Generalising Beenakker Equation to Take into Account Evanescent Modes

Superconductor–normal-metal–superconductor Josephson junctions have been examined extensively in past years. In particular, the case where the normal region is modelled by an ideal normal conductor with an intermittent scattering region can be well described by the celebrated Beenakker equation. Strictly speaking however, this equation is applicable only in the asymptotic plain wave limit, neglecting evanescent modes. However, the situation is changing by recent experimental advances. First, conductor regions are now fabricated with significantly decreased impurity scattering, leading to situations where scattering is dominated by the junction geometry. Second, miniaturisation is advancing such that evanescent modes should no longer be neglected. While this regime has already been captured by numerical methods, we here present an explicit analytical treatment, and generalise the Beenakker equation to short junctions with evanescent modes and geometric scattering in order to investigate the structure of Andreev bound states.