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 $\Phi$, because of quantum interference modulated with periodicity $h/2e$. 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 $1/\Phi$ 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 $\propto 1/\sqrt\Phi$ when the magnetic length $l_m=\sqrt{\hbar/eB}$ 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 $\simeq l_m^2/L$. The 2D lattice vanishes for a circular Fermi surface, when only the usual single row of Josephson vortices remains. 
V. P. Ostroukh, B. Baxevanis, A. R. Akhmerov, C. W. J. Beenakker
cond-mat.supr-con cond-mat.mes-hall