# Superconducting Sub-Terahertz Oscillator with Continuous Frequency Tuning
## 3 Results and Discussion
### Modeling of Suppression Methods and Estimation of Parameters
Reflection ratio from the opposite end is \(r_{2}\). Taking all the effects above into account, we can derive the expression for \(S_{11}\):  
\[S_{11}=r_{1}e^{j\pi}+\frac{r_{2}\big{(}1-r_{1}^{2}\big{)}}{1-r_{1}r_{2}\exp(- 2 xL)\exp(2 ikL)}\, \tag{4}\]  
where \(\kappa\) is the attenuation constant, \(k\) is the wave number \(k=2\pi/\lambda\), and \(L\) is the length of the line; \(e^{j\pi}\) in the first term denotes that the port impedance is larger than that of the line. This expression can be rewritten for the magnitude of \(S_{11}\):  
Figure 6: 3D-models of the simulated geometries. (**a**) FFO without idle region (geometry #0, corresponds to Figure 3a); (**b**) geometry with normal metal layer along the FFO where width is constant (geometry #1, Figure 3b); (**c**) circuit with normal metal layer at the end and along the junction (#2, Figure 3d); (**d**) design with the impedance transformer connected to the end (#3) in order to decrease the reflection. \[\left|S_{11}\right|=\sqrt{\frac{\left(r_{2}-r_{1}-r_{1}^{2}r_{2}\right)^{2}+2 \left(r_{2}-r_{1}-r_{1}^{2}r_{2}\right)r_{1}x\cos(2 kL)+r_{1}^{2}x^{2}}{1-2x\cos (2 kL)+x^{2}}}\, \tag{5}\]  
where \(x=r_{1}r_{2}\exp(-2\kappa L)\). The magnitude of \(S_{11}\) reaches its maxima when \(\cos(2 kL)=1\) and minima when \(\cos(2 kL)=-1\). Corresponding expressions are:  
\[\left|S_{11}\right|_{max}=\frac{r_{1}-r_{2}+r_{1}^{2}r_{2}+r_{1}x}{1+x} \tag{6}\]  
Thus, the difference between maxima and minima is given by formula:  
\[diff=\left|S_{11}\right|_{max}-\left|S_{11}\right|_{min}=\frac{2\ r_{1}r_{2}^{2} (1-r_{1}^{2})\ \exp(-2\kappa L)}{1-r_{1}^{2}r_{2}^{2}\exp(-4\kappa L)} \tag{7}\]  
By taking a natural logarithm of the expression for _diff_ and expanding it into a Taylor series, assuming \(r_{2}^{2}\exp(-2\kappa L)\), being small compared to unity, we obtain:  
\[\ln(diff)\approx 2\ln(r_{2})-2\kappa L+C(r_{1})+\Bigg{[}r_{1}^{2}r_{2}^{2}\exp (-4\kappa L)+\frac{r_{1}^{4}}{2}\Big{(}r_{2}^{2}\exp(-4\kappa L)\Big{)}^{2} \Bigg{]}. \tag{8}\]  
Thus, the magnitude of the oscillations of \(\left|S_{11}\right|\) in this case is almost completely determined (with an error less than 15% for superconducting case and 2% for all the other simulations) by the absorption in the line and reflection ratio from the opposite boundary as the term in square brackets can be neglected. Yet, there is a value \(C(r_{1})\) in the expression above which is the same for all simulated geometries and can be determined by the extrapolation of the curve corresponding the superconducting case to the low-frequency region. This enables us to determine the amplitude of the wave after passing through the structure. Recalculated curves in a format: \(B_{1}/A_{1}=r_{2}^{2}\exp(-2\kappa L))\), where B1 is an amplitude of the wave returned to the port after propagation in the line and reflection from the end, and A1 is the initial amplitude, are shown on Figure 8. For all the geometries, an increase in the attenuation with frequency is observed as expected. This can be explained by two factors. First, the wave number and the attenuation constant depend linearly  
Figure 5:0 \(S_{11}\)-parameter dependence on frequency from simulations in Ansys HFSS. Black curve represents the result for barely superconducting line; red curve depicts the geometry with normal metal in idle region, but with uncovered end; green corresponds to the geometry with normal metal both near the line and at the end (see Figure 6). on frequency [72]. Second, the resistance of the films made either of a superconductor or normal metal increases with frequency [70,71]; this leads to stronger absorption of electromagnetic wave energy in the structures. The introduction of layers of normal metal to the idle region of the junction leads to a decrease in the magnitude of the reflected wave by a factor of 4 (curve Mo #1); an additional normal metal layer at the end of the junction improves this result to more than the order of magnitude (curve Mo #2) in the whole frequency range 200-500 GHz.