## 5 THG Efficiency in Dependence of SOD Coefficients and Phase Mismatching between Interacting Waves
### Dependence of the Crystal Length on \(\Delta_{21}k\) for Reaching Maximum THG Efficiency  
To clarify the dependence of the energy conversion efficiency on the phase mismatching between FW and SH, we provide computer simulations of the problem for different values of \(\Delta_{21}k\), changing from \(-200\) to \(200\), at a propagation of the pulse with its incident duration being equal to \(1\ ps\). Our aim is the definition of the highest conversion efficiency and the corresponding crystal length required for this. The obtained results, computed at a phase matching between TH and FW, are shown in Figure 13.  
One can see that at appropriate choice of the parameters, the highest efficiency can be achieved if the phase mismatching between FW and SH is close to zero, but does not equal to zero: for example, \(\Delta_{21}k=-4\) (Figure 14). The maximum TH intensity achieves a value close to unity, and the TH pulse shape and its spectrum are smooth, as are as the spectra of other pulses. More than \(80\%\) of the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\gamma\) & \(\Delta_{21}k\) & \(D_{1}\) & \(D_{2}\) & \(D_{3}\) & MI & \(z_{MI}\) & \(|A_{3}(z_{MI},\,0.5L_{t})|^{2}\) & \(\eta_{max}\) \\ \hline \(1\) & \(-250\) & \(-0.000032\) & \(-0.000083\) & \(-0.000199\) & Yes & 74 & 0.4225 & 0.34 \\ \hline \(1\) & \(-250\) & \(-0.00032\) & \(-0.00083\) & \(-0.00199\) & Yes & 160 & 0.2894 & 0.51 \\ \hline \(1\) & \(-250\) & \(-0.00064\) & \(-0.00166\) & \(-0.00398\) & No & – & 0.1667 & 0.5 \\ \hline \(\sqrt{2}\) & \(-250\) & \(-0.00064\) & \(-0.00166\) & \(-0.00398\) & Yes & 116 & 0.231 & 0.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Developing of the MI in dependence on incident FW pulse duration and its maximum intensity. Figure 12: SH left sub-pulse shape in the sections \(z=110\) (black solid line), \(114\) (red dashed-dotted line), \(118\) (green dashed line), \(120\) (blue solid line with squares) (**a**) and the comparison between pulse shapes in all four sections (**b**) for the parameters \(\gamma=1\), \(\Delta_{21}k=-250\), \(D_{1}=-0.000032\), \(D_{2}=-0.000083\), \(D_{3}=-0.000199\). incident FW pulse energy transfers to the TH even in the section \(z=6.5\). However, taking into account the phase mismatching between TH and FW (\(\Delta_{31}k=0\)), such a situation may be hardly realised (or it is impossible at all). So, if we could achieve phase matching between FW and SH as well as between FW and TH simultaneously, we could obtain the comb SH and TH generation. However, their sum efficiency will be less than 80%. We see in Figure 13 that for large positive values of phase mismatching between SH and FW, the TH maximum conversion efficiency remains at a high level despite the changing of \(\Delta_{21}k\) in a wide range. However, the crystal length for its achievement also increases because SH is absent; therefore, this mode corresponds to cascading SHG. For a small negative value of the phase mismatching \(\Delta_{21}k\), the maximum efficiency of the FW energy conversion  
Figure 14: Pulse shapes (**a**), FW and TH spectra (**b**) in the section \(z=6.5\). Pulse intensities evolution at their center (**c**), and the frequency conversion efficiency (**d**) along \(z\)-coordinate computed for the parameters \(\gamma=1\), \(\Delta_{21}k=-4\), \(D_{1}=-0.000032\), \(D_{2}=-0.000083\), \(D_{3}=-0.000199\). Figure 13: Dependencies of maximum conversion efficiency (**a**) of SH (dash-dotted line) and TH (solid line), and required crystal length (**b**) on \(\Delta_{21}k\) for the parameters \(\gamma=1\), \(\Delta_{31}k=0\), \(D_{1}=-0.000032\), \(D_{2}=-0.000083\), \(D_{3}=-0.000199\). Inserts show the corresponding characteristics near \(\Delta_{21}k=0\). to TH is about 3-4% higher than for the positive one. However, this requires increasing of the crystal length. Moreover, at large negative values of \(\Delta_{21}k\), the efficiency conversion is restricted by an appearance of the MI, as described above.