  Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz
equation $\Delta u+\omega ^{2}u=0 $ for $ (x,y,z) \in \mathbb{R}^3 $. For the
evolution of such waves along the $z$-axis a Schr\"odinger equation can be
derived through a multiple scaling ansatz. It is the purpose of this paper to
justify this formal approximation by proving bounds between this formal
approximation and true solutions of the original system. The challenge of the
presented validity analysis is the fact that the Helmholtz equation is
ill-posed as an evolutionary system along the $z$-axis.
