Coupled-mode equations for dielectric waveguides based on projection and partition modal amplitudes

There are two objectives in this paper. First, it is pointed out that the coupled-mode theory for parallel dielectric waveguides can have two different formulations, one based on the projection modal amplitudes and the other based on the partition modal amplitudes. The theory is then derived by considering the projection modal amplitudes of the total coupled-system field and including the integrated overlap of the individual waveguide modes. Second, the existing two formulations of the conventional coupled-mode theory are shown to be consistent with each other, following naturally from the discussions in the first part. Coupled-mode equations including the mode overlap integrals are written in terms of the projection modal amplitudes. They are appropriate to power calculations and are equivalent to those equations recently derived by other authors. It is emphasized that using equivalent initial conditions is essential for the different formulations to give identical predictions of power coupling.