## 2 Methodology  
The proposed displacement-measuring interferometer was constructed of two parts: a frequency-locked LD source and a Michelson interferometer; see Figure 1. The I\({}_{2}\) hyperfine structure that spreads from 500 to 905 nm provided a large number of stable references to lock the LD frequency. The hyperfine structurer was detected using a saturated absorption configuration. The frequency of the LD was modulated by modulating the injection current. An isolator was employed to prevent laser feedback to the source that could cause frequency instability. A half-wave plate (HWP) and a polarization beam splitter (PBS1) were used to control the intensity of the reflected and transmitted beams. The reflected beam with high intensity went to the I\({}_{2}\) cell, and the transmitted beam travelled toward the Michelson interferometer. The reflected beam, i.e., a pump beam, passed through the I\({}_{2}\) medium and then bounded to a plate beam splitter. A small part of the pump beam reflected on the plate beam splitter and returned to the I\({}_{2}\) cell on the same path, i.e., a probe beam. A quarter-wave plate (QWP) and PBS2 directed the probe beam to Photo Detector 1 (PD1). The signal from PD1 was demodulated using the lock-in amplifier 1 (LIA1). The demodulated signal was the third harmonic of the absorption signal of the I\({}_{2}\) hyperfine structure. The demodulated signal induced by LIA1 with the zero-crossing was used as the error signal to feed into a proportional-integral-derivative (PID) controller. The PID controller generated a feedback signal and sent it to the laser controller to lock the frequency of the LD to the I\({}_{2}\) hyperfine component. The frequency-locked LD was installed in the Michelson interferometer. On a beam splitter (BS), the beam was divided into two paths: one went to the measurement mirror, which was attached to a piezoelectric transducer (PZT) or a spindle under test; the other beam travelled to the fixed mirror and was reflected to the BS. The two beams recombined and interfered on the BS. The interference signal was detected using PD2. Typically, the measurement accuracy of displacement-measuring interferometers strongly depends on the quality of the interference signals. However, the proposed interferometer used LIAs to detect harmonics from the interference signal and then to calculate the displacement. An LIA detected a specific harmonic whose frequency synchronized with the reference signal and removed anything unsynchronized with the reference signal. Therefore, the LIA could detect a weak signal buried in strong noise, so the effect of an imperfect inference signal could be reduced. The interference signal collected by PD2 was determined as follows [14]:  
\[ \begin{split} I(\Delta L,t)&=I_{0}\Big{\{}1+V\Big{\{} \text{cos}\Big{(}\frac{4\pi n}{\lambda_{0}}\Delta L\Big{)}\ \Big{[}I_{0}(m)+2\sum\limits_{k=1}^{\infty}J_{2k}(m)\cos 4k \pi f_{m}\ t\Big{]}\\ &-2\sin\Big{(}\frac{4\pi n}{\lambda_{0}}\Delta L\Big{)}\sum_{k=1} ^{\infty}J_{2k-1}(m)\sin 2(2k-1)\pi f_{m}\ t\Big{\}}\Big{\}},\end{split}  \tag{1}\]  
where \(I_{0},\ V,\ \lambda_{0}\,n,f_{m}\), \(k\), and \(\Delta L\) are the average intensity, the contrast of a beat signal, the wavelength of the light source, the refractive index, the modulation frequency, an integer, and the displacement of the moving mirror, respectively. The modulation index \(m\) and, hence, the Bessel function \(J_{2k}(m)\) depend on the modulation excursion \(\Delta f\) and the unbalanced length \(L\) of the interferometer, given by:  
\[m=\frac{2\pi\Delta fLn}{c}, \tag{2}\]  
where \(c\) is the speed of light. The signal from PD2 was demodulated by using two LIAs.