Periodic alternating 0- and $\pi$-junction structures as realization of $\phi$-Josephson junctions

We consider the properties of a periodic structure consisting of small alternating $0$- and $\pi$-Josephson junctions. We show that depending on the relation between the lengths of the individual junctions, this system can be either in the homogeneous or in the phase-modulated state. The modulated phase appears via a second-order phase transition when the mismatch between the lengths of the individual junctions drops below the critical value. The screening length diverges at the transition point. In the modulated state, the equilibrium phase difference in the structure can take any value from $-\pi$ to $\pi$ $(\phi$-junction). The current-phase relation in this structure has very unusual shape with two maxima. As a consequence, the field dependence of the critical current in a small structure is very different from the standard Fraunhofer dependence. The Josephson vortex in a long structure carries partial magnetic flux, which is determined by the equilibrium phase.