Analytical model for the optical propagation in a nonlinear left-handed material

Left-handed materials have attracted growing attention in recent years. Although many efforts have been concentrated on the study of linear devices filled with left-handed materials, it was only recently that the nonlinear optical properties and optical response of such materials have been described1,2. However, no analytical description of light propagation in these media has been proposed. In this communication, we develop an analytical model describing the interaction of coherent electromagnetic radiation in structures containing both nonlinear right- and left-handed materials. We study the paraxial propagation of electromagnetic radiation in such structures. Using a multiple time and space scales analysis, we derive a system of two coupled nonlinear Schrödinger equations for the amplitudes of the fields, allowing for both forward and reflected waves to be present in the structure. Diffraction, group velocity dispersion and Kerr nonlinear effects are taken into account. Appropriate boundary conditions are derived. The model that we have obtained is very versatile. As an example of a concrete application, we propose to solve the equations for the propagation in a ring cavity filled with a juxtaposition of two materials with opposite handedness. A numerical investigation shows that the system still exhibits bistable behaviour. We show that diffraction in the right-handed and the left-handed parts can compensate exactly in this system and we investigate the effects of this new physics on the spatial propagation properties.