# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 3 Compensation for the Polarization Mixing of PBS\({}_{1}\)  
The nonlinear error caused by the PBSs in the detection part can be reduced by adjusting the gains of the four photo-detectors, but the polarization mixing caused by the PBS in the interferometer part cannot be eliminated [17]. To solve this problem, an optical compensation method is proposed to eliminate the polarization mixing of the PBS\({}_{1}\) in the interferometer part. Figure 3 shows the optical layout of the homodyne interferometer with an optical compensation part. The separation of the reference and measuring beams, which become two separate parallel beams, at the end of the interferometer part is critical. To achieve this, a right-angle prism is used to replace the planar mirror as the reference mirror. The unwanted s-polarized component in the p-polarized beam and the unwanted p-polarized component in the s-polarized beam are isolated through the P\({}_{2}\) and the P\({}_{3}\), respectively, with a high extinction ratio of approximately 10,000:1. Then, the pure polarized measuring and reference beams are combined by the PBS\({}_{0}\). The polarization mixing of the PBS\({}_{1}\) is theoretically eliminated by adopting this optical compensation. Thus, the inherent polarization mixing of the homodyne interferometer caused by the PBS\({}_{1}\) can be suppressed. Figure 2: The comparison of nonlinearity before and after the gain adjustment: the blue dotted curve—before gain correction; the red solid curve—after gain adjustment. To prevent the generation of the unwanted s- and p-polarized components, the transmission axes of the P2 and the P3 should coincide with the axes of the reference and measuring beams, respectively, thereby ensuring that the measuring pure p-polarized beam and the reference pure s-polarized beam are produced. The ideal Jones matrices of the \(P_{2}\) and the \(P_{3}\) with a high extinction ratio can be expressed as follows:  
\[P_{2}=\left[ \begin{array}{cc}0&0\\ 0&1\end{array} \right]\qquad P_{3}=\left[ \begin{array}{cc}1&0\\ 0&0\end{array} \right] \tag{15}\]  
Assuming that the PBS0 and the M0 are ideal optical components is reasonable because the incident beams passing through the PBS0 are nearly ideal linear polarized beams that originate from the P2 and the P3 with a high extinction ratio. Even if the PBS0 has an imperfect property, it will only induce optical energy loss but will not cause polarization mixing. This nonlinear error can be compensated for by adjusting the gains of the detectors. The ideal matrix Equation (15) of the P2 and the P3 is substituted into Equation (8). The two intensity signals of the homodyne interferometer can be obtained by recalculating Equations (5)-(10) as follows:  
\[\frac{I_{y}}{I_{x}}=\frac{[k_{2}(r_{s2}^{2}+r_{p2}^{2})-k_{1}(t_{p2}^{2}+t_{s2 }^{2})]t_{p1}^{2}r_{s1}^{2}+[k_{2}(r_{s2}^{2}-r_{p2}^{2})+k_{1}(t_{p2}^{2}-t_{ s2}^{2})]t_{p1}^{2}r_{s1}^{2}\sin\varphi}{[k_{4}(r_{s3}^{2}+r_{p3}^{2})-k_{3}(t_{p3}^{2}+t_{ s3}^{2})]t_{p1}^{2}r_{s1}^{2}+[k_{4}(r_{s2}^{2}-r_{p2}^{2})+k_{3}(t_{p2}^{2}-t_{ s2}^{2})]t_{p1}^{2}r_{s1}^{2}\cos\varphi} \tag{16}\]  
As shown in Equation (15), the cross term of \(t_{p1}t_{s1}r_{p1}r_{s1}\) caused by the polarization mixing of the PBS1 has been completely eliminated after using the P2 and the P3 with a high extinction ratio, the PBS2 and the PBS3 have the same parameters as those listed in Section 2, and gain adjustment is applied according to Equation (12). From Equation (15), parameters A and D are equal to zero, whereas parameter C is equal to parameter E. At this point, the signals \(I_{x}\) and \(I_{y}\) have become quadrature signals in Equation (15). That is, all nonlinear errors have been theoretically eliminated. In reality, the linear polarizers with a high extinction ratio are not perfect but experience a small amount of light leakage. Suppose that the position and angle errors of the optical components are disregarded.