## 3 Conclusions and Outlook
The reflectivity of the TM and TE modes is the following [53]:  
\[\ln R_{TE}(\theta_{i})=-\frac{4n_{2}^{2}k_{2}}{n_{1}^{2}\tan(\theta_{i})\sqrt{1 -\frac{n_{2}^{2}}{n_{1}^{2}\sin^{2}\theta_{i}}}\times\left(1-\frac{n_{2}^{2}} {n_{1}^{2}}\right)}, \tag{1}\]  
\[\ln R_{TM}(\theta_{i})=-\frac{4n_{2}^{2}\bigg{(}2-\frac{n_{2}^{2}}{n_{1}^{2} \sin^{2}\theta_{i}}\bigg{)}k_{2}}{n_{1}^{2}\tan\theta_{i}\sqrt{1-\frac{n_{2}^ {2}}{n_{1}^{2}\sin^{2}\theta_{i}}\bigg{(}1-\frac{n_{2}^{2}}{n_{1}^{2}\sin^{2} \theta_{i}}+\frac{n_{2}^{4}}{n_{1}^{2}}\cot^{2}\theta_{i}\bigg{)}}}, \tag{2}\]  
where \(n_{1}=2.12\) is the refractive index of the ATR prism (medium 1) and \(n_{2}+ik_{2}\) (medium 2) is the sample (or air \(n_{2}=1\)) for the THz spectral range. Reflectivity at the interface changes the amplitude of the \(E_{s,p}\) fields, which are detected by the analyser. Additionally, the phase changes upon ATR, depending on the angle, according to the Fresnel formulas defined for the relative refractive index \(n=n_{2}/n_{1}\):  
\[\phi_{TE}(\theta_{i},n) = -2\tan^{-1}\left(\frac{\sqrt{\sin^{2}\theta_{i}-n^{2}}}{\cos \theta_{i}}\right), \tag{3}\] \[\phi_{TM}(\theta_{i},n) = \pi-2\tan^{-1}\left(\frac{\sqrt{\sin^{2}\theta_{i}-n^{2}}}{n^{2} \cos\theta_{i}}\right), \tag{4}\]  
for the \(\theta_{i}>\theta_{c}\), where the critical angle \(\theta_{c}=\sin^{-1}(n)\); the reflection is internal when \(n<1\). The evanescent field depth \(1/\alpha\) is defined by the absorption coefficient \(\alpha=\frac{2\pi}{k}\sqrt{\frac{\sin^{2}\theta_{i}}{n}-1}\) (for \(n<1\)). The Fresnel formulas shown above are plotted in Figure 11. Reflectivity does not change the amplitude of the reflected TE and TM modes at angles of incidence larger than the critical angle \(\theta_{c}\) (Figure 11a). Absorption of the THz beam due to the water humidity can be modelled by increasing the imaginary part of the refractive index \(k_{2}\) (see solid vs. dashed lines in Figure 11a). The phase change upon reflection of TE and TM modes experiences complex changes at the polarising Brewster and critical angles (Figure 11b). Most importantly, the phase differences for the TM and TE modes experience the largest phase change close to the \(\lambda/4\) condition for \(\theta_{i}=35-45^{\circ}\). Typical ATR setups are designed for \(\theta_{i}\approx\pi/4\); hence, a strong phase difference is expected, due to reflection from the sample-diamond interface, depending to the refractive index contrast \(n\); see thinner lines in Figure 11b, which represent phase changes for \(n=n_{2}/n_{1}=1.5/2.12\). Upon reflection from the diamond-sample interface, both amplitudes of the \(E_{z}\) (TM) and \(E_{y}\) (TE) components can be changed because of the anisotropy of absorption, which is additionally affected by the phase change between the TM and TE modes, and is solely defined by the refractive index ratio at the interface. For the coherent E-field addition, the reflected light is elliptically polarised, and can be expressed in s-/p-components, measurable with an analyser (inset in Figure 11b). The ratio of s- and p-components of the E-field \(\tan\beta=E_{p}/E_{s}\) and the phase differ upon reflection \(\delta\). The polarisation ellipsis is expressed by two angular parameters: the orientation angle \(\psi^{\prime}\) and the ellipticity angle \(\chi^{\prime}\) as \(\tan(2\psi^{\prime})=\tan(2\beta)\times\cos(\delta)\) and \(\tan(2\chi^{\prime})=\sin(2\beta)\times\sin(\delta)\). This polarisation change is mainly defined by the phase change \(\delta\) upon reflection. For the incoherent light case, intensities of the two components \(E_{p}^{2}\) and \(E_{s}^{2}\) are added (Equation (4)).