## 2 Peculiarities of ATR Measurements
### Attenuated Total Reflection  
The technique of attenuated total reflection (ATR) spectroscopy has become a popular method to measure the dielectric properties of materials [11; 12]. It relies on total internal reflection within a prism and the resultant evanescent wave that extends beyond the ATR prism surface. A sample is placed on this surface, and the incident energy is absorbed only via the evanescent wave generated between the sample and ATR prism. As such, there is a reduction in the reflected signal intensity at the total reflection angle, when there is no transmitted light into the sample (placed on top of the ATR prism). The advantage of the method lies in the fact that solid and liquid samples can be studied with minimal preparation. This is exceptionally useful for the THz spectral window since free space propagation of THz radiation under normal conditions is strongly attenuated, due to absorption by water vapour. The observed ATR spectrum is noted to be nearly equivalent to that of the transmission mode. If the absorption of the evanescent wave is negligible, the penetration depth of the evanescent E-field (\(1/e\)-level) is given by the following [13]:  
\[d_{p}^{(ex)}=\frac{\lambda}{2\pi n_{1}\sqrt{\sin^{2}\theta_{i}-(\frac{n_{2}}{n _{1}})^{2}}}, \tag{1}\]  
where \(\lambda\) is the wavelength, \(n_{1}\) and \(n_{2}\) are the real parts of the refractive index of the ATR crystal and sample, respectively, and \(\theta\) is the angle of incidence of the incoming radiation; for the beam intensity \(I=E^{2}\), the penetration depth becomes \(d_{p}^{(ev)}/2\). Since \(\sqrt{\sin^{2}\theta_{i}-(n_{2}/n_{1})^{2}}\) has to be a real number, \(n_{2}<n_{1}\sin\theta_{i}\) sets the limit on the refractive index of the sample for a given ATR crystal. ATR finds its use at infrared frequencies of 30 to 400 THz, where the \(d_{p}\) is of the order of \(1\)\(\mu\)m for biological tissue. At these distances, the total absorption of the evanescent wave at distances in the order of the \(d_{p}\) is, indeed, negligible. However, at low terahertz frequencies (0.3 to 3 THz) the wavelength \(\lambda\) is on the order of 1.0-0.1 mm, which yields a \(d_{p}\) on the order of 0.02 to 0.2 mm. The absorption of the evanescent wave becomes not negligible at such depths in biological tissues. A more elaborate equation for the reflected wave intensity, using the complex refractive index \(n^{*}=n+ik\), where \(k\) is the imaginary part of the complex refractive index \(n^{*}\), is required when polarisation becomes important. It is instructive to estimate what are frustrated--the total internal reflection (TIR) transmission predictions. This affects the intensity reaching different depths of the sample as well as accounting for the detected THz power losses when an air gap is formed between the sample and the ATR prism. The fraction of light intensity polarized in the plane of incidence (p-pol) at the wavelength \(\lambda\), transmitted across the gap of width \(d\), with refractive index \(n_{gap}=1\), and in the case of the ATR prism of refractive index \(n\), is given as follows [14] (Figure 1):  
\[T_{p}=\frac{1}{\alpha\sinh^{2}y+1},\ \ \alpha=\left(\frac{n^{2}-1}{2n}\right)^{2} \frac{[(n^{2}+1)\sin^{2}\theta_{i}-1]^{2}}{\cos^{2}\theta_{i}(n^{2}\sin^{2} \theta_{i}-1)},\ \ y=2\pi\left(\frac{d}{\lambda}\right)\sqrt{n^{2}\sin^{2} \theta_{i}-1}, \tag{2}\]  
where \(\theta_{i}\) is the angle of incidence onto the first surface (ATR prism); the same material of refractive index \(n\) is considered on both sides of the gap. This set of formulas also shows that a larger light penetration depth can be achieved if a sample on the ATR prism has high refractive index inclusions. When the inclusion has a larger refractive index, compared to its surroundings, the frustrated TIR conditions applies, and light tunnels through the separation with a lower refractive index. This situation is illustrated numerically in Figure 1b.