Eigenmodes of finite length silicon-on-insulator microring resonator arrays

In this paper the eigenmodes of finite length microring resonator arrays have been systematically studied, both analytically using temporal coupled mode theory (CMT), and numerically using two-dimensional finite element method (FEM). The method for obtaining the values of parameters appearing in simplified CMT model using results of FEM calculations is presented. Calculations were carried out by COMSOL FEM packages for a wide range of distances between the rings. The obtained results reveal that the rotational degeneracy is preserved for a wide range of interrings distances. It is shown how the eigenvalue spectrum depends on the number of cavities in the system. The differences for the cases of odd and even numbers of rings, and its implications on actual applications, are discussed in details. The central branch appearing in odd-number arrays plays significant role for the delay-lines and optical buffering applications. Based on the first order perturbation theory, an analytical expressions for the eigenfrequencies of arbitrary (finite) length linear array of microring resonators are derived. The analytical expressions describing eigenfrequencies are useful for determining positions of the maxima in transfer characteristics in microring arrays with external buses.