Phase integral approach to wave propagation in continuously graded models of flat lenses

We propose a method to determine approximate phase-integral analytical solutions for electric and magnetic fields in flat lenses with the refractive index varying in the radial direction. In our model the gradient of refractive index is approximated by a large number of concentric annuli with step-increasing index. Here the central part contributes to a bulk of the phase transformation, while the external layers act as a graded antireflective structure, matching the impedance of the lens to that of the free space. Such lenses can be modeled as compact composites with continuous permittivity and (if needed) permeability functions which asymptotically approach unity at the boundaries of the composite cylinder. We illustrate the phase-integral approach by obtaining the approximate analytic solutions for the electric and magnetic fields for a special class of composite designs with radially graded parameters.