Analysis of coupled micro rings resonators and coupled Fabry–Pérot resonators with a single physical view

Abstract

Coupled micro ring resonators gained a lot of interest in recent years in the field of silicon photonics. Although it is well recognized that coupled ring resonators and coupled Fabry–Pérot resonators operate with similar physical principles, their analysis in the literature is artificially separated, often each resonator type is analysed with different physical arguments, techniques and nomenclature. As a result, coupled rings resonators and coupled Fabry–Pérot resonators appear as two different problems, and the similarity in the physical principles that govern their operation is blurred. What is more unfortunate is that the established physical intuition, familiar from Fabry–Pérot analysis, is lost in notation when dealing with coupled micro rings. Here, we argue how a lossless boundary between two dielectrics and a lossless waveguide-coupler can be viewed as a single entity: an abstract ‘boundary’ that transmits and reflects optical fields according to Stokes relations. Using this view, Fabry–Pérot and micro rings become a single type of resonator. Accordingly, we calculate effective reflection and transmission coefficients of several configurations of coupled Fabry–Pérot and rings resonators. We calculate these coefficient by the intuitive method of summing reflected fields and by the method of transfer matrix. We illustrate that the effective reflection and transmission coefficients of a coupled ring resonator is similar to 3 layers Fabry–Pérot resonator and discuss the subtle difference. It is hoped that the common nomenclature and analysis used for both types of coupled resonators will give the reader a clear and basic understanding of both types of resonators.