## 3 PSA in a DOF  
PSA implementations commonly involve two configurations: 1. single pump configuration where a non degenerate signal and idler along with a strong pump are launched at the fiber input (see Figure 3), and 2. dual pump configuration, where the signal (or idler) at the input is degenerate and two non degenerate pumps are launched along with it.  
Figure 2: False color plots of (a) \(\beta_{2}\), (b) \(\beta_{3}\) and (c) \(\beta_{4}\) as a function of the core radius and W-ratio \(=b/a\) calculated from the mode solver.  
To study the field propagation in the DOF we use local mode approach [40]. The local mode approach requires slow variation of the fiber core. In a single-mode fiber the criterion for slow variation is \(z_{b}\ll z_{\text{mod}}\), where \(z_{b}=2\pi(\beta-k_{0}N_{3})^{-1}\) is the beat length between the fundamental and cladding mode. Calculation of the beat length gives the value, which does not exceed \(10^{-3}\) m. In our study, the modulation period \(z_{\text{mod}}\) is tens of meters. So the slow variation criterion \(z_{b}\ll z_{\text{mod}}\) is satisfied. We consider the propagation of three stationary copolarized waves with slowly varying complex amplitudes \(A_{0},A_{-1}\) and \(A_{1}\), which represent the electric fields at the frequencies \(\omega_{0}\), \(\omega_{-1}=\omega_{0}-\Omega_{ps}\) and \(\omega_{1}=\omega_{0}+\Omega_{ps}\), where \(\Omega_{ps}\) is the pump-signal angular frequency separation. The total transverse electric field propagating along the single-mode DOF may be written as [35; 40]  
\[E(x,y,z,t)=\frac{1}{2}\frac{f(x,y)}{(nc\varepsilon_{0}A_{\text{ eff}})^{1/2}}\bigg{[}A_{0}(z)\exp\bigg{(}i\int_{0}^{z}\beta(\omega_{0},z^{ \prime})dz-i\omega_{0}t\bigg{)}\] \[+A_{-1}(z)\exp\bigg{(}i\int_{0}^{z}\beta(\omega_{-1},z^{\prime}) dz-i\omega_{-1}t\bigg{)} \tag{6}\] \[+A_{1}(z)\exp\bigg{(}i\int_{0}^{z}\beta(\omega_{1},z^{\prime}) dz-i\omega_{1}t\bigg{)}+\text{c.c.}\bigg{]},\]  
where \(f(x,y)\) is the common transverse modal profile which is assumed to be identical for all three waves along the fiber, \(n\) is the effective index of optical mode, \(c\) is the speed of light in vacuum, \(\varepsilon_{0}\) is the vacuum permittivity, \(A_{\text{eff}}\) is the effective mode area averaged over the fiber length. As discussed in Section 2.2, we neglect the variation of the transverse mode profile along the fiber length. Due to the denominator \((nc\varepsilon_{0}A_{\text{eff}})^{1/2}\) in (6) the values of \(|A_{0}|^{2}\) and \(|A_{\pm 1}|^{2}\) are in terms of Watts. Starting from wave equation (see Chapter 2 in [35]), it is straightforward to derive three coupled ordinary differential equations for \(A_{0},A_{-1}\) and \(A_{1}\)[35; 41]  
\[\frac{dA_{0}}{dz} = i\gamma(|A_{0}|^{2}+2|A_{-1}|^{2}+2|A_{1}|^{2})A_{0}+2i\gamma A _{0}^{*}A_{-1}A_{1}e^{i\xi(z)}-\frac{\alpha}{2}A_{0}, \tag{7}\] \[\frac{dA_{\pm 1}}{dz} = i\gamma(2|A_{0}|^{2}+2|A_{\mp 1}|^{2}+|A_{\pm 1}|^{2})A_{\pm 1}+i \gamma A_{0}^{2}A_{\mp 1}^{*}e^{-i\xi(z)}-\frac{\alpha}{2}A_{\pm 1}, \tag{8}\]  
where \(\alpha\) is the attenuation coefficient. The linear phase mismatch is  
\[\xi(z)=\int_{0}^{z}\beta(\omega_{-1},z^{\prime})+\beta(\omega_{1},z^{\prime}) -2\beta(\omega_{0},z^{\prime})dz^{\prime} \tag{9}\]