## 3 Results and Discussions
Here, we modulate these eight input combinations using four-level signals \(E_{\rm inj1}\), \(E_{\rm inj1}\), \(E_{\rm inj1}\), and \(E_{\rm inj1V}\) (where \(E_{\rm inj1}\), \(E_{\rm inj1}\), \(E_{\rm inj1V}\) and \(E_{\rm inj1V}\) are the four possible values of \(E_{\rm inj}\)). Specifically, \(E_{\rm inj1}\) denotes (0, 0, 0), \(E_{\rm inj1}\)  
Figure 4: The evolutions of the polarization bistability of \(x_{\rm p}\)-PC, \(y_{\rm p}\)-PC, \(x_{\rm S}\)-PC and \(y_{\rm S}\)-PC under different applied electric fields \(E_{\rm A}\). (**a**) The polarization bistability of \(x_{\rm p}\)-PC and \(y_{\rm p}\)-PC under \(E_{\rm A}=11\) kV/mm. (**b**) The polarization bistability of \(x_{\rm S}\)-PC and \(y_{\rm S}\)-PC under \(E_{\rm A}=11\) kV/mm. (**c**) The polarization bistability of \(x_{\rm p}\)-PC and \(y_{\rm p}\)-PC under \(E_{\rm A}=25\) kV/mm. (**d**) The polarization bistability of \(x_{\rm S}\)-PC and \(y_{\rm S}\)-PC under \(E_{\rm A}=25\) kV/mm. represents (0, 0, 1), (0, 1, 0) and (1, 0, 0), \(E_{\rm injIII}\) for (0, 1, 1), (1, 0, 1) and (1, 1, 0) and \(E_{\rm injIV}\) for (1, 1, 1). In this case, the four-level signals remain constant during a bit duration \(T\). Here, \(T\) is equal to 10 ps, which means the data selection logic operates at a speed of 100 Gb/s. We take \(E_{\rm injI}\) = 5.37, \(E_{\rm injI}\)= 5.4, \(E_{\rm injIII}\) = 5.43 and \(E_{\rm injIV}\) = 5.46, which means when \(E_{\rm injI}\), \(E_{\rm inj2}\) and \(E_{\rm inj3}\) are all equal to 1.79, \(I_{1}\), \(I_{2}\) and \(I_{3}\) are all equal to 0, and if \(E_{\rm injI}\), \(E_{\rm inj2}\) and \(E_{\rm inj3}\) are all equal to 1.82, \(I_{1}\), \(I_{2}\) and \(I_{3}\) are all equal to 1. Additionally, when \(E_{\rm A}\) = \(E_{\rm A1}\) = 11 kV/mm, \(C\) = 1, and if \(E_{\rm A}\) = \(E_{\rm A2}\) = 25 kV/mm, \(C\) = 0. The demodulation mechanism for the logic output is described as follows: Suppose that the duration of \(C\) being equal to 0 or 1, denoted as \(T_{1}\), is \(n^{\prime}\) times the bit duration \(T\), i.e., \(T_{1}\) = \(n^{\prime}T\). During the time duration \(T_{1}\), the amplitude of xp-PC, yp-PC, xs-PC and ys-PC is sampled \(M\) times, where \(M\) is equal to \(T_{1}/h\), and \(h\) represents the sampling interval. Within \(T_{1}\), the average amplitudes of xp-PC, yp-PC, xs-PC and ys-PC are respectively denoted by \(A_{\rm Px}\), \(A_{\rm Py}\), \(A_{\rm Sx}\) and \(A_{\rm Sy}\):  
\[A_{\rm Px}=\sum_{j=1}^{M}\frac{E_{P_{x(j)}}}{M} \tag{12}\]  
\[A_{\rm Py}=\sum_{j=1}^{M}\frac{E_{P_{y(j)}}}{M} \tag{13}\]  
\[A_{\rm Sx}=\sum_{j=1}^{M}\frac{E_{S_{x(j)}}}{M} \tag{14}\]  
\[A_{\rm Sy}=\sum_{j=1}^{M}\frac{E_{S_{y(j)}}}{M} \tag{15}\]  
where \(E_{P_{X}(j)}\), \(E_{P_{y(j)}}\), \(E_{Sx(j)}\) and \(E_{Sy(j)}\) respectively represent the \(j\)th amplitude sampling value of xp-PC, yp-PC, xs-PC and ys-PC within \(T_{1}\). By comparing \(A_{\rm Px}\) and \(A_{\rm Py}\), as well as \(A_{\rm Sx}\) and \(A_{\rm Sy}\), we can determine the logic outputs \(O_{1}\) and \(O_{2}\) within \(T_{1}\). The demodulation rules are as follows: when \(A_{\rm Px}>A_{\rm Py}\) and \(A_{\rm Sx}>A_{\rm Sy}\), \(O_{1}=O_{2}\) = 1; if \(A_{\rm Px}\leq A_{\rm Py}\) and \(A_{\rm Sx}\leq A_{\rm Sy}\), then \(O_{1}=O_{2}\) = 0. We calculate the numerical values of \(A_{\rm Px}\), \(A_{\rm Py}\), \(A_{\rm Sx}\) and \(A_{\rm Sy}\) across different time intervals, as presented in Figure 3.0 Figure 5 illustrates the optical data selection logic operations involving two identical logic outputs. Figure 5a depicts the relationship between the applied electric field \(E_{\rm A}\) and the optical injection amplitude \(E_{\rm inj}\). Based on Figure 5b-e, and Table 2, it is evident that during the time interval from 3 ns to 3.04 ns, where \(C\) = 0, the logic inputs consist of (0, 0, 0), (0, 1, 0) and (0, 0, 1).