Directional beams have broad applications in communication and sound reproduction. This paper investigates the theoretical maximum directivity of an infinitely flanged aperture of arbitrary cross-section by means of modal decomposition. We derive a rigorous, algebraic, global maximum solution of the directivity factor, by which a directional beam in a desired direction can be created. Assuming the aperture as the opening of a waveguide, we construct a group of incident modes or a point-source array within the waveguide for synthesizing the theoretical beam obtained in a subspace spanned by all the propagating modes. We elucidate that when the evanescent modes are included in the maximization, the maximum directivity factor increases at the cost of considerable loss in radiation efficiency. Nevertheless, in some cases, the optimum aperture velocity dominated by the lowest-order evanescent components could still be useful for designing metamaterial-lens horn antennas.