## 2 Attenuated Total Reflection
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Frequency**} & \multicolumn{1}{c}{\(\mathbf{\alpha}\)} & \(\mathbf{n_{1}}\) & \(\mathbf{n_{2}}\) & \multirow{2}{*}{**S-Pol.**} & \multicolumn{2}{c}{**Reflectance**} & \multirow{2}{*}{**Non-Pol.**} \\ \hline \multicolumn{2}{c}{**THz**} & \multicolumn{1}{c}{**cm\({}^{-1}\)**} & \multicolumn{1}{c}{**Silicon**} & \multicolumn{1}{c}{**Sample**} & & \multicolumn{1}{c}{**S-Pol.**} & \multicolumn{1}{c}{**P-Pol.**} & \multicolumn{1}{c}{**Non-Pol.**} \\ \hline water & 0.9 & 190 & 3.42 & 2.10 & 0.60 & 0.36 & 0.48 \\ ice & 0.9 & 10 & 3.42 & 1.78 & 0.74 & 0.55 & 0.65 \\ \hline water & 2.0 & 300 & 3.42 & 1.95 & 0.53 & 0.28 & 0.41 \\ ice & 2.0 & 50 & 3.42 & 1.78 & 0.55 & 0.30 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A sample of the \(\mathbf{n}\) and \(\mathbf{\alpha}\) combinations at 0.3 THz and 1.0 THz which deliver the same reflected signal intensity at 45\({}^{\circ}\) incidence. of the incoming radiation beam. A limit to the refractive index of the sample is evident from the equation. Since the expression \(sin^{2}\theta\)\(-(n_{2}/n_{1})^{2}\) has to be a real number, \(sin\)\(\theta\) needs to be \(>\)(\(n_{2}/n_{1}\)). This property sets the limit on the refractive index of the sample that can be examined with attenuated total reflectance. Since it is dependent on \(\theta\) and \(n_{1}\) such limits are specific for a given ATR crystal at a given incident beam angle. When \(sin^{2}\theta\)\(<\)\((n_{2}/n_{1})\), Equation (1) breaks down, and the incoming radiation is partially reflected and partially admitted to the sample at the crystal/sample interface as a traveling wave, i.e., there is no longer \"total reflection\". The practical result is a reduction in the signal reaching the ATR detector, and the reflected signal is now solely dependent on the Fresnel reflection equations. The ATR apparatus in this instance, i.e., where \(sin^{2}\theta\)\(<\)\((n_{2}/n_{1})^{2}\), is now acting in a reflection/transmission mode, where total reflection no longer occurs. This mode is not useless. Rather than delivering a possible set of sample \(n\) and \(\alpha\), it now delivers the real part of the refractive index \(n\) of the sample. There are two common choices for an ATR crystal available for the lower end of the THz spectrum-silicon and diamond. Silicon crystal has an \(n\) of 3.42 [21] at THz frequencies. Using the Equation (1), the maximum \(n_{2}\) is 2.42. This results in most biological materials being in the \"true\" ATR range with an incident angle of 45\({}^{\circ}\), since typical biological tissues at THz frequencies have an \(n\)\(<\) 2.3. Intuitively, it seems that silicon crystal is a good choice, however, contrast becomes a problem. Table 2 notes the reflectance of liquid water and ice (as a proxy for water containing biological tissues), using the Hansen equations. Silicon crystal delivers only modest contrast at 0.90 THz and very poor contrast at 2.0 THz. Diamond crystal has an \(n\) of 2.38 to 2.40 at THz frequencies, [22; 23]. In the case of a diamond crystal and an incident angle of 45\({}^{\circ}\), \(sin^{2}\)\(\theta\) = 0.5. Thus, \((n_{2}/n_{1})^{2}\) needs to be \(<\)0.5, hence \(n_{2}/n_{1}\)\(<\) 0.707, or \(n_{2}\)\(<\) 1.70, for the diamond crystal equipped ATR apparatus to \begin{table}
\begin{tabular}{c c c c c c c} \hline abs coef. (\(\alpha\)) cm\({}^{-1}\) & 10 & 35 & 65 & 10 & 35 & 65 \\ \hline frequency THz & 0.3 & 0.3 & 0.3 & 1.0 & 1.0 & 1.0 \\ \hline sample ref index (\(n_{2}\)) & 1.62 & 1.28 & 1.00 & 1.62 & 1.28 & 1.00 \\ diamond ref index (\(n_{1}\)) & 2.40 & 2.40 & 2.40 & 2.40 & 2.40 & 2.40 \\ \hline reflected intensity (non-pol) & 0.49 & 0.50 & 0.52 & 0.79 & 0.79 & 0.79 \\ \hline \end{tabular} \({}^{\star}\) using the [PERSON] (1965) equations at 0.95 THz and 2.0 THz with a silicon crystal (\(n\) = 3.42).  
\end{table} 
Table 2: The estimated reflectance of liquid water and ice with S, P and non-polarised THz radiation \({}^{\star}\).