Theory of directional pulse propagation: detailed calculations

I construct combined electric and magnetic field variables which independently represent energy flows in the forward and backward directions respectively, and use these to re-formulate Maxwell's equations. The emphasis is on detailed calculations, with a more general overview being published in Phys. Rev. A72 and arXiv. These directional variables enable us to not only judge the effect and significance of backward-travelling field components, but also to discard them when appropriate. They thereby have the potential to simplify numerical simulations, leading to potential speed gains of up to 100% over standard FDTD or PSSD simulations. These field variables are also used to derive both envelope equations useful for narrow-band pulse propagation, and a second order wave equation. Alternative definitions are also presented, along with their associated wave equations.