# All-Optical Simultaneous Frequency Metamorphose Contingent on a Three Parallel SOA-MZIs Copula
## 3 Small-Signal Analysis of Sampling Parallel SOA-MZIs Link  
The frequency up-conversion theoretical responses for three parallel sampling SOA-MZIs mixers are determined by analyzing the small signal in contemplation of comparing its results with the simulation work using the standard modulation scheme. The analysis of the small-signal for each SOA-MZI sampling mixer is subedited by debriefing the powers, gains, and phases as the aggregate of a stationary expression and a variation one. Therefore, three output powers at each SOA-MZI are merged in order to fulfill the overall power, which is the optical power of the simultaneous up-conversion principle established on the three parallel SOA-MZIs link. The analytical study of the mixing process to higher frequencies based on an unattached SOA-MZI is previously explored in [15] for the architectures entitled switching and modulation. In this novel study, we are only interested in the power equality of the output signal to higher frequencies at the external port of the SOA-MZI.  
Figure 2: Electrical spectrums of the mixed signals A, B, and C and the simultaneous mixed signal D after combination.  
In addition, the theoretical explanation of this power is elucidated minutely in [15]. The output mixed power after modulation, mixing, photo-detection, and amplification at the outer port of each SOA-MZI at frequencies \(nf_{s}+f_{m}\) can be prescribed as:  
\[P_{Out,nf_{s}+f_{m}}=-K\frac{p_{m}p_{O,nf_{s}}\overline{G}_{1}\tau_{d}}{1-jw_{m} \tau_{d}} \tag{1}\]  
where  
\(\tau_{d}=\frac{1}{2\pi f_{c}}=25\) ps is called the differential carrier lifetime;  
\(f_{c}\) is the cut-off frequency of the used SOA-MZI;  
\(p_{m}\) is the modulated power at the SOA-MZI input at frequencies \(f_{m}\);  
\(K=2.5\) dB is the attenuation coefficient of the input modulated power;  
\(\overline{G}_{1}=30\) dB is the gain of SOA1 in the upper position of the used SOA-MZI;  
\(r=0.86\) A/W is the responsivity of the used photodiode;  
\(G_{LNA}=33\) dB is the gain of the low-noise amplifier;  
\(R=50\)\(\Omega\) is the load resistance of the photodetector;  
\(p_{O,nf_{s}}=\frac{1}{r}\sqrt{\frac{2p_{nf_{s}}}{G_{LNA}R}}\) is the power of the pulsed signal at the SOA-MZI input;  
\(p_{nf_{s}}\) is the harmonic electrical power at \(nf_{s}\);  
\(p_{f_{s}}=15\) dB is the electrical power at \(f_{s}\); and  
\(p_{5f_{s}}=8\) dB is the electrical power at \(5f_{s}\).  
It is worth bearing in mind that the three SOA-MZIs have the same construction, physical specifications, and running point. The only difference is the frequency of the data signal at \(f_{m}\). For parallel configuration, the relation between the three optical powers is expressed as:  
\[\begin{array}{c}p_{Out,nf_{s}+f_{m}}=-K\frac{p_{1}p_{O,nf_{s}}\overline{G}_{1 }\tau_{d}}{1-jw_{1}\tau_{d}}-K\frac{p_{2}p_{O,nf_{s}}\overline{G}_{1}\tau_{d}} {1-jw_{2}\tau_{d}}-K\frac{p_{3}p_{O,nf_{s}}\overline{G}_{1}\tau_{d}}{1-jw_{3} \tau_{d}}\\ =-Kp_{O,nf_{s}}\overline{G}_{1}\tau_{d}\left(\frac{p_{1}}{1-jw_{1}\tau_{d}}+ \frac{p_{2}}{1-jw_{2}\tau_{d}}+\frac{p_{3}}{1-jw_{3}\tau_{d}}\right)\end{array} \tag{2}\]  
where  
\(\omega_{1}\) is the radian frequency of the first low frequency signal;  
\(\omega_{2}\) is the radian frequency of the second low frequency signal;  
\(\omega_{3}\) is the radian frequency of the third low frequency signal;  
\(p_{1}\) is the modulated power at \(f_{1}\) at the input of the SOA-MZI1 topmost port;  
\(p_{2}\) is the modulated power at \(f_{2}\) at the input of the SOA-MZI2 topmost port; and  
\(p_{3}\) is the modulated power at \(f_{3}\) at the input of the SOA-MZI3 topmost port.