# Superconducting Sub-Terahertz Oscillator with Continuous Frequency Tuning
## 3 Results and Discussion
### Modeling of Suppression Methods and Estimation of Parameters  
In order to confirm the physical principles of the suppression of the resonant modes, to determine the factors contributing to the suppression the most, and to optimize the geometry of absorbers (without a significant decrease in radiation power of FFO), we carried out the calculations of different geometries in Ansys HFSS [68]. Unfortunately, HFSS does not have in-built tools to treat superconducting effects, such as the penetration of the magnetic field into the superconducting electrodes and the change in propagation velocity due to this effect (Swihart velocity) [30]. In the article by [PERSON] et al. [69], the procedure of modeling of the superconducting microstrip line is described. The authors propose to assign special boundary conditions to the corresponding surfaces of the superconductors. In our work, we followed this method. Assigned to the surfaces of superconductors frequency-dependent datasets were generated separately using the expressions of the Mattis-Bardeen theory [70], extended by [PERSON] et al. [71] by taking into account a finite mean free path time of the quasiparticles. In our simulations, we consider a model of Josephson transmission line (JTL). Figure 6 depicts the geometries of the simulated structures. When fabricating the junction, the narrowing regions near its ends were carried out to provide better coupling of the FFO with the outer circuit and to decrease the reflection from the end of the FFO. In geometries 2 and 3, the metal layer overhung the junction and due to this geometry, the field leaked out of the junction. We modeled only the \"input\" part of the FFO where fluxons enter the junction; we assigned a single lumped port for JTL excitation, as shown in Figure 6. This enabled us, on one hand, to investigate the absorption of the electromagnetic waves in the line, and on the other hand, to examine the reflection of the wave from the end of the junction. We could assign the port impedance to match the line in order to avoid reflections, but dispersion in the line at high frequencies due to the change in London penetration depth and geometrical factor made this task almost impossible. As is shown later, it turned out to be much easier to set the impedance of the excitation port much bigger than the characteristic impedance of the line (impedance of the port was set to 100 \(\Omega\) and the characteristic impedance of the line was around 0.3 \(\Omega\)) and then recalculate the data on S\({}_{11}\) in order to eliminate the influence of the port. Figure 5: Comparison of the FFO differential resistance R\({}_{\mathrm{d}}\) dependencies measured as a function of FFO voltage for two different FFO designs presented in Figure 3a,d; see also Figures 2c and 4c. The results of the simulation are shown in Figure 7 where the lines represent the magnitude of S\({}_{11}\)-parameter dependence on frequency. It is clearly seen that the application of the normal metal to the idle region leads to a strong decrease in amplitude of the oscillations which corresponds to the absorption of radiation in the structure. Moreover, when adding normal metal to the end of the FFO, almost complete suppression of the resonances can be obtained (see green curve, which refers to geometry #2 and corresponds to the experimental layout presented in Figure 3d). Here, we derive some formulas in order to eliminate the reflections from the excitation port and quantitatively estimate the level of attenuation in the line and reflection ratio from the ends in the investigated structures. Consider an incident wave coming from the left; it reflects from the outer boundary of the line with the amplitude reflection ratio \(r_{1}\), which is given by the well-known expression:  
\[r_{1}=\big{|}Z_{line}-Z_{port}/Z_{line}+Z_{port}\big{|}, \tag{3}\]  
where \(Z_{line}\) and \(Z_{port}\) denote the characteristic impedance of the line and the impedance of the port. Amplitude of the wave decreases exponentially with the distance due to the absorption in the line.