The study of Fabry-Perot leaky-wave antennas which are properly tapered to optimize their directivity at broadside, is presented in this work. For this purpose, a careful synthesis of the complex aperture illumination (amplitude and phase) induced by the radiating leaky mode is for the first time employed in the synthesis of this type of antennas. The study commences from the one-dimensional (1D) fan-beam Fabry-Perot antennas and extends to two-dimensional (2D) pencil-beam Fabry-Perot antennas. Theoretical and simulated results are reported, illustrating an increase in the antenna directivity at broadside direction in the order of 2dBs for the 1D case, and of 4dBs for the 2D case.