## 6 Nonequivalent Influence of Pulses' GVM on the THG Efficiency under Cascading SHG
This means that we obtain at least the pulse with a picosecond duration if the incident pulse with a duration of \(100\) fs falls on the nonlinear medium. The THG energy grows with the propagation distance. However, the maximum intensity remains at the same level (about \(0.025\)). Figure 20: Pulse shapes and their spectra in the section \(z=85\) (**a,d**) \(z=100\) (**b,e**), \(z=20\) (**c,f**), as well as incident pulse distribution and spectra (dash-dotted line), and the frequency conversion efficiency along \(z\)-coordinate (**g,h,i**) for the incident pulse duration \(\tau_{P}=100\) fs (**a,d,g**), \(1\) ps (**b,e,h**) and \(10\) ps (**c,f,i**). Black solid line denotes FW, green dashed line denotes TH. Insert in (**a**) shows TH intensity distribution. In two other cases, the TH intensity and THG efficiency are high. In fact, at \(\tau_{P}=1\) ps (Figure 20b,e,h), these characteristics of the wave interaction are even better than those achieving at \(\tau_{P}=10\)  ps (Figure 20c,f,i). The possible reason for this is the following: due to rather large GVM between FW and TH, the TH wave is generated and leaves a time domain of wave interaction. Therefore, there is no energy reverse conversion in contrast to Figure 20i, where the TH conversion efficiency oscillations are observed. On the other hand, the maximum energy conversion efficiency is reached at a smaller \(z\)-coordinate if the pulse duration is greater than in the previous case. To clarify the GVM influence on the frequency tripling process, we provide a computer simulation for the coefficients \(\
u_{31}\) (1.02, 0.102, 0.0102, respectively), but for fixed values of SOD coefficients, \(D_{1}=-0.0032\), \(D_{2}=-0.0083\), \(D_{3}=-0.0199\) in all three cases. The computer simulation results are shown in Figure 21 (\(\Delta_{21}k=20\)). The GVM influences both the pulse shapes and the energy conversion efficiency. We see that in Figure 21b, the TH pulse shape is less smooth than in Figure 20b and the energy, converted to TH (Figure 21h), is about 5% lower than the energy shown in Figure 20h. In fact, the lines shown in Figure 21c,f,i are very close to the ones in Figure 5, because the same distance of the maximum THG efficiency achievement (\(z=13\)), as well as its value (about 51%), occurs. A single difference is observed in the pulses shapes (Figure 21c). Thus, the GVM remarkably influences the pulse spectra: in Figure 21e the TH spectrum possesses local minimum in contrast to Figure 20e. Figure 21: Pulse shapes and their spectra in the section \(z=85\) (**a,d**), \(z=40\) (**b,e**), \(z=13\) (**c,f**), as well as incident pulse distribution and spectra (dash-dotted line), and the frequency conversion efficiency along \(z\)-coordinate (**g,h,i**) for the incident pulse duration \(\tau_{P}=100\) fs (**a,d,g**), 1 ps (**b,e,h**) and 10 ps (**c,f,i**). Black solid line denotes FW, green dashed line denotes TH. Insert in (**a**) shows TH intensity distribution. If the phase mismatching between SH and FW is negative (\(\Delta_{21}k=-20\)), then the situation does not differ from (Figure 21) for \(\
u_{31}=1.02\) and \(\
u_{31}=0.0102\) (see Figure 22). However, at \(\
u_{31}=0.102\) the growing of the energy conversion efficiency finishes near the section \(z=40\) due to appearing of MI. If the laser pulse with its incident duration \(\tau_{p}=100\) fs propagates in a medium with anomalous dispersion at FF (\(D_{1}=0.0006\), \(D_{2}=-0.0035\), \(D_{3}=-0.0062\), then at \(\Delta_{21}k=20\) the THG efficiency (Figure 23) increases very slowly with growing of the propagation distance beginning from the certain section of a medium (Figure 23g). At the other two durations of the incident pulse under consideration, the frequency conversion process occurs similar to the case of normal dispersion of a medium at FF. At larger phase mismatching between FW and SH (\(\Delta_{21}k=127\), Figure 24), if all three waves propagate in a medium with normal dispersion, the THG efficiency under large GVM is only about 2% (Figure 24g, \(\tau_{P}=100\,fs\), \(\
u_{31}=1.02\)).