## 7 Discussion  
Let us discuss the advantages of the presented scheme of frequency tripling in comparison with other schemes.  
First of all, a single nonlinear crystal is required for the effective frequency tripling at a phase matching of the process \(\omega(o)+\omega(o)+\omega(o)=3\omega(e)\). Evidently, this setup is simpler than the setup with two nonlinear crystals and the setup using the QPM technique. In accordance with the computation for the crystals presented in Table 1, the phase-matched THG can be achieved for the wavelength of FW which is greater than 903 nm in the ADP crystal, 890 nm in the KDP crystal, 651 nm in the BBO crystal, and 552 nm in the RBBF crystal. Thus, the generation of optical harmonics with a wavelength lower than 200 nm is  
Figure 26: Comparison of the TH pulse shape in the section \(z=1.95\) (a), as well as the TH frequency conversion efficiency along \(z\)-coordinate (b) for the solutions of the problem (1) at the parameters \(\gamma=4\), \(\Delta_{21}k=20\), \(D_{1}=-0.0032\), \(D_{2}=-0.0083\), \(D_{3}=-0.0199\), \(\
u_{21}=0\) and (\(\
u_{31}\), \(\Delta_{31}k)=(0\), \(0\)) (black solid line), (0.1, 0) (red dash-dotted line), (0.1, \(-0.13274362832\)) (green dashed line).  
possible. Figure 2 and Tables 4-6 show that the effective THG can be achieved even if the phase matching condition is not achieved.  
Secondly, because one requires phase matching for the THG and big phase mismatching for the SHG, the highly intensive SH pulse is not required. In all computer simulation results, the SH intensity does not exceed 2-3% of the incident FW intensity. Importantly, the GVM between FW and SH does not remarkably influence the THG process (Figures 17-19).  
Thirdly, the high-efficiency THG can be observed even for rather low intensities (about 1 \(\mathrm{GW/cm^{2}}\) and low) of the incident wave at FF. Hence, pulses with picosecond or even nanosecond durations can be used to minimize the influence of the GVM and the SOD on the frequency tripling. Importantly, the frequency tripling accompanies the TH spectrum broadening (Figures 6-8 and 17), which is absent for the well-known scheme of cascaded THG.  
The sign of effective cubic nonlinearity (see the set of equations (9)) is defined by the phase mismatching \(\Delta_{21}k\), which can be useful in practice. In particular, it allows self-focusing of the pulses even in a medium with a normal SOD.  
The THG efficiency equal to 94.5% for the long pulses (Figure 3) and more than 80% for the short pulses (Figures 8, 13, 15 and 16, Tables 3-5), reached theoretically, is greater than the corresponding value for the scheme based on the QPM technique and the scheme with changing of the SH polarization, and is comparable with the scheme using two crystals.  
It should be emphasized that the length of a crystal in the scheme under consideration should be large--at least 1 cm, as the theoretical estimation showed. That is why in the above mentioned papers which account for cascading SHG, the researchers could not observe a high-efficiency frequency tripling when using the frequency conversion process proposed here because the length of their crystals did not exceed several millimeters.  
The predicted value of 94.5 conversion efficiency was computed without accounting for the absorption of the crystal. The absorption coefficient may change from quite a high value (0.1 cm\({}^{-1}\)) for the ADP crystal and (0.03 cm\({}^{-1}\)) for the KDP crystal to a very low value (<0.001 cm\({}^{-1}\)) for the BBO crystal for the optical pulse with a wavelength \(\lambda_{1}=1064\) nm [68]. At \(\lambda_{3}=353\) nm, the absorption coefficient of the KDP crystal is lower at 0.003 cm\({}^{-1}\), and of the KBBF crystal is 0.1 cm\({}^{-1}\) at the optical pulse wavelength 330 nm [69]. Therefore, the absorption can be neglected in some cases.  
It is important to notice that the proposed scheme for optical frequency conversion can be used for other frequency conversion problems, such as fifth-harmonic generation [70] or frequency down-conversion [71].