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Chris Spencer edited untitled.tex
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\section{codirectional coupling} \section{Introduction}
Coupled-mode theory is concerned with coupling spatial modes of differing polarizations, distribution, or both. To understand codirectional coupling it is useful to have an understand of background material that builds to codirectional coupling. First, consider coupling normal modes in a single waveguide that is affected by a perturbation. Such case is single-waveguide mode coupling. The perturbation in question is spatially dependent and is represented as $\Delta P(r)$, a perturbing polarization. Consider the following Maxwell's equations \[ \nabla x E=i \omega \mu_0 H, \nabla x H=-i \omega \epsilon E-i \omega \Delta P\]
