Slow waves on magnetic metamaterials and on chains of plasmonic nanoparticles: Driven solutions in the presence of retardation

Abstract

Slow waves on chains or lattices of resonant elements offer a unique tool for guiding and manipulating the electromagnetic radiation on a subwavelength scale. Applications range from radio waves to optics with two major classes of structures being used: (i) metamaterials made of coupled ring resonators supporting magnetoinductive waves and (ii) plasmonic crystals made of nanoparticles supporting waves of near-field coupling. We derive dispersion equations of both types of slow waves for the case when the interelement coupling is governed by retardation effects, and show how closely they are related. The current distribution is found from Kirchhoff’s equation by inverting the impedance matrix. In contrast to previous treatments power conservation is demonstrated in a form relevant to a finite structure: the input power is shown to be equal to the radiated power plus the powers absorbed in the Ohmic resistance of the elements and the terminal impedance. The relations between frequency and wave number are determined for a 500-element line for two excitations using three different methods. Our approach of retrieval of the dispersion from driven solutions of finite lines is relevant for practical applications and may be used in the design of metamaterials and plasmonic crystals with desired properties.