Optical pulse propagation with minimal approximations

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 bidirectional equations, after which it is clear that the description can be reduced to a single unidirectional 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 allows a direct term-to-term comparison of an exact bidirectional theory with the approximate unidirectional theory.