## 3 Nonlinear Errors Caused by Optical Elements
)\\ -\text{atan}\Bigl{(}\frac{\sin(\Delta\varphi)}{\cos(\Delta\varphi)}\Bigr{)} \end{array} \tag{11}\]  
\[ \begin{array}{l}here,\left\{\begin{array}{l}a_{p1m}=t_{p1\gamma}s_{1}\approx 1 \\ b_{p1m}=t_{s1}r_{p1\beta}\approx 0\end{array}\right.\end{array}  \tag{12}\]where \(t_{p1}\), \(t_{s1}\) and \(r_{p1}\), \(r_{s1}\) represent the transmission coefficient and the reflection coefficient of PBS1 with respect to the P light and S light, respectively. If the MALDI has \(n\) optical elements, there will be \(3n\) installation errors and \(4n\) manufacturing errors. Therefore, it is time-consuming to adjust the installation errors and test the manufacturing errors one by one. An error sensitivity analysis of the installation errors and the manufacturing errors on the nonlinear errors is proposed to enhance the understanding of which error components are more important or influence the measurement accuracy of the MALDI. The sensitivity coefficient (\(S_{i}\)) of each error component (\(E_{i}\)) can be expressed to be  
\[S_{i}=\frac{\left|\frac{\partial\mathbb{z}}{\partial E_{i}}\right|}{\sum \left|\frac{\partial\mathbb{z}}{\partial E_{i}}\right|} \tag{11}\]  
Each installation error of the optical elements is considered as in the same error range, which is from \(3^{\circ}\) to \(-3^{\circ}\). The sensitivity analysis results of the installation errors are shown in Figure 3. \(\alpha_{i}\), \(\beta_{i}\) and \(\gamma_{i}\) (\(i=p1\), \(p2\), \(p3\), \(q1\), \(q2\), \(q3\), \(n\)) represent the installation errors around \(X\)-, \(Y\)- and \(Z\)-axis of each optical element (PBS1, PBS2, PBS3, Q1, Q2, Q3, NPBS) of the DM. It can be seen from the figure that the sensitive error components of the installation errors on the nonlinear errors are \(\alpha_{p1}\), \(\beta_{p1}\), \(\gamma_{p1}\), \(\alpha_{p2}\), \(\beta_{p2}\), \(\alpha_{p3}\), \(\beta_{p3}\), \(\gamma_{q3}\), \(\alpha_{n}\) and \(\beta_{n}\). The effects of the installation errors of the quarter-waveplate (Q1, Q2, Q3) on the nonlinear errors were less than the effects of the installation errors of the beamsplitter (PBS1, PBS2, PBS3, NPBS) on the nonlinear errors. According to the above analysis, the effects of the installation errors and the manufacturing errors of the optical elements can be reduced by carefully adjusting and selecting the optical elements. However, it is difficult to totally eliminate those effects. Therefore, an algorithm was proposed to eliminate the residual nonlinear errors. The flowchart of the algorithm is shown in Figure 4. The DC offsets and the AC amplitudes are first eliminated by dynamically checking the peak values of \(I_{\sin}\) and \(I_{\cos}\). The phase delay is then eliminated by using the vector summation and subtraction methods [20]. The normalization method is applied to eliminate the new AC amplitude difference between two quadrature signals, which resulted from the vector summation and subtraction methods. Finally, a sinusoidal signal and a cosine signal without the AC amplitude difference, the DC offset and the phase delay were obtained for measuring the displacement. Some simulations were conducted to verify the effectiveness of the method for the elimination of the nonlinear errors. Figure 5 shows the corrected sinusoidal and cosine signals and the corresponding Lissajous figure with respect to Figure 2. It can be seen that the AC amplitudes, the phase delay, the DC offsets and the high-order nonlinear harmonics of the sinusoidal and cosine signals were eliminated. Figure 3: Error sensitivity analysis of the installation errors.