# Frequency Division Multiplexing of Terahertz Waves Realized by Diffractive Optical Elements
## 3 Results
### Focal Spots  
All focal spots have been scanned separately and then inscribed into the common \(xy\) plane. Empty points have been filled with zeros (black color). The registered intensity distributions for the DOE with focal lengths \(f=200\) mm and \(f=400\) mm can be seen in Figure 6.  
First of all, it should be noticed that the focal spots for different frequencies are shifted, as predicted in the simulations. Longer focusing distance also translates into better separation of the focal spots. Although the precise positions of the particular spots differ slightly from the simulated ones, they still remain proportional to them. These positions depend on the focusing distance, which on the other hand comes from the convergence of the illuminating beam. This dependence has been analyzed in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline Frequency [GHz] & 240 & 270 & 300 & 330 & 360 & 390 \\ \hline \multicolumn{6}{c}{Focal spot position [mm] (\(f=200\) mm)} \\ \hline Simulation & 177 & 145 & 126 & 109 & 95 & - \\ \hline Experiment & 162 & 132 & 115 & 102 & 89 & - \\ \hline Ratio & 0.92 & 0.91 & 0.91 & 0.94 & 0.94 & - \\ \hline \multicolumn{6}{c}{Focal spot position [mm] (\(f=400\) mm)} \\ \hline Simulation & 356 & 304 & 260 & 220 & 204 & 176 \\ \hline Experiment & 204 & 177 & 154 & 132 & 116 & 101 \\ \hline Ratio & 0.57 & 0.58 & 0.59 & 0.60 & 0.57 & 0.57 \\ \hline \end{tabular}
\end{table} 
Table 1: Positions of shifted focal spots in the simulation and experiment for each DOE, corresponding to particular source frequencies. For each focal spot, the ratio between the calculated and measured shift value is given. Figure 6: Intensity distributions of the focal spots measured for different radiation frequencies inscribed into the common \(xy\) diagrams illustrating focal plane for the focal length (**a**) \(f=200\) mm and (**b**) \(f=400\) mm. The table presents the comparison of the measured and simulated positions of each focal spot for both DOEs. It is clearly seen that the experimental values differ from those obtained in the simulations. It should be noted, however, that the ratios of the shifts from the optical axis obtained in the experiment and in the simulation are very consistent across the focal spots of the given structure. For \(f=200\) mm, the shift ratios vary from 0.91 to 0.94, while the ratio of the experimental focusing distance to the focal length was equal to 0.90. Thereby, the shift ratios are close to each other and to the proportion of the experimental focusing distance and theoretical focal length. It proves that the redirection angles are constant for given frequencies even for the illumination with the radiation deviating from the plane wave. This observation is very important from the perspective of the potential application, as the structures are not limited by the convergence of the illuminating beam. The same behavior can be observed for \(f=400\) mm. The ratio between the experimental focusing distance and the focal length has been equal to 0.59. In this case, the shift ratios of focal spots have been similar to this value, varying from 0.57 to 0.60. Precise scans of intensity distributions in the focal spots are presented in Figure 7.  
The most circular and the smallest focal spots have been registered for the 300 GHz frequency. This result is expected, as both the structure and experimental procedure have been optimized for this frequency. The geometrical parameters of the focal spots become, however, slightly worse for increasing values of the frequency shift from the DWL. It is connected both with the design of the DOE and the experimental procedure. Firstly, radiation with the frequency differing from the DWL is focused at different distances. As stated before, all measurements have been performed in the single \(xy\) plane; therefore, the scans of all frequencies except 300 GHz have been gathered not exactly in their focal planes.