# Segmented Composite Optical Parametric Amplification
## 3 General Numerical Approach and LiNbO\({}_{3}\) Crystal Simulations
The amplification values are calculated numerically from Equation (1) in the cases when \(A_{1}(0)=0.1\), \(A_{2}(0)=0\), and \(A_{3}(0)=1\) for Figure 4 and \(A_{1}(0)=0.01\), \(A_{2}(0)=0\), and \(A_{3}(0)=1\) for Figure 5. Clearly, a greatly enhanced robustness and frequency bandwidth of the six-segment composite OPA compared to the standard QPM OPA can be recognized. The plots in Figure 5c compare directly the signal intensity amplification spectrum for the optimal range for composite crystal (red line) and optimal range for standard periodic design (blue dash line). These are slices at the pump intensity values of 3.75 GW/cm\({}^{2}\) and 0.8 GW/cm\({}^{2}\) in Figure 5a,b, respectively. Finally, Figure 6 gives the color plots for the signal intensity amplification \(a\) as in Figures 4 and 5, but for the case where the initial signal intensity is as strong as the pump (_r_ = 1, \(A_{1}(0)=1\), \(A_{2}(0)=0\) and \(A_{3}(0)=1\)). This is a highly depleted regime, and the real advantages of the segmented composite approach over the standard QPM in the depleted pump regime can be recognized by the very large red area in the two right panels. In general, a laser beam spot has an intensity distribution with strong intensity in the center of the spot and smaller intensity at the wings (Gaussian beam for example). Therefore, averaging the amplification value for the big red island of the central panel (**b**) in Figure 6 will lead to a significantly higher average amplification than averaging the amplification  
Figure 4: (Color online) Color plots of the signal intensity amplification \(a\) as a function of the input pump intensity \(I_{3}\) and the signal wavelength \(\lambda_{1}\) for the case of MgO:LiNbO\({}_{3}\) and with initial conditions \(A_{1}(0)=0.1\), \(A_{2}(0)=0\), and \(A_{3}(0)=1\) so that \(r\) = 100. The total crystal length is \(L=5\) mm. A pump intensity of 1 GW/cm\({}^{2}\) corresponds to a modified coupling coefficient \(\Omega=1.32\) 1/mm at the central wavelength of 1.55 μm. (**a**) QPM periodic design only. (**b**) Composite crystal with three segments (3 from Table 1). (**c**) Composite crystal with six segments (6b from Table 1). In all cases, the poling period is \(\Lambda=\)29.71 μm. The color code for the amplification \(a\) is given in the color bar on the right-hand side. The three isolines mark intensity amplification levels of 30, 60, and 90.  
values for the oscillation islands seen for standard QPM in Figure 6a. In Figure 6, we also show that the composite approach is very tolerant with respect to errors in the segment lengths, which is typical also of composite pulse techniques in quantum physics. The color plot in Figure 6c is obtained with the sequence 8 of Table 1) like Figure 6b after adding a large random error on each segment length within \(\pm 5\%\) of the segment length itself. After this procedure each segment length was renormalized by a common factor in order to maintain the same total length of 5 mm like for Figure 6b. It can be seen that the large robust red area is maintained nearly unchanged despite for the rather large allowed errors. More realistic random errors of the order of 1-2% give landscapes almost indistinguishable from the one in Figure 6b. Figure 5: (Color online) (**a,b**) Color plots of the signal intensity amplification \(a\) vs. the input pump intensity \(I_{3}\) and the signal wavelength \(\lambda_{1}\) for the case of MgO:LiNbO\({}_{3}\). The initial conditions are \(A_{1}(0)=0.01\), \(A_{2}(0)=0\), and \(A_{3}(0)=1\) so that \(r=10\),000 and the remaining parameters are as in Figure 4. (**a**) QPM periodic design only. (**b**) Composite crystal with six segments (6c from Table 1). The color bar is given on the top. The three isolines mark intensity amplification levels of 3000, 6000, and 9000.