Finite-Differences-Based Solvers for Wave Propagation in Dielectric Waveguides and Rings

Dielectric waveguides and ring resonators are foundational structures in photonics, enabling applications such as optical frequency comb generation, filtering, and sensing. This work presents a complete formulation to analyze electromagnetic wave propagation in these structures, incorporating the shooting method to accurately calculate propagation constants, effective indices, and modal fields. With the help of the coupled-mode theory, we investigate the transmission spectrum of a Si<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf>N<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> waveguide coupled to a Si<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf>N<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ring under the critical resonance condition. Our numerical results agree with the ones generated with COMSOL Multiphysics.