## 3 PSA in a DOF
### Analytical Model of the PSA: Single Pump Configuration
Here the parametric gain coefficient  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]3  
For the fundamental band (\(q=0\)), expression for the gain coefficients (22) represents conventional parametric gain in optical fibers [5; 44; 35]. In assumption that the idler and signal input powers are equal \((|b_{1}(0)|^{2}=|b_{-1}(0)|^{2})\), the total gain for \(q\)-th component of the signal wave can be calculated as follows  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]4  
where \(\Phi_{q}=\arg(b_{1}(0)b_{-1}(0)\mu\
u^{*})=\varphi_{1}+\varphi_{-1}+\varphi_{ \mu}-\varphi_{\
u}\). Here \(\varphi_{\pm 1}=\arg(b_{\pm 1}(0))=\arg(A_{\pm 1}(0))\) are initial phase angles for the signal and idler waves, \(\varphi_{\mu}=\arctan(g^{-1}\kappa\tanh(gz))\) is the phase angle of the complex function \(\mu(z)\), and \(\varphi_{\
u}=\arg(\
u)\) is the phase angle of the complex function \(\upsilon(z)\). Expression for \(\Phi_{q}\) can be rewritten in the following form  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]5  
Equations (23) and (24) represent nonlinear interference of the signal and idler. The interference is phase-sensitive, and the signal and idler can interfere constructively and destructively. The total gain of the signal wave (23) has maximum for \(\Phi_{q}=0\) (parametric amplification) and minimum for \(\Phi_{q}=\pi\) (parametric attenuation). By having signal, idler, and pump waves present at the fiber input and adjusting the relative phase between them, we are able to adjust signal amplification. A remarkable feature of the PSA in the DOF is the dependence of the relative phase \(\Phi_{q}\) on the modulation phase of the fiber dispersion \(\varphi_{\rm mod}\). This feature will be discussed in the Section 3.2. Let us consider the total gain the signal wave when the initial idler is zero, \(b_{1}(0)=0\). From (20) we can find that the idler wave is proportional to the conjugated signal, namely \(\varphi_{1}=-\varphi_{-1}+\varphi_{\
u}\). The gain of the signal wave \(G_{q}=|\mu|^{2}\) does not depend on the phases \(\varphi_{1}\), \(\varphi_{-1}\), \(\theta\), and \(\varphi_{\rm mod}\). In other words, in presence only two input waves we have phase-insensitive amplification. The highest gain occurs for phase-matching condition \(\kappa=0\), see Equation (22). The frequency detuning \(\Omega_{q}\) corresponding to the largest gain coefficient can be found from a polynomial expression:  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]6  
An approximative transformation of Equation (25) may be done from the frequency domain to the more generally used wavelength domain [44]. Derivatives \(\overline{\beta}_{2k}\) are calculated for the frequency of the pump wave \(\omega_{0}=2\pi c\lambda_{p}^{-1}\), where \(\lambda_{p}^{-1}\) is the pump wavelength. We can calculate dispersion coefficients \(\overline{\beta}_{2k+1}^{\rm zero}=d^{2k+1}\overline{\beta}/d\omega^{2k+1}\) at the frequency \(\omega_{\rm zero}=2\pi c\lambda_{\rm zero}^{-1}\) corresponding to the zero-dispersion wavelength \(\lambda_{\rm zero}\). In assumption, that the dispersion coefficients increase linearly with the frequency, i.e., \(\overline{\beta}_{2k}=\overline{\beta}_{2k+1}^{\rm zero}(\omega_{0}-\omega_{ \rm zero})\), Equation (25) becomes  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]7  
where \(\lambda_{s}=2\pi c/\omega_{-1}\) is the wavelength of the signal wave. The parametric gain coefficient \(g\) depends on the pump power \(P_{0}\) and average dispersion coefficients. The analytical treatment of expressions (25) and (26) in the presence of high-order dispersion terms is quite complicated.