  In graphene, a Kekul\'e-Y bond texture modifies the electronic band structure
generating two concentric Dirac cones with different Fermi velocities lying in
the {\Gamma}-point in reciprocal space. The energy dispersion results in
different group velocities for each isospin component at a given energy. This
energy spectrum combined with the negative refraction index in p-n junctions,
allows the emergence of an electronic analog of optical birefringence in
graphene. We characterize the valley birefringence produced by a circularly
symmetric Kekul\'e patterned and gated region using the scattering approach. We
found caustics with two cusps separated in space by a distance dependent on the
Kekul\'e interaction and that provides a measure of its strength. Then, at low
carrier concentration we find a non-vanishing skew cross section, showing the
asymmetry in the scattering of electrons around the axis of the incoming flux.
This effect is associated with the appearance of the valley Hall effect as
electrons with opposite valley polarization are deflected towards opposite
directions.
