## 4 Discussion  
Figure 4 shows performance comparisons of the beam profiles and system stability of the OLB and the GWB. The single-mode waveguide is like a spatial filter, and the end-fire coupling object lenses were used to obtain the low divergence and small spot probe beam. According to the measured beam profiles, as shown in Figure 4a, the GWB has a more confined and symmetrical distribution than the OLB does. The sizes of half-power beam width (HPBW) are around 0.43 and 0.38 mm for the OLB and GWB, respectively. The system stability was then tested in the dual-channel homodyne interferometer. The phase variations after subtraction between two channels were measured, as shown in Figure 4b. In a measuring period of 360 sec, the stabilities based on the values of root mean square (RMS) are 0.009\({}^{\circ}\) and 0.007\({}^{\circ}\) for the OLB and GWB, respectively. In comparing the GWB with the OLB, the system stability is improved slightly by the GWB. To further compare the measurement characteristics of optical displacement for the GWB and OLB using the same CDS and incident positions, the phase responses to displacement when different voltages were applied on the PZT were measured. Figure 5 represents the phase responses for the different applied sawtooth voltages using the GWB at incident positions 1/2H and 3/4H on the CDS of **R** = 10.2 mm. Thus, the incident angles \(\theta_{1i}\) are 15\({}^{\circ}\) and 22.8\({}^{\circ}\) for incident positions 1/2H and 3/4H, respectively. The incident angles \(\theta_{2i}\) are 8.1\({}^{\circ}\) and 13.2\({}^{\circ}\) for incident positions 1/2H and 3/4H, respectively. At position 1/2H, the maximum phase variations are 5.2\({}^{\circ}\), 7.2\({}^{\circ}\), and 9.8\({}^{\circ}\) for the three applied voltages 75, 100, and 150 V, respectively, as shown in Figure 5a. At position 3/4H with the previous three voltages applied, Figure 5b presents the maximum phase variation: 10.7\({}^{\circ}\), 13.7\({}^{\circ}\), and 20.1\({}^{\circ}\). To compare the measurement performance of the GWB, a series of experiments with the same applied voltages, CDS, and incident positions were evaluated by the OLB. The results of phase responses are shown in Figure 6. At position 1/2H, the phase variations are 4.2\({}^{\circ}\), 5.6\({}^{\circ}\), and 7.8\({}^{\circ}\) for the three applied voltages 75, 100, and 150 V, respectively, as shown in Figure 6a. At position 3/4H with  
Figure 4: Performance comparison between the original laser beam (OLB) and guided-wave beam (GWB): (**a**) beam profile and (**b**) system stability. the previous three voltages, Figure 6b presents the phase variations: 10.1\({}^{\circ}\), 12.3\({}^{\circ}\), and 17.4\({}^{\circ}\). Figure 7 summarizes the phase variation versus the voltage of the GWB and OLB at the different incident positions. The results show that the GWB can obtain higher slopes than the OLB can at both incident positions. It is reasonable to assume that the GWB with its low divergence and small spot can improve measurement stability when passing through a curved surface. Figure 5: Phase response for the different applied voltages using the GWB and compact displacement sensor (CDS) of **R** = 10.2 mm at different incident positions: (**a**) 1/2**H** and (**b**) 3/4**H**. Figure 6: Phase response for the different applied voltages using the OLB and CDS of **R** = 10.2 mm at different incident positions: (**a**) 1/2**H** and (**b**) 3/4**H**. Figure 7: Phase vs. applied voltage at the different incident positions (**R** = 10.2 mm) for the GWB and OLB. Based on the confirmed evidence, the CDS with the smaller radius of 7.1 mm was further adopted to test the phase response for the different positions shown in Figure 8. Thus, the incident angles \(\theta_{1i}\) are 17.6\({}^{\circ}\) and 26.9\({}^{\circ}\) for the incident positions 1/2\(\mathbf{H}\) and 3/4\(\mathbf{H}\), respectively. The incident angles \(\theta_{2i}\) are 9.6\({}^{\circ}\) and 16.4\({}^{\circ}\) for the incident positions 1/2\(\mathbf{H}\) and 3/4\(\mathbf{H}\), respectively.