## 2 Attenuated Total Reflection
Figure 1: Illustration of the set of sample \(\alpha\) = 5 cm\({}^{-1}\) to \(\alpha\) = 65 cm\({}^{-1}\) and sample \(n_{2}\) = 1.1 to \(n_{2}\) = 1.65 which deliver a constant normalised reflectance of 0.79 at 1.0 THz with a diamond crystal at 45\({}^{\circ}\) angle of incidence. function in \"true\" attenuated total reflectance mode. It follows that substrates with \(n>1.70\) cannot be examined in attenuated total reflectance mode with the described setup. With \(n_{2}>1.70\), the diamond crystal ATR apparatus becomes a partial reflectance-partial transmission device. In this instance, the reflected portion of the incident radiation is only dependent on the relative refractive indices on either side of the reflective surface. The reflectance can be calculated by using the Fresnel equations, and these are silent about \(\alpha\). According to the Fresnel equations, the reflectance declines rapidly as \(n_{2}\) approaches \(n_{1}\). Given the \(n\) of diamond crystal is \(\sim\)2.40, small changes in \(n_{2}\) in the region of 1.72 to 1.90 will produce large changes in reflection giving good contrast. The relationship of contrast and \(n_{2}\) with diamond crystal are presented in Figure 2.  
When using water and ice THz dielectric properties derived from literature, the Fresnel equations predict an ice to water reflectivity contrast of \(\sim\)300%, i.e., ice is \(\sim\)3\(\times\) more reflective than water when using a diamond crystal. The estimated Fresnel equation based reflectivity is outlined in Table 3. The refractive index of ice is not well established at 2.0 THz, and two values were used in the calculations, \(n=1.78\) and \(n=1.82\) at this frequency. Using the ATR as a reflectance apparatus with a diamond promises better contrast compared to silicon crystal, particularly at 2.0 THz, thus it becomes a more attractive option. There is a dichotomy with a diamond crystal ATR. At THz frequencies, biological fats and oils have an \(n\) of 1.4 to 1.6. Using an ATR diamond crystal with \(n\) of 2.40 results in \"true\" attenuated total reflection at 45\({}^{\circ}\) incident beam with these substances. Reflectance measurements mainly deliver information regarding changes in \(\alpha\). Water based biological compounds have typical n in the order of 1.9 to 2.2, thus a diamond equipped ATR is acting as a partial reflection-partial transmission apparatus, delivering changes in n. The \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Frequency**} & \multicolumn{4}{c}{**Reflectance**} \\ \cline{2-6}  & **THz** & & **S-Polarised** & **P-Polarised** & **Non-Polarised** \\ \hline water & 0.9 & 2.10 & 0.0092 & 0.000085 & 0.0046 \\ ice & 0.9 & 1.78 & 0.27 & 0.073 & 0.17 \\ \hline water & 2.0 & 1.95 & 0.076 & 0.0059 & 0.041 \\ ice & 2.0 & 1.78 & 0.27 & 0.073 & 0.017 \\ ice & 2.0 & 1.82 & 0.20 & 0.040 & 0.012 \\ \hline \hline \end{tabular} \({}^{\star}\) With a diamond crystal \(n=2.40\). Two values for ice used, \(n=1.78\) and \(n=1.82\) at 2.0 THz.  
\end{table} 
Table 3: Reflectance of liquid water and ice using the Fresnel equations at 0.90 THz and 2.0 THz \({}^{\star}\). Figure 2: The normalised reflectance at the diamond/sample interface, diamond, \(n_{1}=2.40\). There is significant change in the reflectance in the region of the refractive index of the sample, \(n_{2}=1.72\) to 1.90 giving good contrast. advantage of minimal sample preparation, and no requirement for thickness measurement (as would be needed for transmission studies), is retained by using the ATR apparatus in both modes. In both cases, only a partial description of the dielectric properties is achieved. If the aim is to characterise the biological sample for the purpose of diagnosis by monitoring temperature dependent variation, then there may only be a need to find a characteristic reflected signal intensity change, rather than the full complex refractive index parameters.