Bound states in the continuum with high orbital angular momentum in a dielectric rod with periodically modulated permittivity

We report bound states in the radiation continuum (BICs) in a single infinitely long dielectric rod with periodically stepwise modulated permittivity alternating from ${\epsilon }_1$ to ${\epsilon }_2$. For ${\epsilon }_2=1$ in air the rod is equivalent to a stack of dielectric disks with permittivity ${\epsilon }_1$. Because of rotational and translational symmetries the BICs are classified by orbital angular momentum $m$ and the Bloch wave vector $\beta$ directed along the rod. For $m=0$ and $\beta =0$ the symmetry protected BICs have definite polarization and occur in a wide range of the radius of the rod and the dielectric permittivities. More involved BICs with $m\ne 0,\beta =0$ exist only for a selected radius of the rod at a fixed dielectric constant. The existence of robust Bloch BICs with $\beta \ne 0,m=0$ is demonstrated. Asymptotic limits to a homogeneous rod and to very thin disks are also considered.