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3$\times$3 transfer matrix modelling of integrated optical devices

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