Superelliptic Josephson Tunnel Junctions

The most important practical characteristic of a Josephson junction is its
critical current. The shape of the junction determines the specific form of the magnetic
field dependence of its Josephson current. Here we address the magnetic diffraction
patterns of specially shaped planar Josephson tunnel junctions. We focus on a wide
ensemble of generalized ellipses, called superellipses, which retain the second-order
symmetry. We analyze the implications of this type of isometry and derive the explicit
expressions for the threshold curves of superelliptic Josephson junctions. A detailed
study is made of their magnetic patterns with an emphasis on the rate of decay of the
sidelobe amplitudes for large field amplitudes.