Klein-Gordon equation approach to nonlinear split-ring resonator based metamaterials: One-dimensional systems

Abstract

The electrodynamics of a one-dimensional split-ring resonator (SRR) based nonlinear metamaterial is studied. The metamaterial is a one-dimensional periodic array of weakly coupled SRRs, with each SRR represented by a nonlinear resistor-inductor-capacitor (RLC) equivalent resonator circuit. Nonlinearity is introduced into the system by the addition of Kerr-type dielectric medium within the SRRs or by the introduction into the system of certain other nonlinear elements (e.g. diodes). In the continuum limit of the system, variations of the charge stored within the capacitive slits of the SRRs in both time and space are shown to be described along the array by a nonlinear Klein-Gordon equation. Analytical solutions of the nonlinear Klein-Gordon equation for various dark and bright envelope, breather, and pulse soliton solutions are obtained and studied. A discussion is given of the relationship between the Klein-Gordon solutions and the solutions of the nonlinear Schr\odinger equation approximation to the nonlinear Klein-Gordon equation. A comparison is made of the Klein-Gordon solutions with intrinsic localized mode (discrete breather) solutions of the discrete system and their continuum limits. An additional continuum limit differential equation for the breather modes of the system is obtained which is not bound by a weak coupling assumption, and its relation to the Klein-Gordon equation is studied. Analytic forms are given for the effects of dissipation in the system on the various bright and dark envelope, breather, and pulse solitons. Discussions are given of the effects of further than first neighbor couplings in the SRR system.