Focusing a beam of light with left-handed metamaterials

We present a theoretical study of a new metamaterial with a gradient negative index of refraction which can focus a collimated beam of light coming from a distant object. Using the Fermat principle for minimum time propagation of a light beam passing an optically inhomogeneous medium we derive a gradient of the negative index of refraction for a slab of the negative refractive metamaterial. A thin metal film pierced with equidistantly spaced circular holes having specific profile of their diameters was used to model a metamaterial with a gradient of the negative index of refraction. Based on the effective medium approximation for the real effective dielectric permittivity of the patterned metal film pierced with round or square holes, we design a profile of the structured film for a given focal length. FDTD calculations confirm the focusing of a plane electromagnetic wave passing through a system of equidistantly spaced holes in a metal slab with decreasing diameters towards the edges of the slab.