## 5 Discussion and Conclusions  
We propose using the cascading process for frequency down-conversion to the mid-infrared (or even THz) range, due to wave generation with difference frequency at multiple frequencies of the interacting pulses. We showed that this method is applicable for frequency down-conversion corresponding to a wavelength of at least 24 \(\upmu\)m, but this is not a strict 'upper bound'. In the framework of a multi-scale method, we derived a set of modified equations and solved these equations without using the energy non-depletion approximation of the pump wave. This solution was supported by numerical experiments. Influence of the GVD--of the pulses on the frequency down-conversion efficiency--was examined. The computer simulation results showed that non-zero incident LFW's energy could be amplified from an initial value of 8% to 75% and the LFW generation with 25% conversion efficiency is possible at the zero-value its incident intensity. It is important (for practice) that the pulse with a larger wavelength possesses a smooth shape and that its spectrum is also smooth. Chirping of the incident HFW pulse could increase the conversion efficiency if the pulse chirp is chosen in an appropriate way. The significant increasing is observed for both up- and down-chirping of the pulse. Concerning the possible experimental setup, based on data from Table 1 and computer simulation results--for the amplification of the optical pulse with the wavelength 4 \(\upmu\)m, we suggest the following configuration: two pulses, approximately with 100 fs durations, and the intensities of 0.612 GW/cm\({}^{2}\) (10% of the sum intensity) and 5.508 GW/cm\({}^{2}\) (90% of sum intensity), with wavelengths \(\lambda_{1}=4\)\(\upmu\)m (75 THz) and \(\lambda_{3}\approx 1.333\)\(\upmu\)m (225 THz), respectively, fall on the AgGaS\({}_{2}\) crystal with the length 4 cm, cut at an angle 49.46\({}^{\circ}\). For amplification of the optical pulse with 10 \(\upmu\)m wavelength, the crystal cut angle should be changed to 79.51\({}^{\circ}\). The duration of the pulses should be increased until 1 ps and the intensities between them should be distributed in the following way: 2.066 GW/cm\({}^{2}\) for the pulse with the wavelength \(\lambda_{1}=10\)\(\upmu\)m (30 THz) and 18.54 GW/cm\({}^{2}\) for the pulse with the wavelength \(\lambda_{3}\approx 3.333\)\(\upmu\)m (90 THz). The experimental setup for a pulse generation with the wavelength 24 \(\upmu\)m can be implemented in the following way: the Te crystal, 2.5 mm in length, should be cut at an angle, 15.78\({}^{\circ}\). Two pulses with 100 fs duration and with intensities of 0.44 GW/cm\({}^{2}\) at the wavelength \(\lambda_{3}=8\)\(\upmu\)m (73.5 THz) and 0.176 GW/cm\({}^{2}\) at the wavelength \(\lambda_{2}=12\)\(\upmu\)m (49 THz) should 'fall' on the crystal. For the frequency down-conversion of the radiation with a wavelength \(\lambda=1.3\) (or 1.1) \(\upmu\)m to the wavelength \(\lambda=24\) (or 30)\(\upmu\)m, we propose, for example, two schemes (Figure 7) that allows us to reach the frequency 12.5 THz (Figure 7a) and 10 THz (Figure 7b), respectively. For this aim, it is necessary to use three crystals. In the first crystal (AgGaS\({}_{2}\)), a radiation with the tripled wavelength 4 \(\upmu\)m is generated. Then, the second similar crystal is used for the generation of a radiation with the wavelength 8 \(\upmu\)m or 10 \(\upmu\)m. The generation of the radiation with wavelength 24 \(\upmu\)m or 30 \(\upmu\)m occurs in the third crystal (Te or Cs\({}_{2}\)HgI\({}_{2}\)Cl\({}_{2}\)). In above consideration, we did not take into account the absorption of laser radiation because our aim was a study of a maximal efficiency of the down-frequency based on the cascaded processes. If this efficiency was to be small, then, evidently, it would not make sense to consider other physical factors that decrease its effectiveness. One of these factor is the crystal's absorption of the laser radiation in the far IR and THz range of frequencies.