# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 2 Nonlinearity in a Homodyne Interferometer
Therefore, the nonlinearity caused by the imperfections of the PBSs at different positions of the homodyne interferometer should be compensated using different means. For the imperfections of the PBS\({}_{1}\) in the interferometer part, the focus is on removing the unwanted s-polarized beam in the transmitted beam and the unwanted p-polarized beam in the reflected beam. For the imperfections of the PBS\({}_{2}\) and the PBS\({}_{3}\) in the detection part, the key is to make the transmitted intensity-to-reflected intensity ratio approach 1. The improvement of the traditional gain-adjustment method can be evaluated by assuming that \(A\), \(B\) and the first terms of \(D\) and \(E\) are all equal to zero. The following conditions for eliminating nonlinear error can be obtained:  
\[ \begin{array}{l}\frac{k_{2}}{k_{1}}=\frac{{t_{s}^{2}+t_{p2}^{2}}}{{r_{s}^{2}+ t_{p2}^{2}}}\\ \frac{k_{4}}{k_{3}}=\frac{{t_{s}^{2}+t_{p3}^{2}}}{{r_{s}^{2}+t_{p3}^{2}}}\\ \end{array}  \tag{12}\]  
Considering the symmetry of the optical layout of the detection part, we select the PBS\({}_{2}\) and the PBS\({}_{3}\) with the same performance parameters, thereby indicating that the reflection and transmission for all the PBSs are the same. The actual parameters of the PBSs are shown in the Table 1.  
Therefore, the relationships of the gains of four photodetectors can be obtained as follows:  
\[ \begin{array}{l}k_{1}=k_{3}\qquad k_{2}=k_{4}\\ \frac{k_{2}}{k_{1}}=\frac{k_{4}}{k_{3}}=0.932\\ \end{array}  \tag{13}\]  
By introducing the preceding modified results into Equation (11), Equation (10) can be rewritten as follows:  
\[\frac{I_{y}}{I_{x}}=\frac{{t_{p1}^{2}}{r_{s1}^{2}}\sin\varphi}{2\cdot{t_{p1}}{ r_{p1}}{t_{s1}}{r_{s1}}+{t_{p1}^{2}}{r_{s1}^{2}}\cos\varphi} \tag{14}\]  
As shown in Equation (14), gain adjustment can only eliminate the nonlinear errors caused by the PBS\({}_{2}\) and the PBS\({}_{3}\) in the detection part. Suppose that the PBS\({}_{2}\) and the PBS\({}_{3}\) are ideal components and only the PBS\({}_{1}\) is not ideal. Then, the result of the two quadrature signals is the same as that of Equation (14), which shows that gain correction cannot compensate for the polarization mixing caused by the PBS\({}_{1}\) in the interferometer part. Here, common commercial PBSs with the properties as mentioned above are chosen, and the nonlinear errors before and after gain correction are simulated using MATLAB software. The results without gain adjustment, i.e., Equation (10), and with gain adjustment, i.e., Equation (14), are introduced \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optical Component** & \({t_{p}}^{2}\) & \({t_{s}}^{2}\) & \({r_{p}}^{2}\) & \({r_{s}}^{2}\) \\ \hline PBS\({}_{1}\) & 0.95 & 0.01 & 0.05 & 0.99 \\ \hline PBS\({}_{2}\) & 0.95 & 0.01 & 0.05 & 0.99 \\ \hline PBS\({}_{3}\) & 0.95 & 0.01 & 0.05 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the optical components. into Equation (4), respectively. Then, the values of the nonlinear errors are simulated, as shown in Figure 2, where the wavelength of the laser is 632.8 nm. The blue dotted curve shows the nonlinear error without any correction, and the peak-to-peak value is approximately 10 nm. The red solid curve shows the nonlinear error when gain correction is applied, and the peak-to-peak value is approximately 4.5 nm. The preceding theoretical analysis and simulation show that reducing the nonlinear error by adjusting only the gains of the detectors for a general PBS is ineffective.