We present the design and analysis of a K-band radially-periodic leaky-wave antenna radiating a circularly polarized broadside pencil beam. The structure is constituted by a metallic strip grating printed on top of a single-layer grounded dielectric slab, and is fed on the bottom by means of a square array of printed surface-wave launchers. The structure is optimized to support the fast n = -1 spatial harmonic, whose behavior has been accurately characterized through a full-wave dispersive analysis developed by means of an in-house method-of-moments (MoM) code. By proper phasing the four independent feeding points, linear or circularly polarized pencil beams can be obtained by exciting a dominant but weakly-attenuated, cylindrical TM leaky-wave mode. In this way, highly-directional broadside pencil beams showing good polarization purity can be obtained. The proposed design is of interest for modern satellite and terrestrial point-to-point communications.