# Segmented Composite Optical Parametric Amplification
## 3 General Numerical Approach and LiNbO\({}_{3}\) Crystal Simulations
By choosing the \(z\)-positions of the frontiers appropriately, one can achieve that the spatial evolutions corresponding to points in the big red area in Figure 2a get very close (nearly \"in phase\") near the end of the device at distance \(L\) with a high final signal wave amplification, as seen in Figure 3a. In contrast, for pure QPM without composite segments such a \"re-phasing\" cannot occur, as can be recognized in Figure 3b. \begin{table}
\begin{tabular}{l l l} \hline \hline \(N\) & **Name** & **Segment Lengths \(I_{1}\); \(I_{2}\); \(\ldots\); \(I_{N}\) in Units of \(L\)** \\ \hline
3 & 3 & 0.373; 0.594; 0.033 \\
4 & 4 & 0.303; 0.522; 0.124; 0.051 \\
6 & 6a & 0.293; 0.258; 0.003; 0.255; 0.124; 0.067 \\
6 & 6b & 0.168; 0.035; 0.345; 0.023; 0.222; 0.207 \\
6 & 6c & 0.223; 0.005; 0.404; 0.175; 0.113; 0.080 \\
8 & 8 & 0.022; 0.064; 0.046; 0.205; 0.270; 0.096; 0.222; 0.075 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerically found segment lengths \(l_{i}\) (in units of total crystal length \(L\)) for composite segmented periodically poled design with \(N\) segments. The given values for \(l_{i}\) are such as to optimize the robustness of the OPA process against variations of the nonlinear coupling coefficient and of the phase mismatch \(\Delta k\). Figure 1: Sign reversal of \(\chi^{(2)}\) nonlinear coefficient for (**a**) standard quasi-phase matching (QPM) technique with local modulation period \(\Lambda\). (**b**) Composite segmented periodically poled design, with example of 3 segments. All segments are periodically poled with the same period \(\Lambda\), however once a new segment begins, the sign reversal order of \(\chi^{(2)}\) is changed. The simulations of Figures 2 and 3 illustrate the general approach for finding the best composite sequence as compared to standard QPM. Next, we will prove the concept further by applying these results to specific practical examples and we make the numerics for a real crystal: 5 mol. % Magnesium Oxide doped Lithium Niobate (MgO:LiNbO\({}_{3}\)). This ferroelectric nonlinear crystal possesses higher damage threshold compared to undoped LiNbO\({}_{3}\), high nonlinear optical coefficient, broad transparency range, and is suitable for domain poling [34]. We compare the standard quasi phase matching with the composite approach for OPA when the three interacting beams share the same phase. The results of the simulations are shown in Figure 3. The results of the simulations are shown in Figure 3.  
same extraordinary polarisation (Type 0 configuration, all beams polarized parallel to crystal _c_-axis) associated to the largest element of the nonlinear tensor \(d_{333}=\chi^{(2)}=27\) pm/V. The color plots in Figures 4 and 5 compare the signal intensity amplification \(a\) for MgO:LiNbO\({}_{3}\) for standard QPM (\(\Lambda=29.71\) um) and for a composite crystal made of six segments. Figure 4 is for intermediate pump-to-signal intensity ratio \(r\) while Figure 5 is for large \(r\). Figure 4 also shows the less optimum case where there are only three composite segments. The nonlinear susceptibility is fixed and the plots are represented for varying input pump intensity (at the fixed wavelength of 1064 nm) and for varying signal wavelength (with center at \(\lambda_{1}=1550\) nm). Note that here, to keep the ratio \(r\) constant for each plot, the input signal intensity changes in the same way as the input pump intensity. Note also that the pump intensity (_y_-axis) takes the role of the coupling in Figure 2. Similarly, as a variation of the signal wavelength with respect to the central one gives rise to a phase mismatch \(\Delta k\); here, the signal wavelength (_x_-axis) takes the role of \(\Delta k\) in Figure 2. The total crystal length is \(L=5\) mm.