## 3 Nonlinear Errors Caused by Optical Elements  
Nonlinear errors, which are induced by installation errors and manufacturing errors, will limit the performance of the MALDI. Here, PBS1 is taken as an example to analyze the nonlinear errors, which are induced by the installation errors and the manufacturing errors. In an ideal case, the Jones matrix of PBS1 can be expressed by [24]  
\[J_{PBS,S}= \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \text{and }J_{PBS,P}= \begin{bmatrix}0&0\\ 0&1\end{bmatrix}  \tag{4}\]  
When PBS1 has an installation error \(\alpha\) with respect to the X-axis, the Jones matrix of PBS1 is revised to be  
\[\left\{ \begin{array}{l}J^{\prime}{}_{PBS,S}=\left[\begin{array}{cc}\cos \arctan(\sin\alpha)&\sin\arctan(\sin\alpha)\\ 0&0\end{array}\right]\\ J^{\prime}{}_{PBS,P}=\left[\begin{array}{cc}0&0\\ -\sin\arctan(\sin\alpha)&\cos\arctan(\sin\alpha)\end{array}\right]\end{array} \right. \tag{5}\]  
The sinusoidal and cosine signals of the MALDI can thus be expressed by  
\[\left\{ \begin{array}{l}I_{\sin p1\alpha}=2E_{\sin p1\alpha}^{2}a_{p1\alpha}b _{p1\alpha}\cos\arctan(\sin\alpha)\sin(\Delta\varphi)\\ I_{\cos p1\alpha}=2E_{\cos p1\alpha}^{2}[a_{p1\alpha}^{2}\cos\arctan(\sin \alpha)\sin\arctan(\sin\alpha)+a_{p1\alpha}b_{p1\alpha}\cos\arctan(\sin\alpha )\cos(\Delta\varphi)]\\ here,\left\{\begin{array}{l}a_{p1\alpha}=(\cos\arctan(\sin\alpha)-\sin \arctan(\sin\alpha))\cos\arctan(\sin\alpha)\\ b_{p1\alpha}=(\cos\arctan(\sin\alpha)+\sin\arctan(\sin\alpha))\cos\arctan( \sin\alpha)\end{array}\right.\end{array} \right. \tag{6}\]  
where \(E_{\sin p1\alpha}\) and \(E_{\cos p1\alpha}\) represent the AC amplitudes of the sinusoidal and cosine signals when PBS1 has an installation error \(\alpha\). Therefore, a nonlinear error \(e_{p1\alpha}\) caused by \(\alpha\) can be expressed by  
\[ \begin{array}{l}e_{p1\alpha}=\frac{\lambda}{4\pi}(\arctan\biggl{(}\frac{a_ {p1\alpha}b_{p1\alpha}\cos\arctan(\sin\alpha)\sin(\Delta\varphi)}{a_{p1\alpha} ^{2}\cos\arctan(\sin\alpha)\sin\arctan(\sin\alpha)+a_{p1\alpha}b_{p1\alpha} \cos\arctan(\sin\alpha)\cos(\Delta\varphi)}\biggr{)}\\ -\arctan\Bigl{(}\frac{\sin(\Delta\varphi)}{\cos(\Delta\varphi)}\Bigr{)} \end{array}  \tag{7}\]  
Similarly, the nonlinear errors \(e_{p1\beta}\) and \(e_{p1\gamma}\) caused by the installation errors \(\beta\) and \(\gamma\) with respect to the \(Y\)- and \(Z\)-axis, respectively, can be expressed by  
\[ \begin{array}{l}e_{p1\beta}=\frac{\lambda}{4\pi}(\arctan\biggl{(}\frac{a_ {p1\beta}b_{p1\beta}\cos\arctan(\sin\beta)\sin(\Delta\varphi)}{a_{p1\alpha}^{ 2}\cos\arctan(\sin\beta)\sin\arctan(\sin\beta)+a_{p1\beta}b_{p1\beta}\cos \arctan(\sin\beta)\cos(\Delta\varphi)}\biggr{)}\\ -\arctan\Bigl{(}\frac{\sin(\Delta\varphi)}{\cos(\Delta\varphi)}\Bigr{)} \end{array}  \tag{8}\]  
\[ \begin{array}{l}here,\left\{\begin{array}{l}a_{p1\beta}=(\cos\arctan(\sin \beta)-\sin\arctan(\sin\beta))\cos\arctan(\sin\beta)\\ b_{p1\beta}=(\cos\arctan(\sin\beta)+\sin\arctan(\sin\beta))\cos\arctan(\sin \beta)\end{array}\right.\end{array}  \tag{9}\]  
\[ \begin{array}{l}e_{p1\gamma}=\frac{\lambda}{4\pi}(\arctan\Bigl{(}\frac{a_ {p1\gamma}\sin(\Delta\varphi)}{a_{p1\gamma}c_{p1\gamma}\cos\arctan(\sin\gamma)+ b_{p1\gamma}c_{p1\gamma}}\Bigr{)}-\arctan\Bigl{(}\frac{\sin(\Delta\varphi)}{\cos( \Delta\varphi)}\Bigr{)})\\ here,\left\{\begin{array}{l}a_{p1\gamma}=(\cos\gamma-\sin\gamma)(\cos \gamma\cos\arctan(\sin\gamma)-\sin\alpha\sin\arctan(\sin\gamma))\\ b_{p1\gamma}=(\sin^{3}\gamma-\sin\alpha\cos^{3}\gamma)(\cos\gamma+\sin\gamma)\\ c_{p1\gamma}=\cos\alpha(\cos^{2}\gamma-\sin^{2}\gamma)(\cos\gamma+\sin\gamma) \end{array}\right.\end{array}  \tag{10}\]  
A nonlinear error \(e_{p1m}\) caused by the manufacturing error of PBS1 can be expressed by  
\[ \begin{array}{l}e_{p1m}=\frac{\lambda}{4\pi}(\begin{array}{l}\text{atan}( \frac{4a_{p1m}^{2}\sin(\Delta\varphi)+(t_{p1}+t_{s1}-r_{p1}-r_{s1})(2a_{p1m}^{2} +4a_{p1m}b_{p1m}\cos(\Delta\varphi))}{(t_{p1}^{2}+t_{p2}^{2})\cos(\Delta\varphi) +2t_{p1}^{2}(a_{p1m}+b_{p1m})^{2}-2t_{s3}^{2}(a_{p1m}-b_{p1m})^{2}}\end{array} \end{array}