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

Abstract

We investigate propagation of a transverse magnetic field through a nonlinear metamaterial slab of sub-wavelength thickness and with a very small and negative linear dielectric permittivity. We prove that, for a given input intensity, the output intensity is a multi-valued function of the field incidence angle so that the transmissivity exhibits angular multi-stability 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 configurations within the slab to be compatible with the transmitted field.Optical hysteresis behavior and bistability are fascinating nonlinear phenomena which have attracted a large research interest in the last decades [1] mainly for their potential photonic applications as optical memories, logic gates and optical computing devices [2]. The possibility of engineering the dielectric permittivity and the magnetic permeability offered by metamaterials [3] has recently allowed to prove that the feedback mechanism supporting bistability is facilitated by the the opposite directionality of the phase velocity and the energy flow in the negative index metamaterial [4]. Nonlinear propagation of light through alternating slabs of positive and negative refractive index materials characterized by a vanishing average refractive index has been considered and bistable switching [5] and gap soliton formation [6] have been predicted. Metal-dielectric multilayer structures have also proved to be media exhibiting interesting bistable behavior [7] since the small average dielectric permittivity combined with the nonlinearity allows the sign of the effective overall dielectric constant to be dependent on the optical intensity [8,9].In this Letter, we investigate electromagnetic transmission through a metamaterial slab of sub-wavelength thickness, characterized by a very small and negative linear dielectric permittivity and exhibiting focusing Kerr nonlinearity. We numerically solve Maxwell equations for the problem of reflection and transmission of an inclined incident transverse magnetic (TM) plane wave and we show that the slab transmissivity is, for a given input intensity, a multi-valued function of the field incidence angle. The novel directional hysteresis behavior is due to the fact that the slab can host the extreme nonlinear regime where the linear and nonlinear contributions to the overall dielectric response can be comparable [8] so that the electromagnetic matching conditions at the output slab boundary allows more than one possible field configurations compatible with the transmitted field.Consider a transverse magnetic (TM) monochromatic field (with time dependence exp(−iωt)) E z (x, z)ê z and H = H y (x, z)ê y propagating through the metamaterial slab, embedded in vacuum, reported in Fig.1(a) of sub-wavelength thickness (along the zaxis) L ...

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