Mathematical refinements of excitation conditions in coupled waveguides

The picture of the electromagnetic field evolution along coupled waveguides in terms of the modes of the isolated primary waveguides and of supermodes of the overall coupler, is critically refined. A transformation matrix method, applied to a supermode set, enables us to express the electromagnetic field in a directional coupler by means of the modes of the isolated waveguides. A vectorial space is associated with the set of solutions of the Maxwell equations, where the orthogonal basis is given by normalized supermodes and another space in which the normalized vector basis is given by the normalized modes. Among the representations of two-coordinate systems, there is an angular rotation that does not maintain the orthogonality of the basis. Therefore, there are two characteristic rotation angles that allow a complete description of the light power exchange between the two guides and of the mode orthogonality. Many examples are included. Finally, a brief extension to N-coupled waveguides is reported.