Klein-Gordon Equation Approach to Non-linear Split Ring Resonator Based Metamaterials: 1D 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 in 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 K-G equation for both dark and bright solutions, breather, and pulse soliton solutions are obtained and studied.  A discussion is given of the relationship between the K-G solutions and solutions of the nonlinear Schrodinger equation (NLSE) approximation to the NLKG equation. A comparison is made of the K-G 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 mode is obtained which is not bound by a weak coupling assumption, and its relation to the K-G equation is studied. Analytic forms are given for the effect of dissipation in the system on the various dark and bright envelope, breather and pulse solitons. Discussions are given of the effects of further than first neighbor couplings in the SRR system.

Keywords: Metamaterials, NLKG equation, Dark & Bright Envelop, Breather and Pulse Solitons.

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For the first time, metamaterials are described by NLKG equation that is found to be a richer variety than NLSE. A detailed comparison shows that both NLKG and NLSE show dark and bright envelop solitons and discrete breathers, but NLKG in addition shows breather pulses, whereas NLSE does not show such pulses. This should trigger further studies on the subject of metamaterials for antenna and other applications in the THz regime.

(Phys. Rev. B, Vol. 84 (2011) 155429)