PHYSICAL REVIEW B 84, 155429 (2011) Klein-Gordon equation approach to nonlinear split-ring resonator based metamaterials: One-dimensional systems Pradipta Giri, 1 Kamal Choudhary, 2 Arnab Sen Gupta, 2,3 A. K. Bandyopadhyay, 4 and Arthur R. McGurn 5,* 1 Dumkal Institute of Engineering and Technology, West Bengal University of Technology, Dumkal, Murshidabad, India 2 Government College of Engineering and Ceramic Technology, West Bengal University of Technology, 73, Abinash Chandra Banerjee Lane, Kolkata-700010, India 3 Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA 4 Ideal Institute of Engineering, Kalyani-741235, Dist. Nadia, India 5 Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008-5252, USA (Received 3 July 2010; published 17 October 2011) 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. DOI: 10.1103/PhysRevB.84.155429 PACS number(s): 78.67.Pt, 05.45.Yv I. INTRODUCTION The electrodynamics of substances with simultaneous negative values of dielectric permittivity (ε) and magnetic permeability (μ) has been a subject of much study, with current interests in possible technological applications. 1 Substances with both negative ε and μ are predicted to posses a negative refractive index and, consequently, to exhibit a variety of optical properties not found in positive indexed materials. Negative index materials, however, do not occur in nature, and only recently has it been shown that they can be artificially fabricated. The experimental realization of such materials was demonstrated by Smith et al. 2 based on theoretical work of Pendry et al. 3 , 4 Smith et al. made a type of metamaterial (MM) as an artificial structure with negative refractive properties. The structure consists of metallic wires responsible for the negative permittivity and metallic split-ring resonators (SRR) responsible for the negative permeability. The optical and electrical properties of the MM are modulated by the proper use of SRRs to give ε, μ< 0 within a region of frequencies, and it is the SRRs that are key in setting negative μ within a material with ε< 0. In particular, unlike naturally occurring materials, the designed MMs show a relatively large magnetic response at THz frequencies. This, in combination with its negative permitivity in the THz, is responsible for an effective negative index in this range of frequencies. From the standpoint of theory, linear and nonlinear SRR have been shown to be described by equivalent resistor- inductor-capacitor (RLC) circuits 5 featuring a self-inductance L from the ring, a ring Ohmic resistance R, and a capacitance C from the split in the ring. Metamaterials with negative refractive properties are then formed as a periodic array of SRR, which are coupled by mutual inductance and arrayed in a material of dielectric constant ε. From the standpoint of experiment, the requirements for the effective electromagnetic application of metamaterials in the THz region introduce the necessity of very high accuracy in the fabrication of SRR- based MMs to produce materials of uniform and consistent properties. The electrodynamics of MM consisting of large numbers of loosely coupled SRRs has recently been studied for discrete lattices, 5 treating the array as a set of capacitively loaded loops 6 (see references therein). It was shown that the system of capacitively loaded loops support wave propagation. Since the coupling between the SSRs is due to induced voltages, these extended wave solutions are referred to as magneto- inductive waves (MI waves) 7–10 and represent a vast area of active research in the field of (a) artificial delay lines and filters, (b) dielectric Bragg reflectors, (c) slow-wave structures in microwave tubes, and (d) coupled cavities in accelerators, modulators, antenna array applications, etc. The RLC configuration of the SRRs cause the MMs based on them to exhibit a resonant frequency for both linear and nonlinear systems, and MI waves are observed to propagate within a frequency band near the resonant frequency. In linear MMs, the SRRs are composed of both linear dielectric and magnetic materials so that the MI dispersion relations do not depend on the EM field intensities. Non- linearity is incorporated into MMs by embedding Kerr-type 155429-1 1098-0121/2011/84(15)/155429(10) ©2011 American Physical Society