## Appendix A Phase Changes in ATR  
The amplitude of reflected ATR signal (for intensity) is defined by the \(R_{s,p}\) coefficients (Equations (2) and (3)). The phase of the reflected \(E\)-field from the ATR (diamond) prism-sample interface is changed and has to be accounted in the analysis of light using the analyser (Figure 1). The phase is changes depending on the angle according to the Fresnel formulas defined for the relative refractive index \(n=n_{2}/n_{1}\):  
\[\phi_{TE}(\theta_{i},n)=-2\tan^{-1}\left(\frac{\sqrt{\sin^{2}\theta_{i}-n^{2}} }{\cos\theta_{i}}\right)\,, \tag{1}\]  
\[\phi_{TM}(\theta_{i},n)=\pi-2\tan^{-1}\left(\frac{\sqrt{\sin^{2}\theta_{i}-n^{ 2}}}{n^{2}\cos\theta_{i}}\right)\,, \tag{2}\]  
for the \(\theta_{i}>\theta_{c}\) where the critical angle \(\theta_{c}=\sin^{-1}(n)\); 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 \(n<1\)); for plotting \(n_{1}=2.12\) is the refractive index of ATR prism (medium 1) and \(n_{2}+ik_{2}\) (medium 2) is the sample (or air \(n_{2}=1\)) for the THz spectral range.  
The Fresnel formulas shown above are plotted in Figure 11. Reflectivity does not change the amplitude of reflected TE and TM modes at angles of incidence larger than the critical angle \(\theta_{c}\) (Figure 11a). Absorption of THz beam due water humidity can be modeled by increasing the imaginary part of the refractive index \(k_{2}\) (see solid vs. dashed lines in (a)). 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 difference of phases for the TM and TE modes experiences 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 reflection from the sample-diamondinterface, depending to the refractive index contrast \(n\); see thinner lines in Figure 11b which represent phase changes for \(n=n_{2}/n_{1}=1.5/2.12\). The large phase change between TE and TM modes is used in Fresnel rhomb polarisers to produce circularly polarised light after two total internal reflections which introduce \(\lambda/4\) (or \(\pi/2\)) phase shift between TE and TM modes.  
Upon reflection from the diamond-sample interface, both amplitudes of the \(E_{z}\) (TM) and \(E_{y}\) (TE) components can be changed because of anisotorpy of absorption. This is affected by the phase change between TM and TE modes, which is solely defined by the refractive index ratio at the interface. The reflected light is elliptically polarised and can be expressed in s-/p-components measurable with an analyser (inset in Figure 11). The ratio of the s- and p-components of the E-field is tan \(\beta=E_{p}/E_{s}\) and the phase difference 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.