Critical coupling, oscillation, reflection, and transmission in optical waveguide-ring resonator systems

Abstract

We generalize an earlier published analysis of critical coupling in a single-ring resonator waveguide system [Electron. Lett.36, 321 (2000)] to multiple-ring resonator all-pass and channel-dropping filter structures. The fundamental analytical relationships among coupling coefficient, internal loss, and oscillation frequency for critical coupling and oscillation conditions are derived for two and three rings coupled to single- and double-bus waveguides using the transfer-matrix method and Sylvester's theorem. Furthermore, we derive the photonic band structure of an infinite periodic ring structure and show that, as the number of rings increases, the reflection band of the double-bus structure approaches the photonic bandgap.