## 3 Results
### Calculation of the Crystals' Coherence Length  
For the convenience of subsequent calculations, the obtained dispersion curves of the terahertz refractive index were approximated in the form of the Sellmeier equations. The optical properties of the crystal in the far IR range are mainly determined by the phonon mode in the vicinity of 46.7 \(\upmu\)m (6.4 THz) [52]. Since the sub-THz range of interest to us is located far enough from this mode, for approximation, it was customary to take the equation in its simplest form:  
\[n_{o}^{2}=A+\frac{B\lambda^{2}}{\lambda^{2}-C} \tag{1}\]  
where \(C\) is the square of the absorption peak center wavelength in micrometers that determines the dispersion. Since there are no detailed data in the literature on the change in the position of phonons in the vicinity of \(\sim\)46.7 \(\upmu\)m with an increase in the sulfur concentration in the composition of the crystal, in our case it was decided to fix this parameter at the value \(C=46.72\)\(\upmu\)m\({}^{2}\approx 2266\)\(\upmu\)m\({}^{2}\). Coefficients \(A\) and \(B\) are fitting parameters, the obtained values of which are presented in Table 1.  
The resulting Sellmeier equations are in good agreement with the data obtained by other authors [41; 52; 53; 54]. Since the resolution in our measurements is not enough, we deliberately omitted the peak in the vicinity of 0.59 THz from the Sellmeier equation. Its amplitude is rather small, and the bandwidth is quite narrow; thus, it affects the accuracy of the equations only in the range of several tens of GHz around it [2; 19]. Moreover, it disappears with an increase in doping concentration [2]. The Equation (2) determines the coherence length, which provides effective NLO interaction of terahertz and femtosecond laser frequencies during crystal propagation [55]:  
\[L_{\text{coh}}(\
u_{\text{THz}})=\frac{c}{2\
u_{\text{THz}}\Big{(}n_{\text{opt }}-\lambda\left.\frac{dn_{\text{opt}}}{d\lambda}\right|_{\lambda}-n_{\text{ THz}}(\
u_{\text{THz}})\Big{)}}, \tag{2}\]  
where \(n_{\text{opt}}\) is the refractive index at the wavelength of \(\lambda=1547\) nm (experimental data from Figure 2); \(n_{\text{THz}}(\
u)\) is the terahertz refractive index (experimental data from Figure 1); \(\lambda\) is the wavelength of laser radiation; \(\
u_{\text{THz}}\) is the frequency of a terahertz wave. It should be noted that in this case, we consider the simplest case of collinear propagation of laser and THz radiation along the \(c\)-axis. In our case, the value of \(dn_{\text{opt}}/d\lambda\) is not strictly determined since we have not measured the dispersion of the refractive index in the vicinity of \(\lambda=1547\) nm for all crystal compositions. These data are also absent in the literature. Let us assume that with an increase in the sulfur concentration in the crystals, the value and dispersion of the refractive index change smoothly within the values corresponding to the extreme compositions of GaSe and GaS. The parameter \(dn_{\text{opt}}/d\lambda\) at \(\lambda\approx 1.5\)\(\upmu\) \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{x}\) & **Coefficient** & **Value** \\ \hline  & \(A\) & 8.06 \\  & \(B\) & 2.54 \\ \hline  & \(A\) & 7.76 \\
0.03 & \(B\) & 2.65 \\ \hline  & \(A\) & 7.70 \\
0.12 & \(B\) & 2.64 \\ \hline  & \(A\) & 7.64 \\
0.16 & \(B\) & 2.48 \\ \hline  & \(A\) & 7.45 \\
0.22 & \(B\) & 2.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficients of expression (1) for GaSe-S crystals. is equal to \(-0.0418~{}\upmu\)m\({}^{-1}\) for GaSe and \(-0.0292~{}\upmu\)m\({}^{-1}\) for GaS [56]. Thus, in the general case, one should expect a deviation of no more than 10% in the denominator of expression (2) for some intermediate value of the sulfur concentration. Taking this into account, for subsequent calculations, we fixed \(dn_{\text{opt}}/d\lambda=-0.04~{}\upmu\)m\({}^{-1}\). Figure 3 depicts the resulting dependences \(L_{\text{coh}}(\
u_{\text{THz}})\).