## 2 Measurement Principle of the MLDI  
Figure 1 shows the schematic diagram of the MLDI, which is composed of a moved part and a fixed part. The fixed part consists of a displacement meter (DM), a laser wavelength meter (LWM) and a laser wavelength stabilizer (LWS). The moved part is composed of a corner cube retroreflector (CR1).  
A laser beam from a laser diode (LD), whose wavelength is measured and stabilized by the LWM and LWS, is divided into two beams by a beam splitter (BS1) after being bent by a mirror (M1). The transmitted beam from BS1 enters into the LWM, which consists of a grating (G), two focus lenses (Fl1 and FL2), two quadrant photodetectors (QPD1 and QPD2), a thermistor (PT100) and two mirrors (M3 and M4), for real-time measuring of the LD wavelength based on the principle of the diffraction grating. The detailed principles of  
Figure 1: Schematic diagram of the MLDI.  
the LWM can be found in our previous research [6] and are not repeated in this paper for the sake of brevity. The LWS is composed of a thermistor, two thermoelectric coolers (TECs) and an LD holder. The principle of the LWS is introduced in the next section. The reflected beam from BS1 is divided again by a polarized beam splitter (PBS1). The reflected beam from PBS1 is projected onto CR2. The transmitted beam from PBS1 is projected onto the CR1 of the DM, which consists of two beam splitters (BSs, BS2 and BS3), three PBSs (PBS1 to PBS3) and four photodetectors (PDs, PD1 to PD4). The reflected beams from CR1 and CR2 are entered into PBS1 again. When CR1 has a motion of \(z\), the count of the interference fringes, which is caused by the reflected beams from CR1 and CR2, will be changed and further cause the output variations of the PDs (PD1 to PD4). According to the outputs of the PDs, the intensity of the sinusoidal and cosine signals in the ideal case can be expressed by [6; 20],  
\[I_{\text{sin}}=\sin(\Delta\varphi)\text{ and }I_{\text{cos}}=\cos(\Delta\varphi), \tag{1}\]  
where \(\Delta\varphi\) is the phase difference between the initial and final phases of the incomplete sinusoidal waves. However, due to the installation errors and the manufacturing errors of the optical elements of the MALDI nonlinear errors, the intensity of the sinusoidal and cosine signals will be varied to  
\[I^{\prime}_{\text{sin}}=E_{\text{sin}}\sin(\Delta\varphi+\delta)+F_{\text{sin} }\text{ and }I^{\prime}_{\text{cos}}=E_{\text{cos}}\cos(\Delta\varphi)+F_{\text{cos}}, \tag{2}\]  
where \(E_{\text{sin}}\) and \(E_{\text{cos}}\), \(\delta\) and \(F_{\text{sin}}\) and \(F_{\text{cos}}\) represent the AC amplitudes, the phase delay and the DC offsets of the sinusoidal and cosine signals, respectively. The sinusoidal and cosine signals in this case are shown in Figure 2a. The Lissajous figure with respect to Figure 2a will thus be distorted, as shown in Figure 2b.  
In addition, the low wavelength accuracy and stability of the LD will influence the accuracy and the stability of the MALDI. Therefore, the moving displacement \(z\) of the moved part can be calculated by [6]  
\[z=\left(N+\frac{\Delta\varphi}{2\pi}\right)\cdot\frac{\lambda}{2n}+z_{en}+z_{ew} \tag{3}\]  
where \(N\), \(\lambda\), \(n\) and \(\Delta\varphi\) represent the integer count of the interference fringes, the wavelength of the LD, the refractive index of the air and the phase difference between the initial phase and the final phase, respectively. \(z_{en}\) and \(z_{ew}\) represent the measurement errors caused by the nonlinear errors and the unstable LD wavelength, respectively. In order to improve the measurement accuracy of the MALDI, \(z_{en}\) and \(z_{ew}\) should be eliminated and compensated.  
Figure 2: Distorted signals: (**a**) sinusoidal and cosine signals and (**b**) Lissajous figure.