## 8 Conclusions  
We proposed and described a new physical scheme of THG based on cascading SHG, which leads to observing of the nonlinear processes inherent to a cubic nonlinear response. Physically, it is based on the consecutive generation of SH and a wave with sum a frequency in the same crystal with quadratic susceptibility under large phase mismatching between the FW and the SH wave. This leads to a possibility of the high-efficiency frequency tripling of optical radiation with the incident power density being enough for the quadratic nonlinear response of a medium to occur. This value can be essentially less than that requiring for the THG in a medium with cubic susceptibility (according to Table 1). The crystal length at which the efficient THG occurs depends on the phase mismatching between the SH and FW. It varies from 2 cm to 4 cm for a pulse with a long duration (Figure 3) and from 1 cm to 20 cm for a pulse with a short duration (Figures 13 and 15). The required crystal length decreases with increasing power density of the incident pulse at FF. The pulse self-modulation at the tripled frequency is absent. Its cross-modulation occurs and, consequently, the TH spectrum is broadened. Therefore, this pulse can further be compressed. These two features distinguish the THG under consideration from cascaded THG through the SHG and SFG as well as from the THG in a medium with cubic susceptibility. The THG conversion efficiency can reach 94% (Figure 3) at the optimal parameters of wave interaction. The high-efficiency THG is reached for both signs of the phase mismatching between FW and SH. The THG can demonstrate a bistable mode under certain conditions. The frequency tripling process via cascading SHG was investigated theoretically. On the basis of the multi-scale method, the modified set of equations was derived. Using its integrals of motion, an analytical solution was developed in the framework of the long-pulse duration approximation. This solution approximates the solution of the Schrodinger equations (1) with high accuracy during the pulse propagation distance, which is many times greater than the distance occurring using the well-known wave energy non-depletion approximation (Figures 3 and 4). The modified equations, as well as the analytical solution, were confirmed by the computer simulation results obtained on the basis of the equations in (1). We studied the influence of other parameters on the efficiency of the frequency tripling, SOD, and the phase mismatching between TH, FW, and GVM between both SH and FW as well as TH and FW. The computer simulation results of frequency tripling accounting for the SOD of the pulses showed the possibility of reaching high THG efficiency for both positive (Figure 5) and negative (Figure 6) phase mismatching between SH and FW. At its negative sign, the MI may appear, and as a result of this, a dramatic decreasing of the THG's efficiency occurs (Figure 11). However, at the beginning stage of the MI, a regular sequence of very short sub-pulses with self-similar shapes may appear (Figure 12). The high THG efficiency can be achieved for both types of a medium: anomalous (Figure 8) and normal (Figure 6) dispersion at the FF. The GVM between SH and FW pulses only influences the pulse shapes without strong changing of the THG efficiency (Figures 17-19). In contrast, the GVM between TH and FW remarkably influences the frequency conversion process (Figures 20-24). To compensate for this negative influence, we proposed and discussed some physical schemes for three-wave interaction (Section 6.3). Moreover, since only quadratic nonlinearity is required for THG through the cascading SHG, it is possible to use pulses with durations of picoseconds or nanoseconds. It allows us to decrease a negative influence of the GVM and SOD. Another way is using so-called \"tilted pulses\" [72,73]. Conceptualization, V.A.T. ; methodology, V.A.T. and D.M.K. ; software, D.M.K. and Z.T. ; validation, V.A.T., D.M.K., M.V.F., Y.Y. and Z.T. ; formal analysis, D.M.K. and M.V.F. ; investigation, V.A.T.