Two-parameters electromagnetic hysteresis of a subwavelength nonlinear epsilon-near-zero slab

We consider propagation of transverse magnetic electromagnetic waves through a nonlinear metamaterial slab, of subwavelength thickness and very small average dielectric permittivity, made up of alternating metal and dielectric nonlinear layers. Exploiting the effective-medium approach, we prove that the output field intensity is a multi-valued function of both the input field intensity and incidence angle so that the transmissivity exhibits a novel kind of multi-stability and a pronounced two-parameter hysteresis behavior. The predicted two-parameter hysteresis physically stems from the fact that, since the linear and nonlinear contributions to the overall dielectric response can be comparable, the slab can host more than one electromagnetic configuration compatible with the incident field and the boundary conditions, regardless of the slab thickness. We investigate slab transmission also through full wave simulations proving the robustness of the two-parameter hysteresis beyond the effective-medium approximation.