Fine tuning topological waveguides using asymptotic analysis

Abstract

We use asymptotic methods to quantify the properties of topological waveguides and show how these concise results can be used very efficiently to design materials with specific, custom specifications. This work uses a general method that we recently developed for studying localised eigenmodes in periodic media with defects. The results of this analysis characterise the existence of localised eigenmodes, determine their eigenfrequencies and quantify the rate at which they decay away from the defect. These results are obtained using both high-frequency homogenisation and transfer matrix analysis, with good agreement between the two methods. This approach is ideally suited to studying problems arising in topological wave physics. We demonstrate the efficacy of our approach by studying materials based on the Su-Schrieffer-Heeger model (a simple example of a one-dimensional crystal supporting topologically protected modes). We show that our theory accurately predicts the localised eigenmodes and we use it to design rainbow devices that separate frequencies according to custom requirements.