## 4 Influence of Pulse Duration of FW on Frequency Conversion Due to Cascading Processes
### Long Pulse Duration Approximation  
In Figure 3a, the conversion efficiency evolution is shown if the phase matching between TH and FW (\(\Delta_{31}k=0\)) occurs. Its maximum value (80%) is achieved in the section \(z=15\). Both lines coincide with each other with high accuracy. However, at phase mismatching \(\Delta_{31}k=-0.0549878\) corresponding to the parameter \(q=3+\frac{9}{\sqrt[3]{2}}-9\sqrt[3]{2}\), the coincidence between both lines occurs only until the pulse propagation distance equaling \(z=12.5\) and the maximum intensity \(|A_{3}|^{2}=0.7\) is achieved at computing based on the original problem (Figure 3b). It occurs because this phase mismatching corresponds to value dividing two modes of the frequency up-conversion. Therefore, for adequate description of the tripling frequency generation, it is necessary either to take into account the next order of the series corresponding to \(O(\Delta_{21}k^{-2})\) or to increase computer simulation accuracy. Its insufficient accuracy leads to the switching between the up-conversion modes because of computation round-off and its accumulation. For confirmation of this hypothesis, we compute additional frequency conversion at phase mismatching which differs a little bit from those. So, at slight changing phase mismatching (\(\Delta_{31}k=-0.053\)), the THG efficiency increases up to 94.5% (Figure 3c) that is essentially greater than in the previous case. The difference in solutions appears at much longer distances of the wave interac  
Figure 3: The TH intensity evolution computed on the base of original problem (solid line), the analytical solution of the modified problem (dashed-dotted line) and the solution of original problem, obtained in the pump’s energy non-depletion approximation (dashed line) at \(\Delta_{31}k=0\) (**a**), \(0.05498780\) (**b**), \(-0.053\) (**c**), \(-0.05\) (**d**). Inserts in (**a**,**c**) demonstrate evolution of the phase difference \(\varphi\) along \(z\)-coordinate.  
tion. At further increasing phase mismatching \(\Delta_{31}k=-0.05\) (Figure 3d), the frequency conversion efficiency decreases insufficiently (93%), but the coincidence between the two solutions improves essentially.  
The insets in Figure 3a,c also confirm the partial compensation of the phase nonlinear incursion by the phase mismatching \(\Delta_{31}k\). The energy conversion from FW to TH occurs if the phase difference \(\varphi\) is negative. In Figure 3a, this takes place if the distance of laser pulse interaction is less than 15 dimensionless units. Meanwhile, that distance is about \(z=45\) in Figure 3c. Both mentioned sections of the z-coordinate corresponds to the phase difference equal to \(\varphi=0\).  
To highlight an advantage of our approach to the solution of the problem, we also show in Figure 3 the original problem's solution in the pump energy non-depletion approximation. One can see that in all cases, the solution of the modified problem approximates the solution of the original problem much better than the pump energy non-depletion solution. In fact, the last one does not even show key features of the problem solution, and is close to two other solutions only when the TH intensity is smaller than 0.2.