## 4 PSA in a DOF: Dual Pump Configuration
### Profile Optimization for Larger PSA Gain  
Here we explore the quest of whether a sinusoidal profile is an optimal profile with respect to providing a large PSA gain as well as reducing SBS due to its varying longitudinal dispersion profile. To this aim, let us first focus our attention in Equation (32). We note that each of the terms in the summation (corresponding to different values of \(n\)) at the right hand side of Equation (32) are as if they are different FWM processes. But they give rise to gain peaks at different pump-signal separations since their phase matching condition is different. This is in fact the origin of high-order MI sidebands in DOF based parametric amplifiers [49; 51]. But now, let us consider the case where instead of one dispersion oscillation wavelength \(z_{\text{mod}}\), there are two, \(z_{m1}\) and \(z_{m2}\). Note, now \(\beta_{2}(z)\) is given by:  
\[\beta_{2}(z)=\overline{\beta}_{2}+\tilde{\beta}_{2}\sin\left(\frac{2\pi z}{z _{m1}}\right)+\tilde{\beta}_{2}\sin\left(\frac{2\pi z}{z_{m2}}\right). \tag{41}\]  
Thus the linear phase mismatch \(\xi(z)\) becomes:  
\[\xi(z)=\Omega_{ps}^{2}\left[\overline{\beta}_{2}z-\beta_{2}^{2}\frac{z_{m1}}{ 2\pi}\cos\left(\frac{2\pi z}{z_{m1}}\right)+\frac{\beta_{2}z_{m1}}{2\pi}- \beta_{2}\frac{z_{m2}}{2\pi}\cos\left(\frac{2\pi z}{z_{m2}}\right)+\frac{ \beta_{2}z_{m2}}{2\pi}\right]. \tag{42}\]  
However, this linear phase mismatch always manifests itself in the exponent as a phase, in the evolution equation of the signal. Thus using the Jacobi-Anger identity, the exponential of \(i\xi\) is written as:  
\[ \begin{split}& e^{i\xi(z)}=\exp\left(i\Omega_{ps}^{2}\overline{ \beta}_{2}z+\frac{i\Omega_{ps}^{2}\tilde{\beta}_{2}z_{m1}}{2\pi}z+\frac{i \Omega_{ps}^{2}\tilde{\beta}_{2}z_{m2}}{2\pi}\right)\\ &\times\Bigg{[}\sum_{n=-\infty}^{\infty}i^{n}J_{n}\left(\frac{- \Omega_{ps}^{2}\beta_{2}z_{m1}}{2\pi}\right)\exp\left(\frac{i2\pi nz}{z_{m1}} \right)\Bigg{]}\times\Bigg{[}\sum_{p=-\infty}^{\infty}i^{p}J_{p}\left(\frac{- \Omega_{ps}^{2}\beta_{2}z_{m2}}{2\pi}\right)\exp\left(\frac{i2\pi pz}{z_{m2}} \right)\Bigg{]}.\end{split}  \tag{43}\]  
From the above equation it is now clear that in the signal evolution equation, now we will start having inter-modulation tones due to the multiplication between the two infinite series. Thus the gain peaks coming from the fundamental frequencies (\(\frac{1}{z_{m1}}\) and \(\frac{1}{z_{m2}}\)) will be now polluted by the inter-modulation frequencies (\(2\frac{1}{z_{m1}}-\frac{1}{z_{m2}}\) and \(2\frac{1}{z_{m2}}-\frac{1}{z_{m1}}\)) that have a different phase matching condition from the fundamental tones. To illustrate this point we run a numerical NLSE simulation of the system considering different cases, i.e., a constant random profile, a DOF with dispersion oscillation wavelength \(z_{m1}\), a profile with \(z_{m1}\) and  
Figure 11: False color plot of \(\beta_{2}\) showing a region where \(\beta_{2}\) is within \(-10\) ps\({}^{2}\)/km and 10 ps\({}^{2}\)/km. Horizontal dashed line indicate W-ratio \(b/a=3\) and vertical dashed lines indicate values of \(\beta_{2}=-5,0,5\) ps\({}^{2}\)/km. \(z_{m2}\) wavelengths and finally a profile with \(z_{m1}\), \(z_{m2}\) and \(z_{m3}\) wavelengths. \(z_{m1}=97\) m, \(z_{m2}=111\) m and \(z_{m3}=122\) m. In Figure 12a we provide the maximum PSA gain as a function of the pump-signal frequency separation and in Figure 12b the corresponding diameter variation along the fiber length is provided. Length of the fiber was 500 m, nonlinear coefficient was 10 (W km)\({}^{-1}\), \(\widetilde{\beta}_{2}=1\) ps\({}^{2}\)/km and pump power was 22 dBm. The maximum PSA gain spectra for these profiles along with the dispersion profiles are shown in Figure 12.  
From Figure 12a, we see that when there is only one dispersion oscillation wavelength (red curve), the maximum PSA gain reaches a larger value compared to when two or three wavelengths are present (blue and green curve).