Robust and non-robust bound states in the continuum in rotationally symmetric periodic waveguides

A fiber grating and a one-dimensional (1D) periodic array of spheres are examples of rotationally symmetric periodic (RSP) waveguides. It is well known that bound states in the continuum (BICs) may exist in lossless dielectric RSP waveguides. Any guided mode in an RSP waveguide is characterized by an azimuthal index m, the frequency ω, and Bloch wavenumber β. A BIC is a guided mode, but for the same m, ω and β, cylindrical waves can propagate to or from infinity in the surrounding homogeneous medium. In this paper, we investigate the robustness of nondegenerate BICs in lossless dielectric RSP waveguides. The question is whether a BIC in an RSP waveguide with a reflection symmetry along its axis z, can continue its existence when the waveguide is perturbed by small but arbitrary structural perturbations that preserve the periodicity and the reflection symmetry in z. It is shown that for m = 0 and m ≠ 0, generic BICs with only a single propagating diffraction order are robust and non-robust, respectively, and a non-robust BIC with m ≠ 0 can continue to exist if the perturbation contains one tunable parameter. The theory is established by proving the existence of a BIC in the perturbed structure mathematically, where the perturbation is small but arbitrary, and contains an extra tunable parameter for the case of m ≠ 0. The theory is validated by numerical examples for propagating BICs with m ≠ 0 and β ≠ 0 in fiber gratings and 1D arrays of circular disks.