# Experimental Study on Partially Coherent Optical Coherent Detection
## Theoretical Analysis  
### Analysis of the Polarization Characteristics of Partially Coherent Light  
Electromagnetic Gaussian Schell-mode beams are a convenient model for studying coherence and polarization theory, and have attracted increasing attention due to their potential applications [15]. The polarization properties of vectored partially coherent beams can be represented by polarization ellipses. Given that the EGSM beam source is located in the \(z>0\) plane, the beam's cross-spectral density matrix is known to be [16]  
\[W_{ij}=(\rho,\rho,z)=\frac{A_{i}A_{j}B_{ij}}{\Delta_{ij}^{2}(z)}\exp\left[- \frac{\rho^{2}}{2\sigma_{i}\sigma_{j}\Delta_{ij}^{2}(z)}\right](i=x,y;j=x,y) \tag{1}\]  
\(\rho\) represents the position vector of any point on the \(z>0\) plane. \(A_{i}\) and \(A_{j}\) represent the amplitudes of the electric field component of the beam in the \(i\) direction and \(j\) direction, respectively. \(B_{ij}\) is a phase correlation factor. \(\sigma\) represents the effective width of the spectral density in the \(i\) and \(j\) directions. The beam spread equals \(\Delta_{ij}^{2}=1+\alpha_{ij}z^{2}\)\((i=x,y;j=x,y)\), where \(\alpha_{ij}=\frac{1}{(kv)^{2}}\bigg{(}\frac{1}{4e^{2}}+\frac{1}{\delta_{ij}^{2}} \bigg{)}\). The coherent length of the field in various directions on the cross-section is characterized by \(\delta_{ij}\). The polarization of the EGSM beam can be characterized using the polarization ellipse, which represents the entire polarization component of the electromagnetic Gaussian Schell beam. It has been proven that the cross-spectral density matrix of an electromagnetic Gaussian Schell beam can be decomposed into its completely polarized part and completely unpolarized part [16]. \[\mathrm{W}(\rho,\rho,z)=\mathrm{W}^{(u)}(\rho,\rho,z)+\mathrm{W}^{(p)}(\rho, \rho,z) \tag{2}\]  
where  
\[\mathrm{W}^{(u)}(\rho,\rho,z)= \begin{pmatrix}A(\rho,\rho,z)&0\\ 0&A(\rho,\rho,z)\end{pmatrix}  \tag{3}\]  
\[\mathrm{W}^{(p)}(\rho,\rho,z)= \begin{pmatrix}B(\rho,\rho,z)&D(\rho,\rho,z)\\ D(\rho,\rho,z)&C(\rho,\rho,z)\end{pmatrix}  \tag{4}\]  
Here \(A(\rho\),\(\rho\),\(z)\), \(B(\rho\),\(\rho\),\(z)\), \(C(\rho\),\(z)\), and \(D(\rho\),\(z)\) can be expressed as  
\[ \begin{array}{ll}A(\rho,\rho,z)&=\frac{1}{2}\big{[}W_{xx}(\rho,\rho,z)+W_{ yy}(\rho,\rho,z)\\ &+\sqrt{\big{[}W_{xx}(\rho,\rho,z)-W_{yy}(\rho,\rho,z)\big{]}^{2}+4\big{|}W_{ xy}(\rho,\rho,z)\big{|}^{2}}\big{]}\end{array}  \tag{5}\]  
\[ \begin{array}{ll}B(\rho,\rho,z)=&\frac{1}{2}\big{[}W_{xx}(\rho,\rho,z)-W_{ yy}(\rho,\rho,z)\\ &+\sqrt{\big{[}W_{xx}(\rho,\rho,z)-W_{yy}(\rho,\rho,z)\big{]}^{2}+4\big{|}W_{ xy}(\rho,\rho,z)\big{|}^{2}}\big{]}\end{array}  \tag{6}\]  
\[ \begin{array}{ll}C(\rho,\rho,z)&=\frac{1}{2}\big{[}W_{yy}(\rho,\rho,z)-W_{ xx}(\rho,\rho,z)\\ &+\sqrt{\big{[}W_{xx}(\rho,\rho,z)-W_{yy}(\rho,\rho,z)\big{]}^{2}+4\big{|}W_{ xy}(\rho,\rho,z)\big{|}^{2}}\big{]}\end{array}  \tag{7}\]  
\[ \begin{array}{ll}D(\rho,\rho,z)=W_{xy}(\rho,\rho,z)\end{array}  \tag{8}\]  
These variables satisfy a quadratic equation, which can be represented as the polarization ellipse, an elliptic equation. \[C(\rho,\rho,z)e_{x}^{(r)2}(\rho,\rho,z)-2\text{Re}D(\rho,\rho,z)e_{x}^{(r)2}( \rho,\rho,z)e_{y}^{(r)2}(\rho,\rho,z)\] \[+B(\rho,\rho,z)e_{y}^{(r)2}(\rho,\rho,z)=\left[\text{Im}D(\rho,\rho,z)\right]^{2} \tag{9}\]  
where Re and Im stand for a complex number's real and imaginary parts, respectively. \[\epsilon_{x}^{(r)}(\rho,\rho,z)=\sqrt{B(\rho,\rho,z)}\cos(\omega t+\delta_{x}) \tag{10}\]  
\[\epsilon_{y}^{(r)}(\rho,\rho,z)=\sqrt{C(\rho,\rho,z)}\cos(\omega t+\delta_{y}) \tag{11}\]  
\[\delta_{y}-\delta_{x}=\arg[D(\rho,\rho,z)] \tag{12}\]  
Let \(a_{1}\) and \(a_{2}\) be the major and minor axes of an ellipse, respectively. The expression for the ellipticity \(\varepsilon\) is as follows:  
\[