# Segmented Composite Optical Parametric Amplification
## 2 Theory  
We start with the simmetrized coupled wave equations for collinear three-wave mixing in the slowly varying envelope approximation [18; 19],  
\[i\partial_{z}A_{1} =\widetilde{\Omega}A_{2}^{*}A_{3}\exp\left[-i\Delta kz\right], \tag{1a}\] \[i\partial_{z}A_{2} =\widetilde{\Omega}A_{1}^{*}A_{3}\exp\left[-i\Delta kz\right],\] (1b) \[i\partial_{z}A_{3} =\widetilde{\Omega}A_{1}A_{2}\exp\left[i\Delta kz\right], \tag{1c}\]  
where \(\widetilde{\Omega}=-(2\chi^{(2)}/\pi c)\sqrt{\omega_{1}\omega_{2}\omega_{3}/n_ {1}n_{2}n_{3}}\) is the effective nonlinear coupling coefficient for first-order QPM, \(z\) is the position along the propagation axis, \(\omega_{j}\) are the frequencies of the three involved waves, and \(n_{j}\) are their refractive indices. Here \(j=1,2,3\) refer to the signal, idler and pump waves, respectively. The quantity \(\chi^{(2)}\) in \(\widetilde{\Omega}\) is the effective second-order susceptibility and \(c\) is the speed of light in vacuum. The amplitudes \(A_{j}\equiv\sqrt{n_{j}/\omega_{j}}\,E_{j}\) in (1) are proportional to the amplitudes \(E_{j}\) of the wave electric fields; \(|A_{j}|^{2}\) is proportional to the number of photons associated to the \(j\)th wave. Note that Equation (1) is written in a form that assumes that quasi-phase-matching is implemented and that the quasi-phase matching period is sufficiently short as compared to the interaction length. Therefore, the phase mismatch parameter \(\Delta k\) already contains the mismatch compensation term associated to the periodic grating, that is,  
\[\Delta k=k_{1}+k_{2}-k_{3}+2\pi/\Lambda\equiv\widetilde{\Delta k}+2\pi/\Lambda. \tag{2}\]  
where \(\Lambda\) is the quasi-phase-matching period, that is, the first-order local poling period in the case of periodically poled crystals. Obviously, for the central operation wavelengths at which the device is designed, one has \(\Delta k=0\). Concurrently, the true phase mismatch \(\widetilde{\Delta k}=k_{1}+k_{2}-k_{3}\), which depends only on the wave-vectors \(k_{j}\) of the three interacting waves, is generally quite far from vanishing. The set of Equation (1) could have been written also by using the quantities \(\widetilde{\Delta k}\) instead of \(\Delta k\), however in this case \(\widetilde{\Omega}\) would need to switch its sign after each distance \(\Lambda/2\) and the term \(2/\pi\) would need to be dropped in the nonlinear coupling coefficient. Depending upon the initial conditions \(A_{j}\left(z=0\right)\), different processes can arise: sum frequency generation (SFG), difference frequency generation (DFG) or OPA. Here, we consider the OPA case, we assume that \(\omega_{3}=\omega_{1}+\omega_{2}\) and we treat first Equation (1) in the limit of validity of the undepleted pump approximation (\(A_{3}=\text{const}\)). In this limit the set of Equation (1) turns into  
\[i\partial_{z}A_{1} =\Omega A_{2}^{*}\exp\left[-i\Delta kz\right], \tag{3a}\] \[i\partial_{z}A_{2} =\Omega A_{1}^{*}\exp\left[-i\Delta kz\right], \tag{3b}\]  
with \(\Omega=\widetilde{\Omega}A_{3}\) being a modified coupling coefficient. When the wave vector mismatch \(\Delta k\) and the coupling coefficient \(\Omega\) can constant, then Equation (3) possess exact analytic solutions [20],  
\[A_{1}\left(z\right) =e^{i\Delta kz/2}\left[A_{1}\left(0\right)\left(\cosh\left(gz \right)-\frac{i\Delta k}{2g}\sinh\left(gz\right)\right)+\frac{i\Omega}{g}A_{2 }^{*}\left(0\right)\sinh\left(gz\right)\right], \tag{4a}\] \[A_{2}\left(z\right) =e^{i\Delta kz/2}\left[A_{2}\left(0\right)\left(\cosh\left(gz \right)-\frac{i\Delta k}{2g}\sinh\left(gz\right)\right)+\frac{i\Omega}{g}A_{1 }^{*}\left(0\right)\sinh\left(gz\right)\right], \tag{4b}\]where \(g=\sqrt{\Omega^{2}-\left(\frac{\Delta k}{2}\right)^{2}}\) is a gain coefficient.