## 1 Introduction
However, the effects of the DoP of the GSM beams propagating along an uplink path and a downlink path in atmospheric turbulence on heterodyne detection have yet to be described. Therefore, the effects of the DoP of a partially coherent Gaussian Schell-model beam on a heterodyne detection system are investigated in this study as the signal beam propagating along an uplink or downlink path in atmospheric turbulence. Several analytical expressions for signal beam and LO beam as polarized and partially coherent GSM beam of the DoP and heterodyne efficiency are derived in our work based on a model of heterodyne detection for a partially coherent beam under the modified Hill spectrum of the refractive index fluctuations. Figure 1: Diagram of laser beam propagation at an uplink or downlink transmission path. Mathematical Model of Heterodyne Detection with Partially Polarized GSM Beam Propagating along an Uplink Path and a Downlink Path in Turbulence  
We suppose the signal beam and LO beam are partially polarized and partially coherent, although the LO beam is commonly polarized and coherent, to obtain a general heterodyne efficiency expression for a heterodyne detection system that includes the coherence and polarization characteristics of the light source. Let us consider that the signal beams and the LO beams are parallel to the optical axis and perpendicular to the detector surface. The instantaneous field of both beams on the detector surface can be expressed, respectively, as [5]:  
\[U_{L}(\mathbf{\rho},t) =\big{[}A_{Lx}(\mathbf{\rho})e_{x}+A_{Ly}(\mathbf{\rho})e_{y}\big{]}e^{i \omega_{L}t} \tag{1}\] \[U_{S}(\mathbf{\rho},t) =\big{[}A_{Sx}(\mathbf{\rho})e_{x}+A_{Sy}(\mathbf{\rho})e_{y}\big{]}e^{i \omega_{S}t} \tag{2}\]  
where \(e_{x}\) and \(e_{y}\) are unit vectors, \(\omega_{L}\) and \(\omega_{s}\) are the central angular frequencies of the fields and \(A_{Lx}\), \(A_{Ly}\) and \(A_{Sx}\), \(A_{Sy}\) are the optical field distribution of the LO beam and the received signal beam in the direction of \(x\) and \(y\), respectively. Based on the model of heterodyne detection, the detected intermediate frequency (IF) power can be evaluated according to the character of a square-law device, and the IF power is expressed as [21]:  
\[P_{IF}=\Re^{2}\iint\iint\Re\big{\{}T_{r}\big{[}\mathbf{W}_{Lij}(\mathbf{\rho}_{1},\mathbf{ \rho}_{2},0)\mathbf{W}_{Sij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)\big{]}\big{\}}d^{2} \rho_{1}d^{2}\rho_{2} \tag{3}\]  
where \(T_{r}\) denotes the trace, \(\mathbf{W}_{Lij}\) and \(\mathbf{W}_{Sij}\) are the beam coherence polarization (BCP) matrix of the LO beam and signal beam, respectively [15], and \(\Re\) is the detector responsivity, which is assumed to be uniform across the detector surface. For coherent detection, a measurement of heterodyne performance is the dimensionless heterodyne efficiency \(\eta_{het}\), which measures the loss in coherent power when the received signal and LO fields are not perfectly matched. The heterodyne efficiency for random fields is defined in an analogous manner in the case of deterministic fields, which can be expressed as [21]:  
\[\eta_{het}=\frac{\iint\iint\Re\big{\{}T_{r}\big{[}\mathbf{W}_{Lij}(\mathbf{\rho}_{1}, \mathbf{\rho}_{2},0)\mathbf{W}_{Sij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)\big{]}\big{\}}d^{2 }\rho_{1}d^{2}\rho_{2}}{\iint\Tr[\mathbf{W}_{Lii}(\mathbf{\rho},\mathbf{\rho},0)]d^{2}\rho \iint\Tr[\mathbf{W}_{Sii}(\mathbf{\rho},\mathbf{\rho},0)]d^{2}\rho} \tag{4}\]  
Equation (4) is the basic mathematical model for the partially coherent beam of heterodyne detection. In the following part, we will establish a mathematical model of a heterodyne detection considering the change of the DoP of the signal beam propagating along an uplink path and a downlink path in atmospheric turbulence.