## 4 Discussion  
The mixer coupled with Antenna 2 has a larger harmonic number than that of the other mixer when coupled with Antenna 1. Even more, the mixer coupled with Antenna 2 shows a better mixing performance with the conversion efficiency of \(-69.6\) dB in the 88 th harmonic mixing, while the other, coupled with Antenna 1, only has a maximum harmonic number of 30 with the conversion efficiency of \(-73.4\) dB. To explore the reasons, we simulated these two microwave circuit models of mixer chips with attached customized Si hyper-hemispherical lens by CST MWS, displayed in Figure 7a. Because the size of the hyper-hemispherical lens is electrically large relative to the structure in the chip, the AR coating is omitted in the model. Simulated input impedances where HTS JJs locate on the mixer chip attached to a Si hyper-hemispherical lens are shown in Figure 7b. The impedance is \(74-\mathrm{j64}\ \Omega\) when the mixer coupled with Antenna 1. However, the other impedance is \(6+\mathrm{j36}\ \Omega\) when the mixer coupled with Antenna 2. Giving an impedance \(Z_{\mathrm{m}}\) of the JJ mixer, the coupling efficiency of the antenna can be computed with standard antenna theory as  
\[\eta_{\mathrm{c}}=\frac{\mathrm{Re}(Z_{\mathrm{m}})}{\sqrt{\left[\mathrm{Re} (Z_{\mathrm{a}})+\mathrm{Re}(Z_{\mathrm{m}})\right]^{2}+\left[\mathrm{Im}(Z_ {\mathrm{m}})+\mathrm{Im}(Z_{\mathrm{m}})\right]^{2}}} \tag{13}\]  
in [42], where \(Z_{\mathrm{a}}\) is the impedance of the antenna. Assuming \(Z_{\mathrm{m}}\approx R_{\mathrm{n}}\)[16], when the mixer with the \(R_{\mathrm{n}}\) of 0.78 \(\Omega\) (Mixer 1) is coupled with Antenna 1, the \(\eta_{\mathrm{c}}\) is only 0.79%. However, when the other mixer with the \(R_{\mathrm{n}}\) of 1 \(\Omega\) (Mixer 2) coupled with Antenna 2, the \(\eta_{\mathrm{c}}\) is 2.7%. Therefore, the coupling efficiency of Mixer 2 is higher than that of Mixer 1. Furthermore, looking back at the surface electric current distribution in Figure 3c, the current where JJ is located in Antenna 2 is stronger than that in Antenna 1. More energy has been transmitted into the junction by Antenna 2. Thus, Mixer 2 coupled with Antenna 2 has a better performance and larger harmonic number.  
If the antenna matches to the superconducting mixer impedance (a very low normal-state resistance), the radiation efficiency of the antenna will be very low because of the low resistance. Thus, further work can focus on other methods to solve the impedance mismatch between the coupling antenna and the JJ.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Mixing Parameters**} & \multicolumn{5}{c}{**Coupled with Traditional Antennas**} \\ \cline{2-6}  & **Log-Periodic** & **Double-Slot** & **Ring-Slot** & **Bowtie 1** \\ \cline{2-6}  & [41] & [16] & [17] & **Antenna 1** & **Antenna 2** \\ \hline \(f_{\mathrm{THz}}\left(\mathrm{GHz}\right)\) & 207 & 642.24 & 614.4 & 207 & 207 \\ \(f_{\mathrm{LO}}\left(\mathrm{GHz}\right)\) & 4.5 & 159.96 & 19.75 & 6.932 & 2.36 \\ \(f_{\mathrm{IF}}\left(\mathrm{Hz}\right)\) & 499 & 2400 & 2150 & 944 & 362.2 \\ \(\eta\) & 46 & 4 & 31 & 30 & 88 \\ \(\eta\) (dB) & \(-71.6\) & \(-18.1\) & - & \(-73.4\) & \(-69.6\) \\ \hline \end{tabular}  
* In this paper.  
\end{table}
Table 1: Performance comparison of HTS JJ mixers coupled with different traditional antennas with the detected maximum harmonic numbers.