# Experimental Study on Partially Coherent Optical Coherent Detection
## Theoretical Analysis
### Polarization States and Mixing Efficiency
With the use of a quarter wave plate, the polarization state of the beam is altered in order to investigate the impact of both circular and elliptical polarization on the operation of the optical mixer. One feature of the quarter wave plate is its ability to modify the polarization state of light without influencing its intensity or signal power. Consequently, by altering the angle \(\theta\) between the fast axis of the plate, a quarter wave plate can be added at the incident end to replicate various polarization states of the signal light and to assess and confirm the intensity of the optical mixer's output signal. A quarter wave plate was chosen to be added to the optical path of the signal light incident on the polarization splitting prism in the original design of the ninety degree mixer. For subsequent calculations, the added quarter wave plate will be represented as  
\[\Lambda_{1/4}=\frac{\sqrt{2}}{2} \begin{pmatrix}1-i\cos(2\theta)&-i\sin(2\theta) \\ -i\sin(2\theta)&1+\mathrm{icos}(2\theta)\end{pmatrix}  \tag{32}\]  
The signal and local oscillator light that is incident on the mixer can be written as  
\[E_{s}^{\prime}=\Lambda_{1/4}E_{s}=\frac{\sqrt{2}}{2} \begin{pmatrix}1-i\cos(2 \theta)&-i\sin(2\theta)\\ -i\sin(2\theta)&1+\mathrm{icos}(2\theta)\end{pmatrix}\begin{pmatrix}k_{1}E_{ s}e^{i\phi(t)}\\ k_{2}E_{s}e^{i\phi(t)}\end{pmatrix}  \tag{33}\]  
\[E_{LO}^{\prime}=\left( \begin{array}{c}k_{3}E_{LO}\exp(i(\frac{\pi}{4}+\psi))\\ k_{4}E_{LO}\exp(i(\frac{\pi}{4}+\psi))\end{array} \right) \tag{34}\]  
When the beam passes through a polarization splitting prism, its reflection and transmittance are considered ideal. The horizontal and vertical components of the signal light and the local oscillator light are separated and sent into the orthogonal and DC channels. \(E_{1}\) and \(E_{2}\) indicate that the beams passing through the two channels are  
\[E_{1}= \begin{pmatrix}k_{3}E_{LO}e^{i\tau}e^{i\psi}e^{i\frac{\pi}{4}}\\ -\frac{\sqrt{2}}{2}ik_{1}E_{s}e^{i\varphi}e^{i\varphi(t)}\sin(2\theta)+\frac{ \sqrt{2}}{2}k_{2}E_{s}e^{i\varphi}e^{i\varphi(t)}(1+i\cos(2\theta))\end{pmatrix}  \tag{35}\]  
\[E_{2}= \begin{pmatrix}\frac{\sqrt{2}}{2}k_{1}E_{s}e^{i\tau}e^{i\varphi(t)}(1-i \cos(2\theta))-\frac{\sqrt{2}}{2}k_{2}E_{s}e^{i\tau}e^{i\varphi(t)}\sin(2 \theta)\\ k_{4}E_{LO}e^{i\varphi}e^{i\varphi}e^{-i\frac{\pi}{4}}\end{pmatrix}  \tag{36}\]  
Then, passing through a half wave plate with a fast axis direction and a horizontal angle of \(22.5^{\circ}\), they become  
\[E_{1}^{\prime}=\Lambda_{1/2}E_{1}=\left( \begin{array}{c}-\frac{1}{2}ik_{1}E_ {s}e^{i\varphi}e^{i\varphi(t)}\sin(2\theta)+\frac{1}{2}k_{2}E_{s}e^{i\varphi}e ^{i\varphi(t)}(1+i\cos(2\theta))+\frac{\sqrt{2}}{2}k_{3}E_{LO}e^{i\tau}e^{i \varphi(t)}e^{i\frac{\pi}{4}}\\ \frac{1}{2}ik_{1}E_{s}e^{i\varphi}e^{i\varphi(t)}(1-i\cos(2\theta))-\frac{1}{2 }k_{2}E_{s}e^{i\varphi}e^{i\varphi(t)}(1+i\cos(2\theta))+\frac{\sqrt{2}}{2}k _{3}E_{LO}e^{i\tau}e^{i\varphi}e^{i\frac{\pi}{4}}\end{array} \right) \tag{37}\]  
\[E_{2}^{\prime}=\Lambda_{1/2}E_{1}=\left( \begin{array}{c}\frac{1}{2}k_{1}E_{ s}e^{i\tau}e^{i\varphi(t)}(1-i\cos(2\theta))-\frac{1}{2}ik_{2}E_{s}e^{i\tau}e^{i \varphi(t)}\sin(2\theta)+\frac{\sqrt{2}}{2}k_{4}E_{LO}e^{i\varphi}e^{i\varphi }e^{-i\frac{\pi}{4}}\\ \frac{1}{2}k_{1}E_{s}e^{i\tau}e^{i\varphi(t)}(1-i\cos(2\theta))-\frac{1}{2}ik_{ 2}E_{s}e^{i\tau}e^{i\varphi(t)}\sin(2\theta)-\frac{\sqrt{2}}{2}k_{4}E_{LO}e^{ i\varphi}e^{i\varphi}e^{-i\frac{\pi}{4}}\end{array} \right) \tag{38}\]  
Figure 3: Mixing efficiency variation with the light vector angle of linear polarization signal. where \(\Lambda_{1/2}\) is the Jones matrix of the half wave plate. \[\Lambda_{1/2}=\frac{\sqrt{2}}{2} \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}  \tag{39}\]  
Using a polarizing beam splitter prism, output light with a 90\({}^{\circ}\) phase difference is generated again.