Analytical expressions for the design of twin junction tuning in SIS mixers

We derive general analytical expressions for matching the complex impedance of the feeding radio frequency (RF) circuit to a twin junction device in Superconductor-Insulator-Superconductor (SIS) mixers. A unique feature of our analysis is that it allows for junctions of different admittance and an RF circuit with complex admittance. This provides important design flexibility and accommodates device fabrication tolerances. We then focus on special cases of identical junctions, in particular those cases that yield simple expressions for matching the twin junction to a real impedance. Finally, we derive design curves that allow the determination of the twin junction parameters for a given complex impedance of the feeding circuit. These curves allow the designer to choose different solutions than the commonly used quarter-wavelength transmission line separating the two SIS junctions, which our analysis shows is not always the optimal solution. Moreover, even when commercial software is used for designing the circuits, our analytical equations will act as an important guide to understanding the physics and guide the designer to the optimal solution.