Two dimensional periodic approach for the study of left-handed metamaterials

Metamaterials exhibiting left-handed properties are nowadays composed of a succession of rods and split-ring resonators, locally organized in a periodic fashion. At a more macroscopic scale, these metamaterials are built in shapes of prisms, slabs, and lenses, each of these geometry allowing for the measurement of different properties: negative refraction index, transmission, and focusing, respectively. The slab, unlike the other geometries, can be studied using a periodic approach, since its cross-section in the longitudinal direction is invariant. This leads to the possibility of studying transmission through slabs using periodic numerical approaches, yielding efficient and accurate numerical codes. This is in particular important when performing parametric studies of split-rings and rods, to identify those frequency bands where a left-handed behavior can be observed. In this paper, we present an integral formulation based on a 2D periodic Green's function, applied to the study of split-rings and rods. Transmission measurements are performed on a structure we have studied previously with a finite-difference time-domain (FDTD) technique, and the prediction of the frequencies were left-handed behavior occurs is shown to be in agreement with our previous results. We use this approach as a first step in the study of split-rings and rods. Ultimately, the geometry is introduced into our FDTD code, which can simulate the dielectric support of the split rings and rods. Experimental results are finally shown for a prism geometry, and clearly show frequency bands where the metamaterials exhibit negative index of refraction. Finally, we verify that the metamaterials thus realized present some isotropy for the magnetic field by performing an addition prism experiment on a rotated structure.