## 1 Introduction
Applying the QPM realized using the electro-optic effect in a quasi-periodic optical super-lattice LiTaO\({}_{3}\) resulted in an 18% frequency conversion efficiency for a 90 ns pulse with an average power of 1.72 W [12]. To date, the highest conversion efficiency (40%) for this approach was obtained in a 10 mm quasi-periodically poled potassium titanyl phosphate (KTiOPO\({}_{4}\) or KTP) crystal for a 15 ps pulse with a 1.5 GW/cm\({}^{2}\) power density [13]. A similar crystal was also used for frequency tripling of a femtosecond (200 fs) pulse with an average power of 2 W [14]. A conversion efficiency of about 8% was reached. This frequency conversion tool was also studied theoretically. The optimal ratio between the nonlinear coupling coefficients and refraction indexes that defines SHG and SFG, allowing to reach 100% conversion efficiency, was discussed theoretically in [15]. This ratio was further used in the numerical simulation of frequency conversion in a bulk medium for a beam with a waist radius of 15 um propagating in a 2 cm super-lattice [16], and a frequency conversion efficiency of about 96% was achieved. The QPM configurations of the various types of waves' interactions were studied in [17]. The disadvantages of this approach are more difficult crystal fabrication and higher cost in comparison with ordinary nonlinear crystals [18]. Furthermore, QPM is harder to implement for short wavelengths, such as those of visible or ultraviolet light, and this technique gives a lower conversion efficiency [19]. Another disadvantage appears during ultraviolet light generation: crystal heating caused by optical energy absorption becomes essential if the light power exceeds a certain value (typically, several watts) [20,21]. Evidently, this restricts the frequency conversion efficiency. One more approach for reaching high-efficiency THG (about 40%) in a single crystal was demonstrated in [18]. An FW pulse with a duration of 40 ps and a beam diameter 10 mm entered the KDP crystal (20 mm \(\cdot\) 20 mm \(\cdot\) 10 mm), in which the SH was generated. The FW energy varied from 0.5 to 4.5 mJ. Then, a quarter-wave plate was used to change the SH wave polarization: this was necessary to achieve phase matching at the SFG. After that, the FW and SH waves, passed through the crystal, were reflected by the mirror and propagated in the same crystal, in which THG occurred. A beam with a sufficiently big diameter was used in this experiment to avoid the influence of its diffraction. This approach was also implemented for Gd\({}_{x}\)Y\({}_{1-x}\)COB [22], ammonium-dihydrogen-phosphate (ADP) [23], and \(\beta\)-barium borate (BBO) [24] crystals with similar results. In our opinion, there is another way to achieve high-efficiency THG in a single crystal with quadratic susceptibility. THG occurs under the phase-matched condition between FW and a wave at tripled frequency; meanwhile, both the SHG and SFG processes occur at large phase mismatching. To achieve this, a birefrigent crystal can be used. This means that the cascaded \(\chi^{(2)}\) process (SHG at large phase mismatching) is used together with SFG (the second process that is named also as cascaded one as we mentioned above). Various aspects of such an interaction of waves and its influence on THG and the measurement of third-order susceptibility were previously discussed in the literature [25; 26; 27; 28; 29; 30; 31; 32; 33; 34] (we discuss these papers below). Let us note that the cascaded \(\chi^{(2)}\) nonlinear phenomenon remains of interest for many years due to its various practical applications. This phenomenon was theoretically predicted in 1974 by [PERSON] and [PERSON] [35]. Beginning in 1991, there has been continued interest in this phenomenon. It allows us to obtain quadratic solitons: the \(\chi^{(2)}\) temporal soliton [36] and the spatio-temporal soliton [37]. In [38], a tilted pulse was used to decrease the negative influence of group velocity mismatching between interacting waves during temporal soliton formation due to SHG in a quadratic medium.