Directivity characteristics of scannable planar arrays

An exactly integrated expression for directivity is used to obtain curves of directivity versus interelement spacing and scan angle for uniformly excited planar arrays composed of discrete identical elements at regular spacings. The expression can be applied to arrays with nonuniform amplitude distributions and with a variety of ideal element factors. Numerical results are given for square lattice arrays for which the element factors were assumed to be isotropic,
\cos \theta, \cos^{2} \theta
, and small dipole in free space. Results are also given for triangular lattice arrays. The effect of the presence of grating lobes on main beam directivity is illustrated. The conditions for which the usual gain formulas approximate the directivity with reasonable accuracy are described quantitatively.