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–Zehnder-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.
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