Multipole approach to metamaterials

Abstract

An analytical description for plane-wave propagation in metamaterials is presented. It follows the usual approach for describing light propagation in homogeneous media on the basis of Maxwell's equations, although applied to a medium composed of metallic nanostructures. Here, as an example, these nanostructures are double (or cut) wires. In the present approach it is assumed that the carriers perform collective oscillations in a single wire. These oscillations are coupled to those in the adjacent wire; thus, the internal carrier dynamics may be described by a coupled-oscillator model. The multipole expansion technique is used to account for the electric and magnetic dipole as well as the electric quadrupole moments of these carrier oscillations within the nanostructure. It turns out that the symmetric normal mode is related to the electric dipole moment whereas the antisymmetric normal mode evokes simultaneously a magnetic dipole and an electric quadrupole moment. It is shown how effective permittivity and permeability can be derived from analytical expressions for the dispersion relation, the magnetization, and the electric displacement field. The results of the analytical model are compared with rigorous simulations of Maxwell's equations yielding the limitations and the domain of applicability of the proposed model

Simple and versatile analytical approach for planar metamaterials

“…(1) 36 . Here we apply the multipole approach to map the displacement onto electric dipole moments, since this is consistent with our previous works 29,33 . Accounting for the contributions of the electric dipole and quadrupole as well as the magnetic dipole moment the wave equation reads as…”

“…This holds as long as the wires are assembled in-plane. For out-of-plane structures higher order multipoles come generally into play 29,33 . Here the spatial coordinate represents the elongation of a negatively charged carrier density driven by an external electromagnetic field, which is the usual assumption in plasmonics 34 .…”

“…(1) γ j is the damping, ω 0j the eigenfrequency, σ ij the coupling constant and q j is the charge for the three oscillators (i, j) ∈ (1, 2, 3), respectively, similarly to those introduced in Ref. 29. The coordinates (x 1 , y 2 , x 3 ) themselves are understood as the displacement of the negatively charged carriers representing oscillating currents or multipole moments where both approaches can be used to determine the effective material response of an artificial medium exhibiting a carrier dynamics as described by Eq.…”

3D THz metamaterials from micro/nanomanufacturing

“…These amplitudes always have to be found from solving the boundary value problem at the interface between a space from which an external illumination hits the interface of the metamaterial. Detrimental to the simplicity of the concept is the unavailability of analytical solutions that provide the amplitude profile of the eigenmode as well as the dispersion relation that relates the Bloch vector to a particular frequency, although analytical models can be developed that allow for an intuitive understanding of the underlying physics [200][201][202]. Therefore, Eq.…”

Self-assembled plasmonic metamaterials

“…Another model to describe the optical response of the unit cell is based on coupled oscillators [ 79 ]. The plasmonic entities of the unit cell are replaced by one or more oscillating currents.…”