# Superconducting Sub-Terahertz Oscillator with Continuous Frequency Tuning
## 1 Introduction  
Superconducting devices have unique characteristics that are unattainable for conventional semiconductor electronics. The ability of superconducting nanostructures to operate at very high frequencies, along with their extremely strong non-linear response, makes them ideal building blocks for state-of-the-art systems designed to detect and to generate terahertz radiation. The development of supersensitive terahertz (THz) receivers and oscillators is currently one of the most intensively and successfully developing areas of superconducting electronics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Superconducting elements have an extremely low intrinsic noise due to their cryogenic operating temperature and quantum nature of mixing [12, 13, 14]; this, in addition to the listed advantages, enables the development of terahertz receivers with unique parameters that are unreachable for devices based on other physical principles. In the frequency range from 100 GHz to 1.4 THz, the most sensitive receiving elements are superconductor-insulator-superconductor (SIS) junctions [14, 15, 16]. The development of superconducting electronics made it possible to create a superconducting integrated receiver (SIR) [17, 18], which combines all receiver components in one chip: a planar antenna, a SIS mixer, a superconducting local oscillator, and an additional harmonic SIS mixer for phase locking of the local oscillator. The high sensitivity and spectral resolution of the SIR were demonstrated while probing the atmospheric limb aboarda high-altitude balloon [19], and in the laboratory for detailed spectral measurements of terahertz radiation emitted by the Bi\({}_{2}\)Sr\({}_{2}\)CaCu\({}_{2}\)O\({}_{8}\) mesa structure, formed by thousands of Josephson junctions [20]. A number of other experiments dedicated to studying the spectra of sub-THz signals were also investigated [21; 22; 23]. The superconducting local oscillator, also known as FFO [24; 25; 26; 27; 28; 29], is based on a long Josephson junction (LJJ), whose length L is much greater than its width W and the Josephson penetration depth \(\lambda_{\rm J}=(\Phi_{0}/(2\pi\ \mu_{0}\ \rm j_{c}\ l))^{1/2}\); where \(\Phi_{0}=\rm h/2e\), \(j_{\rm c}\) is critical current density of the junction, \(\rm l=d+\lambda_{L1}+\lambda_{L2}\) is magnetic thickness of the tunnel junction, \(\rm d\) is the barrier thickness, \(\lambda_{L1,2}\)-magnetic field (London) penetration depth of the superconducting electrodes. In such a long junction, a unidirectional motion of Josephson vortices-fluxons, containing a single magnetic flux quantum \(\Phi_{0}\) arises due to joint action of the applied magnetic field and the transport current, called the bias current \(\rm I_{b}\). When the vortex reaches the end of the junction, electromagnetic radiation is emitted. In accordance with the Josephson relation, the junction biased at voltage V generates electromagnetic oscillations with frequency \(\rm f=(2\pi/\Phi_{0})\) V (that corresponds to 483.6 GHz/mV). The speed and density of the fluxons and, therefore, the frequency and power of the radiation, can be tuned by varying the bias current and/or magnetic field. The magnetic field in the LJJ is created using an external coil or a special integrated control line. The maximum speed of vortex motion is equal to the speed of propagation of electromagnetic waves in the junction, the so-called Swihart velocity \(\rm c_{SW}=c\sqrt{d/(\varepsilon l)}\)[30], where \(\rm c\) is the speed of light in vacuum, \(\rm d\) is the barrier thickness, \(\varepsilon\) is its dielectric constant. For typical FFO parameters, Swihart velocity is 20-30 times less than the speed of light in a vacuum. The operation of the FFO based on LJJ has a number of features associated with various modes of the fluxons motion. At voltages V \(\rm<V_{g}/3\), where \(\rm V_{g}\) is the gap voltage of the tunnel junction, the so-called resonant flow of fluxons is observed.