# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 2 Nonlinearity in a Homodyne Interferometer
\begin{array}{l}E_{r}=R_{1}\cdot Q_{2}\cdot Q_{2}\cdot T_{1}\cdot E_{0}\cdot e ^{i\varphi_{r}},\\ E_{m}=T_{1}\cdot Q_{1}\cdot Q_{1}\cdot R_{1}\cdot E_{0}\cdot e^{i\varphi_{m}} \end{array}  \tag{7}\]  
where \(\varphi_{m}\) and \(\varphi_{r}\) are the phases that correspond to the optical paths of the reference and measuring mirrors, respectively. The vectors of the four interference beams at the end of the detection part can be written as a group of Jones matrices as follows:  
\[ \begin{array}{l}E_{1}=T_{2}\cdot Q_{3}\cdot T_{n}\cdot H\cdot(E_{r}+E_{m}) \\ E_{2}=R_{2}\cdot Q_{3}\cdot T_{n}\cdot H\cdot(E_{r}+E_{m})\\ E_{3}=T_{3}\cdot R_{n}\cdot H\cdot(E_{r}+E_{m})\\ E_{4}=R_{3}\cdot R_{n}\cdot H\cdot(E_{r}+E_{m})\end{array}  \tag{8}\]  
The subscript \(i\) (\(i\) = 1, 2, 3) in \(T_{i}\), \(R_{i}\) and \(Q_{i}\) indicates the number of the PBSs and the QWPs, respectively. \(E_{j}\) is the electric field vector of the four beams that reach the photoelectric detector, where the subscript \(j\) (\(j\) = 1, 2, 3, 4) indicates the number of photoelectric detectors. The intensity signals can be obtained through a preamplifier with an adjustable gain \(k_{j}\), where \(j\) takes values of 1 to 4. Then, the two phase-quadrature signals \(I_{x}\) and \(I_{y}\) that can be obtained by subtracting the value of one detector from that of another detector can be expressed as:  
\[ \begin{array}{l}I_{j}=k_{j}\cdot E_{j}\cdot E_{j}^{*}\\ I_{y}=I_{1}-I_{2}\\ I_{x}=I_{3}-I_{4}\end{array}  \tag{9}\]  
where \(E_{j}\)* is the conjugate transpose matrix of \(E_{j}\). By calculating the Jones matrix, the parameters of the two sinusoidal intensity signals can be obtained as follows:  
\[\frac{I_{y}}{I_{x}}=\frac{A\ +B\text{cos}(\varphi)+C\sin(\varphi)}{D+E\cos( \varphi)} \tag{10}\]  
where \(\varphi(\varphi=\varphi_{m}-\varphi_{r})\) is the phase difference between the reference beam and the measuring beam. Equations (5)-(8) are substituted into Equations (9) and (10), and thus, the parameters of Equation (10) can be written in detail as follows:  
\[ \begin{array}{l}A=\left[k_{2}\left(r_{s2}^{2}+r_{p2}^{2}\right)-k_{1}\left( r_{p2}^{2}+t_{s2}^{2}\right)\right]\cdot t_{p1}^{2}r_{s1}^{2}\\ B=2\left[k_{2}\left(r_{s2}^{2}+r_{p2}^{2}\right)-k_{1}\left(r_{p2}^{2}+t_{s2}^{ 2}\right)\right]\cdot t_{p1}r_{p1}t_{s1}r_{s1}\\ C=\left[k_{2}\left(r_{s2}^{2}-r_{p2}^{2}\right)+k_{1}\left(r_{p2}^{2}-t_{s2}^{ 2}\right)\right]\cdot t_{p1}^{2}r_{s1}^{2}\\ D=\left[k_{4}\left(r_{s3}^{2}+r_{p3}^{2}\right)-k_{3}\left(r_{p3}^{2}+t_{s3}^{ 2}\right)\right]\cdot t_{p1}^{2}r_{s1}^{2}+2\left[k_{4}\left(r_{s3}^{2}-r_{p3 }^{2}\right)+k_{3}\left(t_{p3}^{2}-t_{s3}^{2}\right)\right]\cdot t_{p1}r_{p1}t _{s1}r_{s1}\\ E=2\left[k_{4}\left(r_{s3}^{2}+r_{p3}^{2}\right)-k_{3}\left(r_{p3}^{2}+t_{s3}^{ 2}\right)\right]\cdot t_{p1}r_{p1}t_{s1}r_{s1}+\left[k_{4}\left(r_{s3}^{2}-r_{ p3}^{2}\right)+k_{3}\left(t_{p3}^{2}-t_{s3}^{2}\right)\right]\cdot t_{p1}^{2}r_{s1}^{2} \end{array}  \tag{11}\]Here, the values of \({r_{p}}^{2}\) and \({t_{s}}^{2}\) are extremely small, such that multiplying \({r_{p}}^{2}\) by \({t_{s}}^{2}\) can yield zero. When the gains of the four interference signals are the same (i.e., \(k_{1}=k_{2}=k_{3}=k_{4}\)), parameters \(A\), \(B\) and \(D\) are not equal to zero in Equation (11), whereas parameter \(C\) is not equal to \(E\). Parameters \(A\) and \(B\), the first term of \(D\) and the first term of \(E\), which are induced by the PBS\({}_{2}\) and the PBS\({}_{3}\) in the detection part, can be adjusted to zero and then removed by adjusting the gains of the detectors. However, the cross term of \({t_{p}}{1_{p}}{1_{s}}{1_{s}}{1_{s}}{1_{s}}{1_{s}}\) that corresponds to the polarization mixing of the PBS\({}_{1}\) of the interferometer part cannot be eliminated. Therefore, polarization mixing is the primary factor that causes the nonlinear error for the PBS\({}_{1}\) in the interferometer part. For the PBS\({}_{2}\) and the PBS\({}_{3}\) in the detection part, the unequal separation of optical intensity is the dominant factor that results in the nonlinear error.