## 4 Computer Simulation Results
### Frequency Conversion to Radiation with the Wavelength to 24 \(\mu\)m  
Finally, we consider the frequency down-conversion process in a tellurium crystal at the incident pulse duration, 100 fs, which results in radiation generation with a frequency of 12.5 THz, corresponding to a wavelength of 24 \(\upmu\)m. The phase mismatching between IFW and LFW is small: \(\Delta_{21}k=5.2\). Therefore, the multi-scale method can give non-adequate results. However, some features of those analyses remain valid. The dimensionless SOD coefficients are also large, especially at the wavelength 24 \(\upmu\)m (\(D_{1}=0.218\), \(D_{2}=-0.079\), \(D_{3}=-0.154\)). Consequently, we chose the coefficient of nonlinear coupling equal \(\gamma=2\). Due to the large value of the quadratic susceptibility \(\chi^{(2)}\) of the tellurium crystal, the corresponding sum intensity of the incident pulse can be small enough: 110 MW/cm\({}^{2}\). The computer simulation results are shown in Figure 5.  
Figure 4: An evolution of the frequency conversion efficiency (**a,b,d**) along the \(z\)-coordinate and pulse shapes in the section \(z=40\) (**c**) computed for (\(\gamma\), \(|A_{10}|^{2}\), \(|A_{20}|^{2}\)) = (4, 0.1, 0) (**a**), (4, 0, 0.4) (**b**), (1, 0.1, 0) (**c,d**). As one can see, the high- and low-effective modes of the frequency conversion can be observed in this figure. At the incident intensity \(|A_{10}|^{2}=0.07\) of the LFW, its energy changes from 7% to 34% (Figure 5g) and the pulse duration is about 200 fs (Figure 5a), which approximately corresponds to three periods of wave phase changes. Meanwhile, if the incident intensity slightly increases, up to \(|A_{10}|^{2}=0.08\), then the conversion efficiency \(\eta_{1}\) achieves more than 82% (Figure 5h) and the pulse duration becomes equal, at about 500 fs (Figure 5b), which approximately corresponds to eight periods of wave phase changes. It is very important for practice that the LFW pulse shape is rather smooth (Figure 5a,b) in contrast to the two other waves. If the incident LFW intensity is equal to zero, then the conversion efficiency achieves about 21% (Figure 5i) and the pulse duration is about 250 fs (Figure 5c), which corresponds to 4.5 periods of wave phase changes, approximately. However, the LFW pulse shape contains some local maxima and local minima. Nevertheless, its shape is smooth enough (Figure 5c). In Figure 5d-f, we depict the spectrum of the LFW. In the first two cases (Figure 5d,e), the LFW spectrum is smooth and broadened compared to the incident pulse. Regarding the LFW generation from its incident zero-value intensity (Figure 5f), the LFW spectrum has three local maxima. Thus, one could analyze the three sub-pulses, as could be seen in Figure 5c; they overlap each other in the time-domain. The chirps of the incident pulses:  
\[A_{j0}(t)=A_{j0}\exp(-(t-0.5L_{t})^{2}+ic_{j}(t-0.5L_{t})^{2}),\,j=1,\,2,\,3\]  
can partially improve the frequency down-conversion efficiency. Above \(c_{j}\) equals to \(-1\), \(1\), or \(0\). We made a series of the computer simulations for the parameters: \(|A_{10}|^{2}=0.07\), \(|A_{20}|^{2}=0\), \(\gamma=2\), \(\Delta_{21}k=5.2\), \(D_{1}=0.218\), \(D_{2}=-0.079\), \(D_{3}=-0.154\), at which the frequency down-conversion is relatively small; a quality of the pulse shape is bad for  
Figure 5: Pulse shapes in the section \(z=3\) (**a**), \(7\) (**b**), \(2.5\) (**c**), and the LFW Fourier spectrum (**d–f**), and the frequency conversion efficiency evolution (**g–i**) along the \(z\)-coordinate computed for \((|A_{10}|^{2},\,|A_{20}|^{2})=(0.07,\,0)\) (**a**,**d**,**g**), \((0.08,\,0)\) (**b**,**e**,**h**), \((0,\,0.4)\) (**c**,**f**,**i**). non-chirped pulses (see Figure 5a,d,g).