# Laser Phase Noise Compensation Method Based on Dual Reference Channels in Inverse Synthetic Aldar
## 2 Models and Methods
### Proposed Method Using Dual Reference Channel  
In this paper, we propose an alternative method for RPN compensation. We define a reference differential phase, DP-1, with a delay of \(\tau_{1}\), and a second reference differential phase, DP-2, with a delay of \(\tau_{2}\). For simplicity, we set \(\tau_{1}=\beta\tau_{2},\beta\in N^{*}\). Building on the approach presented in the previous section, we first use DP-1 to decompose the target DPN; then, we proceed to decompose the RPN using DP-2 as follows:  
\[\Phi(t) =\phi_{L}(t-N\tau_{1})-\phi_{L}(t-\tau_{0})\] \[=\phi_{L}(t-N\beta\tau_{2})-\phi_{L}(t-(N\beta+1)\tau_{2})+\phi_{ L}(t-(N\beta+1)\tau_{2})-\phi_{L}(t-(N\beta+2)\tau_{2})+... \tag{11}\] \[+\phi_{L}(t-(M-1)\tau_{2})-\phi_{L}(t-M\tau_{2})+\phi_{L}(t-M\tau _{2})-\phi_{L}(t-\tau_{0}).\]  
Figure 2: Diagram illustrating various compensation methods. Figure 2, in the last row, presents the schematic diagram of the dual reference channel (DRC) compensation method. We utilize the phase from the long delay channel for \(N_{1}\) order CGP, which is followed by the phase from the short delay channel for \(N_{2}\) order CGP. It is observed that the RPN transitions from \(\phi_{L}(t-N\tau_{1})-\phi_{L}(t-\tau_{0})\) to \(\phi_{L}(t-M\tau_{2})-\phi_{L}(t-\tau_{0})\). Researchers usually model laser phase noise as a random walk. Under this widely accepted assumption, the variance of the phase difference over a time interval \(\tau\) is given by  
\[\sigma_{\phi_{L}}^{2}(\tau)=\left\langle\left[\phi_{L}(t)-\phi_{L}(t-\tau) \right]^{2}\right\rangle=\frac{2\mid\tau\mid}{\tau_{C}}, \tag{12}\]  
where \(\tau_{c}\) represents the coherence time, determined by the characteristics of the laser, and \(\langle*\rangle\) denotes the ensemble average [33]. Equation (12) indicates that a smaller time interval correlates with a reduced phase noise. Therefore, employing two differential phases (DPs) with different delays to sequentially decompose the target DPN can significantly reduce the variance of the RPN compared to using a single DPN, as illustrated in  
\[\gamma =\frac{\left\langle\left[\phi_{L}(t-M\tau_{2})-\phi_{L}(t-\tau_{0 })\right]^{2}\right\rangle}{\left\langle\left[\phi_{L}(t-N\tau_{1})-\phi_{L}(t -\tau_{0})\right]^{2}\right\rangle} \tag{13}\] \[=\frac{\tau_{0}-M\tau_{2}}{\tau_{0}-N\tau_{1}}\] \[=1-\frac{M-N\beta}{\frac{\tau_{0}}{\tau}-N\beta}.\]  
For example, let us consider \(\tau_{0}=2.93\;\mu s\), \(\tau_{1}=0.1\;\mu s\), \(\tau_{2}=0.02\;\mu s\). In this case, we find \(M=29,N=146,\beta=5\). After employing the dual reference differential phases (DPs) for decomposition, the variance is reduced to \(\gamma=39.99\%\) of its value with a single reference. Figure 3 illustrates the principles of the two aforementioned RPN estimation methods within the mismatched region. Figure 3a depicts our proposed estimation method, which is based on the concatenated generated phase (CGP) principle and the dual reference channel (DRC) structure. This approach utilizes a piecewise linear approximation, reducing the mismatch delay from \((\tau_{0}-N\tau_{1})\) to \((\tau_{0}-M\tau_{2})\). Figure 3b presents a schematic diagram of the phase linear estimation (PLE) method, as expressed in Equation (10). It is evident that the phase estimate \(\Phi_{est2}(t)\), derived from delay-time proportional inversion, is influenced by the accuracy of extracting phase \(\phi_{\tau_{1}}(t)\). Consequently, the estimation error (indicated by the red area) is significant. Furthermore, this method requires a blind search and optimization across the entire mismatch region, leading to substantial computational demands and compromising the real-time performance of the algorithm in practical applications.