## 4 PSA in a DOF: Dual Pump Configuration
### Numerical Results with Scanning of Dispersion Oscillation Parameters  
Having argued that a DOF (with single dispersion modulation frequency) is the best profile for maximizing the PSA gain, here we plot the PSA simulation results, _viz_, maximum PSA signal gain (top row) and the gain extinction ratio (middle row) in Figures 9 and 10 as false color plots as a function of the pump-signal frequency separation \(\frac{\omega_{m}}{2\pi}\), and the amplitude of \(\beta_{2}\) oscillation, i.e., \(\widetilde{\beta}_{2}\). A DOF of length \(L=1000\) m, pump power \(P=22\) dBm, signal power \(P_{s}=-20\) dBm, period of dispersion oscillation \(z_{\text{mod}}=250\) m was considered. \(\gamma=4\) (W km)\({}^{-1}\) for Figure 9 and 10 (W km)\({}^{-1}\) for Figure 10. Three different values of \(\widetilde{\beta}_{2}\) were considered, i.e., \(-5\) ps\({}^{2}\)/km, 0 ps\({}^{2}\)/km, and 5 ps\({}^{2}\)/km, corresponding to the left, middle and right columns respectively.  
From Figure 9, firstly we see that the fundamental gain spectrum (\(q=0\), or when the pump-signal frequency separation is small, \(<\)0.5 THz) is independent of the dispersion oscillation amplitude \(\beta_{2}\) for an average \(\widetilde{\beta}_{2}\) away from 0. This is because unless the dispersion oscillation amplitude is large compared to the average dispersion, or more specifically, \(\widetilde{\beta}_{2}\)z is comparable to \(\widetilde{\beta}_{2}\frac{\overline{\text{2}}\overline{\text{2}}}{2\pi}\), the effect of dispersion oscillation is negligible as can also be seen from Equation (31). Nevertheless, for \(\widetilde{\beta}_{2}\) away from 0, on increasing \(\widetilde{\beta}_{2}\), the high-order MI sidebands, corresponding to \(q\
eq 0\) start to appear [49]. In this case, since the nonlinear  
Figure 12: (**a**) Numerically computed maximum PSA gain vs. pump-signal frequency separation for four different profiles. A random profile (in magenta), a profile with single dispersion oscillation wavelength \(z_{m1}=97\) m (in red), double dispersion oscillation wavelength, \(z_{m1}=97\) m and \(z_{m2}=111\) m (in blue) and triple dispersion oscillation wavelength, \(z_{m1}=97\) m, \(z_{m2}=111\) m and \(z_{m3}=122\) m (in green). (**b**) Variation of outer diameter as a function of length for the same profiles. \(L=500\) m, \(\gamma=10\) (W km)\({}^{-1}\), \(\widetilde{\beta}_{2}=1\) ps\({}^{2}\)/km and \(P=22\) dBm.  
phase \(\gamma PL\) is about 0.6 rads, the maximum gain of the PSA is about 10 dB. On the contrary, for Figure 10, where the nonlinear phase is about 1.5 rads, the gain is much larger, i.e., above 20 dB. It is also important to note from Figure 10b that a highly flat gain of about 10 dB can be obtained over a large bandwidth of about 1 THz while considering a small \(\bar{\beta_{2}}\approx 0.4\) ps\({}^{2}\)/km.  
We also note by comparing the top and bottom rows of Figures 9 and 10, that for a low nonlinearity, i.e., when the nonlinear phase is 0.6 rads, the analytical 3-wave model agree satisfactorily with the numerical NLSE model. However, when the nonlinear phase is 1.5 rads, the analytical model predicts a much larger gain than the numerical one for \(\overline{\beta_{2}}=-5\) ps\({}^{2}\)/km and \(\overline{\beta_{2}}=5\) ps\({}^{2}\)/km [43; 46]. For \(\overline{\beta_{2}}=0\) ps\({}^{2}\)/km, not only the analytical model overestimates the maximum gain, it also predicts a shorter gain ripple wavelength. This mismatch between the numerical and analytical models is due to the presence of high-order waves that are generated from the FWM processes involving the two strong pumps [49].