## 4 PSA in a DOF: Dual Pump Configuration
### Analytical Model: Three Wave Approach
Thus Equation (32) can be written as:  
\[\frac{dB_{q}}{dz}=i\gamma P\Bigg{[}\Bigg{(}1-\frac{\overline{\beta}_{2}\Omega _{ps}^{2}}{2\gamma P}-\frac{q\pi}{\gamma Pz_{\text{mod}}}\Bigg{)}B_{q}+2B_{q} ^{*}\sum_{n=-\infty}^{\infty}i^{n}J_{n}\Bigg{(}-\frac{\Omega_{ps}^{2}\bar{ \beta}_{2}z_{\text{mod}}}{2\pi}\Bigg{)}\exp\bigg{(}\frac{i2\pi(n-q)z}{z_{\text {mod}}}\bigg{)}\Bigg{]}. \tag{34}\]  
At this point, to get rid of the infinite series in Equation (34), we make the following approximation. We note that the effective contribution of the oscillating terms in the series, over large propagation distances (\(z\gg z_{\text{mod}}\)) is negligible. However, the only terms that are not oscillating, appear when \(n=q\)[14]. Thus neglecting the terms with \(n\
eq q\) in Equation (34), the \(q\)-th order state column vector \(\textbf{B}_{q}(z)=\big{[}B_{q}\quad{B_{q}}^{*}\big{]}^{\text{T}}\) can be written in a matrix form as:  
\[\frac{d\textbf{B}_{q}}{dz}=\textbf{M}_{q}\textbf{B}_{q}, \tag{35}\]  
where \(\textbf{M}_{q}\) is the coefficient matrix for the ordinary differential equation (ODE) system, given by:  
\[\textbf{M}_{q}=i\gamma P\Bigg{[} \begin{pmatrix}1-\frac{\overline{\beta}_{2} \Omega_{ps}^{2}}{2\gamma P}-\frac{q\pi}{\gamma Pz_{\text{mod}}}\\ -2e^{\frac{q\pi}{2}}J_{q}\left(-\frac{\Omega_{ps}^{2}\bar{\beta}_{2}z_{m}}{2 \pi}\right)&-\left(1-\frac{\overline{\beta}_{2}\Omega_{ps}^{2}}{2\gamma P}- \frac{q\pi}{\gamma Pz_{\text{mod}}}\right)\end{pmatrix} . \tag{36}\]  
The solution of \(\textbf{B}_{q}(z)\) can be obtained through the matrix exponential method [5], and is given by:  
\[\textbf{B}_{q}(z)=\bigg{[}\cos(gz)\textbf{I}+\frac{\sin(gz)}{g}\textbf{M}_{q} \bigg{]}\textbf{B}_{q}(0)=\textbf{T}_{q}(z)\textbf{B}_{q}(0), \tag{37}\]  
where **I** is \(2\times 2\) identity matrix, \(\textbf{T}_{q}(z)\) is the transfer matrix that transforms state \(\textbf{B}_{q}(0)\) into \(\textbf{B}_{q}(z)\), and the parametric gain coefficient \(g\)[50] is given by:\[g= \gamma P\Bigg{[}\left(1-\frac{\overline{\beta}_{2}\Omega_{ps}^{2}}{2\gamma P}- \frac{q\pi}{\gamma Pz_{\text{mod}}}\right)^{2}-4J_{q}\left(\frac{-\Omega_{ps}^{2 }\vec{\beta}_{2}z_{\text{mod}}}{2\pi}\right)^{2}\Bigg{]}^{\frac{1}{2}}. \tag{38}\]  
From the above expression of \(g\) in Equation (38) we notice that \(g\) can be either real or imaginary depending on the balance between the first and the second term. As a consequence, when \(g\) is imaginary, the sines and cosines in Equation (37) turn hyperbolic sines and cosines leading to exponential, or in other words high gain situations. It is easily proven that \(\mathbf{T}_{q}(z)\) has a unity determinant. Thus \(\mathbf{T}_{q}(z)\) is a symplectic matrix. We also note here that \(\mathbf{T}_{q}(z)\) has the same form of a transfer matrix of a standard two-mode squeezed state arising from a fiber phase sensitive amplifier [45]. We are typically interested in the case of \(q=0\), or in other words the fundamental gain band of the modulation instability (MI) spectrum of a DOF [14]. As a sanity check, putting \(q=0\) (for the fundamental gain band) and \(\hat{\beta}_{2}=0\) (meaning no dispersion oscillation) in Equation (38) gives us:  
\[g=\gamma P\left[-3-\frac{\overline{\beta}_{2}\Omega_{ps}^{2}}{\gamma P}+\left( \frac{\overline{\beta}_{2}\Omega_{ps}^{2}}{2\gamma P}\right)^{2}\right]^{ \frac{1}{2}}, \tag{39}\]  
which is the parametric gain coefficient for a standard HNLF in a dual-pump configuration with a degenerate signal [43]. The analytical results of the maximum PSA signal gain in the fundamental gain band (\(q=0\)) for a DOF of length \(L=1000\) m, pump power \(P=22\) dBm, \(z_{\text{mod}}=250\) m and \(\gamma=4\) (W km)\({}^{-1}\) and 10 (W km)\({}^{-1}\) are plotted in Figures 9g,h,i and 10g,h respectively. Three different values of \(\overline{\beta}_{2}\) were considered, i.e., \(-5\) ps\({}^{2}\)/km, \(0\) ps\({}^{2}\)/km, and \(5\) ps\({}^{2}\)/km, corresponding to left, middle and right columns respectively.