Pole-zero dynamics of high-order ring resonator filters

We present a pole-zero analysis, based on the transfer matrix method, for ring resonator (RR)-based filters of Type I and Type II, consisting of mutually coupled and side-coupled RRs. The pole-zero plot determines the filter spectral characteristics and is determined primarily by the coupling coefficient between the resonators. The pole-zero dynamics (or root locus) shows how the poles and zeros move in the complex frequency plane as we vary the coupling coefficient. We show that the pole-zero dynamics for the two types of filter are complementary and, furthermore, that the pole-zero plots are related to the conditions of critical coupling and oscillation in the presence of loss or gain. We also show a pole-zero analysis for the apodization of these filters based on an actively tunable Mach-Zelmder-type coupler, where it is shown that apodization corresponds to designing pole-zero pairs with wide separations, whereas the required bandwidth determines the specific values of the coupling coefficient.