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We study the effect of trigonal warping on the focussing of electrons by n-pjunctions in graphene. We find that perfect focussing, which was predicted formassless Dirac fermions, is only preserved for one specific sample orientation.In the general case, trigonal warping leads to the formation of cusp caustics,with a different position of the focus for graphene's two valleys. We develop asemiclassical theory to compute these positions and find very good agreementwith tight-binding simulations. Considering the transmission as a function ofpotential strength, we find that trigonal warping splits the single Dirac peakinto two distinct peaks, leading to valley polarization. We obtain thetransmission curves from tight-binding simulations and find that they are invery good agreement with the results of a billiard model that incorporatestrigonal warping. Furthermore, the positions of the transmission maxima and thescaling of the peak width are accurately predicted by our semiclassical theory.Our semiclassical analysis can easily be carried over to other Dirac materials,which generally have different Fermi surface distortions.

Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene