# Phase-Modulated Continuous-Wave Coherent Ranging Method and Anti-Interference Evaluation
## 3 Ranging Experiments  
According to the mechanism, we built a PhMCW system based on a monostatic configuration to demonstrate the validation of the proposed PhMCW method, as shown in Figure 2. We chose a 1550 nm CW narrow-linewidth laser (Connect CoSF-D) as the light source, followed by an erbium-doped fiber amplifier. The optical phase is modulated by a LiNbO\({}_{3}\) phase modulator, which is driven by an arbitrary function generator (AFG). After that, the modulated laser is split into LO and TX by a 10:90 beam splitter, and the TX signal is collimated and emitted into free space. The RX signal, back-scattered by the target, is routed to a 2 \(\times\) 2 optical coupler and mixed with the LO signal. The mixed optical signal is received by a pair of balanced photodetectors (Thorlabs PDB470C-AC) and transmitted to the electrical IF signal. Finally, the IF signal is monitored, stored, and digitally processed by an oscilloscope (Tektronix MSO54). We chose an off-the-shelf diffused reflection board with a reflectivity of 90% as the target, and the distance to the target was varied from 5 m to a maximum of 50 m.  
Figure 1: (**a**) A schematic of the PhMCW system. PM: phase modulation; BS: beam splitter and OC: optical coupler. (**b**) The optical phase of the LO and RX and the waveform of the IF signal. At each distance, we performed 50 ranging measurements, and the averaged results are shown in Figure 3. The linear relationship between the measured distance and the actual distance indicates the validation of the PhMCW method and the ranging error is as low as 2.2 cm, suggesting good accuracy. In addition to accuracy, precision is another important criterion for LiDAR [1; 11; 12; 18]. Therefore, we calculated the measured results with varying distances, and the distribution results are shown in Figure 4, where the dashed curves indicate Gaussian fits and all distributions confirm a Gaussian distribution. The minimum and maximum standard deviations were 1.7 and 6.2 cm, respectively, demonstrating the exceptional precision of the PhMCW system. Figure 3: Measured distance as a function of actual distance. Inset is the ranging errors of each distance. Figure 2: A schematic of the PhMCW system. LD: laser; EDFA: erbium-doped fiber amplifier; PM: phase modulator; AFG: arbitrary function generator; OC: optical coupler; CIR: circulator; COL: collimator; BPD: balance photodetector; and OSC: oscilloscope. Theoretically, the ranging precision \(\sigma\) is inversely proportional to the square root of the signal-to-noise ratio (SNR), that is, \(\sigma\propto 1/\sqrt[3]{\text{SNR}}\)[33]. In our ranging system, the power of the LO was several orders of magnitude higher than that of the RX. Therefore, the noise in the system is believed to be dominated by the shot noise of the LO. Meanwhile, the IF signal level is proportional to the amplitude of the electric field of the RX signal \(\text{A}_{\text{R}}\), that is, to the square root of the RX power \(\sqrt[3]{\text{P}_{\text{RX}}}\). According to the LiDAR link budget, \(\text{P}_{\text{RX}}\) is primarily affected by the target reflectivity \(\rho\) and the distance to the target d [19]. Therefore, \(\sigma\) is derived to be proportional to \(\sqrt[3]{\text{d}}\) and inversely proportional to \(\sqrt[4]{\rho}\), as given by \(\sigma\propto 1/\sqrt[3]{\text{SNR}}\propto\sqrt[2]{\text{d}}/\sqrt[4]{\rho}\). The target reflectivity is 90%, and, as a result, the ranging precision can be expressed as \(\sigma=\text{A}\times\left(\sqrt[2]{\text{d}}/\sqrt[4]{0.9}\right)\), where A is a coefficient. Figure 5 illustrates the precision values extracted from Figure 4. The precision formulas fitted from Figure 5 are \(\sigma=0.87\text{ cm}/\sqrt[3]{\text{m}}\times\sqrt[2]{\text{d}}/\sqrt[4]{0.9}\). According to such a model, a target located at 100 m would result in a precision of 8.93 cm, which is comparable to commercial LiDARs [32], suggesting promising application prospects for the PhMCW method.