## 2 Problem Statement  
To clarify our scheme of the frequency tripling, let us consider the following frequency conversion processes, shown schematically in Figure 1 for a crystal with quadratic (\(\chi^{(2)}\)) and cubic (\(\chi^{(3)}\)) susceptibilities. A symbol \(\omega\) denotes the FW frequency. Thus, we consider three processes producing a wave with tripled frequency. The first process is a generation of both an optical wave with doubled frequency (\(\omega+\omega=2\omega\), \(\chi^{(2)}\)) and THG based on the SFG (\(\omega+2\omega=3\omega\), \(\chi^{(2)}\)) due to quadratic nonlinear response of a crystal. The second one is the directly tripled frequency wave generation \(\chi^{(3)}\) due to cubic nonlinearity of a crystal. In comparison with the frequency doubling process, this process requires more intensive incident pulse. The third (new) process of frequency tripling is based on frequency doubling at large mismatching, and this process can occur at low intensities of the incident wave in comparison to those occurring at cubic nonlinearity of a crystal. This process is denoted as \(3\omega=\omega+\omega+\omega\) with symbol \(\chi^{(3)}_{cas}\). In Figure 1, we also show another frequency conversion process (\(3\omega=2\omega+2\omega-\omega\), \(\chi^{(3)}\)), which occurs in a cubic medium. Each of these frequency conversion processes is accompanied by own phase matching and own value of the crystal susceptibility at the corresponding frequencies. Therefore, the susceptibilities \(\chi^{(3)}\) and \(\chi^{(3)}_{cas}\) are different. As a rule, only one of various conversion processes can occur at phase matching: in Figure 1 this process is THG (\(\chi^{(3)}\)). It should be emphasized again that the cascading susceptibility \(\chi^{(3)}_{cas}\), including the ratio between \((\chi^{(2)})^{2}\) and phase mismatching between waves at the SHG, was demonstrated in several papers [25; 26; 27; 28; 29; 30; 32]. However, we emphasize again that these studies were made in the framework of the pump non-depletion approximation, and many factors essentially influencing the THG due to induced cubic effects were not taken into account. We suppose that the angle range for achieving the phase matching of the corresponding generation process is small enough to avoid its overlapping with a phase matching angle range of other processes. In this case, it is possible to differ the process of THG, which occurs due to the SFG process (this process occurs at the corresponding phase matching) and due to the cascading SHG which occurs at phase matching between FW and TH and large phase mismatching for others processes of the frequency conversion. We suppose also that the FW possesses an ordinary polarization and the TH has an extraordinary one. Therefore, the phase matching condition of the process \(\omega(o)+\omega(o)+\omega(o)=3\omega(e)\) can be satisfied in nonlinear crystals discussed below. As for the SH wave,  
Figure 1: Refractive index of ordinary FW (circle) and extraordinary TH wave (ellipse) in the uniaxial crystal in dependence from an angle between the crystal axis and wave vector of the FW. Some frequency conversion processes between FW, SH and TH, including the phase-matched process \(\omega(o)+\omega(o)+\omega(o)=3\omega(e)\) (arrows). we believe that both cascaded processes \(\omega(o)+\omega(o)=2\omega(o)\), \(\omega(o)+2\omega(o)=3\omega(e)\) and \(\omega(o)+\omega(o)=2\omega(e)\), \(\omega(o)+2\omega(e)=3\omega(e)\) are possible, so we do not precise the polarization of the SH wave.