## 1 Introduction
### Attenuated Total Reflection  
The technique of attenuated total reflection (ATR) spectroscopy has become a popular method to measure dielectric properties of materials as well as for sensors that use small changes in the refractive index at the interface as a detection mechanism. It relies on the reduction of the reflected signal intensity at the the total reflection angle, i.e., there is no transmitted wave into the sample (placed on top of the ATR prism, see Figure 1). The technique relies on the incident energy being absorbed via an evanescent wave generated in the interface between sample and 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 vapor. The observed ATR spectrum is noted to be nearly equivalent to that of the transmission type, which suggests that the technique is sampling the absorption coefficient of the sample at the given frequency. If the absorption of the evanescent wave is negligible, the penetration depth, \(d_{p}\), of the evanescent wave is given by [25]:  
\[d_{p}=\frac{n_{2}\lambda}{2\pi n_{1}\sqrt{\sin^{2}\theta-(n_{2}/n_{1})^{2}}}, \tag{1}\]  
where \(\lambda\) is the wavelength, \(n_{1}\) and \(n_{2}\) are the real parts of the refractive index of the ATR crystal and sample (from medium 1 to 2), respectively, and \(\theta\) is the angle of incidence of the incoming radiation. Since \(\sqrt{\sin^{2}\theta-(n_{2}/n_{1})^{2}}\) has to be a real number, \(\sin\theta>(n_{2}/n_{1})\), this sets the limit on the refractive index of the sample for a given ATR crystal. ATR has found its use at infrared frequencies from 30 to 400 THz, where the \(d_{p}\) is in the order of 1 \(\mu\)m for biological tissue with the refractive index \(n_{2}\approx 1.5\). At these distances, the total absorption of the evanescent wave is, indeed, negligible. 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 \(d_{p}\) changing from 308 \(\mu\)m to 31 \(\mu\)m for the \(n_{2}=1.52\) and ATR prism \(n_{1}=2.42\). The absorption of the evanescent wave becomes not be negligible at such depths in the 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. The reflection coefficients for s- and p-polarisations can be expressed following ref. [26], i.e., the \(R_{s}\) (a transverse electrical TE-mode):  
\[R_{s}=\frac{\xi-\eta}{\xi+\eta} \tag{2}\]  
where \(\xi=\alpha^{2}+\sqrt{x^{2}+y^{2}}\), \(\eta=\sqrt{2}\alpha\sqrt{\sqrt{x^{2}+y^{2}}-x}\), \(x=\beta+n^{2}k^{2}\), \(y=2n^{2}k\), \(\alpha=\cos\theta\), \(\beta=\sin^{2}\theta-n^{2}\) and the relative refractive index \(n=n_{2}/n_{1}\); the absorption coefficient for intensity is linked to the imaginary part of refractive index \(k\) as \(\alpha_{abs}\equiv 4\pi k/\lambda\) and is usually measured in transmission from the Beer-Lambert law \(I_{out}=I_{int}e^{-\alpha_{abs}d}\) for thickness \(d\); noteworthy, optical losses fitted by the Beer-Lambert law can also be due to scattering that also takes place in biological materials. The \(R_{p}\) (a transverse magnetic TM-mode) is given by [26]:  
\[R_{p}=\frac{u-v}{u+v} \tag{3}\]  
where \(u=\alpha^{2}((\sin^{2}\theta-x)^{2}+\sqrt{x^{2}+y^{2}})\), \(v=2\alpha[(n^{2}-n^{2}k^{2})\sqrt{[\sqrt{x^{2}+y^{2}}-x]/2}]+y\sqrt{[\sqrt{x ^{2}+y^{2}}-x]/2}\). For a given \(\lambda\) and \(n_{1}\) (diamond), this formulae delivers the complex refractive index of the sample \((n+ik)\) for given angle of incidence \(\theta\). Equations (2) and (3) are used to determine \(n\) and \(k\) from the reflectivities \(R_{s}\) and \(R_{p}\). Knowing either \(n\) or \(k\) independently, it is possible to calculate the other parameter.