3x3 transfer matrix modelling of integrated optical devices

Unlike common devices based on ring resonators, the structure in Fig. 1.a involves not only 2x2 couplers but also a 3x3 coupler, which means that a 3x3 transfer matrix approach is required to model the system. To the best of our knowledge, no such a model has been developed before. The only model available in the literature is based on a clever recursive 2x2 transfer matrix model [1], which requires lengthy calculations that depend on the chosen boundary conditions and on the particular geometry chosen. The scope of this document is instead to show how to generalize the standard 2x2 transfer matrix approach to cover any system with 3x3 couplers, and calculate the transfer matrix of any complicated system just as a product of simple 3x3 matrices.