Two-dimensional Josephson vortex lattice and anomalously slow decay of the Fraunhofer oscillations in a ballistic SNS junction with a warped Fermi surface

$ $The critical current of a Josephson junction is an oscillatory function of the enclosed magnetic flux $\mathrm{\Phi }$, because of quantum interference modulated with periodicity $h\mathrm{/}2e$. We calculate these Fraunhofer oscillations in a two-dimensional (2D) ballistic superconductor–normal-metal–superconductor (SNS) junction. For a Fermi circle the amplitude of the oscillations decays as $1\mathrm{/}\mathrm{\Phi }$ or faster. If the Fermi circle is strongly warped, as it is on a square lattice near the band center, we find that the amplitude decays slower $\propto 1\mathrm{/}\sqrt{\mathrm{\Phi }}$ when the magnetic length $l_m=\sqrt{\mathrm{\hslash }\mathrm{/}eB}$ drops below the separation $L$ of the NS interfaces. The crossover to the slow decay of the critical current is accompanied by the appearance of a 2D array of current vortices and antivortices in the normal region, which form a bipartite rectangular lattice with lattice constant $\simeq l_m^2\mathrm{/}L$. The 2D lattice vanishes for a circular Fermi surface, when only the usual single row of Josephson vortices remains.