Transverse limits on the uni-directional pulse propagation approximation

I calculate the limitations on the widely-used forward-only (uni-directional) propagation assumption by considering the effects of transverse effects (e.g. diffraction). The starting point is the scalar second order wave equation, and simple predictions are made which aim to clarify the forward-backward coupling limits on diffraction strength. The result is unsurprising, being based on the ratio of transverse and total wave vectors, but the intent is to present a derivation directly comparable to a recently published emph{nonlinearity} constrained limits on the uni-directional approximation [Kinsler, J. Opt. Soc. Am. B (2007)].