Electrooptic modulation in an arbitrary cross section waveguide

The electromagnetic field of an optical wave of frequency ω, traveling through a long, electrooptic waveguide of arbitrary cross section and composition in the presence of an arbitrary external field of strength E^ext and radio frequency Ω, is derived by a perturbative argument, assuming Ω/ω≪1 and | r_ijkE^ext|≪1, where r _ijk are the linear electrooptic coefficients. An idealized model is solved exactly in the context of rigorous perturbation theory, and the solution is shown to be valid whenever (r_ijk E^ext)²ω/Ω≪1, without restriction on r_ijkE^extω/Ω. The arbitrary cross section formula, when evaluated in the model case, agrees exactly with the rigorous result, so it is argued that the arbitrary cross section formula should also be valid whenever (r_ijkE^ext)² ω/Ω≪1