Interaction of Josephson junction and distant vortex in narrow thin-film superconducting strips

The phase difference between the banks of an edge-type planar Josephson junction crossing the narrow thin-film strip depends on wether or not vortices are present in the junction banks. For a vortex close to the junction this effect has been seen by Golod, Rydh, and Krasnov, \prl {\bf 104}, 227003 (2010), who showed that the vortex may turn the junction into $\pi$-type. It is shown here that even if the vortex is far away from the junction, it still changes the 0-junction to $\pi$-junction when situated close to the strip edges. Within the approximation used, the latter effect is independent of the vortex-junction separation, a manifestation of topology of the vortex phase which extends to macroscopic distances of superconducting coherence.