## 3 Experiments and Results  
### Frequency Stabilized for LD at 1 MHz Modulation Frequency  
The experimental condition and the experimental setup for the locking frequency of the LD are shown in Table 1 and Figure 3, respectively. In the experiment, an LD (6304, New Focus, Irvine, California, USA) at a wavelength of 633 nm was used. The frequency of the LD was modulated by modulating the injection current at 1 MHz and scanned continuously over a range of 1 GHz to detect the hyperfine structure at the transition 6-3, P(33) of \({}^{127}\)I\({}_{2}\). The hyperfine structure was observed by detecting the third harmonic of the saturated absorption signals deduced by LIAs. In this research, a high bandwidth LIA up to 50 MHz was employed (HF2 LI Lock-in Amplifier, Zurich Instruments, Zurich, Switzerland). The I\({}_{2}\) cell was 100 mm long, and the cold-finger temperature of the cell was kept at 15 \({}^{\circ}\)C using a thermal electric cooler (TEC 3-6, Thorlabs Inc., Newton, NJ, USA). In order to obtain a high-frequency stability level for the LD, a high signal-to-noise ratio of the absorption signal was required. The absorption signal depends on the modulation frequency, the modulation excursion, and the physical parameters of the absorber cell (length, pressure, and purity) [25]. Moreover, in the interferometer, the modulation excursion also affects all the harmonic intensities depending on the modulation index. Therefore, a suitable modulation excursion was determined to enhance both the signal-to-noise ratio of the absorption signal and the harmonics intensities. The hyperfine spectrum of the transition 6-3, P(33), the b\({}_{1}\)-b\({}_{21}\) components, was obtained by the LIA, with some values of the modulation excursion being shown in Figure 4.  
Figure 2: Axial error of a high-speed spindle measuring system. PD: photodetector; LIA: lock-in amplifier; FG: function generator; DAQ: data acquisition. Figure 4 shows that the saturated absorption signals of the \({}^{127}\)I\({}_{2}\) hyperfine spectrum at the transition 6-3, P(33) can be determined purely at a modulation excurrent of 25 MHz. The absorption signal with the zero-crossing point from the LIA was used as the error signal to feed to the PID controller to lock the frequency of the LD to the b\({}_{21}\) hyperfine component. To measure the long-term frequency stability of the stabilized LD, a beat frequency measurement with a reference laser set was performed. The set included a standardized He-Ne laser (NEO-92 SI-NF, Neoark Corp, Tokyo, Japan) as a master source. The frequency of the master laser was locked to the a\({}_{13}\) hyperfine component of the R(127), 11-5 transition of I\({}_{2}\), which had an uncertainty of frequency of \(2.5\times 10^{-11}\). The distance between the b\({}_{21}\) component of P(33) 6-3 and the a\({}_{13}\) of R(127) 11-5 was over the bandwidth of the frequency counter. Therefore, an offset-locked He-Ne laser (NEO-OL101K, Neoark Corp., Tokyo, Japan) with the same level of frequency stability, but an unmodulated frequency was used as a slave laser. The avalanche detector 1 (APD1) (C5658, Hamamatsu Photonic, Hamamatsu, Japan) was used to measure the beat frequency between the master and the slave laser to ensure that the reference laser set was operated in the true condition. In this experiment, an offset frequency of 400 MHz was used for the slave laser. The beat signal between the LD and the reference laser set was collected using the APD2 (C5658, Hamamatsu Photonic, Hamamatsu, Japan). The frequency fluctuation (peak-to-peak) in 3 h was less than 4 MHz (Figure 5). The frequency stability of the locked LD was much higher than the normal LDs, whose frequency fluctuation is sometimes at the GHz level for 1 h.