## 3 Results
### Investigations of THz Pulse Time of Flights in Gasoline Solutions with Isopropanol
The leftmost peaks belonging to the THz pulses passing through the air without quartz cuvettes were used to compensate for drifts and fluctuations of the pulses transmitted through gasoline samples. The color code of the arrows above the waveforms is consistent with previous figures. The calculated theoretical dependence of \(\Delta t\) on the mass fraction \(\phi_{1}\) is shown in Figure 9b. It corresponds to the relation predicted by Equation (11). It is, therefore, not surprising that an overlap between the measured and theoretical data shown in Figure 9c is reasonably good. The solid lines on this and the preceding figure (Figure 9b) resemble ab initio calculations with average values of the refractive indices from Table 2. While the correspondence seems better for \(l\) = 1.00 and 2.00 mm cuvettes, there seems to be a systematic difference towards longer measured \(\Delta t\) (as compared to the calculations) for the \(l\) = 5.00 mm cuvette. Since all of the points on this figure represent the time-corrected data, which to a large extent are free from the spectrometer pulse drift, we interpret this visible discrepancy in terms of unequal optical path lengths and longer propagation times of the low-frequency spectral components of THz pulse. As explained in Section 2.2.2, the Rayleigh lengths of the low-frequency components of the pulse can be considerably shorter than the 5.00 mm optical path of the cuvette, resulting in longer propagation distances and times of flight throughout the cuvette as compared to para-axially propagating high-frequency components. As mentioned above, for all cuvette lengths, the ab initio calculations used the same average values of the refractive indices of isopropanol \(<\)\(n_{iso}\)\(>\) and gasoline \(<\)\(n_{gs}\)\(>\). More precisely, the same constant value of \(\Delta n\) difference, \(\Delta n\) = \(n_{iso}\) - \(n_{gs}\), was used according to the functional dependence resulting from the mathematical form of Equation (11). Since these average values do not necessarily represent an optimum, we have additionally fitted the lines to the experimentally measured \(\Delta t\) and calculated the optimal \(\Delta n\) from the slopes of the fits. The fitted functions are shown on the Figure 9c as dash-dotted lines and the \begin{table}
\begin{tabular}{c c c} \hline
**Cuvette Length** & **Isopropanol Concentration** & **Isopropanol Mass** \\  & **(\%wt.)** & **Fraction, \(\phi_{1}\) (ppm)** \\ \hline  & 0.00 & 0 \\  & 0.05 & 500 \\  & 0.20 & 2000 \\  & 0.50 & 5000 \\  & 1.00 & 10,000 \\  & 2.00 & 20,000 \\ \(l\) = 1.00 mm & 5.00 & 50,000 \\  & 6.25 & 62,500 \\  & 12.5 & 125,000 \\  & 25.00 & 250,000 \\  & 50.00 & 500,000 \\  & 75.00 & 750,000 \\ \hline  & 0.00 & 0 \\  & 0.05 & 500 \\  & 0.20 & 2000 \\  & 0.50 & 5000 \\ \(l\) = 2.00 mm & 1.00 & 10,000 \\  & 2.00 & 20,000 \\  & 6.25 & 62,500 \\  & 12.50 & 125,000 \\  & 25.00 & 250,000 \\  & 50.00 & 500,000 \\ \hline  & 0.00 & 0 \\  & 0.05 & 500 \\  & 0.20 & 2000 \\ \(l\) = 5.00 mm \({}^{1}\) & 0.50 & 5000 \\  & 1.00 & 10,000 \\  & 2.00 & 20,000 \\  & 5.00 & 50,000 \\ \hline \end{tabular} \({}^{1}\) Data of the 1 st series only.  
\end{table} 
Table 3: Concentrations and mass fractions of isopropanol in gasoline mixtures. fitted \(\Delta n\) determined in this way are: \(\Delta n\cong\) 0.0576 for \(l\) = 1.00 mm, \(\Delta n\cong\) 0.0660 for \(l\) =2 mm, and \(\Delta n\cong\) 0.1130 (1 st run) and \(\Delta n\cong\) 0.1452 (2 nd run) for \(l\) = 5.00 mm. Figure 9: The measured THz pulse delays in gasoline containing isopropanol as a function of isopropanol concentration.