Focusing: coming to the point in metamaterials

Abstract

This paper reviews some properties of lenses in curved and folded optical spaces. The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We first provide a comprehensive derivation for the equation of geodesics in curved optical spaces, which is a tool of choice to design metamaterials in transformation optics. We then analyse the resolution of the image of a line source radiating in the Maxwell fisheye and the Veselago–Pendry slab lens. The former optical medium is deduced from the stereographic projection of a virtual sphere and displays a heterogeneous refractive index n(r) which is proportional to the inverse of 1 + r 2. The latter is described by a homogeneous, but negative, refractive index. It has been suggested that the fisheye makes a perfect lens without negative refraction [Leonhardt, Philbin arxiv:0805.4778v2]. However, we point out that the definition of super-resolution in such a heterogeneous medium should be computed with respect to the wavelength in a homogenised medium, and it is perhaps more adequate to talk about a conjugate image rather than a perfect image (the former does not necessarily contain the evanescent components of the source). We numerically find that both the Maxwell fisheye and a thick silver slab lens lead to a resolution close to λ/3 in transverse magnetic polarisation (electric field pointing orthogonal to the plane). We note a shift of the image plane in the latter lens. We also observe that two sources lead to multiple secondary images in the former lens, as confirmed from light rays travelling along geodesics of the virtual sphere. We further observe resolutions ranging from λ/2 to nearly λ/4 for magnetic dipoles of varying orientations of dipole moments within the fisheye in transverse electric polarisation (magnetic field pointing orthogonal to the plane). Finally, we analyse the Eaton lens for which the source and its image are either located within a unit disc of air, or within a corona 1 < r < 2 with refractive index . In both cases, the image resolution is about λ/2.