## 2 Peculiarities of ATR Measurements
### Polarisation Analysis with Atr
With the sample, the refractive index contrast (real parts) changes those phase shifts. However, any anisotropy in absorbance \(A_{s,p}\) will change the amplitude of the reflected signal at the corresponding polarisation. This would also cause a change in the polarisation of the outgoing beam at OUT port. Equation (4) accounts for all those changes. Figure 6 illustrates the use of Equation (4) for analysis of beam at the OUT-port of ATR. The analysis is based on two perpendicular field components, \(E_{s}\) and \(E_{p}\), with independent amplitudes. Isotropic and anisotropic absorbance changes are accounted for by the coefficients \(A_{s,p}\). Changes of phases of the s-/p-components with the sample (\(n_{2}\)) placed on top of the ATR prism (\(n_{1}\)) are accounted for by the \(\Delta\psi_{s,p}\). Isotropic absorbance in y- and z-orientations is presented by the curve 2 with \(A_{s}=A_{p}=0.8\) and phase changes corresponding to reflection from the low refractive index material (Figure A1b). Anisotropy along the y- or z-direction in the sample is modelled by a 20% change in coefficient \(A_{s,p}\). As a result, the figure-8 (dipole-like absorbance) is rotated/tilted (an angular shift is introduced). This illustrates that polarisation analysis of the ATR results can be a useful tool for the measurement of anisotropy in the refractive index as well as absorbance by fitting Equation (4) with the experimentally measured data with the polarisation analysis at the OUT port. The TM-mode can experience strong field enhancement at the interface (Figure 4) and can cause additional absorption. Sub-wavelength structures at the sample-ATR prism interface cause depolarisation of the \(E\)-field components, e.g., \(E_{x}=\sqrt{[E_{s}^{2}+E_{p}^{2}]-(E_{y}^{2}+E_{z}^{2})}\) in the sample's frame of reference (Figure 5a); here, the intensity of the incident beam is \(E_{in}^{2}=E_{s}^{2}+E_{p}^{2}\). The above described polarisation changes in the case of ATR should be possible to determine, using polarisation analysis at the OUT port and it will be carried out in future studies. Figure 6: Numerical example of polarisation analysis at the ATR output (OUT) based on Equation (4). The angular dependence of synchrotron THz radiation (at THz beamline) comprised of 20% linear polarisation and 80% isotropic (circular) [28] is shown. (**b**) Sample holder for four-polarisation (4-Pol) ATR measurements: instead of changing the linear polarisation of light beam (THz), the sample orientation can be changed. Sample shown is isotropic THz absorber—black paper. Equation (4) can be fitted by \(Fit(\varphi)=Amp\times\cos(2\varphi+2\varphi)+Offset\), which is used for the 4-Pol method (see the synchrotron radiation angular distribution in Figure 6a, plotted with such dependence). The angle \(2\varphi\) indicates orientation of the sample (e.g., absorbing dipoles) and \(Amp=(I_{max}-I_{min})/2\) and \(Offset=(I_{max}+I_{min})/2\) with \(I_{max,min}\), the maximum and minimum intensities of the transmitted or reflected light. By analysing the OUT-port polarisation and using the 4-Pol fit, it should be possible to establish anisotropy in absorbance (\(A_{s,p}\)) and refractive index ratio \(\frac{n_{2}}{n_{1}}\) defining \(\Delta\psi_{s,p}\). For large samples fitting on 3-mm-window of ATR, sample rotation can be used to realise the 4-Pol method (Figure 6b). Visualisation of the 4-Pol-ATR method (Figure 6) is extendable to the hyperspectral dimension since a broad THz spectrum is recorded. Metallic grid polarisers are polarisation insensitive and by measuring spectra at several angles with the analyser on the OUT port or by rotating the sample on the ATR prism, sample anisotropy can be examined. One can envisage an ATR geometry, where the entire ATR prism with a fixed sample can be rotated (around the z-axis (Figure 5)). Such measurements would require a hemispherical or conical ATR prism and could provide a more flexible setup, e.g., as it is realised for the synchrotron IR microscopy with the Ge-prism [30; 31].