# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 2 Nonlinearity in a Homodyne Interferometer  
A typical homodyne interferometer is shown in Figure 1. The interferometer has a quadrature detector system, and it can be divided into an interferometer part and a detection part. In general,one PBS is located in the interferometer part, whereas two PBSs are in the detection part. A rectangular coordinate system is established, where the propagation direction of the beam is along the Z-axis. The X- and Y-axes are parallel and perpendicular to the paper surface, respectively. The beam from the polarizer P has two orthogonally polarized components: the p-polarized and s-polarized components. This beam is split into two polarized beams by the PBS\({}_{1}\) in the interferometer part, which are then are reflected separately by the reference mirror M\({}_{\rm r}\) and the measuring mirror M\({}_{\rm m}\). Consequently, a phase difference \(\varphi\), which is dependent of the displacement of the measuring mirror, is induced between the two beams. The two polarized beams are combined at the end of the interferometer part. The polarization states of the measuring and reference beams are rotated by 45\({}^{\circ}\) and 135\({}^{\circ}\), respectively, and become 45\({}^{\circ}\) and -45\({}^{\circ}\) after the combined beam passes through the HWP in the detection part. Then, the intensity of the combined beam is evenly split into two beams by the NPBS, and the two beams are split again by the PBS\({}_{2}\) and the PBS\({}_{3}\) in the detection part into four beams to generate interference signals \(I_{1}\), \(I_{2}\), \(I_{3}\) and \(I_{4}\), with a 90\({}^{\circ}\) phase difference between each. The intensity of the four beams is detected by four photoelectric detectors. Two phase-quadrature signals \(I_{0x}\) and \(I_{0y}\) can be obtained from the four signals. Ideally, the phase-quadrature signals \(I_{0x}\) and \(I_{0y}\) should be two sinusoids with a 90\({}^{\circ}\) phase difference. Therefore, the relative phase difference \(\varphi\) with respect to the displacement of the measuring mirror can be derived using the following equations:  
\[ \begin{array}{l}I_{0y}=A\sin(\varphi)\qquad I_{0x}=A\cos(\varphi)\\ \varphi=\arctan(\frac{I_{0y}}{I_{0x}})\end{array}  \tag{1}\]  
where \(A\) is the amplitude of the ideal phase-quadrature signals. The homodyne interferometer is a relative displacement measurement system. \(L_{0}\) represents the relative displacement of the measuring mirror, which is directly proportional to the relative phase difference \(\varphi\) between the reference and measuring beams (assuming that the refractive index of the atmosphere is 1). This relationship can be ideally expressed as:  
\[L_{0}=\frac{\lambda}{4\pi}\varphi \tag{2}\]  
where \(\lambda\) is the wavelength of the laser. Figure 1: Homodyne interferometer with a quadrature detector system. P is the polarizer; PBS\({}_{1}\), PBS\({}_{2}\) and PBS\({}_{3}\) are the polarizing beam splitters; M\({}_{\rm r}\) and M\({}_{\rm m}\) are the mirrors; QWP\({}_{1}\), QWP\({}_{2}\) and QWP\({}_{3}\) are the quarter-wave plates; HWP is the half-wave plate; NPBS is the non-polarizing beam splitter; and D\({}_{1}\), D\({}_{2}\), D\({}_{3}\) and D\({}_{4}\) are the photoelectric detectors. Nevertheless, ensuring that the phase difference between the two quadrature signals is equal to \(90^{\circ}\) is difficult due to the imperfections of the optical components. The relationship between the real displacement \(L\) and the phase difference \(\varphi\) of the interference intensity can be rewritten as follows:  
\[ \begin{array}{l}L=\frac{\lambda}{4\pi}\bigg{[}\arctan\bigg{(}\frac{I_{y}}{I_ {x}}\bigg{)}+m\pi\bigg{]}\\ m=0,1,2,\dots\end{array}  \tag{3}\]  
where \(I_{x}\) and \(I_{y}\) are the quadrature signals in a real case. Therefore, the nonlinear error \(\Delta L\) of the displacement can be expressed as follows:  
\[ \begin{array}{l}\Delta L=\frac{\lambda}{4\pi}\bigg{(}\phi-arc\tan\bigg{(} \frac{I_{y}}{I_{x}}\bigg{)}-m\pi\bigg{)}\\ m=0,1,2,\dots\end{array}