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 (BSCs) 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 discs with permittivity $\epsilon_1$. Because of rotational and translational symmetries the BSCs 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 BSCs have definite polarization and occur in a wide range of the radius of the rod and the dielectric permittivities. More involved BSCs with $m\neq 0, \beta=0$ exist only for a selected radius of the rod at a fixed dielectric constant. The existence of robust Bloch BSCs with $\beta\neq 0, m=0$ is demonstrated. Asymptotic limits to a homogeneous rod and to very thin discs are also considered.