  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.
