Focusing properties of small lenses

The potential use of small substrate lenses for coupling to antennas at millimeter wave frequencies is investigated by analyzing the focusing properties of dielectric spheres with quarter-wave matching layers. The fields and the power density are calculated at various points within the sphere to learn how the focusing deteriorates as the lens is made smaller. The absorption loss of the lens is also calculated. The calculations show that quartz spheres with quarter-wave matching layers, can exhibit good focusing properties down to a radius of about half a free space wavelength. This minimum radius was found to increase almost linearly with the refractive index of the lens. The calculations also indicate that at 94GHz the absorption losses of fused quartz, silicon and gallium arsenide lenses, with radii of one free space wavelength or less, are less than 0.2 dB. As the minimum diameter of a quartz lens is comparable with the spot size in free space, it should be possible to build an imaging lens array in which each lens will act as a separate imaging element.