# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 2 Nonlinearity in a Homodyne Interferometer
 \tag{4}\]  
There are many optical components in the homodyne interferometer with quadrature detection system, which have not only manufacturing errors but also setup errors associated with their angular positioning. All of these form the error sources of nonlinearity errors. For the pair of quadrature signals in Equation (3), the nonlinear errors are mainly embodied in three forms: DC offsets, unequal AC amplitudes and quadrature phase error. Some error sources induce DC offsets of the quadrature signals, others produce unequal amplitudes of the quadrature signals, and some cause quadrature phase error of the quadrature signals. Most optical components have no effect on quadrature phase error. For example, the misalignment of the fast axes and phase retardation error of the wave plate only induce DC offsets and unequal AC amplitudes, which can be easily corrected with hardware circuits or software correction algorithms, but not quadrature phase error, which is more difficult to correct. The polarization mixing of the poor performance of the PBSs is the main source that introduces quadrature phase error. We focus on correction of the polarization mixing of the imperfection of the PBSs, so all of the optical components are assumed to be ideal here, except for the PBSs. Here, the Jones matrix and the Jones vector are used to describe an optical element and a polarized beam, respectively. Suppose that the optical components except for the PBSs are ideal. Then, the angles of the polarized beams are exact. In an ideal case, the Jones matrices of the wave plates, PBS, NPBS and incident beam can be expressed as follows:  
\[ \begin{array}{l}Q=\frac{\sqrt{2}}{2}\left[\begin{array}{cc}1&-i\\ -i&1\end{array}\right]\\ T_{0}=\left[\begin{array}{cc}1&0\\ 0&0\end{array}\right]\hskip 28.452756 ptR_{0}=\left[\begin{array}{cc}0&0\\ 0&-1\end{array}\right]\\ T_{n}=\frac{\sqrt{2}}{2}\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]\hskip 28.452756 ptR_{n}=\frac{\sqrt{2}}{2}\left[ \begin{array}{cc}1&0\\ 0&-1\end{array}\right]\\ E_{0}=\left[\begin{array}{cc}1\\ 1\end{array}\right]E_{0}\end{array}  \tag{5}\]  
where \(Q\) and \(H\) are the ideal Jones matrices of the QWP and HWP, respectively. \(T_{0}\) and \(R_{0}\) are the transmittance and the reflectance matrices of PBS, respectively. \(T_{n}\) and \(R_{n}\) are the transmittance and the reflectance matrices of NPBS, respectively. \(E_{0}\) is the electric field vector of the incident beam, the polarization axis of which is at \(45^{\circ}\). In a real case, the PBSs exhibit different optical characteristics with respect to p- and s-polarized beams. On the one hand, a leakage of an unwanted polarized component exists, which causes the polarization mixing. Therefore, a small part of the s-polarized component can pass through the p-polarized beam direction, and vice versa. On the other hand, a slight optical loss is induced by the absorption of the PBS. That is, the transmissivity-to-reflectivity ratio of the PBS is not equal to 1. To describe the imperfect beam-splitting phenomenon of a PBS, the Jones matrix can be modified as follows:  
\[T=\left[ \begin{array}{cc}t_{p}&0\\ 0&t_{s}\end{array} \right]\qquad R=\left[ \begin{array}{cc}r_{p}&0\\ 0&r_{s}\end{array} \right] \tag{6}\]  
where \(T\) and \(R\) represent the modified matrices of the transmittance and reflectance of the PBS, respectively. In addition, \({t_{p}}^{2}\) and \({t_{s}}^{2}\) denote the transmitted intensity of the p- and s-polarized beams, respectively, whereas \({r_{p}}^{2}\) and \({r_{s}}^{2}\) represent the reflected intensity of the p- and s-polarized beams, respectively. Under ideal conditions, \({t_{p}}^{2}\) and \({r_{s}}^{2}\) are equal to 1, whereas \({t_{s}}^{2}\) and \({r_{p}}^{2}\) are equal to zero. For an actual PBS, however, \({t_{p}}^{2}\) and \({r_{s}}^{2}\) are less than 1, whereas \({t_{s}}^{2}\) and \({r_{p}}^{2}\) are not equal to zero. The electric field vectors of the reference beam \(E_{r}\) and the measuring beam \(E_{m}\) at the end of the interferometer part can be written as follows:  
\[