Theory of intermodulation in a superconducting microstrip resonator

The penetration depth and surface resistance of a superconductor depend upon the superfluid current density. This dependence gives rise to nonlinear mixing in a superconducting microstrip resonator. Here we discuss the problem of intermodulation in which two signals at ${\omega }_1$ and ${\omega }_2$ , laying within the pass band of a microstrip cavity resonance, mix and generate a signal at ${\text{2ω}}_1{\mathrm{-\omega }}_2$ . An expression relating the power generated at ${\text{2ω}}_1{\mathrm{-\omega }}_2$ to the power transmitted at ${\omega }_1$ and ${\omega }_2$ is given. We focus on the high-$T_c$ superconductors where it is believed that the order parameter has $d_{x^2{\mathrm{-y}}^2}$ symmetry. We find for a resonator with a large unloaded Q that intermodulation arises dominantly from the reactive nonlinear inductance of the superconducting film.