Planar Directivity of a Dipole Array

We develop a technique for fast evaluation of superdirective performance of coupled meta-atoms. Traditionally, the directivity of a small antenna is obtained from measurements of the far-field radiated power over a spherical surface. We propose an alternative method of finding the directivity from a faster polar scan over a circular path, by introducing the "planar directivity" (D_2D) for dipole arrays. We derive analytical expressions for maximum achievable values of directivity and D_2D, establish the correlation between them, and identify the range of parameters (inter-element distance, coupling constant, quality factor) for which the conversion of D_2D to D for superdirective dimers is possible.