Lens Shape Equations for Positive and Negative Refractive Index

At 5G and 6G mobile systems, high frequencies such as millimeter and terahertz are used. Multi beam radiation pattern is requesting at a base station. A lens antenna is suitable for high frequency use because of its simple structure. Also, lens antenna can achieve good multi beam performances. As for lens material, by the meta material technology, negative refractive index (n<0) can be possible. In the case of n<0 lens, lens shape becomes concave, and lens thickness is reduced compared to n>0 lens. The thin lens can improve multi beam performances such as beam shape and gain reduction. In order to compare multi beam performances of n>0 and n<0 lens, previously known three types are suitable. The lens shape equation for n<0 condition is not obtained. In this paper, lens shape equations for n<0 lens is derived and compared with n>0 lens equations. In order to ensure correctness of derived equations, ray path data from a feed position to antenna aperture are obtained by a MATLAB ray tracing program. At all antenna types, the rays from a feed are refracted by the lens and aligned parallel to the lens axis at the aperture plane. The ray tracing results agree with the conditions in deriving equations. The correctness of lens shape equations is ensured.