## 3 Results
### Detection of THz Radiation in GaSe:S Crystals
#### 3.3.3 Electro-Optic Coefficient of GaSeS Samples  
Free space electro-optic sampling of THz waves based on the Pockels effect is well described in other works [58; 59]. The terahertz field causes a change in the ellipsoid of the refractive indices of the crystal, which leads to the appearance of birefringence, which is detected by the polarization-optical method using a Wollaston prism (WP) and balanced photodetector. WP divides the beam into two parts with mutually orthogonal polarizations \(I_{1}\) and \(I_{2}\). Thus, the photodiode currents are:  
\[J_{1}=s_{1}I_{1},\]  
\[J_{2}=s_{2}I_{2},\]  
where \(s_{1}\) and \(s_{2}\) are the sensitivities of photodiodes. Taking into account the fact that the photodiodes were manufactured in the same batch, it is permissible to neglect the differences in their sensitivity, i.e., \(s_{1}=s_{2}=s\). In this regard, the current difference is equal to:  
\[\Delta I=(I_{1}-I_{2})s=\Delta Is.\]  
The intensity difference recorded by photodiodes is equal to:  
\[\Delta I=I_{0}\sin\Delta\varphi, \tag{3}\]  
where \(I_{0}\) is the intensity of radiation incident on the detector, \(\Delta\varphi\) is the difference between the phase shift of the ordinary and extraordinary waves, which is expressed in terms of:  
\[\Delta\varphi=\Delta n\frac{\omega}{c}L, \tag{4}\]  
where \(\Delta n\) is the induced birefringence, \(\omega\) is the frequency of laser radiation, \(L\) is the propagation length in an electro-optic medium. The magnitude of \(\sin(\Delta\varphi)\) can be estimated as \(\sin(\Delta\varphi)\)\(\sim\)\(\Delta I/I_{0}\). In our case, measured by the Vega-10A power meter with Ophir head, \(I_{0}=12\) mW. \(\Delta I\) can be calculated using the signal amplitude (Figure 5) and the transimpedance coefficient of the amplifier which is R = 10 k. Thus, \(\sin(\Delta\varphi)\)\(\sim\)\(\Delta I/I_{0}\) = 1.25 \(\times\) 10\({}^{-7}\). Then, here and after, we can put \(\sin(\Delta\varphi)\)\(\sim\)\(\Delta\varphi\) for simplicity, as in other works [58; 59]. Figure 6: Dependence of the peak amplitude of the THz signal detected with GaSe:S, crystal samples on \(x\). Induced birefringence is expressed in terms of a terahertz field that is applied to the crystal with optimal orientation, through:  
\[\Delta n=n_{opt}^{3}rE_{THz}, \tag{5}\]  
where \(n_{opt}\) is the refractive index at the laser wavelength, \(E_{THz}\) is the terahertz field, and \(r\) is the electro-optic coefficient of the detector crystal. Combining (4) and (5) preceding expressions yields:  
\[\Delta\varphi=\frac{\omega}{c}Ln_{opt}^{3}E_{THz}r.\]  
Thus, the difference in radiation intensities detected by photodiodes is represented by the following expression:  
\[\Delta I=I_{0}\frac{\omega Ln_{opt}^{3}E_{THz}r}{c}. \tag{6}\]  
Let us take into account the reflection losses from the crystal faces of optical and terahertz waves. In the case of normal incidence on a dielectric surface, we can calculate the optical reflection coefficient using the Fresnel formula:  
\[R=\bigg{(}\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\bigg{)}^{2}. \tag{7}\]  
Considering the amplitude of the THz field, it is necessary to use the amplitude reflection coefficient for these waves:  
\[J_{2}=s_{2}I_{2},\]0  
Since we only replace the crystal in the system, and the terahertz wave with the probe IR pulse remains the same amplitude and energy, we can compare the electro-optic coefficient of GaSe:S crystals with the well-known one of GaAs. In our case, an antireflection coating with \(\lambda\) = 1.55 \(\upmu\)m is deposited on the face of the GaAs crystal, thus optical reflection is not considered.