Multipole approach to metamaterials

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.

Figure 1

(Color online) Double-wire MM geometry and corresponding suitable charge distributions that support electric dipole, electric quadrupole, and magnetic dipole moments. The dynamics including interactions between the top and the bottom wires is described by a coupled harmonic oscillator model which is indicated by the red (gray) arrows.

Figure 2

(Color online) (a) Geometry of the simulated double-wire meta-atom is shown. (b) Three-dimensional (3D) bulk MM alignment to calculate the dispersion relation of a bulk MM and (c) the slab arrangement which allows additionally the calculation of effective parameters.

Figure 3

(Color online) (a) Dispersion relation obtained from the numerical calculations in an infinite 3D MM as shown in Fig. 2b and the corresponding fitted analytical version (b).

Figure 4

(Color online) Retrieved effective index (a), effective permittivity (d), and effective permeability (g) from FMM simulations of a 2D single-layer metamaterial. Based on the coefficients from fitting the dispersion relation that is equivalent to the refractive index in the 2D case, the resulting effective index (b), the effective permittivity (e), and the effective permeability (h) resulting from the analytical model are shown. (c), (f), and (i) correspond to the analytically determined effective parameters for the 3D infinite bulk metamaterial. Therefore the dispersion relation (Fig. 3) has been fitted by the analytical one.