## 1 Introduction
The residual pump energy can be converted into the TH. Furthermore, in contrast to [3], the changing polarization of the electric field of the SH is not required because only the phase matching between TH and FW should be fulfilled. Based on developed analytical solution of the derived set of equations, we also show that to reach the maximum efficiency of the THG, the phase matching between TH and FW is not required, and weak phase mismatching between these waves is optimal. Our paper is arranged as follows. In Section 2, we write the set of Schrodinger equations which describes three-wave interaction (FW, SH and TH) in a medium with quadratic susceptibility. The parameters of the crystals used for optical frequency tripling are also presented in this part of the paper. In Section 3, we derive the modified problem (Paragraph 3.1) using the multi-scale method presented in the Appendix A. Another, simpler method is presented in this part of the paper. In Paragraph 3.2, the analytical solution of the modified problem is derived. Here, the influence of phase mismatching between interacting waves on THG efficiency is examined theoretically in long pulse duration approximation without using the pump-wave energy non-depletion approximation. In Section 4, the computer simulation results are shown to support the developed analytical solution of the modified problem as well as the conclusions made on its base. This study is provided in Paragraph 4.1. In Section 4.2, a comparison between our approach and the fundamental pulse's energy non-depletion approximation for description of the frequency conversion of short pulses is made. In Section 5, the computer simulation results for an interaction of the shorter pulses are presented mainly for their propagation in a medium with the normal dispersion (Paragraph 5.1). Nevertheless, we also present some results for a medium with the anomalous dispersion at the FF (Paragraph 5.2). It is important to note that the highly effective THG is observed for a wide range of the problem parameters in both cases of the dispersion. In Paragraphs 5.3, 5.5 and 5.7, the influence of the phase matching between both SH and FW and TH and FW on the THG conversion efficiency is studied. In Paragraph 5.4, the developing of the modulation instability (MI) at negative phase mismatching between SH and FW is shown. It strongly decreases the frequency conversion efficiency. In Paragraph 5.6, it is shown that the nonzero phase mismatching between TH and FW can be preferable in some cases to reach higher THG efficiency. An influence of the incident FW intensity on the THG efficiency is also accounted for. In Section 6, an influence of the group velocity mismatching (GVM) on the THG efficiency is examined. It is shown that the GVM between FW and SH does not remarkably influence the THG efficiency (Paragraph 6.1), despite the changing shapes of the FW and TH pulses. In contrast, a large GVM between FW and TH can dramatically decrease the conversion efficiency (Paragraph 6.2). Nevertheless, there are some possibilities which can lead to decreasing of the GVM action. One of them concludes in using of the pulses with a picosecond duration. Another one refers to introducing certain shifts of the FW carrier frequency (Paragraph 6.3). In Section 7, we formulate the main results of the paper, discuss them, and compare them with the results achieved in various schemes of frequency tripling. Section 8 contains the paper's conclusion.