Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity

We investigate propagation of a transverse magnetic field through a nonlinear metamaterial slab of subwavelength thickness and with a very small and negative linear dielectric permittivity. We prove that, for a given input intensity, the output intensity is a multivalued function of the field incidence angle so that the transmissivity exhibits angular multistability and a pronounced directional hysteresis behavior. The predicted directional hysteresis is a consequence of the fact that the linear and nonlinear contributions to the overall dielectric response can be comparable so that the electromagnetic matching conditions at the output slab boundary allow more than one field configuration within the slab to be compatible with the transmitted field.