# Superconducting Sub-Terahertz Oscillator with Continuous Frequency Tuning
## 3 Results and Discussion  
### Traditional FFO  
First, we describe the operation principles of a traditional FFO based on the LJJ. A DC bias current I\({}_{\mathrm{B}}\) is applied through the LJJ in combination with an external magnetic field, which is generated by the current through the superconducting control line parallel to the junction plane. The magnetic field penetrates into the junction at the end: the Josephson phase \(\varphi\)(x), along the junction changes, with the formation of a vortex carrying the magnetic flux quantum \(\Phi_{0}\). The bias current causes this quantum to move, and as a result, a unidirectional viscous flow of Josephson vortices arises in the junction, and each vortex has a length of 2\(\lambda_{\mathrm{I}}\) along the junction plane and 2\(\lambda_{\mathrm{L}}\) in the direction perpendicular to the junction plane. The value of the Josephson penetration depth \(\lambda_{\mathrm{I}}\) depends on critical current density \(\mathrm{j}_{\mathrm{c}}\); typically, \(\lambda_{\mathrm{I}}\) is of about 5 \(\mu\)m for the \(\mathrm{j}_{\mathrm{c}}=5\) kA/cm\({}^{2}\), while the London penetration depth \(\lambda_{\mathrm{L}}\) for niobium films is 85 nm. Moving along the junction under the Lorentz force, the fluxons are repelled from each other and form a chain of vortices, which is quite rigid at high magnetic fields [55; 56]. When a moving fluxon crosses the end of the junction, a voltage pulse V occurs, the integral of which over time is equal to the value of the \"emerging\" magnetic flux quantum:  
\[\int\mathrm{Vdt}=\Phi_{0}=2\;\mathrm{mV}\cdot\mathrm{1}\;\mathrm{ps}. \tag{1}\]  
The speed and density of this fluxon chain, and, consequently, the power and frequency of electromagnetic radiation that occurs when quanta cross the boundary, can be tuned by changing the bias current I\({}_{\mathrm{b}}\) and the current through the magnetic field control line I\({}_{\mathrm{CL}}\). After the vortex collides with the emitting end, a reflected electromagnetic wave arises inside the junction. For a small value of the parameter \(\alpha\), which characterizes the attenuation in the FFO, the wave can reach the opposite end of the junction. At certain frequencies, a standing wave can arise, which will affect the entry of vortices into the junction; as a result, the IVCs in the region of small \(\alpha\) has a pronounced resonant structure. The smaller the \(\alpha\), the steeper the current steps appear on the IVC. At higher operating frequencies, the parameter \(\alpha\) experiences an abrupt increase due to the effect of Josephson self-pumping [33; 34; 35; 36], and a non-resonant mode of motion of a dense flow of vortex chain is formed (see Section 1). The transition between the two operating modes of the FFO is clearly visible on the IVC (Figures 1 and 2) at voltages V = V\({}_{\mathrm{g}}\)/3. For the successful operation of the superconducting oscillator as a local oscillator (LO) in the integrated receiver [17; 18], continuous tuning of the FFO frequency in the entire available frequency range of the device is necessary; in addition, to implement the PLL mode, the linewidth of the FFO emission should not exceed 10-15 MHz. The linewidth of the Josephson junction is determined by the combination of thermal and shot noise in the tunnel junction, which translates into frequency fluctuations at the tunnel junction by the square of differential resistance R\({}_{\mathrm{d}}\); exact expression depends on the specific LJJ model [40; 41; 42; 55]. For a current-biased lumped Josephson tunnel junction at temperature T, the full linewidth at half power \(\delta\)f is given by the expression:  
\[\delta\mathrm{f}=4\;\pi\;\mathrm{k}_{\mathrm{B}}\;\mathrm{T}\;\mathrm{R}_{ \mathrm{d}}{}^{2}\;\mathrm{I}_{\mathrm{b}}/(\Phi_{0}\;\mathrm{V}). \tag{2}\]The noise phenomenon observed in Josephson oscillators has been studied since 1980 [57], in addition to the extra noise that can appear for Josephson junctions with high critical current density [58].