## 4 Influence of Pulse Duration of FW on Frequency Conversion Due to Cascading Processes
### Pulses with Picosecond or Femtosecond Duration  
In the case under consideration, the SOD of the pulses and especially GVM restricts the frequency's conversion efficiency. Nevertheless, a significant part of the FW energy still converts to the wave with tripled frequency. For illustration of the advantage of the derived solution of the problem, we show also the frequency conversion's efficiency computed using the FW energy non-depletion approximation. The incident pulse at the FF possesses the Gaussian distribution of its shape without any chirp:  
\[A_{10}(t)=A_{01}e^{-(t-0.5L_{t})^{2}}, \tag{26}\]  
where \(A_{01}\) is equal to unity due to chosen dimensionless parameters. Computer simulation results are presented in Figure 4. We show FW with red lines, SH with blue lines, and TH with green lines. The computer simulation results of the original problem are shown with solid lines, the computer simulation results of the modified problem with dots, and results of using the wave energy non-depletion approximation with dashed lines. The incident FW pulse distribution is shown by the black line with triangles. Figure 4: Pulse shapes in the section \(z=100\) (**a**) \(z=20\) (**b,c**) and the frequency conversion efficiency along \(z\)-coordinate (**d–f**) for the incident pulse duration and nonlinear parameter \(\gamma\), equal (1 ps, 2) (**a,d**), (100 fs, 2) (**b,e**) and (100 fs, 4) (**c,f**). The problem parameters are equal to \(\gamma=2\), \(\Delta_{21}k=87\), \(\Delta_{31}k=-0.05\). The SOD and GVM coefficients equal \(\
u_{21}=0.07\), \(\
u_{31}=0.21\) and \(D_{1}=-0.0000032\), \(D_{2}=-0.000029\), \(D_{3}=-0.000043\), respectively, corresponding to a 1 ps pulse duration of the FW with a 1064 nm wavelength at its propagation in the KBBF crystal. We show the pulses' amplitudes to show better differences between the problem solutions obtained in various approximations. In Figure 4a, one can see the that coincidence between the solutions of the modified and original problems is perfect. Because of the GVM of presence, the TH pulse moves from the FW one, which results in an asymmetrical shape of the TH pulse. The non-depletion energy approximation for the pump pulse fails to give the adequate description of the wave interaction predicting the super Gaussian pulse shape with much more energy. The only thing that resembles the computer simulation results of the problem is the TH maximum intensity, which is a little bit higher than it should be. As shown in Figure 4d, by using the pulse's energy non-depletion approximation at the wave with FF, the TH energy exceeds initial FW energy near the section \(z=20\), which is completely wrong. In fact, the THG efficiency is about 51%, reaching this value in the section \(z=40\), and then it remains stable. If the pulse duration at the FF is ten times less (100 fs), then the SOD and GVM coefficients increase 100 and 10 times, respectively. In this case, GVM strictly decreases the frequency conversion efficiency: only 11% of the FW energy transfers to TH pulse (Figure 4e) with a small maximum intensity (Figure 4e) and long duration in comparison with the initial FW pulse (Figure 4b). Due to the lower conversion efficiency, the pump wave energy non-depletion approximation works better; however, it is still worse than our approach. It can be seen in Figure 4e, that the dashed line (energy non-depletion approximation) is close to the solid line (numerical simulation of the original problem), but the dotted line (our approach) coincides fully with the solid one. The conversion efficiency may be increased by the growing incident intensity of the FW. For example, if the parameter \(\gamma\) is increased up to 4 (corresponds to increasing incident intensity in four times), then the conversion efficiency reaches 38% (Figure 4f). Its change along the z-coordinate is shown in Figure 4d. The frequency conversion efficiency grows until the section \(z=10\) and then it remains almost constant.