  We predict theoretically that surface of an optical lattice imprinted in
defocusing nonlinear media can support shock, or kink waves. Such new surface
waves contain a modulationally stable pedestal and are strongly localized at
the edge of the optical lattice due to Bragg-type reflection. The kink
steepness and localization degree can be controlled by the lattice depth. We
found two types of kinks, which exhibit distinct stability properties for each
finite gap in the lattice spectrum. Our findings open the way to experimental
observation of optical surface kink waves.
