## 3 Design and Realization  
### Nano-Resolution Laser Interferometry Sensor  
The amplitude of the micro-vibration of an optical platform under environmental random excitation is usually as small as micron scale, so a nano-resolution displacement sensor was needed to monitor and feedback such micro-vibration accurately. The scheme and design of the proposed nano-resolution laser interferometry sensor are shown in Figure 3. A single-frequency circularized beam is output by a collimated laser diode with a wavelength of 635 nm, and it turns into a linear polarized beam through a polaroid. The linear polarized laser beam passes through the first half-wave plate (HWP1) and is split into one reference beam and one measurement beam by the first polarizing beam splitter (PBS1). The reference beam passes through the second quarter-wave plate (QWP2), and then is reflected by a reference mirror (RM) to the PBS1. The measurement beam passes through the first quarter-wave plate (QWP1), and then is reflected by a target mirror (TM) to the PBS1. The reflected reference beam and transmitted measured beam recombine and are incident into the second half-wave plate (HWP2), the recombined incident beam is then evenly split into two beams by a non-polarizing beam splitter (NBS), and the nominal split ratio of the NBS is 50%:50%. One beam passes through the third quarter-wave plate (QWP3) and is split into two beams by the second polarizing beam splitter (PBS2), the other beam is also split into two beams by the third polarizing beam splitter (PBS3). Four photodiodes (PD), named PD1, PD2, PD3, and PD4, are used to detect four interference signals with phases of 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\), respectively. Convex lenses (CL) before PDs and TM are used to gather beams. The intensity of laser beams detected by four PDs can be expressed as  
\[I_{n}=A_{n}+B_{n}cos[\phi+(n-1)\pi/2] \tag{1}\]  
where \(\phi=4\pi\Delta L/\lambda\) is the phase introduced by \(\Delta L\), \(\Delta L\) is the displacement of the TM, \(\lambda\) is the wavelength of laser diode, \(n\) is the channel number and equals 1 to 4. \(A_{n}\) and \(B_{n}\) are the DC bias and AC amplitude of laser intensity. According to Equation (1), it can be obtained:  
\[\left\{ \begin{array}{l}I_{x}=I_{1}-I_{3}=A_{1}-A_{3}+(B_{1}+B_{3})cos\phi\\ I_{y}=I_{2}-I_{4}=A_{2}-A_{4}+(B_{2}+B_{4})sin\phi\end{array} \right., \tag{2}\]  
where \(A_{1}-A_{3}=A_{2}-A_{4}=0\), \(B_{1}+B_{3}=B_{2}+B_{4}\). Equation (2) is a circle function with a radius of \((B_{1}+B_{3})\) and center at \((0,0)\). The displacement of the TM is:  
\[I_{n}=A_{n}+B_{n}cos[\phi+(n-1)\pi/2], \tag{3}\]  
\[\Delta L=\phi/(4\pi/\lambda), \tag{4}\]  
where \(\phi=\arctan I_{y}\!/\!I_{x}\). As shown in Figure 4, a prototype of the laser interferometry sensor was developed, and an experimental setup was set on an optical platform to test its resolution and stability. In the resolution experiment, the laser interferometry sensor was kept still, while the TM is driven away from the sensor with a constant step by a nano-resolution piezoelectric transducer (PZT) stage. The smallest step that causes a perceptible change in the measured displacement is defined as the resolution of the laser interferometry sensor. Experimental results in Figure 5 show that its resolution is better than 2 nm. In the stability experiment, the TM is kept still, and signals of PD1 to PD4 are measured simultaneously and then analyzed. As shown in Figure 6, the stability of the laser interferometry sensor is better than \(\pm\)6 nm/20 min. It can be seen from Figures 5 and 6 that the proposed laser interferometry sensor can achieve the nano-resolution and low sensitivity to environmental noise, which meets our requirements for micron amplitude environmental vibration monitoring and feedback. The maximum sampling rateof the developed nano-resolution laser interferometry sensor was set to 1 kHz, which is sufficient to measure the environmental micro-vibration interference with a frequency mostly less than 50 Hz.