## 3 Results and Discussions
In this scenario, \(A_{\rm Px}=8.74\cdot 10^{-5}<A_{\rm Py}=0.24\), leading to the conclusion that \(O_{1}\) = 0 (as shown in Figure 5f,g); after a delay of \(\tau\) (\(\tau\) = 5 ns), specifically within the time interval from 8 ns to 8.04 ns, it is observed that \(A_{\rm Sx}=0.0078<A_{\rm Sy}=0.92\). Consequently, we conclude that \(O_{2}\) = 0 (as shown in Figure 5h,i). Therefore, it follows that \(O_{2}(t+\tau)=O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). Between the time interval of 3.04 ns and 3.05 ns, the logic input corresponds to (1, 0, 0) with \(C\) = 1. Here, \(A_{\rm Px}\) = 0.25 \(>A_{\rm Py}=5.83\cdot 10^{-4}\), resulting in \(O_{1}\) = 1. After a delay of \(\tau\), specifically within the time interval from 8.04 ns to 8.05 ns, it is observed that \(A_{\rm Sx}\) = 0.56 \(>A_{\rm Sy}=0.35\). Consequently, \(O_{2}\) = 1. Thus, we establish that \(O_{2}(t+\tau)=O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). Between 3.05 ns and 3.06 ns, the logic input transitions to (1, 0, 1) with \(C\) = 0. Here, \(A_{\rm Px}\) = 2.53 \(\cdot 10^{-5}<A_{\rm Py}=0.25\), leading to \(O_{1}\) = 0. Between 8.05 ns and 8.06 ns, it is observed that \(A_{\rm Sx}\) = 0.31 \(<A_{\rm Sy}\) = 0.75. Consequently, \(O_{2}\) = 0. Thus, we conclude that \(O_{2}(t+\tau)=O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). The analysis principle for logic operations in other time intervals follows the same methodology; hence, it is not reiterated here. Based on the aforementioned analysis, it is concluded that \(O_{2}(t+\tau)=O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). This signifies the successful implementation of data selection logic operations and their delay storage. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Time (ns)** & \(A_{\text{Px}}\) & \(A_{\text{Py}}\) & **Time (ns)** & \(A_{\text{Sx}}\) & \(A_{\text{Sy}}\) \\ \hline
3–3.04 & \(8.74\cdot 10^{-5}\) & 0.24 & 8–8.04 & 0.0078 & 0.92 \\
3.04–3.05 & 0.25 & \(5.83\cdot 10^{-4}\) & 8.04–8.05 & 0.56 & 0.35 \\
3.05–3.06 & \(2.53\cdot 10^{-5}\) & 0.25 & 8.05–8.06 & 0.31 & 0.75 \\
3.06–3.1 & 0.25 & \(5.64\cdot 10^{-4}\) & 8.06–8.1 & 0.5 & 0.19 \\
3.1–3.11 & \(4.01\cdot 10^{-5}\) & 0.24 & 8.1–8.11 & 0.19 & 0.79 \\
3.11–3.16 & \(0.25\) & \(5.66\cdot 10^{-4}\) & 8.11–8.16 & 0.48 & 0.14 \\
3.16–3.18 & \(5.21\cdot 10^{-5}\) & 0.24 & 8.16–8.18 & 0.11 & 1 \\
3.18–3.19 & 0.24 & \(3.98\cdot 10^{-4}\) & 8.18–8.19 & 0.5 & 0.33 \\
3.19–3.2 & \(1.02\cdot 10^{-4}\) & 0.25 & 8.19–8.2 & 0.3 & 0.82 \\
3.2–3.24 & 0.24 & \(5.75\cdot 10^{-4}\) & 8.2–8.24 & 0.51 & 0.2 \\
3.24–3.29 & \(7.08\cdot 10^{-5}\) & 0.25 & 8.24–8.29 & 0.04 & 0.9 \\
3.29–3.3 & 0.25 & \(5.83\cdot 10^{-4}\) & 8.29–8.3 & 0.56 & 0.35 \\ \hline \end{tabular}
\end{table} 
Table 2: The numerical values of \(A_{\text{Py}}\), \(A_{\text{Py}}\), \(A_{\text{Sx}}\) and \(A_{\text{Sy}}\) in different time periods. Figure 5: The optical chaotic data selection logic operations. (**a**) The combination of \(E_{\text{A}}\) and \(E_{\text{inj}}\). (**b**) The control logic signal \(C\). (**c**) The logic input \(I_{1}\). (**d**) The logic input \(I_{2}\). (**e**) The optical clock signal \(I_{3}\). (**f**) The combination of \(E_{\text{Px}}\) and \(O_{1}\). (**g**) The combination of \(E_{\text{Py}}\) and \(O_{1}\). (**h**) The combination of \(E_{\text{Sx}}\) and \(O_{2}\). (**i**) The combination of \(E_{\text{Sy}}\) and \(O_{2}\). It is crucial to acknowledge that the fidelity of logic outputs is influenced by various system parameters, such as the bit duration time \(T\) and the noise strength \(N_{\text{oi}}\). These factors can introduce bit errors, potentially compromising the integrity of logic operations. Therefore, the success probability (SP) is employed as a metric to quantify the reliability of data selection logic operations. The SP is calculated as the ratio of the number of correctly identified symbols to the total number of symbols present in the output of the logic operations.