# Segmented Composite Optical Parametric Amplification
## 3 General Numerical Approach and LiNbO\({}_{3}\) Crystal Simulations  
The procedure that we track is the following. The period of the flip sign of the nonlinear susceptibility \(\chi^{(2)}\) is such that the phase mismatch for the OPA process will be zero in Equation (2) due to QPM, resulting in a local modulation period \(\Lambda\) (Figure 1a). However, in contrast to the periodic design, we induce additional sign flips of the coupling coefficient at specific boundaries. In practice, at each segment boundary, there are two domains with the same orientation that merge in a single double-as-long domain, as shown in Figure 1b. The periodic sign switch of \(\chi^{(2)}\) ensures the phase matching for OPA (\(\Delta k=0\)) and the additional sign switches of \(\chi^{(2)}\) at the segment boundaries will change the sign of \(\widetilde{\Omega}\) in the whole crystal segment in a similar fashion as in Shaka-Pines pulses from NMR [12, 13] discussed in the Appendix A. We denote the intervals between two double length domains as \(l_{1},l_{2},l_{3}...l_{N}\), as shown in Figure 1b. The determination of the optimum segment lengths \(l_{k}\) is done using Monte Carlo simulations in the depleted pump regime. In practice, it consists in a maximization of the integral \(Q\) of the (normalized) amplification \(a\) over a surface of interest in the \((\Delta k,\widetilde{\Omega})\)-space, where the values of \(\Delta k\) and \(\widetilde{\Omega}\) are in units of the reciprocal crystal length \(1/L\). The Figure-of-Merit integral \(Q\) is bounded by \(1\) and is given as  
\[Q\equiv\frac{1}{r}\frac{1}{2\widetilde{\Omega}_{max}\Delta k_{max}}\int_{- \Delta k_{max}}^{\Delta k_{max}}\int_{0}^{\widetilde{\Omega}_{max}}a\left( \widetilde{\Omega},\Delta k\right)\ d(\Delta k)\ d\widetilde{\Omega}\enspace, \tag{7}\]where \(r\) is the initial pump-to-signal photon-intensity ratio, \(r\equiv I_{3}(0)/I_{1}(0)\). In our case, we have chosen \(\Delta k_{max}=15/L\) and \(\widehat{\Omega}_{max}=30/L\), the optimization of the integral \(Q\) is done over \(10^{5}\) random sets of the segment lengths \(l_{1},l_{2},l_{3}..l_{N}\). Those corresponding to the best solutions are listed in Table 1. We have found that the use of a small number of composite segments (two, three, and partly four) do not lead to any strong improvement with respect to the standard QPM case. In contrast, already, for a moderate number of segments between six and eight, we find a significant improvement of the robustness of the amplification process. In this case, our analysis shows that there are different solutions for the optimum segment configurations (given in Table 1), which works better depending on the initial amplitude of the signal wave \(A_{1}(0)\) (as compared to the pump wave amplitude taken as \(A_{3}(0)=1\)). Figure 2 illustrates the performance of the composite crystals with six segments (6a from Table 1) compared to standard periodic design. The amplification values are calculated numerically from Equation (1) in the cases when \(A_{1}\left(0\right)=0.1\), \(A_{2}\left(0\right)=0\), and \(A_{3}\left(0\right)=1\), and thus a pump-to-signal photon-intensity ratio \(r=100\). Figure 2 shows clearly that the region of high signal intensity amplification expands strongly for the segmented composite crystal compared to standard periodic design. In other words, the composite crystals exhibit much broader acceptance bandwidths compared to a standard quasi-phase-matching. The working principle of the composite concept can be recognized directly with the help of Figure 3, which depicts the evolutions of the signal wave intensity for the conditions associated to the three positions P1, P2, and P3 given in Figure 2. Clearly, each segment boundary gives rise to a \"kick\" for such evolutions.