Synthesis of a Luneburg lens with a flat surface using quasi-conformal transformation optics

In modern systems of radiolocation, navigation and communication, the requirements for antennas are becoming higher requirements every year, namely: operation in a wide frequency range, the ability to change of direction of the main lobe of the radiation pattern. Antenna systems with similar characteristics can be built using dielectric antenna beamforming structures. One of these structures is the Luneberg lens, the peculiarity of which is its spherical symmetry. However, the curved surface of this lens significantly complicates the placement of transmitting and receiving elements along it, which increases the complexity of constructing the entire antenna system. This paper proposes an algorithm for constructing a Luneberg lens with a flat surface. The lens was synthesized using the method of quasi-conformal optical transformations, the mathematical algorithm of which is also described in this work. The paper also presents the results of mathematical modeling of the antenna system using a Luneberg lens with a flat surface at different positions of the emitter relative to the center of the lens, as well as different cut angles. The simulation results show that the synthesized lens can be used to construct a multi-beam antenna system that allows the direction of the main lobe of the antenna radiation pattern to be rearranged over a wide range of angles. However, the scanning angles of this system are limited by the lens geometry, the larger the maximum scanning angle we choose, the more significant the influence of the side lobes on the radiation pattern becomes.