Equivalent Planar Lens Ray-Tracing Model to Design Modulated Geodesic Lenses Using Non-Euclidean Transformation Optics

This article describes a design procedure that enables a time-efficient evaluation of the focusing properties of modulated geodesic lenses using ray tracing on the equivalent gradient-index planar lens. The method uses transformation optics to define the equivalent planar relative permittivity distribution of axially symmetric surfaces and a ray-tracing model to evaluate the phase distribution in the aperture of the lens. This approach is of interest to optimize modulated geodesic lenses having polynomial profiles, reducing their height while preserving their wideband behavior and wide angular focusing properties. The approach is validated with a specific lens design. The profile is optimized at 30 GHz, while the focusing properties are monitored over the complete Ka up-link frequency band allocated to satellite communications (i.e., 27.5-31 GHz). The manufactured prototype produces 21 beams equally spaced every 7.5 degrees over the extended angular range of +/- 75 degrees. The ray-tracing model results are compared in detail with the corresponding full-wave model results and experimental data. The manufactured design has return loss better than 15 dB over a fractional frequency bandwidth larger than 30%, in line with the predictions. Excellent scanning properties are demonstrated over an angular range of +/- 60 degrees with scan losses below 1 dB and good pattern stability, including on sidelobe levels. A height reduction by a factor of 4, when compared to a conventional geodesic lens, is demonstrated with this specific design.