Perturbation theory of injection locking for Josephson junction oscillator

The system consisting of a Josephson element inside a resonant cavity biased by a current source has been analyzed from the circuit point of view. The dynamical equations for an ideal Josephson element shunted by a resistance and an external linear circuit in series with a rf voltage source are solved by a systematic perturbation theory. When the rf voltage is zero, these solutions describe the cavity induced step and for a non zero rf voltage, these solutions indicate the phenomenon of phase locking. The lock range is obtained as a function of the cavity Q, the coupling of the element to the cavity, the critical junction current and the incident power. The results are compared with the Longacre and Shapiro theory of the magnitude of the cavity induced step, the general phase locking theory of Adler and the experimental results of Stancampiano and Shapiro.