## 2 Attenuated Total Reflection  
The technique of attenuated total reflection (ATR) spectroscopy has become a popular method to evaluate the dielectric properties of materials. It relies on the reduction in the reflected signal intensity at an angle of incidence where total reflection occurs at the crystal/sample interface. The technique relies on some of the incident energy being absorbed when an evanescent wave is generated in the sample. The advantage of the method lies in the fact that solid and liquid samples can be studied with minimal preparation, since the only requirement is that a sample comes into secure contact with the crystal. The equations governing the reflection of an incident wave in ATR are outlined in Hansen, [19]. The Hansen equations take account of the angle of incidence and the complex refractive index of the sample (\(\mathbf{n_{2}}^{*}\)) and the crystal (\(\mathbf{n_{1}}^{*}\)). The complex refractive index (\(\mathbf{n}^{*}\)) is defined as \(\mathbf{n}^{*}=\mathbf{n}+\mathbf{i}\mathbf{k}\), where \(\mathbf{n}\) and \(\mathbf{k}\) are the real and imaginary part of the refractive index, respectively. The term \(\mathbf{k}\) is related to the absorption coefficient (\(\mathbf{\alpha}\)) via the equation \(\mathbf{k}=\mathbf{\alpha}\mathbf{\lambda}/\mathbf{4}\pi\), where \(\mathbf{\lambda}\) is the wavelength of the radiation. It is apparent from the equations that there is a set of refractive index/absorption coefficient combinations which can satisfy a given reflectance; no unique pair of \(\mathbf{n}\) and \(\mathbf{\alpha}\) can be deduced from reflectance alone if a synchrotron-like continuous, non-polarised beam is used. The combinations are dependent on frequency and angle of incidence. Examples of the combinations of sample refractive index \(\mathbf{n_{1}}\) and sample \(\mathbf{\alpha}\) at 0.3 THz and 1.0 THz, at 45\({}^{\circ}\) angle of incidence and a diamond crystal (\(\mathbf{n_{2}}\) = 2.40) are set out in Table 1.  
The example of the change from \(\mathbf{n_{2}}\) = 1.1 to \(\mathbf{n_{2}}\) = 1.65 with a change in the absorption coefficient from \(\mathbf{\alpha}\) = 5 cm\({}^{-1}\) to \(\mathbf{\alpha}\) = 65 cm\({}^{-1}\) to deliver a constant normalised reflected signal intensity of 0.79 at 1.0 THz is illustrated in Figure 1.  
The changes in reflected signal intensity can come about with changes in either parameter, \(\mathbf{n_{2}}\) or \(\mathbf{\alpha}\). Furthermore, changes in \(\mathbf{n_{2}}\) and \(\mathbf{\alpha}\) can either act to produce a synergistic reflectance change (when both \(\mathbf{n_{2}}\) and \(\mathbf{\alpha}\) are simultaneously increasing or decreasing) or negate any reflectance change when one parameter is increasing and the other decreasing. Attenuated total reflection produces an evanescent wave in the sample. The penetration depth (\(\mathbf{d_{p}}\)) of the evanescent wave is given by [20]:  
\[\mathbf{d_{p}}=\frac{\mathbf{\lambda}}{2\pi\mathbf{n_{2}}\sqrt{\sin^{2}\!\theta-\left( \frac{\mathbf{n_{2}}}{\mathbf{n_{1}}}\right)^{2^{\infty}}}} \tag{1}\]  
where \(\mathbf{\lambda}\) is the wavelength of the incident radiation in air, \(\mathbf{n_{1}}\) and \(\mathbf{n_{2}}\) are the _real_ parts of the refractive index of the ATR crystal and sample, respectively, and \(\theta\) is the angle of incidence