  We derive a coupled mode theory for the interaction of an optical cavity with
a waveguide that includes waveguide dispersion. The theory can be applied to
photonic crystal cavity waveguide structures. We derive an analytical solution
to the add and drop spectra arising from such interactions in the limit of
linear dispersion. In this limit, the spectra can accurately predict the cold
cavity quality factor (Q) when the interaction is weak. We numerically solve
the coupled mode equations for the case of a cavity interacting with the band
edge of a periodic waveguide, where linear dispersion is no longer a good
approximation. In this regime, the density of states can distort the add and
drop spectra. This distortion can lead to more than an order of magnitude
overestimation of the cavity Q.
