Transverse Coupling in Fiber Optics Part I: Coupling Between Trapped Modes

There has been a renewed interest during the last few years in the evaluation of the transverse* coupling between two parallel open waveguides in connection with integrated optics circuitry 2 ' 3 and longdistance optical communication by bundles of glass fibers. The coupling between two open waveguides can be obtained by replacing the field of one waveguide by an equivalent current and evaluating the perturbation caused by this current on the other waveguide. 4 A more direct and slightly more general (but essentially equivalent) derivation, based on Lorentz's reciprocity theorem, is given in this paper. A related result, applicable only to lossless fibers, has been used to evaluate the coupling between dielectric rods with circular cross section. 6 The perturbation formula derived in this paper involves an integral along a contour located between the two waveguides. A seemingly different perturbation formula has been recently proposed t h a t involves a surface integral over the cross section. 6 The two formulas are shown to be in fact identical. We will not discuss in detail other coupling formulas such as the ones proposed in Refs. 7 or 3. In Ref. 7, the coupling is obtained by applying the Rayleigh-Ritz * The word "transverse" is used here to distinguish the problem of two dielectric waveguides lying side by side, where the transfer of power takes place in transverse directions, and the axial coupling between two waveguides placed end to end, where the transfer of power takes place along the z axis (the later arrangement is discussed, for instance, in lief.