## 3 PSA in a DOF
### Analytical Model of the PSA: Single Pump Configuration  
In this section we focus on a single pump configuration. Single pump configurations are often simpler in terms of theoretical treatment since they do not contain high-order pumps that are generated through cascaded four-wave mixing between the pumps in a dual pump configuration [42; 43]. Figure 3: An illustration of the signal, pump and idler for a single pump PSA configuration. \(\Omega_{ps}\) is the pump-signal angular frequency separation. \(0\), \(1\) and \(-1\) refer to the indices of the waves. (not to scale). Considering the Taylor series expansion of \(\beta(\omega)\) till the \(2p\)-order, the linear phase mismatch (9) is given by:  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]  
where \(\beta_{2k}(z)\) are even coefficients of the Taylor series expansion of \(\beta(\omega)\). Integrating Equation (5), we get  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\bigg{\{}\overline{\beta}_ {2k}z-\bar{\beta}_{2k}\frac{z_{\rm mod}}{2\pi}\bigg{[}\!\cos\!\left(\frac{2\pi z }{z_{\rm mod}}+\varphi_{\rm mod}\right)-\cos(\varphi_{\rm mod})\bigg{]}\bigg{\}}. \tag{11}\]  
In Equations (7) and (8) the amplitudes \(A_{-1}\), \(A_{1}\), and \(A_{0}\) correspond to idler, signal and pump waves. We can analyze Equations (7) and (8) by assuming that the pump power \(P_{0}=|A_{0}|^{2}\) is much larger than the power of signal and idler. In a lossless assumption (\(\alpha=0\)), the amplitude of the pump can be written as \(A_{0}(z)=a_{0}\exp(i\gamma P_{0}z)\). We then insert this into Equation (8), and substitute \(A_{\pm 1}(z)=a_{\pm 1}(z)\exp[i\gamma P_{0}z-iz\sum_{k=1}^{p}\bar{\beta}_{2k} \Omega_{ps}^{2k}((2k)! )^{-1}]\) to obtain coupled equations:  
\[\frac{da_{1}}{dz} = i\imath_{1}\bigg{(}\gamma P_{0}+\sum_{k=1}^{p}\frac{\overline{ \beta}_{2k}\Omega_{ps}^{2k}}{(2k)!}\bigg{)}+i\gamma a_{0}^{2}a_{-1}^{*}R(z) \tag{12}\] \[\frac{da_{-1}^{*}}{dz} = -ia_{-1}^{*}\bigg{(}\gamma P_{0}+\sum_{k=1}^{p}\frac{\overline{ \beta}_{2k}\Omega_{ps}^{2k}}{(2k)!}\bigg{)}-i\gamma a_{0}^{*2}a_{1}R^{*}(z), \tag{13}\]  
where  
\[R(z)=\exp\!\left(i\frac{z_{\rm mod}}{2\pi}\sum_{k=1}^{p}\frac{2\bar{\beta}_{2 k}\Omega_{ps}^{2k}}{(2k)!}\bigg{[}\!\cos\!\left(\frac{2\pi z}{z_{\rm mod}}+ \varphi_{\rm mod}\right)-\cos(\varphi_{\rm mod})\bigg{]}\right)\!. \tag{14}\]  
We expand the function \(R(z)\) as a Fourier series with fundamental period \(z_{\rm mod}\)  
\[R(z)=\sum_{q=-\infty}^{\infty}c_{q}\exp\!\left(iq\frac{2\pi}{z_{\rm mod}}z \right)\!, \tag{15}\]  
\[c_{q}=J_{q}\Bigg{(}\frac{z_{\rm mod}}{2\pi}\sum_{k=1}^{p}\frac{2\bar{\beta}_{ 2k}\Omega_{ps}^{2k}}{(2k)!}\Bigg{)}\exp\!\left(-i\frac{z_{\rm mod}}{2\pi}\sum_ {k=1}^{p}\frac{2\bar{\beta}_{2k}\Omega_{ps}^{2k}}{(2k)!}\cos(\varphi_{\rm mod} )+iq\Big{(}\varphi_{\rm mod}+\frac{\pi}{2}\Big{)}\right)\!, \tag{16}\]  
where \(J_{q}\) are Bessel functions of the first kind. We introduce \(b_{\pm 1}(z)=a_{\pm 1}(z)\exp(-iq\pi z/z_{\rm mod})\). Then equating the coefficients of \(\exp(-iq\pi z/z_{\rm mod})\) gives  
\[\frac{db_{1}}{dz} = i\kappa b_{1}+i\chi b_{-1}^{*}, \tag{17}\] \[\frac{db_{-1}^{*}}{dz} = -i\chi^{*}b_{1}-i\kappa b_{-1}^{*}, \tag{18}\]  
where \(\chi=\gamma c_{q}P_{0}\exp(i2\theta)\), \(\theta=arg(a_{0})\) is the initial phase angle of the pump wave, and  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]0The solution of Equations (17) and (18) can be written as  
\[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]1 \[\xi(z)=\sum_{k=1}^{p}\frac{2\Omega_{ps}^{2k}}{(2k)!}\int_{0}^{z}\beta_{2k}(z^{ \prime})dz^{\prime}, \tag{10}\]2  
where \(\mu(z)=\cosh(gz)+i\kappa g^{-1}\sinh(gz)\), and \(\upsilon(z)=i\chi g^{-1}\sinh(gz)\).