Frequency perturbation theory of bound states in the continuum in a periodic waveguide

Abstract

In a lossless periodic structure, a bound state in the continuum (BIC) is characterized by a real frequency and a real Bloch wave vector for which there exist waves propagating to or from infinity in the surrounding media. For applications, it is important to analyze the high-$Q$ resonances that either exist naturally for wave vectors near that of the BIC or appear when the structure is perturbed. Existing theories provide quantitative results for the complex frequency (and the $Q$ factor) of resonant modes that appear or exist due to structural perturbations or wave vector variations. When a periodic structure is regarded as a periodic waveguide, eigenmodes are often analyzed for a given real frequency. In this paper, we consider periodic waveguides with a BIC and study the eigenmodes for a given real frequency near the frequency of the BIC. It turns out that such eigenmodes near the BIC always have a complex Bloch wave number, but they may or may not be leaky modes that radiate out power laterally to infinity. These eigenmodes can also be the so-called complex modes that decay exponentially in the lateral direction. Our study is relevant for applications of BICs in periodic optical waveguides, and it is also helpful for analyzing photonic devices operating near the frequency of a BIC.