## 1 Introduction
Using a GaSe crystal with a length of 8 mm and a down-conversion frequency process, a source of the tunable frequency, belonging to a frequency range of 0.21-3 THz, and conversion efficiency equal 0.00168%, was created [25]. A review of the sources of the THz radiation based on the frequency down-conversion process is presented in [26]. In turn, a review of quantum cascade lasers using the the difference-frequency generation process for obtaining THz radiation is found in [27]. It should be noted that a process of the difference-frequency generation was also used for developing THz CW spectrometers based on lithium niobate channel-waveguides [28], which is important regarding the problem of substance detection and identification. A crystal AgGaS\({}_{2}\) is widely used for IR radiation generation [29; 30; 31; 32; 33]. IR radiation generation with wavelength intervals at 3.9 \(\upmu\)m and 9.4 \(\upmu\)m, at incident pulse durations in picosecond ranges, was discussed in [29]. The achieved conversion efficiency of the energy transform was a few percent for the incident pulse energy of 10 mJ. The difference-frequency generation was chosen as a base of the IR spectrometer [30]. The pulses possessing the energy (10 \(\upmu\)J) and duration (50 ns), and wavelength intervals at 3-12 \(\upmu\)m, were obtained in [31]. The femtosecond pulse interaction scheme used for obtaining pulses with a tunable frequency in the mid-infrared range and output energy, equaling several mJs, was discussed in [32]. The source of pulses, constructed using the difference-frequency generation with the efficiency 0.6%, with tunable wavelength intervals at 5-12 \(\upmu\)m, and a duration 8 ns, was presented in [33]. In past years, the frequency up-conversion in the tellurium chains have been widely discussed (see, for example, [34]); thus, one may expect the frequency down-conversion process in this crystal will be effective. The tellurium crystal has a big transmittance range in the infrared range of the wavelengths (at least until 24 \(\upmu\)m, according to [35]), and a large second-order susceptibility [36]. The frequency down-conversion to the range of the mid-infrared frequencies with a peak power of 100 \(\upmu\)W, achieved in tellurium crystal with a length 11 \(\upmu\)m, is presented in [37]. In the current study, we propose a new approach for the frequency down-conversion to infrared (or even THz) range using the difference-frequency generation at wave interactions with multiple frequencies \(\omega\) (low-frequency wave, LFW), \(2\omega\) (intermediate frequency wave, IFW), \(3\omega\) (high-frequency wave, HFW), in the case of large phase-mismatching between two interacting waves. To our knowledge, such a scheme, regarding the frequency down-conversion, has not been discussed in previous literature. We present the computer simulation results for the frequency down-conversion of femtosecond pulses, with duration equal to 100 fs for the pulses with corresponding wavelengths, 4 \(\upmu\)m (75 THz), 10 \(\upmu\)m (50 THz; belongs to the infrared frequencies range), and 24 \(\upmu\)m (12.5 THz; belongs to the THz frequencies range), if the LFW and HFW interact in the phase-matched mode and the phase mismatching between IFW and LFW is large enough. Therefore, it allowed us to apply the multi-scale method to derive the modified set of equations and to solve those equations analytically in the long pulse duration approximation framework and plane wave approximation, but without using the energy non-depletion approximation of HFW. The computer simulation is provided for two well-known crystals--AgGaS\({}_{2}\) and tellurium. Our calculations, based on the results of the Sellmeier equations, taken from study [38], show that it is possible to achieve phase matching between LFW and HFW in this crystal at the LFW wavelength, up to 10 \(\upmu\)m in the crystal AgGaS\({}_{2}\).