In the frame of Coupled-Mode Theory (CMT), for a given eigen-state of polarization, a two-mode waveguide can be formally interpreted as a set of two identical, mutually coupled single-mode waveguides, where the even/odd... [ view full abstract ]

In the frame of Coupled-Mode Theory (CMT), for a given eigen-state of polarization, a two-mode waveguide can be formally interpreted as a set of two identical, mutually coupled single-mode waveguides, where the even/odd supermodes play the roles of the symmetric (slow) and antisymmetric (fast) modes of the two-mode waveguide. Within this formalism, the relevant parameters are the phase mismatch and the coupling constant: symmetry ensures that the former is zero, whereas the latter is defined by the half-difference between the propagation constants of the supermodes.

In a ternary system made of three identical, phase-matched, mutually coupled single-mode waveguides, the propagation constants of the supermodes are equidistant in the k-space. In a three-mode waveguide, however, the three modes (slow, “neutral” and fast) are not equidistant. It is therefore impossible to proceed to the same identification, with one coupling constant as the sole modelling parameter.

Nevertheless, a formal equivalence is still relevant, provided we also take into account the “diagonal perturbation” exerted on the central waveguide by its two neighbours. As it happens, the three-mode waveguide can be interpreted as a set of three mutually coupled single-mode waveguides, where the middle one appears slightly detuned. We propose to show how to derive the equivalent mismatch and the equivalent coupling constant from the three propagation constants.

Alternatively, another picture is possible, with three identical, phase-matched, mutually coupled single-mode waveguides, where this time the coupling is not limited to the nearest neighbours. Once again, two distinct parameters are required (two coupling constants). Both approaches lead to similar results in terms of mode indices. This means that, if a formal identification is actually possible in the frame of CMT, the solution is not unique.

We’ll illustrate our findings on planar waveguides, where exact calculations are readily available, not only for the propagation constants but also for the transverse distribution of the optical fields on each mode (see Figure).

This approach can prove of interest for the analytic investigation of photonic integrated structures involving Modal Division Multiplexing (MDM), notably in terms of design and tolerances.