## 2 Principle of Standard and Differential Modulation Modes
As we can display from the electrical spectrum of the mixed signal achieved after filtering at \(\lambda_{s}\), the data signals were up-converted from \(f_{m}\) to \(nf_{s}\pm f_{m}\). The differential modulation solved two main issues together: the enhancement of mixed signals and up-conversion process of a number of simultaneous IF signals at the SOA-MZI output. For the counter-direction in Figure 2, we interchanged between the signals at the OP and the MP in comparison with the co-direction. The operation of both principles  
Figure 2: SOA-MZI in the counter-direction configuration. (**a**) Standard modulation mode, (**b**) Differential modulation mode. of standard and differential modulation modes possessed the same function as the co-direction. The results at the SOA-MZI output showed that the sampled signal at \(nf_{s}\pm f_{m}\) diminished slightly in comparison with the co-direction for both modes. It is worth noting that all the physical parameters of the SOAs in the SOA-MZI are the same for Figures 1 and 2. Besides, the most significant SOA parameters are shown in Table 1. The structure of the real SOA-MZI used to obtain a frequency conversion to higher frequencies by all-optical sampling is illustrated in Figures 1 and 2, while its static and dynamic characteristics are studied in order to identify the best operating point used in the up-conversion process. The nominal operating point was chosen as follows: the bias currents of SOA1 and SOA2 were both biased at 350 mA and the phase shifters were controlled in order to obtain the minimum optical power at the SOA-MZI output. Table 1 lists the parameter values of SOA-MZI used in our simulations for the conversion gain and EVM results. The differential carrier lifetime \(\tau_{d}\) was estimated by measuring the SOA-MZI bandwidth (cutoff frequency) using a pump-probe technique. Besides, we observed that the conversion gain was related to the carrier lifetime \(\tau\), which depends on the SOA physical properties. Additionally, for a given SOA-MZI, \(\tau_{d}\) and \(\tau\) were inter-dependent and associated to its operating condition. Therefore, the effect of each critical SOA-MZI parameter depended on the exact operating point of the SOA-MZI. As a result, \(\tau_{d}\), \(\tau\), and the bias currents of the SOAs were chosen in order to obtain the best operating point of the SOA-MZI. This leads to the better quality of the mixed signal at the SOA-MZI output as well as the optical transmission system. The theoretical study of the frequency up-conversion function based on single SOA-MZI has already been studied in [36] for the switching architecture in co-propagating direction. In our work, we were only interested in the equation of the optical power obtained at the SOA-MZI output, while its derivation was demonstrated in detail in [36]. The optical modulation power at the output port (OP) of the SOA-MZI at mixing frequencies \(nf_{s}\pm f_{m}\) can be written as:  
\[P_{Out,nf_{s}\pm f_{m}}=-K\frac{p_{m}p_{O,nf_{s}}\overline{G}_{1}\tau_{d}}{1- jw_{m}\tau_{d}} \tag{1}\]  
where  
\(\tau_{d}\) is the differential carrier lifetime,  
\(K\) is the attentaion coefficient of \(p_{m}\),  
\(\overline{G}_{1}\) is the SOA1 gain,  
\(p_{m}\) is the optical input modulation power at a frequency \(f_{m}\),  
\(p_{O,nf_{s}}=\frac{1}{\gamma}\sqrt{\frac{2p_{nf_{s}}}{\zeta_{U,n}R}}\) is the optical power of the sampling signal,  
\(p_{nf_{s}}\) is the electrical power at \(f_{s}\),  
\(r=0.8\) A/W is the responsitivity of the PIN photodiode,  
\(G_{LNA}=33\) dB is the gain of the low noise amplifier,  
\(R=50\)\(\Omega\) is the load resistance of the photodetector. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Linewidth enhancement factor & 4 & \\ Differential Carrier Lifetime & 26 & ps \\ Carrier Lifetime & 69 & ps \\ SOA Saturation Power & 15 & dBm \\ SOA1 Gain & 27 & dB \\ SOA2 Gain & 28 & dB \\ SOA1 Current & 350 & mA \\ SOA2 Current & 350 & mA \\ \hline \hline \end{tabular}
\end{table}