## 2 Peculiarities of ATR Measurements
### Polarisation and Field Enhancement at the Interface of ATR Prism  
A unique feature of ATR geometry is that the maximum intensity of the incident beam, regardless of its polarisation, is localised at the interface (see details below and Figure 4). This is defined by the conditions when light propagates from high-to-low refractive index (even at low angles of incidence when the ATR condition is not fulfilled). For normal beam propagation from low-to-high index (incidence from air), the intensity maximum is \(\lambda/4\) above the surface due to interference between the incident and reflected beams. Maximum intensity at the interface using the back-side illumination is used in Raman scattering sensing and light-induced back-side wet etching (LIBWE) in subtractive machining [21; 22]. The polarisation of the THz beam is not controlled, as the beam is delivered to the ATR prism. This is the condition when maximum intensity is obtained on the sample. The experimentally determined polarisation of synchrotron radiation at THz/far-IR beamline of the Australian Synchrotron (see Appendix B) is 90% linear and 10% circular, due to the sum of the linear and circular polarisations at \(<\)100 cm\({}^{-1}\). This translates to the prevailing s-pol. on the sample placed on the ATR window for the measurements. For wavenumbers larger than 100 cm\({}^{-1}\), the ratio linear-to-isotropic polarisation is 20-80%. Figure 3: Temperature dependence of butter sample (2-mm-thick) at 1 and 2 THz measured with Au top-mirror [15]; wavelength of 1 THz is 0.3 mm and 2 THz is 0.15 mm. The insets show the reflectance intensity vs. the wavenumber; air (no Au top-mirror) reflectance is taken as the reference. Dotted-lines (black) are eye guide splines of the experimental data. For ATR measurements, the depth of the evanescent field is larger for the p-polarisation as compared with s-pol (Figure 2c; Equation (3)). It is important in the case of the frustrated total internal reflection (TIR) modelled for the non-polarised light by Equation (2) (see Figure 1b). In addition, E-field enhancement for the component of the E-field perpendicular to the interface between two dielectrics can be harnessed. Indeed, the boundary conditions of the Maxwell equations require that the normal component of the displacement \(D_{n}=\varepsilon E_{n}\) be continuous across the boundary (when there are no surface charges), and \(\varepsilon\) is the permittivity at the corresponding wavelength--the \(E_{n}=E_{p-pol}\cos\theta_{i}\). Hence, the \(D_{n}\) in both the air and diamond are the same, which defines the E-field enhancement \(E_{air}=\frac{\varepsilon_{airair}}{\varepsilon_{airair}}\times E_{diamond} \approx 2.4^{2}E_{diamond}\) since \(n=\sqrt{\varepsilon}\). Note that here, we assume the E-field inside the ATR prism as a reference to calculate enhancement at the air-diamond interface (on the air side). For comparison, reflection at the normal incidence from the vacuum onto the diamond prism with \(n=2.4\) has \(R=\frac{(n-1)^{2}}{(n+1)^{2}}\approx 17\%\) defining the transmitted \(T=1-R\) fraction of THz radiation (would be inside the ATR prism). The penetration depth of the evanescent field depends on the wavelength of light in the ATR geometry (see the bottom row in Figure 4). The intensity distribution in the case of frustrated TIR depends on the normalised gap \(d/\lambda\) (Equation (2); Figure 1b). It shows that depending on the angle of incidence and refractive indices, light transmission can take place only when gaps are sub-wavelength. In standard ATR measurements, such frustrated TIR conditions can take place due to the presence of structures and different inclusions of  
Figure 4: Maxwell’s scaling (change nm to μm for the same refractive index): numerical finite difference time domain (FDTD; numerical solutions) calculations for optical wavelengths of \(\lambda=1\) μm – 0.65 μm to simulate evanescent field at the ATR geometry used (at THz radiation) and at the angle of incidence of \(\theta_{i}=45^{\circ}\) (ATR interface is \(45^{\circ}\)-tilted).