## 4 PSA in a DOF: Dual Pump Configuration
### Analytical Model: Three Wave Approach  
In this section we describe the standard three-wave (two pumps and a degenerate signal) model [23] for studying the signal gain in a DOF. In this model we neglect the fiber attenuation and consider the pumps to be undepleted along the length of the fiber. We also set the phase of the dispersion modulation \(\varphi_{\text{mod}}=0\) rad. The differential equations governing the evolution of the signal and the two pumps are Equation (7) and Equation (8) respectively. Setting \(\alpha=0\) in Equation (8) and considering the two pumps to be undepleted along the length of the DOF, their slowly varying complex amplitudes denoted by \(A_{\pm 1}\) are given by:  
\[A_{\pm 1}=\sqrt{p}e^{\beta_{\gamma}P_{2}}, \tag{29}\]  
where \(P\) is the undepleted pump power. Here we assumed the input phases of the pumps to be \(0\) rad. Using Equation (29) in Equation (7), and considering the pump waves to be much stronger than the signal, hence neglecting self-phase modulation, the evolution of the slowly varying complex amplitude of the signal \(A_{0}(z)\) is thus given by:  
\[\frac{dA_{0}}{dz}=i2\gamma P\Big{[}2A_{0}+A_{0}^{*}e^{i\phi_{\gamma}P_{2}}e^{i \xi(z)}\Big{]}. \tag{30}\]  
Considering Equation (10) for \(\varphi_{\text{mod}}=0\) and \(\overline{\beta}_{j}=0,j=4,6,\dots\), we get:  
\[\xi(z)=\Omega_{ps}^{2}\left[\overline{\beta}_{2}z-\beta_{2}\frac{z_{\text{mod }}}{2\pi}\cos\left(\frac{2\pi z}{z_{\text{mod}}}\right)+\frac{\bar{\beta}_{2} z_{\text{mod}}}{2\pi}\right]. \tag{31}\]  
Figure 8: An illustration of the degenerate signal and the two pumps considered in a three-wave model. Notations are same as Figure 3.  
Figure 7: Phase-spectral distribution of the gain \(G_{S}\) for \(q=1\) (**a**) and \(q=2\) (**b**). Parameters are same as in Figure 6.  
We note from the above expression that the dispersion oscillation can only have a non-negligible effect only when \(\bar{\beta}_{2}\) is much larger than \(\overline{\beta}_{2}\). This is typically the case for \(\overline{\beta}_{2}\approx 0\). One might argue that when \(\overline{\beta}_{2}\approx 0\), the high order dispersion terms neglected in the Taylor series expansion of \(\beta_{2}\) might have a significant impact on the gain dynamics. But, we should also remember that in our case \(\overline{\beta}_{2}\approx 0\) does not mean that \(\beta_{2}\approx 0\) throughout the fiber length, since \(\beta_{2}\) depends on \(z\). In fact, there are only few (as many as the number of cycles of the dispersion oscillation) regions in the fiber where \(\beta_{2}\approx 0\). In all the other regions, the absolute local value of \(\beta_{2}\) is larger, which depends on \(\bar{\beta}_{2}\). Numerical simulations are utilized to validate this assumption, as we discuss later. Using Equation (31) in Equation (30), the evolution of \(A_{0}\) is given by:  
\[ \begin{split}&\frac{dA_{0}}{dz}=i\gamma P\Bigg{[}4A_{0}+2A_{0}^{*} \exp\Bigg{(}i\delta\gamma Pz+i\Omega_{ps}^{2}\overline{\beta}_{2}z+\frac{i \Omega_{ps}^{2}\bar{\beta}_{2}z_{\text{mod}}}{2\pi}\Bigg{)}\\ &\times\sum_{n=-\infty}^{\infty}i^{n}I_{n}\Bigg{(}\frac{-\Omega _{ps}^{2}\bar{\beta_{2}}z_{\text{mod}}}{2\pi}\Bigg{)}\exp\Bigg{(}\frac{i2\pi nz }{z_{\text{mod}}}\Bigg{)}\Bigg{]},\end{split}  \tag{32}\]  
where \(J_{n}()\) is the Bessel function of the first kind, of order \(n\). Here we have used the Jacobi-Anger expansion, to convert the cosine in the exponential term in Equation (30) into an infinite series of exponentials with linear exponents. In order to solve for \(A_{0}\), we perform the following transformation of variable:  
\[A_{0}=B_{q}\exp\Bigg{[}i\Big{(}3\gamma P+\frac{\overline{\beta}_{2}\Omega_{ps }^{2}}{2}+\frac{q\pi}{z_{\text{mod}}}\Big{)}z+\frac{i\Omega_{ps}^{2}\bar{\beta }_{2}z_{\text{mod}}}{4\pi}\Bigg{]}, \tag{33}\]  
where \(q\in\mathbb{Z}\) is a parameter that corresponds to the DOF induced sideband order [17].