Lens equation for flat lenses made with hyperbolic metamaterials

This study aims to give a general theory that enables the design of flat lenses based on hyperbolic metamaterials. We derive a lens equation that is demonstrated to involve the curvature of the dispersion relation. Guided by this theory, hyperbolic lenses of focal length ranging from zero to a few wavelength are simulated. High transmission efficiency is also obtained by reducing the amount of metal compared to the dielectric material.