# Compensation of Frequency Drift in Frequency-Sweep Polarization-Modulation Ranging System
## 3 Experimental Results and Discussion  
In order to verify the effectiveness of above methods, the experimental system shown in Figure 1 was established and several experiments were conducted. In the system, a He-Ne laser (Newport R-32734) served as the light source, a high-frequency phase modulator (Newport, 4431, V\({}_{\pi}\) is 40 V) served as the modulator, a signal generator (Keysight, N5171B) provided the modulation signal, the preset sweeping range was set from 2.500 to 2.520 GHz, with a sweep step size of 1 kHz. The residence time at each frequency was 20 ms. An avalanche detector (Thorlabs, APD410A) was used for the photoelectric conversion. A PXI oscilloscope (NI, 5171R) served as the acquisition and processing module. The target was approximately 80 m away from the crystal. To evaluate the method of reciprocating sweep in compensation of thermally induced phase delay, an \"in phase\" frequency was obtained by different methods and results are shown in Figure 4. In the figure, black and red dots respectively denote the results obtained through forward and backward sweep, and blue dots denote the average of measurement results obtained by the reciprocating sweep. Results obtained through forward and backward sweeps drifted in opposite directions, and the forward sweeps had a larger statistical average than that of the backward sweeps. Through reciprocating sweeps, the results had a repeatability of 1727 Hz, which was much smaller than that of forward sweep (3130 Hz) and backward sweep (2503 Hz). Figure 4: Experimental results of “in phase” frequency. To validate our measurement method of N, N was tested with multi-frequency intervals. The \"in phase\" frequency \(f_{x}\) near the lower edge of band was first obtained in the reciprocating sweep method, then, the \(u\)th \"in phase\" frequency on its upper side was obtained, and \(f_{\rm b}\) and N were respectively determined according to Equations (10) and (11). \(u\) was set to change from 1 to 10 in seven groups of tests, each group of tests contained 10 independent measurements; the statistical results are shown in Figure 5. In the single-frequency interval method, the mean squares of \(f_{\rm b}\) and N were 2384 Hz and 1.5, respectively. As \(u\) increased from 1 to 8, the mean square of \(f_{\rm b}\) was reduced to 332 Hz, and the mean square of N was reduced to 0. Further increase of \(u\) would result in further reduction of the standard deviation of \(f_{\rm b}\), but in our experiment, the frequency interval number of 8 was sufficient for an accurate measurement of N.  
With the accurate measurement of N, distance could be determined through the \"in phase\" frequency and the corresponding N; results are shown in Figure 6. The statistical average of the forward sweeps was 6 \(\upmu\)m larger than the backward sweeps. The statistical uncertainties of the forward sweeps and reciprocating sweeps were 87.1 \(\upmu\)m and 44.1 \(\upmu\)m, respectively. Finally, an uncertainty test was conducted using a dual-frequency laser interferometer. The target retroreflector was moved from 79.29 to 80.29 m by ten steps at an increment of 100 mm. A commercial heterodyne interferometer (5519, Keysight) was used to measure the displacement of the target. By the application of linear fitting, the residual error between the reference displacement and the measured distance could be obtained as shown in Figure 7. The residual error of the forward sweep was from \(-\)85.9 to 113 \(\upmu\)m. With the reciprocating sweep method, the max residual error was 54.4 \(\upmu\)m. The relative error was attributed to the random measurement error of frequency, the temperature dependence of crystal properties, the air-path distance variation, and the instability of frequency reference. The random measurement error of frequency could be further reduced by reducing the sweep step size and increasing the signal-to-noise ratio of the system.