Radiation pattern of planar antennas with optimum and arbitrary illumination

R. C. Spencer has shown that the optimum illumination which minimizes the normalized second moment of the far-field power pattern is given by the fundamental mode solution of the Helmholtz´ Equation for the aperture shape. Optimum illuminations and their far-field power patterns have been obtained for various aperture shapes including a rectangle, circle, annulus and ellipse. A method is presented to calculate the far-field power pat terns for apertures of such shapes with an arbitary illumination by expanding the given illumination into a series of characteristic mode solutions of the Helmholtz´ Equation for the given aperture shape. The far-field pattern for each characteristic mode can then be treated as the case of optimum illumination.