Propagation equations for optical pulses are needed to assist in describing applications in ever more extreme situations -- including those in metamaterials with linear and nonlinear magnetic responses. Here I show how to derive a single first order
propagation equation using a minimum of approximations and a straightforward factorization mathematical scheme. The approach generates exact coupled bi-directional equations, after which it is clear that the description can be reduced to a single uni-directional first order wave equation by means of a simple slow evolution approximation, where the optical pulse changes little over the distance of one wavelength. It also also allows a direct term-to-term comparison of an exact bi-directional theory with the approximate uni-directional theory.
