Design and modelling of ring resonators used as optical filters for communications applications

Abstract

This project is a theoretical study of multiple coupled ring resonators, which offer potential applications as demultiplexing filters in DWDM optical transmission systems. The rings can be fabricated as integrated optical structures or they can be formed using micro- or nano-optical fibres. Our approach is analytical, which provides detailed predictions with minimal computer resources. The ideal filter spectral profile for most applications is as close as possible to a rectangle (known as “box-like”) and in order to achieve this we design and model multiple ring resonators. We formulate the compound ring resonator theory with complex field equations to account for phase and amplitude. Then we calculate the transfer functions. We do it in two ways: one way is using linear equations and the other is by matrix theory. We apply both methodologies to one-, two- and three-ring resonators and we show how the matrix formalism can be extended to model arrays of N identical rings. By using the transfer functions we provide detailed physical interpretations of the spectra which are required to design good filter characteristics. We show that rings of equal circumferences provide the best profiles and we derive simple analytical formulas, called “degeneracy condition”, to predict the required coupler ratios for two- and three-ring resonators. It is thus possible to provide a transfer function with single peaks of equal and unity magnitude and a depth of modulation that we choose. Provided that the couplers within the rings conform to the degeneracy condition, we can predict the finesse of a double-ring transfer function. We further extend the ring resonator matrix theory to N identical rings by using a method called “diagonal decomposition”. The amplitude transfer function for N rings can thus be derived with this more advanced mathematical technique. The result that we obtain is in a format that can be extended in future more extended studies. Throughout this project our aim is to provide tangible design guidelines for compound ring resonators, with their potential application to telecommunications networks in mind.