## 1 Introduction  
Third-harmonic generation (THG) attracts permanent attention from researchers because it is one of the effective ways to obtain laser radiation with a frequency in the ultraviolet range. Recently, the frequency tripling process was applied in the National Ignition Facility laser system, in which the first ever controlled laser fusion with a positive output energy balance was achieved [1]. There are well-known schemes for reaching the high-efficiency THG. Among them, we first emphasize a scheme using two crystals with quadratic susceptibility \(\chi^{(2)}\). The first crystal is used for phase-matched second-harmonic generation (SHG), and the second one is used for phase-matched sum frequency wave generation (SFG). Let us note that such a scheme of THG is often called the cascaded generation of the third harmonic (TH). Using this scheme, a conversion efficiency of 80% has been reached in physical experiments [2, 3]. With this aim, two Type II potassium dihydrogen phosphate (KDP) crystals were used, each 70 mm in diameter and 12 mm thick, polished to a flatness of \(\lambda/4\) at 0.5 \(\upmu\)m, and with a wedge of 7 arcsec. It should be noted that the two-crystal scheme required a certain ratio between the fundamental wave (FW) and the second-harmonic (SH) intensities in the input of the second crystal which is equal to 1:2. Another obstacle is the influence of group velocity mismatching between FWand SH, which can strongly decrease the frequency conversion efficiency of a short pulse. A possible solution was proposed in [4], where a doubled frequency was generated using a pulse with a duration of 1 ps at the fundamental frequency (FF). Then, the SH pulse was compressed until a duration of 200 fs (its power density reached 40 GW/cm\({}^{2}\)) and was mixed with the pulse at the FF. The frequency conversion efficiency reached about 40%. Another experimental setup was proposed in [5], where a KDP crystal was used to delay the extraordinary FW with a pulse duration of 0.6-1 ps and the incident power density was varied in the 6-75 GW/cm\({}^{2}\) range. THG has also been widely studied in a single crystal with quadratic susceptibility. A review of different approaches can be found in [6]. Nevertheless, many papers devoted to this problem have been published during passed years, and we briefly discuss them here. One of the most popular ways to carry out frequency tripling for optical radiation is via frequency conversion using the quasi-phase matching (QPM) technique, in which both the SHG and SFG processes are quasi-phase-matched (the description of this technique can be found, for example, in [7]). In such a structure, various materials are used to reach a high efficiency in frequency conversion. It should be emphasized that lithium niobate (LiNbO\({}_{3}\)) is the most popular crystal. So, a 2D periodically poled LiNbO\({}_{3}\) crystal was used for frequency tripling of a 118 fs pulse at a 1568 nm wavelength with a maximum power density of 3.2 GW/cm\({}^{2}\)[8]. An 8% conversion efficiency was reached. The same frequency conversion efficiency was also achieved in a 16 mm chirped periodically poled LiNbO\({}_{3}\) crystal [9] for a 3.5 ns pulse with average power changing between 5 and 20 mJ. THG in a 12 mm short-range-ordered crystal was also obtained in [10] for a 16 ps pulse with 1.73 uJ energy and 12% frequency conversion efficiency. Another popular nonlinear crystal used for QPM is the lithium tantalate (LiTaO\({}_{3}\)) crystal. A frequency conversion efficiency of about 27% was reached in a 15 mm quasi-periodic optical super-lattice for an 1444 nm wavelength and with an average power of 20 mW [11]. An 18 mm periodically poled LiTaO\({}_{3}\) crystal was used for frequency tripling of a 30 ns pulse with a 1342 nm wavelength to reach the frequency conversion efficiency of about 19%.