# Experimental Study on Partially Coherent Optical Coherent Detection
## Theoretical Analysis
### Coherent Detection Principle  
The sensitivity of direct detection technology is greatly affected by factors such as the noise of photodetectors, so its signal-to-noise ratio is relatively low. In addition, due to its limitations in detection area and response speed, direct detection technology is not very suitable for some high-speed and long-distance transmission fields. In order to achieve quantum noise limitation detection and detect weak light signals, it is necessary to use highly sensitive coherent detection methods. Coherent detection technology can place light detection systems under quantum noise limitations, achieving the efficient detection of weak light signals.  
Photodetectors cannot directly obtain the frequency and phase information of light, but interference occurs when two beams of light overlap with each other. By detecting changes in interference intensity, the phase information of light can be inferred, and then the frequency information of that light can be obtained. Therefore, photodetectors utilize interference phenomena to obtain the phase and frequency information of light waves, in order to achieve coherent detection. In Figure 1, the principle diagram of coherent detection is shown. The signal light and local oscillator light are mixed in the optical mixer, and then the photodetector converts the optical signal into an electrical signal. After circuit processing, the required information is extracted [17].  
Figure 1: Coherent detection principle diagram.  
Let the signal light and local oscillator light fields be  
\[\left\{\begin{array}{l}E_{S}=A_{S}\exp[i(\omega_{S}t+\varphi_{S})]\\ E_{L}=A_{L}\exp[i(\omega_{L}t+\varphi_{L})]\end{array}\right. \tag{15}\]  
Here, \(A_{S}\), \(\omega_{S}\), and \(\varphi_{S}\) represent the amplitude, angular frequency, and initial phase of the signal light, respectively; \(A_{L}\), \(\omega_{L}\), and \(\varphi_{L}\) represent the amplitude, angular frequency, and initial phase of the local oscillator light. Assuming that the laser beam sizes of the signal light and the local oscillator light are the same, and the two beams overlap well with each other. The light intensity of the intermediate frequency signal on the detection surface is  
\[\begin{array}{ll}I=&A_{S}^{2}+A_{L}^{2}+A_{S}A_{L}\{\exp[i(\omega_{S}-\omega_ {L})t-(\varphi_{S}-\varphi_{L}))]\\ &+\exp[-i(\omega_{S}-\omega_{L})t-(\varphi_{S}-\varphi_{L}))]\}\end{array} \tag{16}\]  
Simplifying Equation (16) yields  
\[I=A_{S}^{2}+A_{L}^{2}+2A_{S}A_{L}\cdot\cos[(\omega_{S}-\omega_{L})t+(\varphi_ {S}-\varphi_{L})] \tag{17}\]  
where \(\Delta\omega=\omega_{S}-\omega_{LO}\) and \(\Delta\varphi=\varphi_{S}-\varphi_{LO}\), and the intermediate frequency current returned by the detector is  
\[i(t)=\Re\cdot\left\{A_{S}^{2}+A_{L}^{2}+2A_{S}A_{L}\cdot\cos[(\omega_{S}- \omega_{L})t+(\varphi_{S}-\varphi_{L})]\right\} \tag{18}\]  
Here, \(\Re\) is the responsivity of the detector. From Equation (18), it can be seen that the first two terms in the intermediate frequency current response of the detector are DC terms, while the third term is a signal term. Coherent detectors can detect the intensity, frequency, and phase changes of light waves through this signal term. This detection method is very flexible, and suitable modulation methods can be selected to avoid interference, thus achieving high-precision optical signal detection and measurement. In addition, due to the fact that the energy of local oscillator light is several orders of magnitude higher than that of signal light during the mixing process, the energy of the signal term will increase, thereby improving the sensitivity of the detection.