Theoretical formalism for collective electromagnetic response of discrete metamaterial systems

TL;DR

This paper presents a comprehensive theoretical framework for understanding the collective electromagnetic behavior of discrete metamaterial systems, accounting for near-resonant interactions, radiative couplings, and cooperative responses of resonators.

Contribution

It introduces a Lagrangian and Hamiltonian formalism for coupled electromagnetic and resonator dynamics, including recurrent scattering effects, in a unified quantum electrodynamics approach.

Findings

Demonstrates collective eigenmodes in metamaterials due to strong radiative interactions.

Derives equations capturing recurrent scattering processes to all orders.

Shows cooperative response in a split ring resonator array with modified resonance properties.

Abstract

We develop a general formalism to describe the propagation of a near-resonant electromagnetic field in a medium composed of magnetodielectric resonators. As the size and the spatial separation of nanofabricated resonators in a metamaterial array is frequently less than the wavelength, we describe them as discrete scatterers, supporting a single mode of current oscillation represented by a single dynamic variable. We derive a Lagrangian and Hamiltonian formalism for the coupled electromagnetic fields and oscillating currents in the length gauge, obtained by the Power-Zienau-Woolley transformation. The response of each resonator to electromagnetic field is then described by polarization and magnetization densities that, to the lowest order in a multipole expansion, generate electric and magnetic dipole excitations. We derive a closed set of equations for the coherently scattered field and normal mode amplitudes of current oscillations of each resonator both within the rotating wave approximation, in which case the radiative decay rate is much smaller than the resonance frequency, and without such an assumption. The set of equations includes the radiative couplings between a discrete set of resonators mediated by the electromagnetic field, fully incorporating recurrent scattering processes to all orders. By considering an example of a two-dimensional split ring resonator metamaterial array, we show that the system responds cooperatively to near-resonant field, exhibiting collective eigenmodes, resonance frequencies, and radiative linewidths that result from strong radiative interactions between closely-spaced resonators.