## 4 Experiment and Analysis  
Figure 6 shows the setup of our experiments. It includes two parts: the FSI-based absolute distance measurement system, and the interferometer (Renishaw XL-80). The ECL has been used for five years, and mode-hopping of the ECL (NewfocusTLB-6728) always happens. In experiments, the power of the ECL was 8 mW, and the scanning range of the ECL was 1515-1535 nm. The optical frequency was set to sweep as a triangle at tuning rate of 100 nm/s. The sampling frequency of the DAQ (Gage CSE161G4) was 5 MHz, and the number of sampling points was \(1\times 10^{6}\). The measured target was a 10-cent coin, and the incident angle of the measurement signal was 12 degrees, so the backward echo signal was very weak. Figure 5: Comparison between the ideal frequency and the simulated result. The He-Ne interferometer was used to verify the accuracy of our method. The cube-corner prism of the He-Ne interferometer and the target of the FSI system were both fixed on a slider. The slider moved along a precise guiderail, and the range of the guiderail was 1000 mm. The measured result in Figure 4 consists of two parts. One was the distance from the ECL to the fast steering mirror (FSM) of our FSI-based system, and it can be obtained when the FSM was horizontal. Another was the measured distance from the FSM to the measured target, such as the results in Figures 7 and 8. In our experiments, the former was 8385.432 mm. The measured distance in Figures 7 and 8 was the measured result minus 8.385.432 mm. Figure 7 shows the comparison of the experiment results between the method of [PERSON] et al. [17] (part a) and the adaptive filtering method and the mosaic algorithm (part b) with the same experiment data. It proves the necessity and effectiveness of our method to deal with the noises and the mode-hopping signals. Figure 6: Experiment setup. Figure 7: Part (**a**,**b**) respectively represent the experiment results of not using andusing the adaptive filtering method and the mosaic algorithm with the same data. In order to analyze the stability of our method, we measured the coin 20 times at one position and used our adaptive filtering method and mosaic algorithm to progress the experiment data. As shown in Figure 8, the blue line denoted the results of 20 measurements, and the green line denoted the residuals. When there were noises, the mode-hopping signals and the interfering signals in our system, the statistical uncertainty of our method was 5.7 \(\upmu\)m over 20 measurements, which was the same order as the result of [PERSON] et al. [16]. To evaluate the accuracy of our method, the coin moved 62.45 mm every time, and totally moved seven times. The FSI-based system and the He-Ne interferometer simultaneously measured the displacement of the slider. So, our FSI-based system and the laser interferometer could use the same air refractive index. The measured results of the laser interferometer were used as the truth value of the moving distance. Figure 9 showed the difference between the measured results of our systemand the laser interferometer. The max relative error of our method was 9.6 \(\upmu\)m. The relative error was attributed to the Abbe error, the air-path distance variation, and the optical dispersion error. In our experiment, the moving distance was less than 500 mm, so the Abbe error between the laser interferometer and our FSI-based systemwas less than 0.5 \(\upmu\)m. The air-path distance variation was mainly caused by the variation of the air refractive index. The temperature, pressure and humidity in the experiments were basically stable, so the error caused by the air-path distance variation was within 1 \(\upmu\)m. The optical dispersion error was originated from the optical frequency sweep of the ECL, and was compensated by Equation (7). According to Equation (7), the moving distance was 0.06245 m, \(\beta\) was \(-\)23 ps\({}^{2}\)/km, and the tuning bandwidth was 10 nm, so the distance offset was 2.3 \(\upmu\)m.  
The CRLB was used to evaluate the performances of our system.