Abstract
In this paper, we present a theoretical study of electronic transport in planar Josephson Superconductor-Normal Metal-Superconductor (SN-N-NS) bridges with arbitrary transparency of the SN interfaces. We formulate and solve the two-dimensional problem of finding the spatial distribution of the supercurrent in the SN electrodes. This allows us to determine the scale of the weak coupling region in the SN-N-NS bridges, i.e., to describe this structure as a serial connection between the Josephson contact and the linear inductance of the current-carrying electrodes. We show that the presence of a two-dimensional spatial current distribution in the SN electrodes leads to a modification of the current-phase relation and the critical current magnitude of the bridges. In particular, the critical current decreases as the overlap area of the SN parts of the electrodes decreases. We show that this is accompanied by a transformation of the SN-N-NS structure from an SNS-type weak link to a double-barrier SINIS contact. In addition, we find the range of interface transparency in order to optimise device performance. The features we have discovered should have a significant impact on the operation of small-scale superconducting electronic devices, and should be taken into account in their design.