## 2 Theory and Model
In our experimental setup, the applied electric field \(E_{\rm A}\) used to modulate the control logic signal \(C\) is a square wave signal characterized by distinct low and high levels, denoted as \(E_{\rm A1}\) and \(E_{\rm A2}\), respectively, i.e., \(C=0\) when \(E_{\rm A}=E_{\rm A1}\), and \(C=1\) if \(E_{\rm A}=E_{\rm A2}\). Under the appropriate conditions of \(E_{\rm A}\) and utilizing a suitable logic output demodulation mechanism, data selection logic operations can be effectively implemented, i.e., \(O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). According to the theory of generalized chaos synchronization, when the S-VCSEL undergoes strong light injection, its output polarization light exhibits high correlation with the polarization light emitted from the PPLN. Therefore, it is feasible to establish \(O_{2}(t+\tau)=O_{1}(t)=I_{1}(t)\overline{I_{3}}(t)+I_{2}(t)I_{3}(t)\). This expression signifies the successful implementation of delay storage in data selection logic operations. Under the influence of spontaneous emission noise, based on the SFM of VCSEL, the rate equation of M-VCSEL subjected to external optical injection can be expressed as follows:  
\[ \begin{array}{rl}\frac{dE_{\rm Ar}(t)}{dt}=&k(1+ia)[N_{M}E_{Mx}(t)+in_{M}E_ {My}(t)-E_{Mx}(t)]-i(\gamma_{p}+\Delta\omega)E_{Mx}(t)\\ &-\gamma_{a}E_{Mx}(t)+\sqrt{\beta_{sp}\gamma_{e}N_{M}^{\ast}\varsigma_{x}}+k_ {x}E_{inj}\\ \end{array}  \tag{1}\]  
\[ \begin{array}{rl}\frac{dE_{My}(t)}{dt}=&k(1+ia)[N_{M}E_{My}(t)-in_{M}E_{Mx}( t)-E_{My}(t)]+i(\gamma_{p}-\Delta\omega)E_{My}(t)\\ &+\gamma_{a}E_{My}(t)+\sqrt{\beta_{sp}\gamma_{e}N_{M}}\varsigma_{y}+k_{y}E_{inj} \\ \end{array}  \tag{2}\]  
\[ \begin{array}{rl}\frac{dN_{M}(t)}{dt}=&-\gamma_{c}[N_{M}(t)(1+|E_{Mx}(t)|^{2 }+|E_{My}(t)|^{2})]+\gamma_{c}\mu_{M}\\ &-i\gamma_{e}\mu_{M}[E_{My}(t)E_{Mx}^{\ast}-E_{Mx}^{\ast}E_{My}^{\ast}(t)]\\ \end{array}  \tag{3}\]  
Figure 1: Schematic diagram of high-speed optical chaotic data selection logic operations. \[ \begin{array}{ll}\frac{dn_{M}(t)}{dt}=&-\gamma_{s}n_{M}(t)-\gamma_{e}n_{M}(t)(|E_{ Mx}(t)|^{2}+\big{|}E_{My}(t)\big{|}^{2})\\ &-i\gamma_{e}N_{M}(t)[E_{My}(t)E_{Mx}^{*}(t)-E_{Mx}(t)E_{My}^{*}(t)]\end{array}  \tag{4}\]  
In the above formulas, the subscript \(M\) represents M-VCSEL; the subscripts \(x\) and \(y\) represent x-PC and y-PC, respectively; \(N\) is the total carrier concentration, while \(n\) is the difference in concentration between carriers with spin-up and spin-down; \(E\) is the normalized amplitude, and \(E=\sqrt{\texttt{g}/\gamma_{e}}A\), where g is the differential material gain; \(A\) is the slowly varying amplitude; \(\gamma_{e}\) is the nonradiative carrier relaxation rate; \(a\) is the linewidth enhancement factor; \(k\) is the field decay rate; \(\gamma_{s}\) is the spin-flip relaxation rate; \(\gamma_{a}\) is the linear dichroism; \(\gamma_{p}\) is the linear birefringence; \(\mu_{\text{M}}\) is the normalized bias current of the M-VCSEL; \(\sqrt{\gamma_{e}N_{M}\beta_{sp}}\) denotes the noise term, with \(\beta_{sp}\) being the spontaneous emission factor, and noise strength \(N_{\text{o}i}=10\text{lg}(\sqrt{\gamma_{e}N_{M}\beta_{sp}})^{2}\). \(\xi_{x}\) and \(\xi_{y}\) are two independent Gaussian white noises with a mean value of 0 and a variance of 1; \(k_{x}\) and \(k_{y}\) are the injection strength for x-PC and y-PC, respectively; \(E_{inj}=\sqrt{\texttt{g}/\gamma_{e}}A_{inj}\) is the sum of \(E_{\text{inj1}}\), \(E_{\text{inj2}}\) and \(E_{\text{inj3}}\), where \(A_{inj}\) is the slowly varying amplitude of the injected field; \(\Delta\omega=\omega_{inj}-\omega_{ref}\) is the detuning between the frequency of the injected field and the reference frequency of the M-VCSEL; \(\omega_{inj}\) is the angular optical frequency of the injected field; the reference optical frequency \(\omega_{ref}\) is defined as \((\omega_{x}+\omega_{y})/2\), where \(\omega_{x}=-\gamma_{p}+\alpha\gamma_{a}\) and \(\omega_{y}=\gamma_{p}-\alpha\gamma_{a}\) are the optical frequency of the x-PC and the y-PC of the free-running VCSEL.