  In this paper, we investigate the beamforming performances of holographic
surfaces implemented as lossless antenna arrays with less than half-wavelength
spacing. We first develop a method to quantify the mutual coupling effect among
the antennas in an array. The developed coupling model is general and
applicable to arrays with arbitrary distribution of any type of antennas with
arbitrary structure, physical size and radiation power pattern. In particular,
it reduces to a neat analytical expression for arbitrarily deployed isotropic
antenna arrays. We then discuss the beamforming design for holographic
surfaces, and in particular provide analytical beamforming characterizations
for arrays with two arbitrarily spaced isotropic antennas. Numerical results
indicate that, by accounting for the mutual coupling effect between antennas,
the array densification by packing more antennas in a given surface aperture
can significantly enhance both the beamforming gain and spatial resolution of
the system. The beamforming gain enhancement and beamwidth reduction can be
several dBs higher than, and more than half of, those achieved by the
conventional half-wavelength spaced antenna arrays in the same surface
aperture. The gains of densification become saturated when the antenna spacing
is below a critical value, and the saturated gain reduces as the surface
aperture increases.
