  In a lossless periodic structure, a bound state in the continuum (BIC) is
characterized by a real frequency and a real Bloch wavevector 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 wavevectors 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/exist due
to structural perturbations or wavevector 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 given real frequencies near the frequency of the BIC.
It turns out that such eigenmodes near the BIC always have a complex Bloch
wavenumber, but they may or may not be leaky modes that radiate out power
laterally to infinity. These eigenmodes can also be complex modes that decay
exponentially in the lateral direction. Our study is relevant for applications
of BICs in optical waveguides, and it is also helpful for analyzing photonic
devices operating near the frequency of a BIC.

Optics