## 5 THG Efficiency in Dependence of SOD Coefficients and Phase Mismatching between Interacting Waves
### Developing of FW MI at Negative \(\Delta_{21}k\)  
The THG efficiency can be higher at a negative phase mismatching between SH and FW than its corresponding value achieving at a positive one. However, the frequency conversion efficiency may be restricted by MI developing. For example, this occurs if the pulse duration is equal to 1 ps.  
The MI develops firstly at the wave with FF (Figure 11). So, despite the pulse being smooth in the section \(z=68\) (Figure 11a), the weak oscillations are already observed in the section \(z=74\) (Figure 11b) near the pulse center. Then, the intensity oscillations enhance and a certain periodic structure appears in the section \(z=82\) (Figure 11b) with about 40 sub-pulses with the dimensionless duration 0.03 (approximately 30 fs). Nevertheless, the TH pulse shape remains smooth at it propagates along certain distance, during which the THG efficiency grows continuously despite the development of the oscillating process. Furthermore, the situation changes dramatically in the section \(z=100\) (Figure 11d), for example, the TH maximum intensity becomes weak. In contrast, the SH intensity begins to grow and we observe two sub-pulses which leave the area of the wave interaction. Their velocities of motion are constant, and their phase distributions are linear. Moreover, the sub-pulses propagate without any changes to their shape nor maximum intensities as one can see in Figure 12b. Thus, we see the soliton propagation of a wave at doubled frequency.  
To prove that the MI causes these oscillations, we present the computer simulation results in Table 2. As is known, MI's development depends on the SOD values. So, if the influence of the SOD increases and the other parameters remain the same, then MI does not appear. With this goal, we increase the SOD coefficients ten times. However, the MI is still observed, but the required distance for its appearing increases up to \(z=160\). As a consequence, the THG efficiency becomes higher: about 51%. After that, we choose SOD coefficients 20 times greater, and the MI does not appear at all. However, because the SOD influence enhances in comparison with the previous case, then the THG efficiency is a bit lower (50%). Increasing \(\gamma\) by the factor \(\sqrt{2}\) (which means that the effective cubic nonlinearity is two times greater) results in the MI appearing again. Due to enhancing the nonlinear coefficient, the THG efficiency grows (62%) at the decreased length of a medium (see the fourth row in Table 2). Thus, if the phase mismatching between FW and SH is  
Figure 11: Pulse shapes in the sections \(z=68\) (**a**), 74 (**b**), 82 (**c**), 100 (**d**). Pulse intensities evolution at their center (**e**), and the frequency conversion efficiency (**f**) along \(z\)-coordinate for the parameters \(\gamma=1\), \(\Delta_{21}k=-250\), \(D_{1}=-0.000032\), \(D_{2}=-0.000083\), \(D_{3}=-0.000199\). Inserts in (**b**,**c**) demonstrate FW intensity near pulse center.  
negative, one must carefully choose the incident pulse intensity to avoid the MI developing and to achieve high conversion efficiency.  
It should be emphasized briefly that the physical reason for MI appearing differs from the corresponding one occurring in a medium with the quadratic nonlinear response [65; 66], because in our case it is observed in a medium in which all three pulses experience decompression. This means that a medium possesses the normal dispersion at all frequencies. However, this condition does not restrict MI's appearance, and it may be observed at various combinations of SOD types.