## 1 Introduction  
It is well known that optical heterodyne (coherent) detection technology is widely used in free-space optical communications (FSOCs), broadcasting, radar systems and other fields since the reception sensitivity of this technology is nearly 20 dB higher than that of intensity modulation/direct detection methods [1, 2, 3, 4, 5]. Heterodyne detection has been widely used in satellite-satellite [6], satellite-ground [7, 8] and ground-ground [9] communication links in recent decades. The most significant effect on the performance of FSOC systems in the latter two applications is atmospheric turbulence [10], since the stochastic motion of the turbulent medium affects laser beam propagation. This propagation distortion causes a mismatch in the amplitude, phase, polarization and other characteristics of the signal beam and LO beam, reducing spatial coherence and the performance of the coherent optical communication system [11]. As a result, the polarization state and distorted wavefront of the signal beam should be corrected before heterodyne detection can be performed. Polarization state is a vector characteristic of electromagnetic light fields that is commonly employed in polarization control, free-space optical communications, particle capture and target detection and identification. The degree of polarization (DoP) is a physical quantity that describes the polarization state of a vector partly coherent beam. The signal beam propagating in the satellite-ground field (i.e., Figure 1) is less affected by atmospheric turbulence than the ground-ground communication links. The signal beam propagates through the atmosphere through an uplink or downlink path to reach the receiving terminal, as shown in Figure 1. The distance from the ground to the lower terminal and the vertical height of the receiving terminal, as illustrated in Figure 1, are \(h_{0}\) and \(H\), respectively. \(\theta\) is the path zenith angle. The uplink and downlink transmission paths of the signal beam distorted by turbulent atmosphere are indicated by the red solid and dotted lines, respectively. There has been an increased interest in the effect of atmospheric turbulence on the performance of heterodyne detection in recent years. Using the cross-spectral function of a GSM beam propagating through atmospheric turbulence, for example, [PERSON] et al. [12] derived a closed-form expression of heterodyne efficiency. [PERSON] et al. [5] developed a mathematical model for a heterodyne detection system for partially coherent Gaussian Schell beams under the Tatarskii turbulence spectrum in 2015. Furthermore, some of the authors of this paper investigated the effect of signal beam propagation in atmospheric turbulence on heterodyne detection, e.g., angle-of-arrival fluctuation [3], beam mode of the partially coherent Gaussian Schell-model beam [1; 2], irradiance and phase fluctuations [13] and wavefront correction [10]. However, these works were based on the vector theory of partially coherent light fields or Gaussian beam. In electromagnetic fields, partial polarization is a well-known phenomenon [5]. [PERSON] et al. investigated the polarization characteristics of the GSM beam [14] and electromagnetic Gaussian Schell-model vortex (EGSMV) beam in atmospheric turbulence [15]. In 2017, [PERSON] et al. [16] analyzed the effects of polarization aberrations on the output polarizations states. The signal-to-noise ratio (SNR) and the heterodyne efficiency for the partly polarized and coherent beam in free space were both expressed using the beam coherence-polarization (BCP) matrix, which was developed by [PERSON] et al. [17; 18]. [PERSON] et al. analyzed the polarization properties of the mixing of statistically independent optical fields in a coherent optical field mixing scheme [19]. [PERSON] et al. [20] investigated the impact of beam polarization on the effectiveness of the heterodyne detecting system.