## 2 Materials and Methods
### Antenna Optimization
Figure 6 shows the reflection coefficient as a function of frequency, for several values of \(Z_{J\!I}\). Simulation results show that the system performance deteriorates rapidly for \(Z_{J\!I}\) values different from those required for optimal impedance matching conditions. The system efficiency drops from 99.75% for \(Z_{J\!I}\cong 2\)\(\Omega\) to 95% for \(Z_{J\!I}\cong 1\)\(\Omega\) and down to 80% for \(Z_{J\!I}\cong 0.6\)\(\Omega\), excluding ohmic losses. The mismatch is also reflected in a shift of the central frequency. Figure 5: Antenna reflection coefficient S\({}_{11}\) vs. frequency for various matching strips lengths. W = 4.6 \(\upmu\)m.  
Figure 6: Reflection coefficient S\({}_{11}\) vs. frequency for various values of [PERSON] junction characteristic impedance \(Z_{J\!I}\) for bow-tie antenna. Figure 7 shows the real (continuous blue line) and the imaginary (red dots) parts of the antenna input impedance \(Z_{in}\) for a structure with matching strips 4.6 \(\upmu\)m wide and 108 \(\upmu\)m long, as a function of frequency. Clearly, \(Z_{in}\) is highly correlated to the junction impedance \(Z_{II}\) around 155 GHz, reaching high impedance matching at this frequency. Figure 8 shows the radiation pattern and gain power, 2-D far-field pattern, at the central frequency of 155 GHz. The antenna gain is around 2.8 dBi in the \(-Z\) axis, better than the gain of 2.5 dBi along the opposite direction. Figure 9 shows the results of simulations for the reflection coefficient S\({}_{11}\), as a function of frequency, for antennas with a strips length of 108 \(\upmu\)m, for various values of matching strips width W. Here, the lowest reflection coefficient was obtained for an antenna with a strip width of 4.6 \(\upmu\)m. This simulation renders a bandwidth of \(\sim\)4 GHz, at 2.615:1 VSWR, with an efficiency of up to 99.75%,  
Figure 8: Simulation of 2-D radiation pattern at 155 GHz, phi is the angle perpendicular to the \(x\) axis (Figure 4). Figure 7: Antenna input impedance \(Z_{in}\) vs. frequency for L = 108 \(\upmu\)m, W = 4.6 \(\upmu\)m.  
not including ohmic losses. The reflection coefficient obtained is \(-26\) dB at a central frequency of 155 GHz. A very strong dependence of reflection coefficient on the width of the matching strips is observed. This correlation is more pronounced in determining the central frequency of the detection system, which emphasizes the need for a great precision in the production process of such a detection system. To improve the directivity of the antenna, a silicon (Si) hemispherical lens, 1 mm in diameter, was added in the simulations, as demonstrated in Figure 10[39; 40]. The lens was placed on the backside \(-Z\) direction of the MgO substrate, to prevent potential damage to the junction located in the center of the antenna. Figure 11 shows the simulation results for the reflection coefficient, obtained for the antenna with a Si lens, with matching strips length L of 108 \(\upmu\)m, for various values of strips width W. As in the previous case, the lowest reflection was obtained for an antenna with a width of 4.6 \(\upmu\)m. The simulation renders a bandwidth of \(\sim\)3.4 GHz, at 2.615:1 VSWR, with an efficiency of up to 99.2%, not including ohmic losses. The reflectivity obtained is \(-21\) dB at the central frequency of 148.2 GHz. Figure 10: Structure of antenna with a Si lens. Figure 9: Reflection coefficient S\({}_{11}\) vs. frequency for various matching strips widths without a lens. Figure 12 shows the radiation curve and gain power, 2-D far-field pattern, at the central frequency of 148.2 GHz. The radiation pattern indicates the lower gain power along the \(Z\) axis, i.e., the antenna performance is inferior in that direction, as compared with the gain of 6.2 dBi along the \(-Z\) axis. In order to obtain high performance, the detector should be positioned pointing in the \(-Z\) direction relative to the radiation source.