Commensurability Effects in Large Josephson Junctions
  Two types of commensurability effects are possible in a large Josephson junction patterned with columnar defects. The first occurs for a periodic array of pins when the mean fluxon spacing (tuned by the magnitude of the applied in--plane magnetic field) is a rational fraction of the defect spacing. We show that this effect leads, under fairly general conditions, to a mapping of the behavior of the Josephson junction near the commensurate field values to that of a zero field junction with an effective Josephson penetration depth. The second occurs for more general arrangements of pinning sites, when the orientation of the Josephson vortex lattice (tuned by the direction of the applied field) nearly matches the orientation of the defects. We investigate this tilt response in the limit of a single Josephson vortex. The results are compared, where possible, to recent experiments. As an aside from our main analysis, we prove that, contrary to recent claims in the literature, the critical current density vanishes in the thermodynamic limit, even in the presence of (non--pathologically distributed) pinning disorder 
Leon Balents and Steven H. Simon