\) obtained from the displacement measurement results of the NMM-1 system and the MMI kit. The relationship between the measured laser wavelength \(\lambda_{mean}\) and the variation of drift +1 st order diffraction angle detected by AC2 was also plotted in Figure 16. For the sake of comparison, \(\lambda_{cal}\) and \(\lambda_{mean}\) are listed in Table 1. It is clearly seen that an order of \(10^{-6}\) accuracy was achieved in measured wavelength by the designed wavelength correction. It means that, if the 3 DOFLM system measures a distance of 20 mm, the maximum measurement error, on average, would be estimated as about 20 nm. Figure 14: Stability of output signals of the autocollimator (AC2) in the wavelength corrector kit. Figure 15: Calibration results of AC2 in the wavelength corrector kit. From the above experiments, it can be realized that a special feature of this 3 DOFLM system is that it only needs to calibrate and store the nominal wavelength and the corresponding initial diffraction angle once using a reference. With these stored parameters, the system can automatically correct the wavelength by itself for the remaining measurements. The comparison results between the NMM-1 and 3 DOFLM systems with wavelength correction are given in Figure 17. It can be seen that, as expected, the measurement error of the 3 DOFLM system falls within \(\pm\) 25 nm, which was much smaller than that without wavelength correction in Figure 12. The corresponding standard deviation of the measurement results was evaluated to be between 2.2 and 13.5 nm. The effectiveness of the wavelength corrector kit was, thus, verified.