## 3 Results and Discussion
### ATR Experimental and Numerical Data  
A total of 160 scans at each sample thickness were performed with synchrotron ATR and the results were averaged. The radiation was not polarised and the beam spot size on the sample was \(\sim 1\) mm in diameter, positioned at the center of the 3-mm-top window (Figure 1b). The spacer defining the sample thickness was set at \(d=0.05,0.1,0.15,0.2\), and 0.3 mm successively. The Au-mirror was placed over the lard sample at the defined separation \(d\) from the surface of ATR prism; the spacers were located so as not to intercept the THz beam in the measurement area. The thickness of lard was considered to correspond to the defined thickness \(d\) since the ATR was measured under an considerable applied force using the ATR clipping mechanism. Each measurement was carried with a new identical lard sample and a fresh region of Au-coated slide glass. The experimental results are plotted in Figure 5 in logarithmic-linear presentation of the normalised THz intensity vs. thickness of the lard sample. The linear slope in this presentation corresponds to a single-exponential decay of intensity, hence the absorption coefficient can be determined. The absorption depth was found corresponding to \(l_{abs}=1/\alpha_{abs}\approx 0.37\) mm (intersection of linear slope defined by the first points and the \(1/e\)-level). The spectral region where the normalised transmission is larger that one are not meaningful due to low reflectivity used for normalisation. It is noteworthy that for the gold-coated mirror, a surface plasmon polariton (SPP) can be launched at the dielectric-gold interface, however, it was not investigated here due to the fixed 45\({}^{\circ}\) angle of incidence. For comparison, FDTD simulations were conducted using 0.95 THz single frequency radiation, with the spacer \(d\) set at the same values as in the experiment. Separate simulations were conducted for s- and p-polarisations. The FDTD results of power density were normalised for comparison with the experimental data. The maximum intensity of unity corresponds to a gold mirror directly placed over the diamond crystal. The results are shown in Figure 5 and follow qualitatively similar behaviour: intensity is reduced for larger sample thickness. As expected from the above discussion on polarisation in the THz beam at \(\sim\)1 THz, the closest match between experiment and theory was obtained for s-pol. The difference between experiment and FDTD simulations can be explained by lard properties used in the model, which might be slightly different mainly due to water presence in THz beam path. Since the FDTD detector, which was measuring reflected power (Figure 2) has a fixed location, the local intensity distribution had an inherent oscillating behaviour due to back-reflected light from the top-mirror. We verified that the mesh was adequate and did not cause spurious changes of the detected power with the sample thickness. Figure 4: **(a)** Raw data: THz spectra measured through the ATR setup with lard of different thickness \(d\) on top of the ATR prism (color coded). Each spectral line was plotted from four scans; several scans show uncertainty range. **(b)** The same spectra presented in frequency after normalisation of the transmitted light to the ATR prism transmission in air (dashed lines) and with top Au-mirror (solid lines). There is a strong water absorption band at \(\sim 0.6\) THz [19] and reflection normalisation (to mirror attached to ATR prism) is affected at the vicinity of the absorption band. There was an apparent difference in the slope of optical power loss with thickness in FDTD data (Figure 5). For the \(\theta=45^{\circ}\) angle of incidence, the ratio of penetration depth for p-(\(\parallel\)) and s-pol. (\(\perp\)) is equal \(d_{p}^{\parallel}/d_{p}^{\perp}=2\)[29], which is valid for an isotropic sample such as rendered pork lard. This explains a stepped power loss for the p-pol. that penetrates deeper into the sample for shallow depths \(d<0.15\) mm.