Effective Medium Theory of Left-Handed Materials

We analyze the transmission and reflection data obtained through transfer matrix calculations on metamaterials of finite lengths, to determine their effective permittivity $ϵ$ and permeability $\mu$. Our study concerns metamaterial structures composed of periodic arrangements of wires, cut wires, split ring resonators (SRRs), closed SRRs, and both wires and SRRs. We find that the SRRs have a strong electric response, equivalent to that of cut wires, which dominates the behavior of left-handed materials (LHM). Analytical expressions for the effective parameters of the different structures are given, which can be used to explain the transmission characteristics of LHMs. Of particular relevance is the criterion introduced by our studies to identify if an experimental transmission peak is left or right handed.

Figure 1

(a) Frequency dependence of the transmission coefficient of a lattice of SRRs (solid line) and of closed SRRs (dashed line). (b) As in (a) for a metamaterial of SRRs plus wires (solid line) and of closed SRRs plus wires (dashed line), with the wires next to the SRRs. The dotted line shows the transmission for wires only. (c) As in (b) but with the wires in the opposite face of the SRRs. The insets show the geometry of the unit cell. The relative permittivity for the metals is taken to be ${ϵ}_m=(-3+i5.88){10}^5$ and for the dielectric board ${ϵ}_b=12.3$. All relative permeabilities are one. The unit cell size is $6a\times 14a\times 14a$. $a$ is the discretization length and $c$ the light velocity in air.

Figure 2

Each row shows the response of each one of five periodic systems to electromagnetic wave. This response is shown through the frequency dependence of transmission coefficient (left panels), dielectric function (middle panels), and magnetic permeability (right panels).