Transition from thin-film to bulk properties of metamaterials

Effective material parameters for metamaterials are usually extracted for a single thin layer only. We show that, in general, these parameters cannot directly be assigned to a multilayered (bulk) metamaterial. The principal issue is the presence of higher-order Bloch modes. These modes are usually negligible for a single layer but can have considerable influence on the results for a periodically stacked structure. In particular, we show examples where a stack of single-layer negative refractive index metamaterials exhibits a positive index. Furthermore, on the basis of the dispersion relation of the respective Bloch modes, we derive criteria for the validity of the effective parameter approach.

Figure 1

(Color online) Outline of the geometry under consideration.

Figure 2

(Color online) Transmitted amplitudes for a MM consisting of (a) metallic wires and (d) SRRs as a function of the number of layers. The real parts of the effective material parameter (permittivity or permeability) altered by each unit cell are shown in (b) and (e). Their respective imaginary parts are shown in (c) and (f). The same colors and types of lines as used in (a) appear in all subsequent figures to denote the number of layers (one $\text{layer}=\text{solid}$ blue line; two $\text{layers}=\text{solid}$ green line; three $\text{layers}=\text{dashed}$ blue line; four $\text{layers}=\text{dashed}$ green line; five $\text{layers}=\text{dashed}$-dotted blue line).

Figure 3

(Color online) Transmission for a medium whose unit cell comprises an SRR and a thick metallic wire (see inset).

Figure 4

(Color online) (a) Real and (b) imaginary parts of the effective refractive index retrieved from a medium whose unit cell comprises an SRR and a thick metallic wire (see inset).

Figure 5

(Color online) Ratio of the amplitudes of the first-order Bloch mode, which is evanescent, and the zeroth-order Bloch mode for a medium made of SRRs and thick wires in (a) and for a medium made of the same SRRs but with thinner wires in (b).

Figure 6

(Color online) (a) Real and (b) imaginary parts of the effective refractive index and (c) real and (d) imaginary parts of the effective impedance retrieved for a medium whose unit cell comprises SRRs and metallic wires, as shown in the inset. The parameter retrieval was performed for several numbers of layers. The red dots in (a) and (b) denote the refractive index as deduced from the dispersion relation of the zeroth-order Bloch mode.

Figure 7

(Color online) Effective propagation constant of the two lowest-order Bloch modes propagating in the unit cell comprising [(a) and (b)] the SRR and the thick wire and [(c) and (d)] the SRR and the thin wire. Real parts of the propagation constants are shown in (a) and (c) and imaginary parts are shown in (b) and (d). Blue circles correspond to the zeroth-order Bloch mode and green stars to the first-order Bloch mode.

Figure 8

(Color online) (a) Real and (b) imaginary parts of the effective refractive index retrieved for a medium whose unit cell comprises SRRs and thick metallic wires, as shown in the inset. Geometrical details of the elements in the unit cell were not altered but only the period was increased by $200\mathrm{nm}$ as compared to the results shown in Fig. 4; hence, the period was $600\mathrm{nm}$. The parameter retrieval was performed for several numbers of layers the MM consists of.

Figure 9

(Color online) (a) Real and (b) imaginary parts of the effective refractive index retrieved for a fish-net structure as a function of the number of the layers (one $\text{layer}=\text{solid}$ blue line; two $\text{layers}=\text{solid}$ green line; three $\text{layers}=\text{dashed}$ blue line; four $\text{layers}=\text{dashed}$ green line). The coordinate system, the chosen polarization, and the geometry are shown on top of the figure. Geometrical details of the structure were adopted from values published in literature (Ref. 23) and are detailed in the text.