## 3 Experiments and Results
### Frequency Stabilized for LD at 1 MHz Modulation Frequency
\begin{table}
\begin{tabular}{c c} \hline Laser source & 6304 \\ Wavelength & 632–634 nm \\ Modulation frequency for LD & 1 MHz \\ Modulation amplitude for LD & 25 MHz \\ Temperature of cold finger & \(15\pm 0.1\)\({}^{\circ}\)C \\ Temperature; pressure; humidity & \(20\pm 0.1\)\({}^{\circ}\)C; \(1013\pm 0.02\) kPa; \(33\pm 0.01\) RH \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental conditions of the locking frequency of the laser diode (LD). Figure 3: Experimental setup of the frequency-stabilized LD system. LD: laser diode; PD: photodetector; APD: avalanche detector. The frequency-locked LD was then used for the Michelson interferometer to measure the displacement. In order to verify the improvement of the new system with our previous one [14], the triangular form of displacement was generated using the same PZT (P750.20, Physik Instrumente, Karlsruhe, Germany). The conditions of the new and old experiment are shown in Table 2 in the new experiment and old experiment column, respectively. The interference signal of the new system is shown in Figure 6. The phase shift deduced from the displacement of the target mirror was measured using the first and second harmonics of the interference signal. In our previous work, there was some noise that could be seen from the harmonic signals [13,14]; hence, the measurement accuracy was reduced. In this experiment, the digital LIA (HF2 LI, Zurich Instruments, Zurich, Switzerland) was employed to detect the first and second harmonics (Figure 7a). A Lissajous diagram showing the movement direction using these harmonics is shown in Figure 7b. To determine the  
Figure 4: Hyperfine spectrum of the transition 6–3, P(33) of \({}^{127}\)I\({}_{2}\) obtained by third-harmonic detection with a modulation frequency of 1 MHz: (**a**) a modulation excursion of 10 MHz; (**b**) a modulation excursion of 15 MHz; (**c**) a modulation excursion of 20 MHz; (**d**) a modulation excursion of 25 MHz. Figure 5: Frequency stability of the locked LD for 3 h.  
measurement resolution, the displacement of the target mirror was simultaneously measured by both the interferometer and a capacitive sensor (Micro Sense Model 4083 with Probe Model 2806 L, MicroSense, Lowell, MA, USA). The capacitive sensor can operate in a range of \(\pm 10\) um and at a measurement resolution of 0.7 nm. The displacement measurement results are shown in Figure 8a. A small shift in the displacement results in the vertical direction can be seen in the figure. This shift was induced by the air gap between the probe of the capacitive sensor and the target mirror and the initial phase of the interferometer. However, a PZT displacement of 1.5 um and a period of 1 s were successfully measured using the interferometer. The difference in the displacement-measuring results between the proposed interferometer and the capacitive sensor is shown in Figure 8b. A difference of about 20 nm due to environmental disturbance can be seen in the figure. A comparison of the difference between the new system and our previous one (see Figure 8d [14]) is shown in Figure 9. The difference in the displacement measuring result using the new system was less than a half of the previous result. \begin{table}
\begin{tabular}{c c c} \hline  & **New System** & **Old System** \\ \hline Operating frequency of PZT (Hz) & 1 & 1 \\ Amplitude of PZT stage (μm) & 1.5 & 1.5 \\ Modulation frequency for LD (kHz) & 1000 & 300 \\ Cut-off frequency of low pass filters (kHz) & 900 & 100 \\ Measuring time (s) & 10 s & 10 s \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental conditions of the displacement measurements. Figure 6: Interference signal of the displacement-measuring interferometer. Figure 7: First and second harmonics detected by the lock-in amplifiers (LIAs) (**a**) and illustrated in a Lissajous diagram (**b**). In order to verify the performance of the interferometer at a higher measurement speed, the interferometer was employed to measure the axial error of a spindle. The spindle (Chuo Precision Industry Co., Tokyo, Japan.)