Josephson flux-flow oscillators in nonuniform microwave fields

We present a simple theory for Josephson flux-flow oscillators in the presence of nonuniform microwave fields. In particular we derive an analytical expression for the $I-V$ characteristic of the oscillator from which we show that satellite steps are spaced around the main flux-flow resonance by only even harmonics of the rf frequency. This result is found to be in good agreement with our numerical results and with experiments.