Three‐dimensional single‐shell dielectric lens design using complex coordinates and local axis‐displaced confocal quadrics

Abstract

This study proposes an analytical formulation in complex coordinates to synthesise threedimensional single-shell dielectric lens surfaces. An exact formulation based on geometrical optics is developed, and the synthesis problem is modelled as a non-linear second-order partial differential equation of the Monge-Ampère (MA) type. An alternative numerical scheme based on axis-displaced confocal quadrics is used to locally represent the surface of dielectric lens. The advantage of this approach is to analytically obtain the partial derivatives present in the MA equation. To validate the proposed methodology, numerical results are obtained by an iterative algorithm that allows the simultaneous control of the transmitted beam width and the power density amplitude in the far-field region. To illustrate the synthesis effectiveness, two examples of lens antennas are simulated and analysed.