# Laser Phase Noise Compensation Method Based on Dual Reference Channels in Inverse Synthetic Aldar
## 2 Models and Methods
### Analysis of the CGP Method with Single Reference Channel (SRC)  
The DPN in the signal \(I_{1}(t)\), characterized by a long delay difference, can be decomposed into the sum of several differential phases (DP) \(\phi_{\tau_{1}}(t)=\phi_{L}(t)-\phi_{L}(t-\tau_{1})\) with shorter reference delay \((\tau_{1})\) and similar forms, denoted as:  
Figure 1: Block diagram of the coherent lidar system with direct intermediate frequency sampling structure. \[ \begin{split}\phi_{\tau_{0}}(t)&=\phi_{L}(t)-\phi_{L}(t- \tau_{0})\\ &=\phi_{L}(t)-\phi_{L}(t-\tau_{1})+\phi_{L}(t-\tau_{1})-\phi_{L}(t-2 \tau_{1})+...\\ &+\phi_{L}(t-(N-1)\tau_{1})-\phi_{L}(t-N\tau_{1})+\phi_{L}(t-N \tau_{1})-\phi_{L}(t-\tau_{0})\\ &=\sum_{n}^{N-1}\phi_{\tau_{1}}(t-n\tau_{1})+\phi_{L}(t-N\tau_{1 })-\phi_{L}(t-\tau_{0}),\end{split}  \tag{6}\]  
where \(N=floor(\frac{\tau_{0}}{\tau_{1}})\) is the integer multiple of \(\tau_{1}\) closest to \(\tau_{0}\), and \(\Phi(t)=\phi_{L}(t-N\tau_{1})-\phi_{L}(t-\tau_{0})\) represents the residual phase noise (RPN) after decomposition. The concatenated generated phase (CGP) method can be regarded as the reverse process of decomposition [30]. In other words, the differential phase with a shorter delay can be efficiently obtained by applying the delayed self-heterodyne (DSH) method and the Hilbert transform. An estimate of the target DPN can be concatenated-generated by time shifting and accumulating these phases. This estimate approaches the actual value when accumulated to the \(N\)-th order. We can implement this process by convolving a short delay differential phase with a delay-and-sum (DAS) filter [28]:  
\[ \begin{split}\hat{\phi}(t)&=h_{D}(t)^{*}\phi_{\tau _{1}}(t)\\ &=\sum_{n=0}^{N-1}\delta(t-n\tau_{1})^{*}\phi_{\tau_{1}}(t).\end{split}  \tag{7}\]  
By constructing the estimate in a conjugate form and multiplying it with the target signal, it is possible to compensate for most of the DPN as follows:  
\[I_{cont}=I_{tar}\cdot\exp[-j\hat{\phi}(t)]. \tag{8}\]  
Ideally, we assume that the target delay \(\tau_{0}\) is an exact integer multiple of the reference delay \(\tau_{1}\), i.e., \(\phi_{\tau_{0}}(t)=0\), meaning the DPN is perfectly compensated. However, in practical applications, it is nearly impossible for the two delays to maintain an exact multiple relationship, especially as the distance may vary for moving targets. Consequently, in the traditional CGP method, a mismatch region (MR) exists between the \(N\)-th and \((N+1)\)-th orders, with a mismatch path delay length of \(\Delta\tau_{0}=\tau_{0}-N\tau_{1}\), where \(0\leq\Delta\tau_{0}<\tau_{1}\). To reduce the scope of the mismatch region, it seems reasonable to employ a differential phase \(\phi_{\tau_{2}}(t)=\phi_{L}(t)-\phi_{L}(t-\tau_{2})\) with a shorter reference delay \(\tau_{2}\)\((<\tau_{1})\):  
\[\phi_{\tau_{0}}(t)=\sum_{m=0}^{M-1}\phi_{\tau_{2}}(t-m\tau_{2})+\phi_{L}(t-M \tau_{2})-\phi_{L}(t-\tau_{0}), \tag{9}\]  
where \(M=floor(\frac{\tau_{0}}{\tau_{2}})\) is the integer multiple of \(\tau_{2}\) closest to \(\tau_{0}\), and \(M>N\). However, this approach proportionally increases the cascade order. Due to the randomness of the laser phase, the CGP method cannot extract the original phase without bias, resulting in a phase estimation error. As the cascade order increases, the estimation error compounds, making it increasingly difficult to counteract the influence of laser phase noise. Therefore, the reference delay should not be excessively short. In the existing literature [30], this delay is generally set near the coherence length of the laser used. The first three rows of Figure 2 illustrate the uncompensated DPN and the CGP method based on SRC. The second row indicates that using the phase extracted from a single reference channel (SRC) with a long delay for \(N\)-order concatenation may lead to an extended mismatch region (MR). In the third row, the phase extracted using the shorter delay SRC is concatenated in M orders, resulting in excessive cascade orders.