# Compensation Method for Polarization Mixing in the Homodyne Interferometer
## 1 Introduction  
The requirements for dimensional measurement accuracy are increasing with the rapid development of nanotechnology. A homodyne interferometer has become one of the most important tools for nanometre measurement because of its high measurement accuracy and traceability to international length standards [1]. However, the measurement accuracy of a homodyne interferometer with a quadrature detector system is restricted by nonlinear errors [2]. Several compensation methods for reducing the nonlinearity have been suggested. An ellipse-fitting method was proposed firstly by [PERSON] [3], which was later improved by [PERSON] and [PERSON] [4; 5]. In this method, two raw signals are fitted into a general elliptical equation using the least-squares method and then the nonlinear coefficients are calibrated from the calculated parameters of the general elliptical equation. Therefore, the total nonlinear errors including the interferometer itself and the signal detection system can be corrected, thereby proving that this approach is a comprehensive method. However, this method is time-consuming and unsuitable for real-time compensation because the coefficients of the elliptical equation should be calculated prior to the real measurement, which requires considerable computational effort. Then, [PERSON] devised a fine correction method involving Heydemann correction [6]. The proposed method suppresses the nonlinearity in a homodyne interferometer to0.18 nm, which would be 1.49 nm with conventional Heydemann correction. However, the step-by-step fine-ellipse fitting is quite time-consuming and is not suitable for real-time measurement. [PERSON] further studied the Heydemann correction and proposed a new ellipsefitting method based on the minimization of the geometric distance between the measured and fitted signal values [7, 8]. The statistical uncertainty of the Heydemann correction is analyzed in detail, which has a high dependency on the signal-to-noise ratio of signals. Methods using hardware-based correction have been proposed to eliminate the three primary errors (DC offsets, unequal AC amplitudes and quadrature phase error). A real-time correction technique based on the use of a microcontroller and computational analogue circuits has been developed, and the better the quality of the quadrature signals, the more accurate the method is [9]. Meanwhile, [PERSON] provides a real-time compensation of the variable cyclic errors utilizing a field-programmable gate array in a homodyne laser interferometer [10]. Others have proposed their solutions to improve the optical configurations. [PERSON] proposed an optical structure that consists of two homodyne interferometers in which two orthogonal single-frequency beams share a common reference arm and partial measurement arm [11, 12]. A series of experiments demonstrate that the proposed interferometer is able to realize sub-nanometer accuracy without nonlinearity for nanometer displacement and nanometer accuracy for millimeter displacement. [PERSON] has proposed an optical layout based on a non-polarizing beam splitter (NPBS) and balanced interference between two circularly polarized beams [13]. Besides the replacing polarizing beam splitters (PBSs) of the detection part with Wollaston prisms (WPs) with a high extinction ratio (\(1\times 10^{5}\)), the PBS in the interference part is replaced with an NPBS, the half-wave plate (HWP) and the quarter-wave plate (QWP) in the detection part are removed. The optical layout is simple, but the cost of the WP is more than the cost of the PBS. For a homodyne interferometer, existing research indicates that the quadrature phase error is the main effect that produces nonlinearity error, and the polarization mixing introduced by the poor performance of the PBSs is the main error source that causes the quadrature phase error [13, 14, 15, 16]. The nonlinearity caused by the imperfections of the PBSs has been studied by some researchers, and several compensation methods have been proposed.