Accurate paraxial Maxwell beams

Abstract: In this work, the paraxial version of Maxwell equations is derived with the use of two Riemann-Silberstein vectors. Exact solutions of these equations are then obtained representing the paraxial electromagnetic fields. These fields satisfy the full Maxwell equations up to the order of $(\lambda/w_0)^2$. The solutions contain some additional terms, which turn out to be relevant for polarization, especially in the case of beams endowed with orbital angular momentum.