  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.
