## 2 Peculiarities of ATR Measurements
### Polarisation Analysis with Atr
The most general expression of the THz power measured after the ATR prism would be the addition of two perpendicular contributions along the s- and p-polarisations:  
\[Power^{(OUT)}(\varphi)=A_{s}\times E_{s}^{2}\cos^{2}(\varphi+\Delta\psi_{s}) +A_{p}\times E_{p}^{2}\sin^{2}(\varphi+\Delta\psi_{p})\,, \tag{4}\]  
where \(A_{s,p}\) are the arbitrary amplitudes accounting for both absorption (losses) and changes in amplitude due to reflection; \(\Delta\psi_{s,p}\) are the phases for s- and p-polarisations, respectively. The angle \(\varphi\) defines the orientation of the linear polarisation in the incident beam, i.e., for \(\varphi=0^{\circ}\), only \(E_{s}\) polarisation is present, and \(\varphi=90^{\circ}\) corresponds to pure \(E_{p}\). When pure \(E_{s}\) polarisation is incident on the sample and there are no changes in the phase of the reflected beam, the orientational output power will have a horizontally aligned figure-8 (Figure 5) and, consequently, for the pure \(E_{p}\) case, a vertical figure-8. When the incident beam has both \(E_{s}\) and \(E_{p}\) fields, which are absorbed along the z-(evanescent) and y-directions, the emerging field can be modelled with the Equation (4), where changes of phases are also set as fitting parameters. The phase change upon reflection of s-/p-polarisations (TE/TM modes) are solely defined by the real part of the refractive index. The THz beamline receives \(\sim\)20% of a linear \(E_{L}\) (along the slit of the first mirror) polarisation, contributing to \(E_{s}\), and the remaining 80% is isotropic \(E_{I}\) (circular), contributing to the \(E_{s}\) and \(E_{p}\) at frequencies larger than \(\sim\)3 THz [28] (see Appendix B for experimental details of the synchrotron beamline). The transmitted power through a metallic linear-grid analyser at the entrance port \(1N\) to ATR prism is \(Power^{(IN)}(\varphi)=\frac{1}{2}(E_{I}^{2}+E_{L}^{2})+\frac{1}{2}E_{I}^{2} \cos(2\varphi)=\frac{1}{2}(0.2+0.8)+\frac{1}{2}0.2\cos(2\varphi)\equiv 0.5+0.1 \cos(2\varphi)\) (shown in Figure 6a). This is an expected angular distribution at the \(IN\) port if there is no polariser. After the ATR, with no absorption for s- and p-components (\(A_{s,p}\)) and no phase change  
Figure 5: **(a)** Geometry and conventions of ATR; photo of the Bruker Alpha unit setup on the THz-beamline compartment. ATR conventions: incident beam (IN) has \(E_{s,p}\) polarisations selected by a wire grid polariser incident on the ATR diamond (\(n_{1}=2.42\)[29] prism; plane of incidence is xz. The reflected beam is elliptically polarised due to phase and amplitude changes incurred upon reflection (e.g., air case \(n_{2}=1\)). The sample is contributing with absorption and phase change for the output beam (OUT). At the interface (sample) the evanescent field \(ET_{TM}\) (or \(E_{z}\)) and \(ETE_{TE}\) or (\(E_{y}\)) defines the light–matter interaction. **(b)** The reflected beam can be expressed via \(E_{s,p}\) projections measured analyser and given by Equation (4). The angle \(\varphi\) is the orientation of the wire grid polariser in the incidence side (IN); \(\varphi=0^{\circ}\) corresponds to pure \(E_{s}\) component incident onto the ATR prism. The amplitude span in the polar plot is [0–1]. The two components of the \(E_{p}\) and \(E_{s}\) are shown separately by the dipole-like figures (each term in Equation (4)). (\(\Delta\psi_{s,p}\)) upon reflection (see Equation (4)), the same angular dependence is expected. However, even without any sample, the beam reflected from the ATR prism will change its polarisation, due to phase changes for s-/p-polarisations \(\Delta\phi_{s,p}\), respectively (Equation (4)); see Figure A1b where the s-pol. (TE) experiences an advance of phase by \(\sim\)\(\lambda\)/5, while the p-pol. (TM) reduces the lag of the 2\(\pi\) phase by a similar amount at the \(\theta_{i}=\pi/4\) angle of incidence on the ATR prism. As a result, just by reflection without any sample, there is a change in polarisation at the OUT port.