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A Method for Optimising Superdirectivity of Coupled Meta-Atoms via Planar Directivity Evaluation

We propose a simple and rapid way of optimising directivity in metamaterial-inspired endfire antenna arrays with strong inter-element coupling. We introduce, in addition to the traditional 3D directivity, also its planar equivalents defined as the ratio of the power density in the desired endfire direction to the average power density calculated either in the horizontal (azimuthal) or vertical (elevation) plane. Using dimers of magnetically coupled split-ring resonators with only one element driven by an external source, we derive conditions that must be satisfied in order to realise superdirective current distributions. The superdirective conditions link the quality factor of the resonators and their coupling constant to the array size and the operating frequency. We demonstrate that a rapid measurement in the azimuthal plane can be used as a reliable indicator for whether the superdirective conditions for the 3D directivity are satisfied. Analytical calculations are verified by CST simulations in the MHz frequency range for meta-atoms of circular and square shape. Our method can be extended to arrays comprising larger number of meta-atoms of various shape and would enable rapid prototyping of 3D-printed meta-atoms with desired radiation properties.