## 2 Peculiarities of ATR Measurements
### A Useful Condition When Atr Becomes Invalid  
When measuring the reflection of THz radiation in the ATR geometry at \(\theta_{i}=45^{\circ}\), the ratio of the refractive indices of the diamond prism \(n_{1}=2.42\) and sample \(n_{2}\) is important since the ATR occurs at \(n_{1}\sin\theta_{i}>n_{2}\) (Equation (4)). This sets \(n_{2}<1.71\) and is critically important in the characterisation of bio-materials close to this ATR condition (a heating/cooling compatible ATR unit is shown in Figure 2a,b). With water \(n_{2}\approx 2.1\) and ice \(n_{2}\approx 1.7\) at \(\approx 1\) THz, direct transmission into the sample takes place (Figure 2c). This change in the refractive index can be exploited for better depth characterisation of materials, due to the larger penetration depth of the evanescent field and direct transmission into the sample. The latter case requires modification of the measurement technique via the addition of a top reflecting mirror to redirect THz radiation back to the detection direction along the ATR geometry (Figure 2c) as we introduced recently [16]. The optical path length \(l_{T}\) in the case of the transmission mode in the ATR setup is defined by the distance, \(H\), to the mirror (see drawing in Figure 2c) as \(l_{T}=2H/\cos(\theta_{2})\) (from [PERSON]'s law \(n_{1}\sin(\theta_{1})=n_{2}\sin(\theta_{2})\) and the Pythagorean identity of \(\sin\) and \(\cos\) functions, one finds \(\cos\theta_{2}=\sqrt{1-\left(\frac{n_{1}}{n_{2}}\sin\theta_{1}\right)^{2}}\)). Precise determination of \(H\) is important since the surfaces of the ATR prism and sample housing unit usually have height differences of tens-of-micrometers; Figure 2a,b. Power reflected at the first prism-sample interface is \(I(R_{1})=RI_{in}\), where \(R\) is the energy reflection coefficient and \(I_{in}\) is the incident power. Considering a perfect reflection \(R_{Au}=1\) at the top-mirror, the power re-directed back to the detector is \(I(R_{2})=(1-R)^{2}I_{in}\exp(-al_{T})\). The experimentally measured ratio \(X_{exp}\equiv\frac{Power_{Sample}/Dionend}{Power_{Air}/Dionend}=(I(R_{1})+I(R_{2} ))/I_{in}=R+(1-R)^{2}\exp(-\alpha l_{T})\). The absorption (optical losses) coefficient is then \(\alpha=-\frac{\cos\left(\theta_{2}\right)}{2H}\)\(\ln\left(\frac{X_{exp}-R}{(1-R)^{2}}\right)\). The reflectivity coefficient (power, intensity) is defined by the angle of inci  
Figure 1: **(a)** Photo and schematics of ATR setup used for temperature dependent absorbance spectroscopy at the THz/Far-IR beamline [15; 16]. **(b)** Visualisation of frustrated TIR (Equation (2)) for different angles of incidence and refractive indices \(n\) of the slabs. The index \(n=1.6\) also corresponds to the case when two \(n=2.4\) (diamond) slabs are separated by \(n_{gap}=1.5\) (bio-sample), i.e., \(\frac{n}{n_{gap}}=\frac{2.4}{1.5}=1.6\). dence \(\theta_{i}\) and the complex refractive indices of sample and ATR-prism \(\tilde{n}=n+i\kappa=\sqrt{\varepsilon}\), where \(\varepsilon\) is the permittivity. For the normal incidence and lossless ATR diamond prism \((n_{1},\kappa_{1}=0)\), the reflectivity from the sample \((n_{2},\kappa_{2})\) is \(R=[(n_{1}-n_{2})^{2}+\kappa_{2}^{2}]/[(n_{1}+n_{2})^{2}+\kappa_{2}^{2}]\) and the absorbed part of energy is \(A=1-R\) since there is no transmission \(T=0\). The Beer-Lambert law defines the transmitted part \(T=e^{-al_{T}}\) in a single pass through thickness \(l_{T}\); \(al_{T}\equiv\ln(10)OD\), where the optical density (absorbance) is calculated from transmittance \(OD=-\lg T\) (note that this expression can be used for back-reflected signals in the ATR setup with a top mirror when the transmission is collected in the reflection direction and the ATR condition is not satisfied). For an arbitrary angle of incidence \(\theta_{i}\), the reflectivity for s-/p-pol.