# Fast Reconfigurable SOA-Based Wavelength Conversion of Advanced Modulation Format Data +
## 2 Theory
### FWM with Dual Co-Polarized Pumps  
We next look at the vector theory of FWM in nonlinear SOA. The states of polarization of the two co-polarized pumps and signal are shown in Figure 2a. Pump1 and pump2 are co-polarized. For simplicity, we consider a linearly polarized input signal at an arbitrary angle \(\theta\) with respect to the pumps. The output spectrum of this FWM scheme is given in Figure 2b. The converted idlers generated by the beating between the dual co-polarized pumps have optical frequencies \(\omega_{p1}-\omega_{p2}+\omega_{s}\) and \(\omega_{p2}-\omega_{p1}+\omega_{s}\). We show that the FWM wavelength conversion scheme preserves the polarization properties of the signal within the idler at frequency \(\omega_{p2}-\omega_{p1}+\omega_{s}\). The electric field of the two co-polarized pumps (\(\overset{\rightarrow}{E}_{p1}\) and \(\overset{\rightarrow}{E}_{p2}\)) and signal (\(\overset{\rightarrow}{E}_{s}\)) are given by,  
\[\overset{\rightarrow}{E}_{p1}=\ A_{p1}\exp\bigl{(}j\bigl{(}\omega_{p1}t+ \phi_{p1}\bigr{)}\bigr{)}\overset{\rightarrow}{x} \tag{5}\]  
\[\overset{\rightarrow}{E}_{p2}=\ A_{p2}\exp\bigl{(}j\bigl{(}\omega_{p2}t+\phi_ {p2}\bigr{)}\bigr{)}\overset{\rightarrow}{x} \tag{6}\]  
\[\overset{\rightarrow}{E}_{s}=\ A_{s}\exp\bigl{(}j(\omega_{s}t+\phi_{s}) \bigr{)}\bigl{[}\cos(\theta)\overset{\rightarrow}{x}+\sin(\theta)\overset{ \rightarrow}{y}\bigr{]} \tag{7}\]  
where A\({}_{p1}\), A\({}_{p2}\), and A\({}_{s}\) are the amplitudes of pump1, pump2, and the signal, respectively. The idler at a frequency of \(\omega_{i}=\ \omega_{p2}-\ \omega_{p1}+\omega_{s}\) is generated by the resultant of two beating processes given by [38; 39; 40],  
\[\overset{\rightarrow}{E}_{i}=r\bigl{(}\omega_{p2}-\ \omega_{p1}\bigr{)} \biggl{(}\overset{\rightarrow}{E}_{p2}\overset{\rightarrow*}{E}_{p1}^{*} \biggr{)}\overset{\rightarrow}{E}_{s}+r\bigl{(}\omega_{s}-\omega_{p1}\bigr{)} \biggl{(}\overset{\rightarrow}{E}_{s}\overset{\rightarrow*}{E}_{p1}^{*} \biggr{)}\overset{\rightarrow}{E}_{p2}, \tag{8}\]  
where \(r\bigl{(}\omega_{p1}-\ \omega_{p2}\bigr{)}\) is the efficiency of the beating process between the two pumps and \(r\bigl{(}\omega_{s}-\ \omega_{p1}\bigr{)}\) is the efficiency of the beating between the signal and the pump1 [41].  
In the case of co-polarized pumps scheme, we tune the wavelengths of the two pumps such that the input signal is far enough from the pumps in order to avoid their interaction, which means \(\bigl{|}\omega_{p2}-\omega_{p1}\bigr{|}\ll\bigl{|}\omega_{s}-\omega_{p1} \bigr{|}\). Therefore, the electrical field of the idler can be given by,  
\[\overset{\rightarrow}{E}_{i}=r\bigl{(}\omega_{p2}-\omega_{p1}\bigr{)} \biggl{(}\overset{\rightarrow}{E}_{p2}\overset{\rightarrow}{E}_{p1}^{*} \biggr{)}\overset{\rightarrow}{E}_{s} \tag{9}\]  
Substituting Equation (5) to Equation (7) in Equation (9) gives,  
\[\overset{\rightarrow}{E}_{i}=\ r\bigl{(}\omega_{p2}-\omega_{p1}\bigr{)}\ A_{p1}^{*}\ A_{p2}^{\ \signal. Assuming the SOA is polarization independent, the dual co-polarized pumping scheme can be used to convert a signal with arbitrary polarization with a constant conversion efficiency and single nonlinear SOA.