## 2 FMCW LiDAR
### FMCW LiDAR Principle of Operation  
Figure 2 shows the basic ranging principle diagram of FMCW LiDAR. FMCW LiDAR uses a modulated continuous laser signal to measure the distance and speed of the target by mixing the transmitted light and the received light. The system uses a linear frequency modulated chirped laser as the transmission signal and the local oscillator signal to mix the weak scattered signal with the strong local oscillator signal on the detector. The signal is significantly amplified by this mixing process, allowing relatively noisy detectors to be used. Due to the use of coherent detection, FMCW LiDAR can directly measure the velocity of moving objects by sensing the Doppler shift of light, and because the detector can only detect light close to the frequency of the local oscillator light, it is almost unaffected by interference from ambient light. Although more sophisticated implementations may replace the balanced detector with a full coherent receiver and introduce additional components to sample multiple polarizations, this does not have a major influence on the basic physics of operation. The system combines the light from the two input A and B on a coupler and then detects it by two photodiodes. The photocurrent difference \(i(t)\) between the two photodiodes is used as the output of the receiver; the system can obtain the frequency difference through the photocurrent difference, thereby measuring the distance and speed of the target. The electric field \(E_{1}\) of the first photodiode and the corresponding photocurrent \(i_{1}(t)\) are  
\[ \begin{array}{l}E_{1}=\frac{1}{\sqrt{2}}(E_{A}+E_{B})=\frac{1}{\sqrt{2}}(A \,\cos\omega_{A}t+B\,\cos\omega_{B}t),\\ i_{1}(t)=\frac{1}{2}(i_{A}+i_{B})+\sqrt{I_{A}t_{B}}\,\cos\left(\omega_{A}- \omega_{B}\right)t,\end{array}  \tag{1}\]where \(E_{1}\) represents the electric field of the first photodiode, \(i_{1}(t)\) is the photocurrent of the first photodiode, \(E_{A}\) and \(E_{B}\) represent different input light fields, A and B are the amplitude of \(E_{A}\) and the amplitude of \(E_{B}\), \(i_{A}\) and \(i_{B}\) represent different input time-averaged photocurrents, and \(\omega_{A}\) and \(\omega_{B}\) represent different light frequencies. The second photodiode electric field \(E_{2}\) and the corresponding photocurrent \(i_{2}(t)\) are  
\[ \begin{split} E_{2}=\frac{1}{\sqrt{2}}(E_{A}-E_{B})=\frac{1}{ \sqrt{2}}(A\,\cos\omega_{A}t-B\,\cos\omega_{B}t),\\ i_{2}(t)=\frac{1}{2}(i_{A}+i_{B})-\sqrt{i_{A}i_{B}}\,\cos\,( \omega_{A}-\omega_{B})t,\end{split}  \tag{2}\]  
where \(E_{2}\) represents the electric field of the second photodiode, \(i_{2}(t)\) is the photocurrent of the second photodiode, \(E_{A}\) and \(E_{B}\) represent different input light fields, A and B are the amplitude of \(E_{A}\) and the amplitude of \(E_{B}\), \(i_{A}\) and \(i_{B}\) represent different input time-averaged photocurrents, and \(\omega_{A}\) and \(\omega_{B}\) represent different light frequencies. The output current \(i(t)\) is the difference between the first photodiode photocurrent \(i_{1}(t)\) and the second photodiode laser photocurrent \(i_{2}(t)\):  
\[ \begin{split} i(t)=& i_{1}(t)-i_{2}(t)=2\sqrt{i_{A} i_{B}}\,\cos\,(\omega_{A}-\omega_{B})t\,\end{split}  \tag{3}\]  
where \(i_{1}(t)\) is the photocurrent of the first photodiode, \(i_{2}(t)\) is the photocurrent of the second photodiode, \(i_{A}\) and \(i_{B}\) represent different input time-averaged photocurrents, and \(\omega_{A}\) and \(\omega_{B}\) represent different optical frequencies. Therefore, the optical heterodyne detector directly measures the frequency difference \(\omega_{A}-\omega_{B}\) between the two input beams, and the FMCW LiDAR directly uses this to measure the target distance, which we will discuss in Section 2.3.1. The basic working principle of FMCW LiDAR is to emit a laser beam with linear frequency modulation and receive the light signal reflected by the target. The frequency of the emitted light changes linearly over time to form a frequency modulated signal.