This report focuses on a design method for gradient index (GRIN) lens antennas with controllable aperture field distributions. First, we derive differential equations representing optical paths in a gradient index medium with two optical surfaces by using geometrical optics, and then we formulate a novel design method for GRIN lens antennas based on these equations. The Levenberg-Marquardt algorithm is applied as a nonlinear least squares method to satisfy two conditions-focusing and shaping the aperture field distribution-thus realizing a prescribed radiation pattern. The conditions can be fulfilled by optimizing only the index (or permittivity) distribution, whereas the shapes of the optical surfaces remain as free parameters that can be utilized for other purposes, such as reducing reflection losses that occur on the surfaces, as illustrated in this report. A plano-concave GRIN lens is designed as an example, applying the proposed method, to realize a sidelobe level of -30 dB pseudo Taylor distribution, and a maximum sidelobe level of -29.1 dB was observed, indicating it is sufficiently accurate for practical use. In addition, we discuss the convergence of this method considering the relationship between the number of the initial conditions and the differential order of the design equations, factoring in scale invariance of the design equations.

We present a review of several types of microwave antennas made of metamaterials, including the resonant electrically small antennas, metamaterial-substrate patch antennas, metamaterial flat-lens antennas, and Luneburg lens antennas. In particular, we propose a new type of conformal antennas using anisotropic zero-index metamaterials, which have high gains and low sidelobes. Numerical simulations and experimental results show that metamaterials have unique properties to design new antennas with high performance.