This communication presents a rigorous method to synthesize radiating aperture field distributions that produce monopulse difference patterns. This method is based on the modal expansion of the electromagnetic fields at the aperture, and thanks to the use of a convex formulation, it is guaranteed that the achieved result is the optimal solution for the addressed problem. Besides, the synthesis method is valid for any aperture geometry and desired polarization. First, the theory of the synthesis method is presented, and then, several numerical examples with different aperture geometries and polarizations are shown to demonstrate the capabilities of the proposed technique. Finally, the formulation is generalized to allow the synthesis of radiation patterns with nulls and lobes of arbitrary amplitude and phase in any desired direction.