## 2 Materials and Methods
### Influence of System Parameter Mismatch on the Transmission of LFF Waveforms  
In practice, it is difficult to guarantee that both lasers have identical parameters. Thus, in this section, the effect of the mismatch of various physical parameters on the system performance is investigated. The pump current is studied as the external operating parameter. Firstly, the error of the word probability between the drive and response laser is defined as follows,  
\[d=\frac{|P_{\mathrm{R}}-P_{\mathrm{D}}|}{P_{\mathrm{D}}}, \tag{9}\]  
where \(d\) represents the event probability error, and \(P_{\mathrm{R}}\) and \(P_{\mathrm{D}}\) are the probability of a specific word of the response laser and drive laser, respectively. Figure 6: Parameter relationships that can make the response laser generate the LFF dynamic. (**a**) The relationship of \(I_{\mathrm{R}}\) and \(I_{\mathrm{D}}\) and (**b**) the relationship between \(\kappa_{\mathrm{R}}\) and \(\kappa_{\mathrm{D}}\). From Figure 7a, certain transmission errors occur for all words when \(I_{\text{R}}\) is smaller than \(I_{\text{D}}\). When \(I_{\text{R}}\) is between 14.75 mA and 14.8 mA, the transmission of word 021 has the biggest error, and when \(I_{\text{R}}\) is larger than 14.85 mA, the transmission of word 210 always has the largest error. When \(I_{\text{R}}\) gradually increases to 15 mA, all of the errors converge to the zero point, indicating that the LFF waveform from the drive laser is successfully transmitted to the response laser. In addition, from Figure 7b, the red and black lines are \(\text{rate}_{\text{D}}\) and \(\text{rate}_{\text{R}}\), respectively, and \(\text{rate}_{\text{D}}\) is fixed at 41.97 MHz. The pulse rate of the response laser increases gradually with the injection current and increases rapidly when the \(I_{\text{R}}\) is 14.5 mA and 14.6 mA, indicating that the \(\text{rate}_{\text{R}}\) is greatly influenced by the injection current at this time. When \(I_{\text{R}}\) increases from 14.7 mA to 15 mA, the curve increases smoothly to 41.97 MHz. What is more, Figure 7c shows the overall increase in the correlation coefficient \(CC\) with the increasing current. The mismatch of the internal physical parameters, such as the linewidth enhancement factor \(\alpha\), can also induce a derivation on the LFF transmission. From Figure 8a, the transmission of the word 021 always has a large error when \(\alpha_{\text{R}}\) is not equal to 5. For instance, the error is larger than 0.7 when \(\alpha_{\text{R}}\) is 4.8 and 5.2, and reaches the largest value when \(\alpha_{\text{R}}\) is larger than 5.8. Meanwhile, the transmission of word 210 also always has a large error in the whole range of \(\alpha_{\text{R}}\). In addition, the error of the transmission of word 012 is always small and less than 0.3. In Figure 8b, \(\text{rate}_{\text{R}}\) increases rapidly and gradually approaches \(\text{rate}_{\text{D}}\) when \(\alpha_{\text{R}}\) increases from 4.2 to 5, while in the range 5 to 6, \(\text{rate}_{\text{R}}\) decreases steadily and gradually deviates from \(\text{rate}_{\text{D}}\). This means that when \(\alpha_{\text{R}}\) is smaller than \(\alpha_{\text{D}}\), the LFF dynamic from the drive laser can hardly be mapped to the response laser. While for \(\alpha_{\text{R}}>\alpha_{\text{D}}\), the LFF dynamic can be better transmitted. This can also be proven by the correlation coefficient. In Figure 8c, \(CC\) changes steeply when \(\alpha_{\text{R}}\) increases from 4.2 to 5, indicating that the LFF transmission is seriously affected in this region. When \(\alpha_{\text{R}}>\alpha_{\text{D}}\), the system is more sluggish to parameter mismatch. These phenomena are consistent with the trend  
Figure 7: Effect of pump current mismatch on system transmission. (**a**) Event probability error, (**b**) pulse rate, and (**c**) correlation coefficient. of the spike rate. Thus, \(\alpha_{\text{R}}<\alpha_{\text{D}}\) will cause a greater error in the transmission of the LFF waveform.