## 1 Introduction
Using a femtosecond pulse with a duration of 300 fs and a power density changing between 25 and 250 GW/cm\({}^{2}\) allowed to reach the frequency conversion efficiency about 6% in a BBO crystal with a length of 1 mm, while phase mismatching equals \(-\)240 or 330 mm\({}^{-1}\)[28; 29]. Evidently, at such a pulse intensity, the cubic susceptibility also played a significant role in the frequency tripling. As shown in [30; 31] for a KTP crystal with a length of 1.1 mm, the cubic susceptibility provides major contribution to the conversion efficiency, which achieved 2.9%, for a pulse of 22 ps with an incident power density of 45 GW/cm\({}^{2}\) at the phase mismatching between SH and FW as low as \(-\)127 mm\({}^{-1}\). An 11% conversion efficiency was reached in a 3 mm bismuth triborate (BiB\({}_{3}\)O\({}_{6}\) or BiBO) crystal for the pulse with a duration of 80 fs and an incident power density of 700 GW/cm\({}^{2}\) with an SHG phase mismatching of several hundred mm\({}^{-1}\)[32]. From the obtained results [25; 26; 27; 28; 29; 30; 31; 32], it follows that the frequency conversion efficiency is smaller than those reached using other setups. However, this situation can be changed. First of all, in all mentioned papers, the crystal length was too small (a few millimeters) for practical implementation of cascading processes occurring at large phase mismatching between FW and SH. Secondly, the theoretical studies made in those papers, as well as in [33], were provided in a framework of the pump-wave energy non-depletion approximation. Evidently, this approximation can be applied only at weak frequency conversion. Moreover, another condition often used in these papers was \(|A_{1}|>>|A_{2}|>>|A_{3}|\), where \(A_{j}\), \(j=1\), 2, 3 are the complex amplitudes of FW, SH and TH, respectively. A main feature of this condition concludes in lower intensity of the TH in comparison with those for other waves. Although this condition was not used in [33], the analytical derived formulas were very complicated ones and their use is difficult. The numerical simulation was made in [34] for the crystal length equal to 1 mm that is not sufficient for observing frequency tripling due to the physical mechanism proposed above. Nevertheless, from our point of view, one of the main results of these theoretical studies was the demonstration of the appearance of \(A_{1}^{3}\) in the solution, describing the amplitude of TH, together with \((\chi^{(2)})^{2}\) and phase mismatching \(\Delta_{21}k\) between SH and FW upon using the pump-wave energy non-depletion approximation. However, other terms inherent to the cubic nonlinear response of a medium leading to self- and cross-modulation are not taken into account. Thus, one can conclude that the authors of these papers did not account for all processes that occur at the cascaded THG under the condition of large phase-mismatching between SH and FW. Moreover, the used approach (the pump-wave energy non-depletion approximation) and length of the crystal did not allow us to perform deep analysis of the possibilities of the frequency conversion physical mechanism discussed below. The current study removes this gap because we present the study of THG in the crystal with quadratic susceptibility via the cascading SHG and SFG processes under the condition of the of large phase-mismatching between SH and FW without using the above mentioned restrictions. Based on multi-scale method (see, for example, [57]), we demonstrate that the cascaded SHG in a crystal with the quadratic susceptibility leads to a response of a medium similar to the cubic susceptibility allowing to observe THG. It should be emphasized that this method was applied in [51] in which a self-modulation of the pulse with FF was demonstrated under cascaded SHG. As is shown theoretically below, SHG efficiency is very low for the conditions under consideration--about 2-3%, or less than is comparable with losses using a mirror [2; 3] without special features allowing to decrease its reflection on both wavelengths.