## 5 THG Efficiency in Dependence of SOD Coefficients and Phase Mismatching between Interacting Waves
### Normal Dispersion of a Medium at FW Wavelength  
First of all, we consider the frequency tripling of the pulse with wavelength \(\lambda=800\) nm and propagating in a BBO crystal with the normal dispersion, which corresponds in our notation to negative values of the dimensionless SOD coefficients. In Figure 5a-c, the pulse shapes are shown for the frequency up-conversion at a relatively small phase mismatching between SH and FW (\(\Delta_{21}k=20\)) and at the incident pulse duration of about 100 fs. It should be stressed that a positive sign of phase mismatching \(\Delta_{21}k\) corresponds to the FW pulse compression under its propagation in a medium with the normal SOD if the THG is not essential.  
We see a good coincidence between the FW and TH pulse shapes, computed using the original problem solution and the modified one, until the section \(z=13\). Despite a medium possessing only the quadratic nonlinearity, the TH intensity is high: it achieves 0.75 dimensionless units at the pulse center. This growth is accompanied by the spectral broadening of the FW and TH pulses in comparison with the incident FW pulse spectrum. However, the FW pulse spectrum has two local minimums in this section of a medium.  
After the distance equals 10 units (Figure 5e), the FW intensity begins its growth. This happens due to the sub-pulse formation at the FF. Indeed, in the section \(z=7\) (Figure 5a), the FW pulse shape is smooth, and it becomes close to plate distribution near the pulse center. After this section of a medium, the intensity grows in this part of the pulse (Figure 5b), and its increase becomes very remarkable (Figure 5c).  
The conversion efficiency achieves 51% and, at the same distance, the TH spectra (Figure 5d) has a smooth shape without any local minima. The analytical solution (21) perfectly describes the wave interaction until the section \(z=7.5\). Then, we see that the FW pulse self-focusing and TH efficiency, predicted by the theory, are slightly higher than the corresponding values obtained from computer simulation results. Nevertheless, the set of equations (9) correctly describes a self-focusing of the pulses, as well as the presence of the effective cubic nonlinearity under wave interaction occurring at the cascading SHG.  
Figure 5: Pulse shapes in the sections \(z=7\) (**a**), 10 (**b**), 13 (**c**), and FW and TH spectra in the section \(z=13\) (**d**). Pulse intensities evolution at their centers (**e**), and the frequency conversion efficiency evolution (**f**) along \(z\)-coordinate computed for the parameters \(\gamma=1\), \(\Delta_{21}k=20\), \(D_{1}=-0.0032\), \(D_{2}=-0.0083\), \(D_{3}=-0.0199\).  
If the sign of the phase mismatching between FW and SH is negative (\(\Delta_{21}k=-20\)), then the energy conversion from the FW to TH also occurs with high efficiency (Figure 6). Nevertheless, there are new essential features which accompany this process.  
Therefore, this sign of the phase mismatching corresponds to FW pulse decompression due to the induced cubic nonlinear response in a medium with normal dispersion. Therefore, the pulse duration of the FW and TH is greater than those with positive phase mismatching (compare Figures 5 and 6). However, despite the FW pulse decompression, the bandwidth of the TH pulse spectrum is greater than the FW incident pulse spectrum. This means that the essential spectral broadening of the TH pulse occurs and this pulse can be additionally compressed. Moreover, the energy conversion reaches the value of about 60% at increased crystal length in comparison with the previous case: the crystal length must be about 3  cm. Other features of the THG process are similar to the generation process upon positive phase mismatching.