The possibility of achieving directive fan-beam radiation with planar Fabry-Pérot cavity antennas constituted by an upper ferroelectric thin film and a lower ground plane having ultrasubwavelength thickness is studied by means of a simple transverse-equivalent-network approach and a cylindrical leakywave analysis, deriving simple design formulas. The performance of the proposed antenna is investigated in terms of power density radiated at broadside and directivity in the principal planes, pointing out the main limitations and tradeoffs associated with the reduced thickness.