Matrix Analysis of Coupled Microring Resonator Polygons

The resonant properties of a photonic molecule, composed by N microring resonators forming a regular polygon, are for the first time determined analytically using the transfer matrix method. It is found that the transfer matrix between rings n, n + 2, n + 4 , ... is independent of the polygon vertex angle, allowing the application of Floquet theorem for periodic propagation in a cylindrically symmetric structure. Corresponding to even or odd N, the molecule possesses 1 + N/2 or 1 + N discrete resonances, which satisfy the dispersion equation of the straight coupled-resonator optical waveguide (CROW) with infinite rings. The field amplitudes in the rings are determined as eigenvectors of the corresponding eigenvalue problem. By incorporating the molecule into a channel dropping filter system, the presence of these resonances in the transmission spectrum is verified.