## 2 Peculiarities of ATR Measurements
### Feasibility Test  
The s- and p-pol. polarisation ratio of the synchrotron THz beam is not the same on the sample, due to multiple reflections caused by mirrors in the ATR unit (Figure 7a). The THz beam reflected from the sample-prism interface experiences further reflections until it reaches the OUT port. In total, from the IN to the OUT port, there are ten reflections, including one from the the sample. The beam inside the ATR unit is focused by parabolic mirrors and experiences a change in the s-/p-portions, due to the change in the direction of propagation inside the unit. Due to focusing onto the sample (a 3 mm window), a range of incidence angles are covered, which contributes to different Fresnel reflection coefficients and a change in the phase. The polarisation response of the ATR unit without sample (in air ambient) is analysed. A mesh grid polariser is set at the IN port and an analyser is set at the OUT-port. Linear (horizontal) polarisation is set to enter the ATR unit, which corresponds to the linear pol. of the synchrotron radiation (\(\varphi=0^{\circ}\) in Figure 7b). A spectral window of 30-630 cm\({}^{-1}\) is selected with the beamsplitter, and the detector is a Si bolometer. A mesh-grid polariser/analyser has the same extinction ratio \(E_{r}=\frac{T_{max}}{T_{min}}\) over the all wavelengths used; \(T_{max,min}\) is the transmission at the maximum and minimum orientation of the polariser, respectively. Figure 7: **(a)** Schematic of the ATR unit (Pike) used in this study. From the port IN to OUT, there are ten reflections (mirrors m1–10). **(b)** Cross-polarised performance of the ATR unit. The polarisation is set horizontal (\(\varphi=0^{\circ}\)) at the IN port. This corresponds to the maximum of linear polarisation of synchrotron radiation extracted by the first mirror (slotted mirror at the bending magnet of the synchrotron). THz spectrum over 30–630 cm\({}^{-1}\) (0.9–18.9 THz) is integrated (spectral range is defined by the mylar beamsplitter, and the detector is a Si bolometer). The water vapour absorption coefficient at 1 atm over the same spectral window [36] is shown in the inset. **(c)** Co-rotation of the mesh-grid polariser and analyser (at the IN and OUT ports): dots are experimental data and fits by \(E_{s}^{2}\propto\cos^{8}(\varphi+\Delta\psi_{s})\) and \(E_{p}^{2}\propto\sin^{8}(\varphi+\Delta\psi_{p})\) intensity components (see text for details). Four angles separated by 45\({}^{\circ}\) degrees are selected, where the local maxima or minima are observed (Angles 1–4). **(d)** Polarisation analysis of signal at port OUT for Angles 1–4 (at the IN port): experimental data by markers 1–4 and fit by Equation (4). The legend shows the fit functions explicitly. The measured ATR reference signal without a sample is only affected by the THz active absorbers/scatterers in air; hence, mostly water at the spectral window is used. The calculated water spectrum over this spectral window in shown in the inset of Figure 7b. The evanescent field extends \(\sim\)\(\lambda/4\) into the air and experiences absorption and phase changes of s-/p-pol., corresponding to the refractive index ratio \(n_{2}/n_{1}\approx 1/2.42\) (without the sample with \(n_{2}^{(air)}=1\)). The result of the change in polarisation between the s- and p-pol. and beam focusing (range of incidence angles) is a low extinction ratio for the entire unit \(E_{r}\approx 2.5\). There is strong light leakage from the ATR unit, even in the crossed polariser-analyser conditions shown in Figure 7b; the \(E_{r}\) of the mesh grid polarisers is \(\sim\)\(10^{3}\). The influence of the input polarisation and internal inter-change between the s- and p-pol. is revealed by co-rotating the mesh grid polarisers at the IN and OUT ports (Figure 7c). More angularly narrow s- and p-pol. lobes are observed, as compared to the extinction ratio measurements shown in Figure 7b and the cross vs circle markers in Figure 7c. There are two maxima and two minima per \(\pi\)-range of the \(\varphi\) rotation, marked as Angles 1-4 in Figure 7c.