## 3 PSA in a DOF
### Role of Pump Phase \(\theta\) and Modulation Phase \(\varphi_{\text{mod}}\) on Signal Gain  
To demonstrate the effect of the dispersion oscillation on the PSA we solve Equations (7) and (8) numerically. In these calculations, we assume that the initial power of the pump wave \(P_{0}\) to be \(0.5\) W. The initial powers of the signal and idler waves are \(|A_{-1}(0)|^{2}=|A_{1}(0)|^{2}=10^{-6}\) W. Since for a PSA, the total gain depends on the common phase (\(\varphi_{1}+\varphi_{-1}-2\theta-q\varphi_{\text{mod}}\)) (see Equations (23) and (24)), the signal and idler phases at the input of the fiber are considered to be zero. Simulations are carried out for different input pump phase \(\theta\) and modulation phase \(\varphi_{\text{mod}}\). The dispersion modulation profile of the considered DOF with single modulation frequency follow Equation (5).  
In the case of PSA in DOFs, the high-order sidebands (\(q\
eq 0\)) provide interesting perspectives in terms of applications. To evaluate the signal gain \(G_{S}\) we scan the pump-signal frequency separation near the two resonant frequencies \(\
u_{1}\) and \(\
u_{2}\) (i.e., \(q=1\) and \(2\)), for different input phases \(\theta\) and different dispersion modulation phases \(\varphi_{\text{mod}}\) (see Figure 5).  
At \(\theta=-0.6\) rad, a broadband amplification peak is observed. A change in the phase of the pump wave leads to the emergence of a narrow parametric attenuation gap (anti-gain dip) with a high extinction of about 30 dB [see Figure 5a]. As the phase is varied from \(\theta=-0.6\) rad to \(\theta=1.2\) rad, the anti-gain dip moves through the gain spectrum from low to high frequency pump signal separation. The dip's shift occurs synchronously for both the first \((q=1)\) and second \((q=2)\) gain band. This is an interesting feature of the PSA system that would allow preferentially filtering out unwanted frequencies in a wavelength multiplexed system, just by tuning the phase of the input signal.  
Contrary to a phase insensitive amplification with a DOF [12], we further found out that the position of the anti-gain dip depends not only on the phase of the pump wave \(\theta\), but also on the phase of the dispersion modulation \(\varphi_{\text{mod}}\). From Figure 5c,d, we see that for the modulation phase \(\varphi_{\text{mod}}=-1.2\) rad the first band (\(q=1\)) remains unaffected as the modulation phase is increased from \(\theta=-0.2\) rad to \(\theta=0.2\) rad. At the same conditions the second band (\(q=2\)) show the characteristic dip in the gain spectrum.  
The behavior of anti-gain dip can be explained using the analytical model given in Section 3.1. The gain bandwidth of the \(q\)-th spectral component is given by the frequency dependency of the gain coefficient (22). The parametric attenuation within selected \(q\)-th band is realized under the condition \(\cos(\Phi_{q})=-1\) in Equation (23). The value of \(\Phi_{q}\) depends on the phases as \((\varphi_{1}+\varphi_{-1}-2\theta-q\varphi_{\text{mod}})\). This value does not depends on the \(q\) when \(\varphi_{\text{mod}}=0\). As a result, anti-gain dips appear in the first \((q=1)\) and second \((q=2)\) bands synchronously (Figure 5a,b). Effect of the harmonic number \(q\) on the condition \(\cos(\Phi_{q})=-1\) appears only for nonzero \(\varphi_{\text{mod}}\). When \(\varphi_{\text{mod}}=-1.2\) rad the condition \(\cos(\Phi_{q})=-1\) is not reached within the first band \((q=1)\) (Figure 5c). The first band remains unchanged for \(\theta=-0.2,0\), and 0.2 radians. While for the second band \((q=2)\), the frequency corresponding to the condition \(\cos(\Phi_{q})=-1\) moves across the entire band (Figure 5d).  
Figure 5: Plot of gain of the signal as a function of the pump-signal frequency separation for different values of the initial phase \(\theta\) in radians. (**a**,**b**) \(\varphi_{\text{mod}}=0\) rad; (**c**,**d**) \(\varphi_{\text{mod}}=-1.2\) rad. \(q=1\) in left column and \(q=2\) in right column. Other fiber parameters are the same as in Figure 4b,c.